UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Symmetric flow past orthotropic bodies : single and clusters Masliyah, Jacob Heskel 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1970_A1 M38.pdf [ 24.57MB ]
Metadata
JSON: 831-1.0059123.json
JSON-LD: 831-1.0059123-ld.json
RDF/XML (Pretty): 831-1.0059123-rdf.xml
RDF/JSON: 831-1.0059123-rdf.json
Turtle: 831-1.0059123-turtle.txt
N-Triples: 831-1.0059123-rdf-ntriples.txt
Original Record: 831-1.0059123-source.json
Full Text
831-1.0059123-fulltext.txt
Citation
831-1.0059123.ris

Full Text

SYMMETRIC  FLOW P A S T ORTHOTROPIC  BODIES:  S I N G L E AND C L U S T E R S  by JACOB HESKEL MASLIYAH B. S c U n i v e r s i t y C o l l e g e , L o n d o n , M.Sc, U n i v e r s i t y o f New B r u n s w i c k ,  A T H E S I S SUBMITTED  I N P A R T I A L F U L F I L M E N T OF  THE R E Q U I R E M E N T S DOCTOR OF in  1964 1966  FOR THE D E G R E E  OF  PHILOSOPHY  t h e Department of  CHEMICAL ENGINEERING  We a c c e p t  t h i s t h e s i s as conforming required standard  THE U N I V E R S I T Y OF B R I T I S H  February,  1970  to the  COLUMBIA  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  by  his  of  this  thesis at  it  may  representatives.  written  for  freely  permission  purposes  thesis  partial  the U n i v e r s i t y  make  tha  in  is  financial  for  gain  Depa r t m e n t  Date  j  f  ^  J  8.  Columbia  !f7Q  of  Columbia,  British for  extensive by  shall  the  that  not  requirements I  agree  r e f e r e n c e and copying  t h e Head o f  understood  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  of  available  be g r a n t e d  It  fulfilment  of  this  be a l l o w e d  or  that  study. thesis  my D e p a r t m e n t  copying  for  or  publication  without  my  i  ABSTRACT  Numerical was s u c c e s s f u l l y  s o l u t i o n of the Navier-Stokes accomplished,  l a x a t i o n technique of Jenson,  u s i n g an a d a p t a t i o n o f t h e r e f o r axisymmetric  o b l a t e and p r o l a t e s p h e r o i d s a t p a r t i c l e up t o 100. 0.999  The a s p e c t r a t i o  ticity of  distributions  a sphere.  flow past  Reynolds  single  numbers  o f t h e s p h e r o i d s v a r i e d between  (nearly p e r f e c t sphere) For low a s p e c t r a t i o s  equation  and 0.2. t h e s u r f a c e p r e s s u r e and v o r -  showed a m a r k e d d i f f e r e n c e  The a p p e a r a n c e o f t h e wake b u b b l e  from  those  behind  a  s p h e r o i d was f o u n d t o be a s t r o n g f u n c t i o n o f t h e p a r t i c l e shape. Numerical  s o l u t i o n s were a l s o o b t a i n e d f o r two-  d i m e n s i o n a l symmetric the f l o w p a r a l l e l  flow past e l l i p t i c a l  t o the major a x i s  0.995 t o 0.2 a t R e y n o l d s  f o r aspect r a t i o s of  numbers up t o 9 0 , and w i t h t h e f l o w  parallel  t o the minor a x i s  Reynolds  numbers up t o 40.  found  cylinders, with  f o r an a s p e c t r a t i o o f 0.2 a t The n u m e r i c a l s o l u t i o n  was  t o be l e s s s t a b l e t h a n t h e c o r r e s p o n d i n g t h r e e -  dimensional axisymmetric  case.  The v a r i a t i o n o f t h e t o t a l d r a g c o e f f i c i e n t Reynolds  with  number f o r t h e s p h e r o i d s and t h e e l l i p t i c a l  l i n d e r s o f v a r i o u s a s p e c t r a t i o s was n o t much from t h a t o f a sphere  and a c i r c u l a r  cylinder,  cy-  different respectively.  The r e s u l t s f o r b o t h t h e s p h e r o i d s and t h e e l l i p t i c a l linders  showed a s t e a d y t r e n d w i t h R e y n o l d s  number  cy-  from  i i  Stokes  and/or Oseen Happel's  zero  vorticity  creeping of  flow  free cell  flow past  surface c e l l model were  constant  packed  beds were  deviate  model  flow.  and  Kuwabara's  employed f o r the study  swarms o f a l i g n e d s p h e r o i d s  aligned elliptical  Kozeny  t o boundary l a y e r  from  cylinders.  Large  i t s commonly  found  significantly  by b o t h  i n shape from  clusters  deviations of the  assumed  models  and  of  value  of 5 f o r  for particles  a sphere  or a  which circular  cylinder. In lower  g e n e r a l , Happel's  total  vorticity clusters  drag  model  and  coefficients f o r both  of e l l i p t i c a l  Contours  free  s u r f a c e model p r e d i c t e d  than  d i d Kuwabara's  t h e swarms o f s p h e r o i d s  zero and t h e  cylinders.  of the streamlines, e q u i - v o r t l c i t y  equi-velocity  lines  are  presented.  lines  i i i T A B L E OF CONTENTS Page ABSTRACT  i  L I S T OF T A B L E S  v i i  L I S T OF F I G U R E S  ,  v i i i  L I S T OF DIAGRAMS  xv -  ACKNOWLEDGEMENTS  .. x v i  CHAPTER I INTRODUCTION  1  CHAPTER I I R E V I E W OF P R I O R WORK  3  1. R E V I E W OF A N A L Y T I C A L WORK  3  A.  Single Bodies  A.l  Spheroids  '  A. 2 E l l i p t i c a l B.  3  Cylinders  5 ..  Swarms  17 23  2. R E V I E W OF N U M E R I C A L WORK  39  CHAPTER I I I FORMULATION  OF F I N I T E D I F F E R E N C E E Q U A T I O N S AND  BOUNDARY C O N D I T I O N S A.  Spheroids  A.l  Formulation  o f t h eF i n i t e  Difference  .  .  Conditions  B.  Elliptical  B.l  Formulation  ,  Cylinders o f the Finite  Equations B.2 B o u n d a r y  43 43  Equations A. 2 B o u n d a r y  <  43 50 57  Difference 57  Conditions  6l  iv Page  CHAPTER  IV  FORMULATION  OF DRAG C O E F F I C I E N T E Q U A T I O N S  AND  THE  PRESSURE D I S T R I B U T I O N A.  Spheroids  B.  Elliptical  CHAPTER  67 67  Cylinders  75  V  D I S C U S S I O N AND  P R E S E N T A T I O N OF R E S U L T S  1. A P P L I C A B I L I T Y OF THE N A V I E R - S T O K E S AND  C H O I C E OF THE N U M E R I C A L  79 EQUATIONS  TECHNIQUE  79  2. R E L A X A T I O N PROCEDURE  80  3. COMPUTER PROGRAM  87  4.  V A L I D I T Y OF THE N U M E R I C A L  5.  R E S U L T S FOR  WORK  88  S I N G L E SPHEROIDS  101  6. R E S U L T S FOR  SINGLE E L L I P T I C A L CYLINDERS  157  7. R E S U L T S FOR  SWARMS OF P A R T I C L E S I N C R E E P I N G FLOW  212  A.  Swarms o f S p h e r o i d s  B.  Clusters  CHAPTER  212  of E l l i p t i c a l  Cylinders  222  VI  CONCLUSIONS  AND  RECOMMENDATIONS  2'35  L I S T OF R E F E R E N C E S  237 .  NOMENCLATURE  245  APPENDIX 1 BOUNDARY L A Y E R THEORY AND  P O T E N T I A L FLOW D E V E L O P -  MENT  1-1  A.  Spheroid  B.  Elliptical  1-1 Cylinders  1-4  V  Page APPENDIX I I DATA FOR SINGLE SPHEROIDS, SINGLE ELLIPTICAL CYLINDERS, SWARMS OF SPHEROIDS AND CLUSTERS OF E L L I P TICAL CYLINDERS  .  S i n g l e Oblate S p h e r o i d s  I I - l  I I - l  S i n g l e P r o l a t e Spheroids  11-19  Single E l l i p t i c a l  C y l i n d e r s ,with O.R.>  1 .......  Single E l l i p t i c a l  C y l i n d e r s w i t h O.R.< 1  11-30 11-38  Swarms o f O b l a t e S p h e r o i d s , Happel's Model .... I I - 4 1 Swarms o f O b l a t e S p h e r o i d s , Kuwabara's Model ..  11-49  Swarms o f P r o l a t e S p h e r o i d s , Happel's Model ...  11-57  Swarms o f P r o l a t e S p h e r o i d s , Kuwabara's Model .  II-63  C l u s t e r s o f E l l i p t i c a l C y l i n d e r s w i t h O.R.>  1,  Happel's Model C l u s t e r s of E l l i p t i c a l  II-69 C y l i n d e r s w i t h O.R.>  1,  Kuwabara's Model  II-76  • C l u s t e r s o f E l l i p t i c a l C y l i n d e r s w i t h O.R.<  1,  Happel's Model  11-83  C l u s t e r s o f E l l i p t i c a l C y l i n d e r s w i t h O.R.<  1,  Kuwabara's Model  II-90  APPENDIX I I I 1. EVALUATION OF THE KOZENY CONSTANT FROM BASIC DEFINITION A.  Spheroids  B.  E l l i p t i c a l Cylinders  ••  III-l III-l .  III-3  vi  Page  2. E V A L U A T I O N OF THE KOZENY CONSTANT U S I N G  THE  T O R T U O S I T Y METHOD 3. E V A L U A T I O N OF TICAL  A P / L FOR A C L U S T E R  C Y L I N D E R S AT CONSTANT Rew  III-4 OF  ELLIP•  IH-6  APPENDIX I V COMPUTER  PROGRAMS  IV-1  1. S P H E R O I D S  IV-2  2. E L L I P T I C A L C Y L I N D E R S  IV-6  vii  L I S T OF T A B L E S Table  Page  1  Values  of the coefficient A  14  2  Relaxation  factors  f o roblate  3  Relaxation  factors  f o r prolate  4  Relaxation factors c y l i n d e r w i t h O.R.  n  spheroids spheroids  83 84  for elliptical ^ 1.0  85  Relaxation factors f o r e l l i p t i c a l c y l i n d e r w i t h O.R. < 1.  86  6  Some s e l e c t e d  data  f o rsingle  91  7  Some s e l e c t e d cylinders  data  f o rsingle  5  8  spheroids elliptical  95  E f f e c t o f g r i d s i z e on C a n d a comparison with analytical solutions f o r Re =0.01 D T  96  v i i i  L I S T OF F I G U R E S Page 1.  Oblate  s p h e r o i d a l mesh  2.  Oblate  spheroid  3.  Elliptical  c y l i n d r i c a l mesh  4.  Elliptical  cylinder  5-  Variation diameter.  6. 78. 9.  with  of total  system.  i t s outer  with drag  envelope.  coefficient  60 envelope. with  62  t h e mean 98  V a r i a t i o n o f t h e t o t a l drag number f o r a s p h e r e .  coefficient  with  Reynolds 102  V a r i a t i o n o f t o t a l drag c o e f f i c i e n t with Reynolds number f o r v a r i o u s a s p e c t r a t i o s o f a n o b l a t e s p h e r o i d s .  104  V a r i a t i o n o f t o t a l drag c o e f f i c i e n t w i t h Reynolds number f o r v a r i o u s a s p e c t r a t i o s o f a p r o l a t e s p h e r o i d .  105  Variation of C spheriods.  107  D S  /C  D p  with  aspect aspect  ratio ratio  f o r oblate f o r prolate,  Variation of frontal oblate spheroids.  stagnation pressure  12. V a r i a t i o n o f f r o n t a l prolate spheroids.  stagnation pressure  13. S u r f a c e 14.  51  system.  i t s outer  10. V a r i a t i o n o f G p p / G p g w i t h spheroids. 11.  47  pressure  with  A.R.  108 f o r 109  distribution  Surface pressure d i s t r i b u t i o n R e y n o l d s number.  with  A.R.  f o r 110  f o r a s p h e r e a t Re=10. o f a sphere  112  at high 113  15. S u r f a c e p r e s s u r e Re=1.0.  distribution  f o r oblate  spheroids at  16. S u r f a c e p r e s s u r e Re=5-0.  distribution  f o r oblate  spheroids at  17. S u r f a c e Re=10.  pressure  distribution  18. S u r f a c e Re=20.  pressure  114 115  f o r oblate  spheroids at 1  distribution  f o r oblate  1  6  spheroids at 117  ix 19.  Surface pressure d i s t r i b u t i o n f o r o b l a t e spheroids at Re=50.  20.  118  Surface pressure d i s t r i b u t i o n f o r oblate spheroids at Re=100.  21.  119  Surface pressure d i s t r i b u t i o n f o r p r o l a t e spheroids a t Re=1.0.  121  22. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r p r o l a t e s p h e r o i d s a t Re=5.0.  23.  122  Surface pressure d i s t r i b u t i o n f o r p r o l a t e spheroids a t Re=10.  123  24. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r p r o l a t e s p h e r o i d s a t Re=20.  25.  124  Surface pressure d i s t r i b u t i o n f o r p r o l a t e spheroids a t Re=50.  •  125  26. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r a p r o l a t e s p h e r o i d a t Re=100.  126  S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an o b l a t e s p h e r o i d w i t h A.R. = 0.999 ( S p h e r e ) .  127  S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an o b l a t e s p h e r o i d w i t h A.R. = 0.9.  128  S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an o b l a t e s p h e r o i d w i t h A.R. = 0.5-  129  30. S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an o b l a t e s p h e r o i d w i t h A.R. = 0.2.  130  31..Surface v o r t i c i t y d i s t r i b u t i o n f o r a p r o l a t e spheroid w i t h A.R. = 0 . 9 .  131  32. S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r a p r o l a t e s p h e r o i d ; w i t h A.R. = 0 . 5 .  132  27. 28. 29. ;  33.  Surface v o r t i c i t y d i s t r i b u t i o n f o r a p r o l a t e spheroid w i t h A.R. = 0.2.  133  34. V a r i a t i o n o f wake l e n g t h w i t h Re f o r s p h e r o i d s .  135  35-  136  Location of separation point f o r spheroids.  36. S t r e a m l i n e s f o r a n e a r l y s p h e r i c a l o b l a t e s p h e r o i d (A.R. = 0 . 9 9 9 ) .  37. S t r e a m l i n e s f o r an o b l a t e s p h e r o i d w i t h a s p e c t 0.9.  ratio  138  139  X  38.  Streamlines  f o r an o b l a t e s p h e r o i d w i t h a s p e c t  ratio  0.5.  39.  Streamlines  f o r an o b l a t e s p h e r o i d w i t h a s p e c t  ratio  0.2. i 4 l  40. V o r t i c i t y (A.R.  lines  f o r a nearly s p h e r i c a l oblate spheroid  .  = 0.999)-  41. V o r t i c i t y  1  lines  f o r an o b l a t e s p h e r o i d w i t h a s p e c t  ratio  Vorticity 0:5.  lines  f o r an o b l a t e s p h e r o i d w i t h a s p e c t  ratio  43. V o r t i c i t y 0.2.  lines  f o r an o b l a t e s p h e r o i d w i t h a s p e c t  ratio  4  2  143  0.9.  42.  140  -. hl  . 1 ^ 5  44. S t r e a m l i n e s  f o r a p r o l a t e spheroid w i t h aspect  ratio  0.9.  146  45.  Streamlines  f o r a p r o l a t e spheroid w i t h aspect  ratio  0.5.  147  46. S t r e a m l i n e s  f o r a p r o l a t e spheroid with aspect  ratio  0.2.  148  47. V o r t i c i t y 0.9-  lines  f o r a p r o l a t e spheroid with aspect  ratio  48. V o r t i c i t y  lines  f o r a p r o l a t e spheroid with aspect  ratio  lines  f o r a p r o l a t e spheroid with aspect  ratio  0.5.  49. V o r t i c i t y 0.2.  50. V e l o c i t y l i n e s  f o r spheroids  a t Re-100.  149 150 151  153  51.  S u r f a c e p r e s s u r e d i s t r i b u t i o n a t h i g h Re f o r a n o b l a t e s p h e r o i d w i t h A.R. = 0.999 ( s p h e r e ) .  154  52.  Surface pressure d i s t r i b u t i o n s p h e r o i d w i t h A.R. = 0.2.  155  53.  S u r f a c e p r e s s u r e : d i s t r i b u t i o n a t h i g h Re s p h e r o i d w i t h A.R. = 0.2.  a t h i g h Re f o r a n o b l a t e for a prolate  54. V a r i a t i o n o f t h e f r o n t a l s t a g n a t i o n p r e s s u r e w i t h number f o r o b l a t e s p h e r o i d s o f h i g h a s p e c t r a t i o .  Reynolds  55. V a r i a t i o n o f t h e f r o n t a l s t a g n a t i o n p r e s s u r e w i t h number f o r o b l a t e s p h e r o i d s o f l o w a s p e c t r a t i o .  Reynolds  56. V a r i a t i o n o f t h e f r o n t a l s t a g n a t i o n p r e s s u r e w i t h number f o r p r o l a t e s p h e r o i d s w t t h A.R. = 0.9.  Reynolds  156 -^g 159 160  57. V a r i a t i o n o f t h e f r o n t a l s t a g n a t i o n p r e s s u r e w i t h ' R e y n o l d s number f o r p r o l a t e s p h e r o i d s o f l o w a s p e c t r a t i o . l 6 l  xi  58.  V a r i a t i o n o f the boundary l a y e r n u m b e r f o r s p h e r o i d s a t n = TT/2. Drag  60.  V a r i a t i o n of t o t a l drag c o e f f i c i e n t number f o r e l l i p t i c a l c y l i n d e r s .  62. 63. 64.  for circular  with  Reynolds 162  59.  61.  coefficients  thickness  cylinders. with  164 Reynolds 166  V a r i a t i o n of t o t a l drag c o e f f i c i e n t with Reynolds number f o r f l o w a l o n g t h e m a j o r a x i s , w h i c h i s t a k e n the c h a r a c t e r i s t i c l e n g t h .  169  Variation of the f r o n t a l stagnation pressure orientation ratio f o r e l l i p t i c a l cylinders.  170  c  c  with  f o r a nearly  Surface pressure d i s t r i b u t i o n c y l i n d e r w i t h O.R. = 1 0 / 9 .  f o r an  Surface pressure d i s t r i b u t i o n c y l i n d e r w i t h O.R. = 2.  f o r a n e l l i p t i c a l <-  67.  Surface pressure d i s t r i b u t i o n c y l i n d e r w i t h O.R. = 5.  f o r an  elliptical  68.  Surface pressure d i s t r i b u t i o n c y l i n d e r w i t h O.R. = 0.2.  f o r an  elliptical  66.  69.  circular !71  elliptical 1  70.  Surface v o r t i c i t y d i s t r i b u t i o n c y l i n d e r w i t h O.R. = 1 0 / 9 .  f o r an  elliptical  71.  Surface v o r t i c i t y d i s t r i b u t i o n c y l i n d e r w i t h O.R. = 2.  f o r an  elliptical  72.  Surface v o r t i c i t y d i s t r i b u t i o n c y l i n d e r w i t h O.R. =5.  f o r an  elliptical  73.  Surface v o r t i c i t y d i s t r i b u t i o n c y l i n d e r w i t h O.R. = 0.2.  f o r an  elliptical  74.  Variation cylinder. Location  length with  174  circular 1  point  7  7  178  179  180  181  Re f o r e l l i p t i c a l 1  of separation  2  !75 f o r a nearly  o f wake  7  173  Surface v o r t i c i t y d i s t r i b u t i o n cylinder (O.R. = 1 / 0 . 9 9 5 ) .  75.  168  V a r i a t i o n o f n s / r j p w i t h R e y n o l d s number f o r v a r i o u s orientation ratios of e l l i p t i c a l cylinders.  Surface pressure d i s t r i b u t i o n cylinder (O.R. = 1 / 0 . 9 9 5 ) .  65.  as  for elliptical  cylinders.  °  2  183  xii -76.  77. 78. 79. 80. 81.  C o m p a r i s o n o f s u r f a c e p r e s s u r e d i s t r i b u t i o n on e l l i p t i c a l c y l i n d e r s a t Re=40 w i t h p o t e n t i a l f l o w . "Variation of the f r o n t a l s t a g n a t i o n pressure with R e y n o l d s number f o r e l l i p t i c a l c y l i n d e r s o f A.R.=0.2. V a r i a t i o n of the f r o n t a l s t a g n a t i o n pressure R e y n o l d s number f o r e l l i p t i c a l c y l i n d e r s .  with 187  188  Flow c h a r a c t e r i s t i c s O.R.-= 2 a t R e = 0 . 0 1 ,  190  Streamlines  f o r an  elliptical  for a nearly circular  cylinder  with  cylinder  = 1/0.995).  Vorticity  (O.R.  lines  191  for a nearly circular  cylinder  = 1/0.00.5).  192  83a.  S t r e a m l i n e s f o r an at Re=l and 5.  83b.  S t r e a m l i n e s f o r an e l l i p t i c a l at Re=15 and 40.  c y l i n d e r w i t h O.R.=10/9  S t r e a m l i n e s f o r an e l l i p t i c a l at Re=90.  c y l i n d e r w i t h O.R.=10/9  83c.  elliptical  cylinder  with  O.R.=10/9  c y l i n d e r w i t h O.R.=  2  84b. S t r e a m l i n e s f o r an e l l i p t i c a l at Re=15 and 50.  c y l i n d e r w i t h O.R.=  2  85a.  S t r e a m l i n e s f o r an e l l i p t i c a l at Re=l and 5.  c y l i n d e r w i t h O.R.=  5  S t r e a m l i n e s f o r an e l l i p t i c a l at Re=20 and 40.  c y l i n d e r w i t h O.R.=  5  S t r e a m l i n e s f o r an e l l i p t i c a l at Re=l and 5.  c y l i n d e r w i t h O.R.=  S t r e a m l i n e s f o r an e l l i p t i c a l at Re=15 and 40.  c y l i n d e r w i t h 0,R.=  86a. 86b. 87a. 87b.  193 194  84a. S t r e a m l i n e s f o r an e l l i p t i c a l a t ' R e = l and 5.  85b.  186  V a r i a t i o n of the boundary l a y e r t h i c k n e s s w i t h R e y n o l d s n u m b e r f o r e l l i p t i c a l c y l i n d e r s a t n=ir/2  (O.R. 82.  185  195 196 197 198 199  0.2 200 0.2 201  V o r t i c i t y l i n e s f o r an e l l i p t i c a l O.R. = 10/9 at Re=l and 5 .  cylinder with  V o r t i c i t y l i n e s f o r an e l l i p t i c a l O.R. = 10/9 at. Re=15, 40 and 90.  cylinder with  2 0 2  2  03  xiii 88a, V o r t i c i t y l i n e s f o r a n e l l i p t i c a l O.R. = 2 a t R e = l a n d 5.'  cylinder  with  88b. V o r t i c i t y l i n e s f o r a n e l l i p t i c a l O.R. = 2 a t Re=15 a n d 5 0 .  cylinder  with  89a. V o r t i c i t y l i n e s f o r a n e l l i p t i c a l O.R. = 5 a t R e = l a n d 5.  cylinder  with  89b. V o r t i c i t y l i n e s f o r a n e l l i p t i c a l O.R. = 5 a t Re=20 a n d 40.  cylinder  with  90a. V o r t i c i t y l i n e s f o r a n e l l i p t i c a l O.R. = 0.2 a t R e = l a n d 5.  cylinder  with  90b. V o r t i c i t y l i n e s f o r a n e l l i p t i c a l O.R. = o.2 a t Re=l-5 a n d 4 0 .  cylinder  with •  204 2  °5  206  2  91a. V e l o c i t y l i n e s f o r a n e l l i p t i c a l O.R. = 0.2. -  cylinder  91b. V e l o c i t y l i n e s f o r a n e l l i p t i c a l O.R. = 10/9.  cylinder  0  7  208  2  °9  with 2  1  0  2  1  1  with  92.  Variation of velocity r a t i o with concentration f o r o b l a t e s p h e r o i d s : H a p p e l ' s m o d e l , Re=0.01.  213  93.  Variation of velocity r a t i o with concentration f o r o b l a t e s p h e r o i d s : K u w a b a r a ' s m o d e l , Re=0.01.  214  94.  Variation of velocity r a t i o with concentration f o r p r o l a t e s p h e r o i d s : H a p p e l ' s m o d e l . Re=0.01.  215  95.  Variation of velocity r a t i o with concentration f o r p r o l a t e s p h e r o i d s : K u w a b a r a ' s m o d e l , Re=0.01.  216  96.  V a r i a t i o n o f t h e Kozeny c o n s t a n t w i t h c o n c e n t r a t i o n for oblate spheroids.  97.  V a r i a t i o n o f t h e Kozeny c o n s t a n t w i t h for prolate spheroids.  concentration  .„  "  98.  Flow p a t t e r n s  99.  V a r i a t i o n o f t h e t o t a l drag c o e f f i c i e n t w i t h conc e n t r a t i o n f o r e l l i p t i c a l c y l i n d e r s w i t h 0,R.>< 1.0: H a p p e l ' s m o d e l , Re=0.01.  223  V a r i a t i o n o f t h e t o t a l drag c o e f f i c i e n t with conc e n t r a t i o n f o r e l l i p t i c a l c y l i n d e r s w i t h O.R.^ 1.0: K u w a b a r a ' s m o d e l , Re=0.01.  224  100.  f o r an o b l a t e s p h e r o i d a l c e l l  A.R. = 0.2, c = 0.1832.  model,  221  XIV  101.  102.  103.  " V a r i a t i o n o f t h e . t o t a l drag c o e f f i c i e n t with conc e n t r a t i o n f o r e l l i p t i c a l c y l i n d e r s w i t h O.R. ^ 1.0: K u w a b a r a ' s m o d e l , Re=0.01.  225  V a r i a t i o n o f the t o t a l drag c o e f f i c i e n t with conc e n t r a t i o n f o r e l l i p t i c a l c y l i n d e r s w i t h O.R. £-1.0:' K u w a b a r a ' s m o d e l , Re=0.01.  226  V a r i a t i o n o f Kozeny c o n s t a n t w i t h O.R. .< 1.0 a t Re = 0.01.  227  for elliptical '  cylinders  104.  V a r i a t i o n o f Kozeny c o n s t a n t w i t h c o n c e n t r a t i o n f o r e l l i p t i c a l c y l i n d e r s w i t h O.R.> 1.0 a t Re=0.01.  228  105.  V a r i a t i o n o f AP/L w i t h o r i e n t a t i o n t i c a l c y l i n d e r s , Rew=0.01.  229  106.  Square  107.  Comparison between the by Kuwabara's z e r o v o r t a s i n g l e row o f e l l i p t i extended t o a square c e  108.  cell  model  for elliptical  ratio for  ellip-  cylinders.  t o t a l drag c o e f f i c i e n t obtained i c i t y c e l l model and f l o w past c a l c y l i n d e r s , a l s o by Kuwabara, l l m o d e l : Re=0.01.  Equi-vorticity lines for elliptical O.R. = 5, c =-0.200.  cell  231  232  models, 233  XV  LIST OF DIAGRAMS  Diagram 1  Page A c t i o n o f the normal and t a n g e n t i a l s t r e s s e s on an element of a r e a of t h e oblate spheroid  69  A c t i o n o f the normal and t a n g e n t i a l s t r e s s e s on an element of a r e a o f the e l l i p t i c a l cylinder  76  Appendix I I I 111.1  Oblate s p h e r o i d  111.2  E l l i p t i c a l cell.model  III-3  111.3  Main and average f l o w d i r e c t i o n s f o r packed beds  III-4  E v a l u a t i o n of a n g l e 6  III-5  111.4  c e l l model  III-l  xvi  ACKNOWLEDGEMENT  I  gratefully  encouragement this  f r o m D r . Norman E p s t e i n ,  investigation Thanks  uous  interest I  to  u n d e r whose  direction  was c o n d u c t e d .  i n this  am i n d e b t e d  port.  Thanks  Centre  f o rt h e i r  support  guidance and  a r e e x t e n d e d t o Dr.. Z e e v R o t e m f o r h i s c o n t i n -  the National  I  acknowledge the h e l p f u l  work. to the University  Research Council  are also  am a l s o  o f Canada  Columbia and  f o rfinancial  d u e t o t h e m e m b e r s o f t h e U.B.C.  cooperation indebted  throughout this  of British  Computing  and forbearance.  t o my w i f e  work.  sup-  Esther  f o rh e r c o n t i n u a l  CHAPTER I Introduction  The engineering fluid  wide  range  o f systems  and suspended  of applications  i n chemical  i n which  takes  solid  a greater understanding The dynamics  occur  particles from  of their  i n situations  One common a p p l i c a t i o n  pheric  pollution,  solid  smelters  represents another water,  chemical  from  dusts  where  i n which  not only  and from  leaving  transfer  a n d mass t r a n s f e r ,  at which  these  transfer  fluidized  can take  processes  of the f l u i d  o f atmosair.  The  d r y e r s and Seepage  and s e d i m e n t a t i o n interactions. unit  operation i n  bed r e a c t o r s a r e  reaction,  but also  place i n the bed.  occur  liquid  idealiza-  application.  of solid-fluid  chemical  collection  o f such  h a s become a n i m p o r t a n t  industries,  the mechanics  and m i s t  particles  flow of o i l i n wells  designed  by  Dust  examples  important  a f e w m o r e common e x a m p l e s Fluidization  the  interaction of  as i n t h e c l e a n i n g o f v e n t i l a t i n g  of valuable by-products  are  of particle  i s i n the elimination  recovery  a  characteristics.  applications  are simple p r a c t i c a l  tion.  of underground  underlying  where mutual  of fine  p l a c e between  has l e dt o t h e need f o r  as n e g l i g i b l e .  suspensions  i n gases  particles  straightforward  c a n be t a k e n  dilute  drops  most  motion  and process  are strongly  f l o w , knowledge  heat Rates  influenced  of which  i s basic  2  in  the fundamental  relationship  study  between them  Gravitational are  also  received  the attention  still More  of  4,  of chemical  designed  separation processes  particle-fluid  industry  and m i n i n g  on a f a i r l y work w i l l  interactions,  empirical  t h a t t h e d e s i g n o f any d e v i c e w h i c h  actions  c o u l d b e made m o r e r a t i o n a l  of the laws  governing  them.  as one  over  devices  settling  are  basis.  give a better  a n d i t seems  have  engineers  by h i n d e r e d  suppose  ledge  and  s e d i m e n t a t i o n and t h i c k e n i n g  of particles  theoretical  and t h e  8).  and c e n t r i f u g a l  However,  the c l a s s i f i c a t i o n  usually  ( 1 , 2,  w i d e l y employed i n the chemical  a long period. and  o f t h e t r a n s p o r t phenomena  understanding  reasonable  to  i n v o l v e s such a c q u i r e s more  interknow-  3  CHAPTER' I I Review of . an  The  Work .  study  incompressible  rigid of  Prior  body  the  the  only  Payne and  can  equations  Pell  (7).  Brenner 1.  Review of  A.  Single  fluid  continuity  renders  known:  result  (5),  Lamb  less  a large part obtained  Brenner  (9).  Analytical  Work  of  Dryden  in this  theory  of  the  difficult, rather  (6),  flow  regarding  to  of  s o l u t i o n s of  complicated the  subject  very  however, the  assumptions  in a  solution  non-linearity  exact  of  impermeable  equations,  solution  s m a l l number o f  an  simultaneous  The  their  about  flow  and  is  such  i t s character  mathematical viscous  flows  manner, c . f . Happel  and  Bodies  The bodies  and  fluid  I n many c a s e s ,  of problems (8)  isothermal laminar  conditions.  simplifying  Indeed,  consists  the  c h a r a c t e r are  made w h i c h  problem.  past  boundary  reasonable be  and  a relatively  specialized  steady  t h e r e i n r e q u i r e s the  Navier-Stokes  Navier-Stokes  that  the  viscous Newtonian  immersed  prevailing  and  of  mathematical  i s the  first  a n a l y s i s of  stage  interactions.  i n the  In the  orthotropic bodies,  study'of  and  elliptical  circular  are  each  cylinders.  a special  case  of  of  isolated particle-  symmetric  c l a s s e s of bodies  namely, spheroids cylinder  flow past  understanding  present  two  the  were  flow  considered,  A  sphere  and  a  spheroid  a  and  an  elliptical defined  cylinder,  respectively.  as a body which  symmetry p l a n e s . special  case  Navier-Stokes  , Zj  v  =h h h  r  1  , h  :  2  and h  2  E  h 2  =  i o  3  3  h  a  L  g  >  ) - 2E  0  viscosity  linear  because  H  o f the presence  form  of the Navier-Stokes  J  -  p i s the density, y dynamic p r e s s u r e  F  )  h3  -  R  R  (1)  coefficients,  X1  (2)  h i 8n J The  elliptical,  are discussed  order  i nV  and i t i s n o t  on t h e r i g h t -  effects.  Another  i s g i v e n by  (3)  i s t h e v i s c o s i t y , p»  and  i s t h e dimen-  v' i s t h e d i m e n s i o n a l — 2  and  V  with  (8) a n d b y G o l d s t e i n  VP*  —  The o p e r a t o r s V  T  stream  3 2  o f the terms  equation  =  2  2  systems  c o n t r i b u t e d by t h e i n e r t i a l  p v ' . V v' -y .V v'  • ^'>  * ; _  metric  and Brenner  1 i s of the fourth  side,  i n orthogonal  3 (  coordinates.  by H a p p e l  hand  state a x i -  2  <M  2  clarity  Equation  ,2-  3  the curvilinear  (10).  f o r steady  . h hj 3 . 3 h ' 3n  excellent  vector.  and  and  and s p h e r o i d a l c o o r d i n a t e  sional  a  cylinders  of the dimensional  spherical  where  elliptical  are the dimensional  L  3  i n terms  (  3  h  -  (£,n) b e i n g  perpendicular  (spheroids being  flow, expressed  9 (ijAE  i s the kinematic  ,  equation  or two-dimensional  curvilinear coordinates t function ^ , i s  where h  right  particle i s  p a r a l l e l e p i p e d s (8).  The  v E  three mutually  I t includes ellipsoids  o f an e l l i p s o i d ) ,  rectangular  symmetric  possesses  An o r t h o t r o p i c  a r e g i v e n by  velocity  5  (4)  and V  2  = hi h  2  h  r3  3  +  l 3qj  +  3  h  3q  hi 2  h  2  h  and  ( i  i  ) the unit  A.l  Spheroids  2  ,  3  }  q  The o l d e s t so-called the of  "Stokes  inertial  >Q  2  3  flow  vectors.  formulation flow".  effects  i s very  characteristic  of flow  are negligible  small.  flow  that  velocity  and/or  ( 3 ) j i n 1851,  the i n e r t i a l  then reduces  the Reynolds  fluid,  terms  with  those  while  appear-  and/or  considering t h e motion  of  seems t o h a v e b e e n t h e f i r s t  to  of the Navier-Stokes equation, which  to  (6)  = 0 Such a s i m p l i f i c a t i o n o f t h e e q u a t i o n o f m o t i o n stantial,  that  number o f  t h e body d i m e n s i o n  kinematic v i s c o s i t y i s large (8).  omit  by t h e a s s u m p t i o n  number a r e s u f f i c i e n t l y s m a l l ,  i n a viscous  i s the  T h i s s i t u a t i o n i s o b t a i n e d when t h e  the  sphere  an o b s t a c l e  i n comparison  i n the Reynolds  a  past  I t i s defined  ing  Stokes  2  ) being the c u r v i l i n e a r coordinates  v i s c o s i t y , o r more p r e c i s e l y ,  the  3q  (5)  1  i  3  -I  h  (q  ,  h hi  2  3  3  respectively, x  3q  9Qi  3  f o r the equation of motion  i s very  t h u s becomes  sub-  linear.  6 means, f o r e x a m p l e , t h a t i f ij/ a n d  This  satisfy Exact  the aforementioned  solutions  geometries, to  linear  in  spherical  are then  equation, feasible  v i a classical  partial  t|/  each  2  then  separately  s o d o e s ty\ + ty\ .  even f o r r e l a t i v e l y  complex  superposition techniques applicable  differential  equations.  A general  c o o r d i n a t e s has been developed  solution  by Sampson  using  the separation of variables  technique.  given  independently  and by Haberman and  by S a v i c  (13), a n d h a s b e e n r e v i e w e d The c o m p l e t e functions  i s given  iMr,6) =  +  by H a p p e l  solution  I t has a l s o  and Brenner  of equation  been  Sayre  (8).  6 f o r the  stream  r  r  by  X  |  (12)  (11)  (Anr " + B  (A  n  r  n  + B  n  n  r  _+  n + 1  r  _  n  +  1  C  n  + C  + D  n + 2  r  n  n  +  2  n  + D  n  ) I  n + 3  r  _  n  +  3  )  (O  n  H  n  (7) where and term  (r,0) a r e t h e s p h e r i c a l  Dn a n d t h e c o r r e s p o n d i n g  primed  terms  5 i s d e f i n e d b y t, = c o s 8. I ( C ) a n d n  functions  o f order n anddegree  respectively.The boundary the  c o o r d i n a t e s , w h i l e A n , B n , Cn  streaming  H. (0n  of the f i r s t  are evaluated  conditions.For the no-slip  uniform  function  constants  -1/2  are constants.  from  a r e Gegenbauer and second  type,  the applicable  c o n d i t i o n at the sphere  condition at i n f i n i t y  as g i v e n by e q u a t i o n  The  7 becomes  , the Stokes  and  stream  (?)  ij/  = % U a  2  2 + 2 ^)  (| - ^  sin e  (8)  2  where a i s t h e r a d i u s o f t h e s p h e r e a n d U i s t h e v e l o c i t y o f the u n d i s t u r b e d  stream,  of the undisturbed  the l i n e  p U  2  being  i n the d i r e c t i o n  stream.  The d i m e n s i o n a l P* = I  6=0  surface pressure  i s then  g i v e n by  (1 + p | c o s e )  (9)  where Re i s t h e R e y n o l d s number d e f i n e d as Re = ^ The d r a g F  (10)  force experienced 4ITU D  =  (11)  2  where D 2 i s a c o n s t a n t 75- a U . F  by t h e s p h e r e i s g i v e n by  from e q u a t i o n  7 which i s equal t o  Hence =  6Try  aU  (12)  T h i s i s t h e w e l l known S t o k e s '  law o f r e s i s t a n c e .  The g e n e r a l s o l u t i o n o f t h e c r e e p i n g f l o w e q u a t i o n 6 f o r s p h e r o i d a l c o o r d i n a t e s i s g i v e n by Sampson ( 1 1 ) :  V =C  0  +  I3  +  DjT  ) {  (C )  + ^ B3  D3  +  + \  (C ) { B* H  x =  cosh 4  2  {  B T 2  + C  2  H (x ) + B* 3  (T ) +  D,  H,  (T  X,  (T)  +  D  2  H,  ( ) + E*.H T  Hg (T ) }  ) + B* H,  (T ) }  (13)  where (14)  ( )} T  8  and =  z,  For  cos n  a prolate spheroid  ip' =  replacing  +  2  2  spheroid  C  I .  2  2  (T ) +  D  H  2  (x ) }  2  function  16 b y IX,  stream  functions  given  transformed  by e q u a t i o n  16  spheroid  2  {(T +1)/(T -1)} COth"^-{T/(T 2  a.  and  stream and by  become r e s p e c t i v e l y  1 -  c (x -l)(l-? ) 2  simply  surface  f o r a prolate spheroid  f o r an o b l a t e  2  by  where X i s sinh£ .  c o n d i t i o n at i n f i n i t y , the Stokes  form  2  (16)  i s given  the no-slip condition at the spheroid  a uniform  U  obtains  the stream  x i n equation  For  ib' = 1/2  one  BT  I (C) {  For an o b l a t e  its  (15)  -1)}  2  (17)  d.  {(X +1)/(T -1)} 2  COth  2  a  a  _ 1  T-{T /(T a  g_  d.  -1)}  2  d.  and  ip'= 1/2 U  c (A2+D(l-C ) 2  1 -  2  {X/(X +1)}-{(X 2  -D/(A  2  ci  {X  /(X a.  where  subscript  length  potential  a  by u s i n g  given  -D/(A  2  a.  denotes  cot  _ 1  A  2  +1)}  ci  the surface  Payne  cot  (18)  and P e l l  Weinstein's  _ 1  X, <•  and  c  the focal  (7) a r r i v e d a t t h e same  generalized axially  symmetric  theory.  The  F  a.  +1)}  d,  +1)}-{(X  of the spheroid.  expressions  then  2  2  drag  force  (7*8)  experienced  by t h e s p h e r o i d i s  by  = 6 TTU U a „ K  (19)  9  a  where  i s t h e e q u a t o r i a l r a d i u s o f t h e s p h e r o i d and  Q  =—  K  —  —  oblate  (20) { A'. a  (A  a  -1) c o t  x  xa }  while K , . = prolate  1  ZZZZZZ |  Al  -1 { ( T  2  (21)  +1) C O t h '  1  x  a  -  T > a  A comparison o f t h e drag f o r c e e x p e r i e n c e d by a s p h e r o i d as g i v e n by e q u a t i o n 19 w i t h t h a t f o r a sphere, a q u a t i o n 12, shows t h a t the r a t i o o f the two drag f o r c e s I s g i v e n by t h e f a c t o r K.  D e s p i t e a m i s p r i n t i n t h e i r e x p r e s s i o n f o r ^•Q-^j.ate' ^ P P l a  e  and Brenner (8) p r o v i d e an a c c u r a t e t a b u l a t i o n o f K as a f u n c t i o n o f aspect r a t i o A.R. f o r both o b l a t e and p r o l a t e s p h e r o i d s , A.R. b e i n g d e f i n e d as t h e r a t i o o f minor a x i s t o major a x i s o f t h e s p h e r o i d .  F o r a t h i n c i r c u l a r d i s c (A.R. =  0 ) , t h e t a b u l a t e d v a l u e o f K i s 0.84883-  A circular disc .is, ;  of c o u r s e , t h e l i m i t i n g case o f an o b l a t e s p h e r o i d whose minor axis i s zero.  The drag f o r c e e x p e r i e n c e d by a d i s c can be  o b t a i n e d from e q u a t i o n s F =  19 and 20 as  \ -y 0, g i v i n g a  16 y a U  (22)  e  S i m i l a r e x p r e s s i o n s f o r the drag f o r c e a r e g i v e n by Sampson (11),  Ray ( 1 4 ) , Roscoe (15) and Gupta ( 1 6 ) . The d i m e n s i o n l e s s  surface pressure d i s t r i b u t i o n f o r  Stokes f l o w p a s t an o b l a t e s p h e r o i d i s 6 K p  =  1 +  oblate Re  /  +  1  2 , -, \  1 x* +cos n / * a ' 2  2  cos n  (22a)  10 and f o r a p r o l a t e s p h e r o i d ,  ^a \! -f p-)cos n \ X + s i n n ./ a 1  6K prolate P = 1 + Re  C 22b)  Once a g a i n t h e u n i t y term o f e q u a t i o n s 22a and 22b comes from c o n s i d e r a t i o n o f t h e i n e r t i a l terms i n t h e N a v i e r - S t o k e s e q u a t i o n s , s i n c e - f o r s t r i c t l y c r e e p i n g f l o w i t i s absent. I t s h o u l d a l s o be mentioned t h a t Oberbeck (17) i n 1876 s o l v e d t h e problem o f t h e steady t r a n s l a t i o n o f an e l l i p s o i d i n a v i s c o u s l i q u i d w i t h o u t t h e use o f t h e stream f u n c t i o n (due t o l a c k o f symmetry), u s i n g i n s t e a d t h e v e l o c i t y components expressed ellipsoid.  i n terms o f t h e g r a v i t a t i o n a l p o t e n t i a l o f t h e  F o r t h e e l l i p s o i d s o f r e v o l u t i o n one o b t a i n s t h e  same r e s u l t s as g i v e n by e q u a t i o n s 17 and 18.  Lamb (5) d i s -  cusses Oberbeck's t e c h n i q u e i n d e t a i l . The  case when t h e i n e r t i a l e f f e c t s o f t h e f l o w a r e  s m a l l b u t not n e g l i g i b l e was f i r s t u n s u c c e s s f u l l y t r e a t e d by Whitehead (18), who attempted  t o extend S t o k e s '  original  s o l u t i o n f o r a t r a n s l a t i n g sphere t o h i g h e r Reynolds numbers using a simple p e r t u r b a t i o n technique.  This technique  starts  with the Stokes' s o l u t i o n of the creeping flow equation, V V v„ - v Pj = 2  and t h e c o n t i n u i t y V . v' = 0  0  ( 2 3 )  equation,  0  ( 2 l j )  f o r t h e boundary c o n d i t i o n s o f no, s l i p a t t h e s u r f a c e and an undisturbed p a r a l l e l flow at i n f i n i t y .  This s o l u t i o n , given  11 —t  by  '  ( v , P ) , i s then used f o r t h e i n e r t i a l term o f t h e N a v i e r 0  0  Stokes e q u a t i o n as f o l l o w s : y V  2  v[  * • fi v  x  - V pj  =p VQ.V  (25)  VQ  = o  (  2  b e i n g t h e improved v e l o c i t y v e c t o r o f t h e f l o w f i e l d .  though e q u a t i o n  )  Even  25 i s now l i n e a r and hence e a s i e r t o s o l v e than  the complete N a v i e r - S t o k e s no e x p o n e n t i a l l y d e c a y i n g  equation, there e x i s t s unfortunately s o l u t i o n t h a t can s a t i s f y t h e boun-  dary c o n d i t i o n s , v , = 0 a t t h e s u r f a c e and v from the s u r f a c e .  = U f a r away  Moreover, as Happel and Brenner (8) p o i n t  o u t , t h e next a p p r o x i m a t i o n at i n f i n i t y .  6  for velocity, v  2  , becomes  infinite  The r e s u l t s g i v e n by such p e r t u r b a t i o n a r e r e -  f e r r e d t o as Whitehead's paradox. Oseen (19, 20) showed t h a t Stokes' ized Navier-Stokes"equation  s o l u t i o n of the l i n e a r -  i s o f the. form v = U + Ua  0(l/ ). v  Hence a t l a r g e d i s t a n c e s from the s u r f a c e o f t h e sphere t h e i n -' - - ' 2 i e r t i a l term p v„.V v becomes p U a 0 ( / 2 ) and t h e v i s c o u s term -2 - ' i y V v becomes y Ua 0 ( / 3 ) . The r a t i o o f t h e i n e r t i a l t o t h e v i s c o u s term then becomes o f t h e o r d e r This r a t i o i n P 1  r  0  ±  r  0  creases i n d e f i n i t e l y w i t h r however s m a l l U/v may be. F o r t h i s r e a s o n Stokes'  c r e e p i n g f l o w e q u a t i o n cannot be c o n s i d e r e d as ;  v a l i d a t p o i n t s f a r away from the sphere u n l e s s Re+0. At p o i n t s near t h e sphere t h e i n e r t i a l f o r c e s tend t o v a n i s h while the viscous forces are o f the order ^f^-. Stokes' a p p r o x i m a t i o n  T h i s means t h a t  f o r t h e f l o w f i e l d i s not u n i f o r m l y  v a l i d throughout b u t breaks down a t a l a r g e d i s t a n c e from t h e  sphere, and hence v the  Q  does not g i v e an a c c u r a t e e s t i m a t e o f  i n e r t i a l term a t such l a r g e d i s t a n c e s .  Consequently i t  cannot be used f o r t h e f i r s t a p p r o x i m a t i o n o f t h e i n e r t i a l terms as proposed by Whitehead. _i  _i  Oseen suggested t h a t t h e i n e r t i a l term p v .V v _  be  _ _i  r e p l a c e d by pU .V v , g i v i n g yv  2  v' -  VP*  (27)  = p U • vv'  V.v' = 0  (28)  The i n e r t i a l terms a r e thus t o some e x t e n t t a k e n i n t o a c c o u n t , the  p r e v i o u s a p p r o x i m a t i o n b e i n g much improved a t i n f i n i t y  but somewhat i m p a i r e d near t h e s u r f a c e .  The d r a g f o r c e  e x p e r i e n c e d by a sphere i s t h e n g i v e n by P =  where  a  6TT a U (1 + (3/16) Re)  (29)  u  i s t h e r a d i u s o f t h e sphere and Re i s t h e Reynolds  number based on t h e sphere d i a m e t e r . Oseen's o r i g i n a l s o l u t i o n o f e q u a t i o n s 27 and 28, a l s o developed i n d e p e n d e n t l y by Burgers e t a l . (21) o n l y an approximate s o l u t i o n .  i n 1916, i s  The complete a n a l y t i c a l  of Oseen's e q u a t i o n s was f i r s t d e r i v e d by G o l d s t e i n  solution  (22, 23),  g i v i n g t h e drag f o r c e as a s e r i e s i n Reynolds number: F =  6 iryU  f  19 Re 71 Re 1 + j j - R e - - y ^ - +.  3  3  -  30179 Re 344 06400  122519 R e 560742400 " 5  +  (30) The f i r s t term o f t h e above e q u a t i o n i s t h e v a l u e g i v e n by  13  S t o k e s , w h i l e t h e second term i s t h a t g i v e n by Oseen and by Burgers e t a l . F o r v a l u e s o f Re up t o and i n c l u d i n g 2, t h e drag can be c a l c u l a t e d from t h e s e r i e s as g i v e n by e q u a t i o n 30.  F o r Re up t o 20, G o l d s t e i n p r e s e n t e d a t a b u l a t i o n o f F  a g a i n s t Re o b t a i n e d by s o l v i n g one o f t h e i n t e r m e d i a t e s t e p s of Oseen*s e q u a t i o n s n u m e r i c a l l y . Tomotika and A o i (24) c o n s i d e r e d Oseen's e q u a t i o n s and o b t a i n e d t h e d i m e n s i o n a l stream f u n c t i o n as </ = _ u a Y |  +  2  ^  a  - ^ r  2  In t h e l i m i t when Re-»- 0  1 6  cos 9}  +  3  R e  2  (4 " •  sin 6  (3D  2  t h e above e q u a t i o n degenerates  into  the w e l l known Stokes stream f u n c t i o n as g i v e n by e q u a t i o n 8. t  Based on t h e above e x p r e s s i o n f o r \p , Tomotika and A o i c l a i m e d t h a t a s m a l l v o r t e x i s formed behirid a sphere even f o r Re as low as 1.0.  Pearcey and MoHugh  d e t a i l e d computation  (25a) , however, i n t h e i r  o f Oseen's e q u a t i o n s , p o i n t e d out t h a t  no v o r t e x i s a t t a c h e d t o t h e r e a r o f t h e sphere even a t Re = 10, and t h a t e q u a t i o n 31 I s o n l y v e r y approximate  as i t  was o b t a i n e d by an e a r l y t r u n c a t i o n o f t h e s e r i e s f o r one o f the c o n s t a n t s i n the s o l u t i o n .  The d i m e n s i o n l e s s s u r f a c e  p r e s s u r e d i s t r i b u t i o n , P, g i v e n by Pearcey and McHugh (25a) ' c o u l d be put i n t h e form P = ,  =  F  1/2 2 P U  2 f  *  A  n  P  n +  i cos 6  (32)  where P ] _ • i s a Legendre p o l y n o m i a l o f t h e f i r s t k i n d . n +  A  14 t a b u l a t i o n o f t h e c o e f f i c i e n t A . i s g i v e n i n T a b l e 1. n  Table 1. V a l u e s o f t h e c o e f f i c i e n t Re n  Ref  1 (25a)  4 Ref (25a)  Ref  10 (25a)  Ref  (25b)  0  3.52627  1.21748  0.7169  0.13225  1  0.45078  0.38926  0.32761  0.23725  3  0.2216 x 10"•2 •4 -0.7004 x 10"  4  0.1726 x 10"•5  0.861  5  -0.3496 x 10"•7  -0.421  2  6  0.6 x 10~  9  0.15192 x 1 0 -0.1323 x 1 0 x 10  _ 1  _i|  x 10~  0.14 x 10~  5  - 1  0.3363 x 10" 3 2 -0.402 x 10" x 10" 3 -4 -0.1 x 10 0.22  0.1600 -0.0055 -0.05 0.003 0.0270  6  7  -0.002  8  -0.0165  9  0.002  10  0.012  I t i s i n t e r e s t i n g t o note t h a t Tomotika and A o i (24) observed Oseen's s o l u t i o n t o g i v e a c o n s t a n t r a t i o o f t h e form to f r i c t i o n  (pressure)  (skin) drag c o e f f i c i e n t whatever t h e v a l u e o f t h e  Reynolds number. The s o l u t i o n o f Oseen's e q u a t i o n s f o r t h e case o f a s p h e r o i d was e f f e c t e d by an approximate method by Oseen h i m s e l f Aoi  (26) .  (27) j u s i n g an exact a n a l y t i c a l t e c h n i q u e as o u t l i n e d by  Goldstein  (22, 23) , o b t a i n e d i d e n t i c a l e x p r e s s i o n s t o Oseen 's  f o r both t h e form and.. the ..skin drag forces...  In order t o  maintain conformity with the corresponding creeping flow s o l u t i o n s , t h e e x p r e s s i o n f o r t h e t o t a l drag f o r c e becomes  15  6  F =  TT  +  u U a K { 1 e  3  }  I* lb  R e  (33)  where K i s g i v e n by e q u a t i o n 20 f o r an o b l a t e s p h e r o i d and by e q u a t i o n 21 f o r a p r o l a t e s p h e r o i d , w h i l e Re i s based on t h e e q u a t o r i a l diameter  o f the s p h e r o i d .  The r a t i o o f the form  drag t o the s k i n drag f o r a p r o l a t e and an o b l a t e s p h e r o i d by A o i (27) 2  as developed *  i s g i v e n by  1 - (Ta - 1) ( a c o t h  S  T  1  T  a - 1)  and P /F p  P  q  - J ^  S  + »  1 -  U  - *a o o f  + 1) (1 - X  a  1  oot-^^)  f o r a p r o l a t e and an o b l a t e s p h e r o i d , r e s p e c t i v e l y .  Accord-  i n g t o the above e x p r e s s i o n s , the r a t i o o f the drag f o r c e s are independent of Re. While Oseen's l i n e a r i z e d i n e r t i a l term seems a s a t i s f a c t o r y approximation Stokes  t o the t r u e i n e r t i a l term i n the N a v i e r -  e q u a t i o n at g r e a t d i s t a n c e s from the body, c o n t r o v e r s y  has been aroused approximation  by the f a c t t h a t i t appears t o be a poor  i n the neighbourhood o f the body.  The  u n s a t i s f a c t o r y s t a t u s o f the Oseen e q u a t i o n s p e r s i s t e d u n t i l the work o f Lagerstrom and Pearson  (30)  and Cole  (28) , Kaplun  and Kaplun and Lagerstrom  (29) , Proudman  (32) , who  suggested  t h a t one should abandon the attempt t o o b t a i n p e r t u r b a t i o n f i e l d s which are u n i f o r m l y v a l i d throughout the f l o w and seek i n s t e a d t o f i n d s e p a r a t e a s y m p t o t i c  field  s o l u t i o n s which  are l o c a l l y v a l i d i n the s e p a r a t e r e g i o n s near t o , and f a r from, the body. These " i n n e r " and " o u t e r " s o l u t i o n s a r e each d e t e r -  16 mined by a s y m p t o t i c a l l y matching them i n t h e i r common domain of v a l i d i t y . (30)  The e x p r e s s i o n o f Proudman and Pearson  f o r the  drag f o r c e e x p e r i e n c e d by a sphere p l a c e d i n a u n i f o r m  stream-  i n g f l o w i s g i v e n by 6 ir y D a (1 + ^  P =  .Re +  Re  (36)  In- (Re/2))  2  The f i r s t two terms a r e i d e n t i c a l t o Oseen's e x p r e s s i o n , e q u a t i o n 29.  The above e x p r e s s i o n g i v e s b e t t e r agreement w i t h  e x p e r i m e n t a l d a t a than Oseen's e x p r e s s i o n , up t o Re o f 2. 2, t h e l o g a r i t h m i c term o f e q u a t i o n  U n f o r t u n a t e l y , f o r Re>  36 becomes l a r g e and thus t h e e q u a t i o n d i v e r g e s and can no l o n g e r be used t o e v a l u a t e t h e drag f o r c e .  extended t h e t e c h n i q u e o f Proudman and Pearson o t h e r terms,  Re •, 9^  3  2  +  —  t o i n c l u d e some  giving  6 T P U a (1 + l ! T  F =  (31b)  Breach  (  4u-  +  X  l  R e  +  1^0~  323 n  2  R e 2  l  n  27  - T 6 W - )  +  W -  ^'  R e / 2 )  3 R  e  l n  Re 2")  , +  ''-  ( 3 7 )  p  where and Re  y = 0.5772... 3 Re  i s Euler's constant.  l n — ^ ~ a r e new.  The terms i n Re  The new drag e q u a t i o n does not p r o v i d e  any improvement on t h e p r e v i o u s r e s u l t s by Pearson and Proudman and cannot be used f o r Re  >  0.7.  Breach  (31a)  g e n e r a l i z e d t h e above r e s u l t s t o render them a p p l i c a b l e t o spheroids.  These s o l u t i o n s a r e a g a i n l i m i t e d by t h e n a t u r e  of t h e a p p r o x i m a t i o n s  and prove t o be inadequate  f o r Re > 2.  The e x p r e s s i o n f o r the drag f o r c e on s p h e r o i d s can be r e w r i t t e n i n t h e form  !7 P = 6 i T y U a  e  2 + | - R e K + --^|Q R e 9  K { l  1  2  In~r  }  08)  Once a g a i n the f i r s t term i n the above e q u a t i o n i s f o r c r e e p ing  f l o w , the second b e i n g Oseen's e x t e n s i o n w h i l e the t h i r d  i s new.  For a s p h e r e , K = 1 and e q u a t i o n 38 becomes i d e n t i c a l  to t h a t o f Proudman and  A.2  Pearson.  E l l i p t i c a l Cylinders F o r the t w o - d i m e n s i o n a l  problem  of streaming flow past  a c i r c u l a r c y l i n d e r i n an i n f i n i t e medium, the c r e e p i n g f l o w e q u a t i o n 6 has no v a l i d s o l u t i o n t h a t can s a t i s f y b o t h the n o - s l i p c o n d i t i o n at the s u r f a c e of the c y l i n d e r and s t r e a m i n g f l o w at i n f i n i t y . Stokes  (65)  T h i s s i t u a t i o n was  d i s c o v e r e d by  and i s u s u a l l y r e f e r r e d t o as "Stokes  Krakowski and Charnes  (66)  uniform  Paradox".  ' g e n e r a l i z e d S t o k e s ' Paradox t o  i n c l u d e any t w o - d i m e n s i o n a l infinite i n a l l directions.  body p l a c e d i n a medium which i s I n v a l i d s o l u t i o n s which  defy  S t o k e s ' paradox f o r the s t r e a m i n g motion p e r p e n d i c u l a r t o the a x i s o f a c i r c u l a r c y l i n d e r are g i v e n by Wilton  (68)  Berry and Swain  (67)  ,  and H a r r i s o n (69) .  Lamb (5)  a p p l i e d Oseen's e q u a t i o n s t o the case of f l o w  past a c i r c u l a r c y l i n d e r , and the d i f f i c u l t y of s a t i s f y i n g a l l the boundary c o n d i t i o n s d i s a p p e a r e d .  H i s approximate  solution  to Oseen's e q u a t i o n s g i v e s the drag f o r c e per u n i t l e n g t h o f the c y l i n d e r as  18  The v a l i d i t y o f Lamb's e q u a t i o n i s w e l l e s t a b l i s h e d by the e x p e r i m e n t a l r e s u l t s o f W i e s e l s b e r g e r and Jayaweera and Mason (145).  (147), Finn  (146),  The e x a c t s o l u t i o n o f  Oseen's e q u a t i o n s was o b t a i n e d by Tomotika and A o i ( 2 4 ) . T h e i r e x p r e s s i o n f o r t h e d i m e n s i o n a l stream f u n c t i o n i s ij/= a U {A (a- - -r) 2 2 ^2 ~ 2 a r {  C  §  where A = f B = C  and a  )  2  +  D  B -a l n ( r / a ) } s i n e 2 ( a l  n  r  /  )  }  ( l n Re - 2.0022) > D  5f  =  s l n  2 9  ( 4 0 )  '  - B C ,  i s the c y l i n d e r r a d i u s .  the a u t h o r s  a  +  As f o r t h e case o f Sphere,  (24) c l a i m t h a t the above e x p r e s s i o n p r e d i c t s  a vortex at the r e a r of t h e . c y l i n d e r .  I t i s very  likely  t h a t t h i s e x p r e s s i o n i s t o o approximate i n i t s d e r i v a t i o n to put any weight  on the s t r e a m l i n e s g i v e n by i t .  The f o r c e  per u n i t l e n g t h i s g i v e n by  F = 2uyU  Z B o  m  ,  and the e v a l u a t i o n of t h e c o n s t a n t s B^, B numerically. expressed S  (41)  r  and B were done 2  I n a f u r t h e r paper Tomotika and A o i (70)  the drag f o r c e p e r u n i t l e n g t h i n s e r i e s terms o f  and Re:  19 P  47rUy  S  (42)  102 where S  0.5 -Y - I n (Re/8)  Y being Euler's constant.  (43) The f i r s t term i n e q u a t i o n  42 i s t h a t o b t a i n e d by Lamb ( 5 ) . e x p r e s s i o n i s q u i t e adequate.  F o r Re < 0.5, Lamb's The drag f o r c e p e r u n i t  l e n g t h o b t a i n e d by e q u a t i o n 42 agrees q u i t e w e l l w i t h t h e numerical  s o l u t i o n of equation  4 l up t o Re = 2.0.  Using  Oseen's e q u a t i o n , Tomotika and A o i found t h a t t h e r e l a t i v e c o n t r i b u t i o n s o f t h e form drag and t h e s k i n drag t o the t o t a l drag were t h e same and independent o f Reynolds number.  Kaplun (29) a p p l i e d t h e i n n e r and o u t e r ex-  p a n s i o n s f o r t h e f l o w past a c i r c u l a r  c y l i n d e r and ob-  t a i n e d t h e f o r c e p e r u n i t l e n g t h as  where  S i s g i v e n by e q u a t i o n 43. The  bounded f l u i d  f l o w p a s t an e l l i p t i c a l falls  c y l i n d e r i n an un-  i n t h e r e a l m o f t w o - d i m e n s i o n a l f l o w and  thus no v a l i d s o l u t i o n c o u l d be o b t a i n e d f o r t h e c r e e p i n g flow equation.  I n v a l i d s o l u t i o n s which contravene Stokes  paradox a r e d i s c u s s e d by B e r r y and Swain  (67).  20 Oseen's e q u a t i o n s  were s o l v e d a p p r o x i m a t e l y by  H a r r i s o n (69) and by B a i r s t o w e t a l . (71). per u n i t l e n g t h on t h e e l l i p t i c a l  The drag  force  c y l i n d e r whose major  a x i s , a, i s p a r a l l e l t o t h e stream i s g i v e n by  4rruU  F = —; ± (1 + w  h  e  r  e  0  =  (45)  ;  a) - y - I n Re/8  1 - b/a l + b/a  , a  n  d  R e  (1 + a)  D =  ~  2pUa  The drag f o r c e p e r u n i t l e n g t h e x p e r i e n c e d  by an e l l i p -  t i c a l c y l i n d e r whose minor a x i s , b, i s p a r a l l e l t o t h e stream i s g i v e n by  p  — i ^ l U  =  _  (  1  _  CT)  _ _ Y  _  m  8  (  1  +  q  )  For a c i r c u l a r c y l i n d e r a=b and e q u a t i o n s i n t o equation  39, g i v e n by Lamb.  (46)  45 and 46 c o l l a p s e  When a=1 e q u a t i o n s  45 and  46 g i v e t h e drag f o r c e p e r u n i t l e n g t h on a f l a t p l a t e f o r f l o w p a r a l l e l and p e r p e n d i c u l a r t o t h e p l a t e , r e s p e c t i v e l y . Tomotika and A o i (73), u s i n g an exact s o l u t i o n o f Oseen's e q u a t i o n , o b t a i n e d t h e drag force per u n i t l e n g t h f o r t h e f l o w a l o n g and p e r p e n d i c u l a r t o t h e major a x i s o f an e l l i p t i c a l c y l i n d e r as  21  4TTU I -j^-  F =  ,  Re  1  \  L  - x  2  2  2  ]L+^[15-4a-l8a  E  2  4  0  2  2  4  -12a - a ]} -  2  3  4  [31a+10a -6a +10a +a ]L ~[4-3a -2a ]L +  — M {L -k32 [l+aJ L R  —  32[l+a] L  {L -|[l+2a-a 0  2  5  3  4  2  5  1  2  3  3  2  [280+ll85a-987a -2065a -90a +525a +85a +3a ] L 2  rirr 2304  where  [  3  2,  5  6  -  7  225- 12Oa-524a -2l6a +39OaV44Oa +l80a +24a +a ]} '(47) 2  3  L = | ( l + a ) - Y - l n  6  5  7  8  Re/8(1 + a)  and  v  -  1 _  ^  R  e  2  32[l+a]  2  L  { L - | [ l - 2 a - a ] L+^-[15+4a-l8a +12a -a ]} 2  2  2  3  4  —n—p(L -ip[4-3a +2a ]L -i7 [31a+10a -6a +10a +d ]L + 32^[l+a] IT Re  ?  2  ij  q  1  3  4  2  3  4  0  3  4  ^ ^[280-ll85a-987a +2065a -90a -525a +85a -3a ]L 2  3  i|  5  2  T  5  6  -  7  8  ^Q -[225-120a-524a -2l6a +390a +440a +l80a +24a +a 8 ]}J 2  3  i|  5  6  7  1|  (48)  where  L = -| (1 - a) - y - I n Re/8(1 + a) , r e s p e c t i v e l y  C l e a r l y the f i r s t term o f e q u a t i o n 47 and 48 I s e q u i v a l e n t to e q u a t i o n s  45 and 46 r e s p e c t i v e l y .  Tomotika and A o i (73)  have i n d i c a t e d t h a t t h e  s o l u t i o n s o b t a i n e d by S i d r a k (75)  f o r t h e drag f o r c e on (74)  an e l l i p t i c a l c y l i n d e r , and t h a t by Davies p l a t e , are i n e r r o r .  fora flat  P i e r c y and Winny (72) o b t a i n e d an  e x p r e s s i o n f o r t h e drag f o r c e on a f l a t p l a t e up t o terms 2  i n Re  which i s i n agreement w i t h t h e c o r r e s p o n d i n g ex-  p r e s s i o n o f Tomotika and A o i .  The drag force as g i v e n  by e q u a t i o n 47 and 48 can be assumed t o be v a l i d up t o Re ^ 2.  Imai (76)  c o n s i d e r e d t h e u n i f o r m f l o w p a s t an  e l l i p t i c a l c y l i n d e r a t an a r b i t r a r y angle o f i n c i d e n c e , his  r e s u l t s f o r t h e s p e c i a l cases o f f l o w p a r a l l e l t o t h e  major and f l o w p a r a l l e l t o t h e minor a x i s b e i n g i n agreement w i t h e q u a t i o n 47 and 48. For h i g h Reynolds numbers, Oseen s T  asymptotic  t h e o r y f o r Re -* °° can reproduce t h e p a t t e r n s o f r e a l f l o w to a r a t h e r s u r p r i s i n g e x t e n t , a c c o r d i n g t o M u l l e r Oseen (26)  a p p l i e d the asymptotic  (77).  t h e o r y t o t h e case  of a c i r c u l a r c y l i n d e r and a f l a t p l a t e .  Stewartson  (25b)  a p p l i e d t h e t h e o r y f o r f l o w p a s t a sphere, w h i l e H o c k i n g (78,  79)  s t u d i e d the flow past a c i r c u l a r d i s c s e t p e r -  p e n d i c u l a r t o the f l o w .  Tamada and M i y a g i (80)  studied  the f l o w p a s t a f l a t p l a t e s e t p e r p e n d i c u l a r t o t h e stream and o b t a i n e d an approximate f o r m u l a f o r t h e drag f o r c e p e r u n i t l e n g t h o f t h e p l a t e a t f i n i t e Re:  23  P = \ p U where  2  L [TT + 8 r ( \ ) R e "  3 7 4  ]  (49)  L i s t h e p l a t e w i d t h , Re = u L/v and  Gamma f u n c t i o n . asymptotic  r i s the  The f i r s t term o f e q u a t i o n 49 i s t h e  v a l u e o f P f o r Re -*• °° .  An improvement on  e q u a t i o n 49 was a l s o made by M i y a g i ( 8 l a , 8 l b ) f o r Re o f 4, 8, 1 2 , 24 and 96.  Kuo ( 2 ) improved B l a s i u s '  s o l u t i o n (83) f o r t h e drag f o r c e e x p e r i e n c e d by a f l a t p l a t e s e t along the streaming flow a t intermediate Reynolds numbers, u s i n g boundary l a y e r t h e o r y . B.  Swarms I t i s n a t u r a l t o suppose t h a t t h e d i s t u r b a n c e  due t o a body immersed i n a f l o w i n g f l u i d spreads i n t h e f l o w f i e l d and t h a t t h e presence o f more than one body i n an o t h e r w i s e u n i f o r m f l o w g i v e s r i s e t o a mutual i n t e r a c t i o n between t h e immersed b o d i e s . been made t o develop  Many attempts  a reliable theoretical  have  relationship  between t h e c o n c e n t r a t i o n of - such a "swarm" and t h e f o r c e e x e r t e d on a g i v e n body w i t h i n t h e swarm, i n o r d e r t o p r e d i c t , f o r example, t h e s e t t l i n g r a t e i n t h e case o f s e d i m e n t a t i o n , o r t h e p r e s s u r e drop a c r o s s a packed bed o f particles. The b a s i c e q u a t i o n f o r c r e e p i n g f l o w through a g r a n u l a r bed i s g i v e n by Darcy's (94) e m p i r i c a l e q u a t i o n o f p e r m e a b i l i t y , which i s now commonly w r i t t e n as  W  (50)  L'  where U i s t h e s u p e r f i c i a l v e l o c i t y  (empty tube  K i s the p e r m e a b i l i t y o f the bed and drop a c r o s s the bed depth L'.  velocity),  AP'is the pressure  Emersleben (95) and S l i c h t e r  (96) t r i e d , w i t h not much s u c c e s s , t o j u s t i f y  Darcy's  e q u a t i o n on t h e o r e t i c a l grounds. B l a k e (98) a c c e p t e d D u p u i t s 1  that the i n t e r s t i t i a l  (97) assumption  v e l o c i t y through a g r a n u l a r bed  equals U/e,where e i s t h e p o r o s i t y o f t h e bed (volume o f pore space p e r u n i t volume o f bed) and by d i m e n s i o n a l a n a l y s i s extended  k vi S  e q u a t i o n 50 t o  • L  f  where k has become known as Kozeny's c o n s t a n t and S i s t h e p a r t i c l e s u r f a c e a r e a p e r u n i t volume o f t h e bed.  Kozeny  (99) s u b s e q u e n t l y d e r i v e d t h e above e q u a t i o n by assuming t h a t a packed bed c o n s i s t s o f a group o f s i m i l a r p a r a l l e l  channels  ( c a p i l l a r i e s ) such t h a t t h e t o t a l i n t e r n a l s u r f a c e and t h e t o t a l i n t e r n a l volume a r e e q u a l t o t h e p a r t i c l e s u r f a c e and the pore volume, r e s p e c t i v e l y  (100).  I n t r o d u c i n g the  • h y d r a u l i c r a d i u s , m, d e f i n e d as t h e r a t i o o f t h e volume o c c u p i e d by t h e f l o w i n g f l u i d t o the wetted s u r f a c e , g i v e s  S  = e / m  (52)  combining K = m  e q u a t i o n s 50, 51 and 52 e / k  2  Letting S S = S  yields  y  (53)  be the s p e c i f i c s u r f a c e of the p a r t i c l e s ( 1 - e )  y  (54)  and s u b s t i t u t i n g e q u a t i o n 54 i n t o e q u a t i o n  U  = C  K  ]  L  then  ^  51,  5-5  (55)  y k ( 1 - e r v  E q u a t i o n 55 i s known, as the Carman-Kozeny e q u a t i o n .  Kozeny  r e a l i z e d t h a t , owing t o t h e t o r t u o u s c h a r a c t e r of the f l o w through packed beds, t h e l e n g t h of the e q u i v a l e n t s h o u l d be Lg , where L where k  0  e  > L  T  .  Hence k = k  i s the c a p i l l a r y c o n s t a n t ; k  channel. Hatch (103)  Davies  (101),  Q  Q  ( L  channel / L'  e  ) , 2  = 2 for a circular  P i e r c y et a l . (102), P a i r  and Ratkowsky and E p s t e i n (104)  and  showed t h a t  k  D  f o r v a r i o u s n o n - c i r c u l a r c h a n n e l shapes c o n s i d e r e d l i e s i n g e n e r a l , between 1.7 the t o r t u o u s i t y .  The  and 3.0.  The r a t i o L ' / L* i s termed e  Kozeny c o n s t a n t k i s a f u n c t i o n o f  the p o r o s i t y of the bed, as w e l l as of the p a r t i c l e shape and o r i e n t a t i o n .  The p r i n c i p a l shortcomings  of the Carman-  Kozeny e q u a t i o n are t h a t i t o v e r c o r r e c t s f o r Ae due to changes i n p a r t i c l e p r o p e r t i e s as opposed to Ae due t o changes i n degree o f p a c k i n g  (105)  and t h a t i t i s i n v a l i d f o r h i g h  Reynolds numbers and f o r h i g h p o r o s i t i e s  (105).  A review  on the Carman-Kozeny e q u a t i o n and some twenty o t h e r e m p i r i c a l c o r r e l a t i o n s i s g i v e n by E p s t e i n (105).  26 For low Reynolds numbers f l o w , the  technique  of r e f l e c t i o n s can be used f o r d i l u t e , assemblages or swarms of p a r t i c l e s .  T h i s method, i n s i m p l e terms, i n v o l v e s a  p i e c e - w i s e matching of boundary c o n d i t i o n s on the boundary s u r f a c e s , i n c l u d i n g the c o n t a i n e r w a l l s , u s i n g a s e t o f partial solutions.  A g e n e r a l d i s c u s s i o n on t h i s t e c h -  nique i s g i v e n by Happel and Brenner ( 8 ) . (84,  85,  Smoluchowski  86) c o n s i d e r e d an assemblage of n f a l l i n g  h a v i n g a c u b i c arrangement both i n an i n f i n i t e medium and i n a c o n t a i n e r .  spheres  stagnant  F o r the l a t t e r c a s e , the  drag  f o r c e e x p e r i e n c e d by a g i v e n sphere i n the assemblage i s g i v e n by P = 6n y a U ( 1 + 2.6  a/h  )  (56)  where h i s the d i s t a n c e between the c e n t r e s of two i n the assemblage.  Burgers  spheres  (87), McNown and L i n (88)  L i n (89) o b t a i n e d a p p r o x i m a t e l y  and  the same e x p r e s s i o n as  above f o r a d i l u t e c u b i c assemblage ( a/h  >>1)  .  The  c o n c e n t r a t i o n o f the s p h e r i c a l bodies i n a c u b i c assemblage i s g i v e n by c =  4TT  3  ( a / h )  3  (57)  and e q u a t i o n 56 t h e r e f o r e becomes P = 6 i y U a ( 1 + 1.61  c 1/3  (58)  27  Pamularo (92) extended  the t e c h n i q u e of r e f l e c t i o n s and  o b t a i n e d the drag f o r c e e x p e r i e n c e d by a sphere i n a d i l u t e assemblage of c u b i c , rhombohedral and random s u s pensions as F = 6ir p U a ( 1 + 1.91  c  1 / 3  )  (59)  F = 6i n U a ( 1 +  c  1 / 3  )  (60)  c  1 / 3  )  (61)  1.79  and P =  6TT  u U a  respectively.  C  1 + 1.3  The r e f l e c t i o n t e c h n i q u e becomes  extremely  t e d i o u s f o r c o n c e n t r a t e d suspensions and i s t h e r e f o r e not commonly used f o r such s u s p e n s i o n s , i t s a p p l i c a b i l i t y confined to d i l u t e Another  being  systems. i m p o r t a n t t e c h n i q u e used t o e v a l u a t e  the drag f o r c e on a p a r t i c l e i n a swarm of o t h e r p a r t i c l e s i s t h e " c e l l method".  T h i s method i s based on t h e concept  t h a t an assemblage o f p a r t i c l e s can be d i v i d e d i n t o a number o f i d e n t i c a l c e l l s , one p a r t i c l e occupying each c e l l . T h i s method i s more a p p l i c a b l e than t h e method o f r e f l e c t i o n s t o c o n c e n t r a t e d s u s p e n s i o n s , e s p e c i a l l y where t h e w a l l e f f e c t due t o t h e v e s s e l c o n t a i n i n g t h e s u s p e n s i o n i s negligible. Cunningham (118) used the c e l l model t o p r e d i c t the drag f o r c e F f o r a swarm o f s p h e r e s .  He c o n s i d e r e d t h a t  the swarm c o u l d be r e p r e s e n t e d by a r i g i d sphere by a r i g i d o u t e r envelope.  surrounded  T h i s model p r e s e n t s t h e  d i f f i c u l t y t h a t t h e p o s i t i o n o f t h e o u t e r envelope i s empirically  determined.  28 U c h i d a (90) i n v e s t i g a t e d a c u b i c assemblage u s i n g a c u b i c model, and o b t a i n e d f o r a d i l u t e swarm o f spheres P = 6TT  U. a ( 1 + 2.1 c  U  1  /  (62)  )  3  Kawaguchi (91) used a n o t h e r c e l l model f o r spheres by c o n s i d e r i n g a sphere w i t h i n a f r i c t i o n l e s s c y l i n d e r , and obtained F = 61 p U a ( 1 -  °  1 / 3  +  °-7968 c )  -  (63)  1  0.9095. f o r a c u b i c assemblage and M = 1.1458  where M =  f o r a body-centred l a t t i c e .  For h i g h d i l u t i o n , i . e .  0 , e q u a t i o n 63 becomes  c  F = 6n y U a ( 1 + Happel and Ast  c  l  /  3  )  (125) o b t a i n e d a s i m i l a r  (  6  4  )  expression.  Hasimoto (93) c o n s t r u c t e d a p e r i o d i c a r r a y by repeating a basic c e l l .  He r e p l a c e d each sphere by a  p o i n t f o r c e r e t a r d i n g the f l o w and o b t a i n e d e x p r e s s i o n s f o r P u s i n g F o u r i e r s e r i e s f o r s i m p l e , b o d y - c e n t r e d and f a c e c e n t r e d c u b i c arrangements.  His r e s p e c t i v e e x p r e s s i o n s  f o r the c r e e p i n g f l o w regime a r e : F = 6-rr y U a ( 1 - 1.76 c F = 6TT y U a ( 1 F  = 6TT  V  1  Ua  ( 1  1  1.791c -  1.791c  /  1/3  1 / 3  3  + c - 1.5593 c + c - 0.329 c  T  2  + c - 0.302 c  r  2  2  )  1  (  6  6  5  6  )  1  (  )  1  (67)  29 When c 0 . , e q u a t i o n 65 becomes  P = 6 T T y U a ( l  + 1.76  c  1  /  3  )  (68a)  )  (68b)  and e q u a t i o n 66 and 67 becomes F = 6 u y U a ( l +  1.79 c  1  /  3  I t seems c l e a r t h a t t h e g e n e r a l e x p r e s s i o n f o r t h e drag f o r c e i n d i l u t e s u s p e n s i o n s has t h e form P = 6ir y U a ( 1 + q c  1  /  3  (69)  )  The v a l u e o f q does n o t v a r y much between a u t h o r s and geometric arrangements, i n d i c a t i n g t h a t t h e p a r t i c l e ment i s not very i m p o r t a n t i n u n i f o r m d i l u t e  arrange-  suspensions.  Hasimoto (93), u s i n g t h e same t e c h n i q u e as f o r s p h e r e s , o b t a i n e d an e x p r e s s i o n f o r t h e f o r c e p e r u n i t l e n g t h on a square a r r a y o f c i r c u l a r E =  cylinders:  i ^ p — - ± l n ( c A ) - 1.3105 + c A y  (70)  U  When c ->• 0,the above e x p r e s s i o n becomes F  1 ln(c/Tr) - 1.31 "2  U  1  J  Brinkman (106) o b t a i n e d a r e l a t i o n s h i p between the p o r o s i t y e and t h e r e l a t i v e s e t t l i n g v e l o c i t y a p p l i c a b l e to both d i l u t e and c o n c e n t r a t e d systems.  The drag f o r c e  a c t i n g on a sphere i n t h e swarm is. g i v e n by P = 6TT y U a [1 + J ( 1 - e ){1 - (8/(1-e ) - e )  1 / 2  > I"  1  ( 7 2 )  30  When e -< 1/3,  Brinkman's model f a i l s .  T h i s c e l l model  c o n s i d e r e d the f l o w p a s t a sphere embedded i n a packed bed.  The p r e s s u r e drop i s summed from two  sources,  namely, t h a t g i v e n by Darcy's e q u a t i o n and t h a t g i v e n by the c r e e p i n g f l o w e q u a t i o n , so t h a t VP' = -  + yV  Brinkman reasoned  v'  (73)  t h a t when the p o r o s i t y e-^ 1.0  the se-  cond term of e q u a t i o n 73 becomes the dominant one, and when e i s 01 5- | the f i r s t term i s the dominant one.  I n terms  of the d i m e n s i o n a l stream f u n c t i o n , e q u a t i o n 73 c o u l d be reduced E%'  t o the form = (constant) E i|/  (74).  2  S i n c e Darcy's e q u a t i o n i s e m p i r i c a l , I t f o l l o w s t h a t Brinkman's r e s u l t s are not t h e o r e t i c a l l y r i g o r o u s . An e x t e n s i o n o f Brinkman's model was made by Spielman  and Goren (107)  arrangements:  a)  f o r cylinders i n four d i f f e r e n t  A l l c y l i n d e r s normal to f l o w ,  c y l i n d e r s p a r a l l e l to f l o w ,  c) , t w o - d i m e n s i o n a l  t r i b u t i o n i n p l a n e s p a r a l l e l t o f l o w and d) s i o n a l random d i s t r i b u t i o n . 0.7  All  random d i s -  three-dimen-  The range of e g i v e n  was  < e < 0.997-  R i c h a r d s o n and Z a k i (108) for  b)  sedimenting  spheres.  s e t t l i n g i n such a way o t h e r i n hexagonal-type  employed a c e l l  model  P a r t i c l e s were assumed t o be  t h a t they are a l i g n e d one tubes.  above t h e  They c o n s i d e r e d two p a t t e r n s :  c o n f i g u r a t i o n I assumed t h a t t h e spheres  i n adjacent  h o r i z o n t a l l a y e r s a r e t h e same d i s t a n c e a p a r t  vertically  as h o r i z o n t a l l y ; c o n f i g u r a t i o n I I assumed t h a t t h e spheres i n a d j a c e n t h o r i z o n t a l l a y e r s touch each o t h e r .  The r e -  s u l t s o f c o n f i g u r a t i o n I I a r e i n good agreement w i t h Brinkman's model f o r 0.6<e<0.95. For t h e case o f c y l i n d e r s t h e method o f r e f l e c t i o n s cannot be used d i r e c t l y due t o S t o k e s ' paradox. The c e l l models o f Hasimoto (93) and o f Spielman and Goren (107) f o r c i r c u l a r c y l i n d e r s have been noted  earlier.  Emersleben (95) o b t a i n e d an approximate s o l u t i o n f o r l a m i n a r f l o w p a r a l l e l t o c i r c u l a r c y l i n d e r s i n square a r r a y u s i n g a second o r d e r E p s t e i n Zeta f u n c t i o n (120,8). Sparrow and L o e f f e r (119) s u b s e q u e n t l y  o b t a i n e d an exact  a n a l y t i c a l s e r i e s s o l u t i o n f o r l o n g t i t u d i n a l laminar flow a l o n g c i r c u l a r c y l i n d e r s i n e q u i l a t e r a l t r i a n g l e and square arrays. For an i s o l a t e d row o f p a r a l l e l c i r c u l a r c y l i n d e r s M i y a g i (121), u s i n g S t o k e s ' a p p r o x i m a t i o n , and Tamada and F u j i k a w a  (122), u s i n g Oseen's a p p r o x i m a t i o n , ,  o b t a i n e d f o r f l o w p e r p e n d i c u l a r t o t h e c y l i n d e r s a t low Reynolds number  - l n ( a / h ) - 1.33 + •(. i r / 3 ) (a/h)  where h i s t h e d i s t a n c e between the c e n t r e s o f n e i g h b o u r i n g cylinders.  F o r a/h<<l , e q u a t i o n 75 becomes v e r y c l o s e t o  t h a t g i v e n by Hasimoto, e q u a t i o n 71» where f o r a square a r r a y c = Tra /h •  T h i s shows t h a t f o r d i l u t e  arrays  the p r e s s u r e drop a c r o s s s e v e r a l rows of c y l i n d e r s i s equal t o the sum o f p r e s s u r e drops a c r o s s i s o l a t e d rows. For h i g h v a l u e s o f Re, Tamada and F u j i k a w a n u m e r i c a l a n a l y s i s t o o b t a i n F.  F o r a dense arrangement  of c y l i n d e r s i n an i s o l a t e d row, K e l l e r p = _?JLJLU_ 2 /2 Kuwabara  r ! _  2 a / h  employed  (123) g i v e s  ]"5/2  ( 7 6 )  (124), u s i n g Oseen's a p p r o x i m a t i o n s ,  treated  the case o f flow- p a s t an i s o l a t e d row o f p a r a l l e l  ellip-  t i c a l c y l i n d e r s f o r the f o l l o w i n g s i t u a t i o n s : a)  major a x i s p a r a l l e l t o f l o w  F = 4 T r y U [ S  b)  ) -  2  ! J — + 2 d Re  T  Q  + 2  T  C  r  1  (77)  minor a x i s p a r a l l e l t o f l o w  F = c)  + i ( l + a  4TT  y U [ S + | ( 1 - a ) 2  ]  _  1  (78)  circular cylinders  F = 4 T r y U [ S  + | -  ^L__  +  2 T  0  ]  _  1  (79)  d)  f l a t p l a t e s set perpendicular to flow  F =  4TT  y U [ S M  d Re  = 4IT y U [ l n ( d/7T  e)  2 T ] . . °  _  1  J  ) - 2 s(3)  (Re d / l 6 f r )  2  + ...]  -  1  (d Re/l67r) + . . . ]  _  (80)  f l a t p l a t e s set p a r a l l e l to flow  F.=  4 T r y U [ S  + l  -i - ^ -  + 2 T  0  ]"  = 4TT y U [ 1 - ln(ir/d) - 2 5(3) where. S = l n [ 8 ( l + a ) / R e ] 2  -y  1  2  , Re =. 2(l+a 2 )  1  (8l)  Up/y  p  Y i s Euler's constant,(1+a 2  ) i s the major  (1+a  ) i s the minor s e m i - a x i s , 2 f u n c t i o n , d = h / ( l + a )and T  G  c(x)  semi-axis,  i s Riemann's z e t a  = 27r/(dRe)+ | l n ( d Re/l67T) + £ - 5(3) + |  ? ( 5 ) Cd R e / l 6 7 r )  4  -  C(3) and 5(5) are b o t h of 0|1|  (d Re/l67r) + 2  (82)  .  Equations  77 - 81 are a p p l i c a b l e when d i s l a r g e .  Numerical  c a l c u l a t i o n s are g i v e n by Kuwabara f o r d =  10.  Another c e l l model was  (126)  developed by Happel  t o p r e d i c t the drag f o r c e e x e r t e d on a sphere f o r a s e d i menting swarm.  A random assemblage i s c o n s i d e r e d  to  c o n s i s t of a number of c e l l s , each of which c o n t a i n s a s p h e r i c a l envelope,  such t h a t the r e l a t i v e volume of  fluid  t o s o l i d i n a c e l l i s the same as i n the e n t i r e assemblage. Such an envelope i s assumed t o be f r i c t i o n l e s s . s o l u t i o n of the c r e e p i n g f l o w e q u a t i o n  ( e q u a t i o n 6) i n  s p h e r i c a l c o o r d i n a t e s , as g i v e n by e q u a t i o n 7,  ij/ =  The  | U sin 6  ( A r  2  constants  + B r  2  2  A,  B,  2  C  2  + C /  2  and D  2  2  2  The  '+ D r 2  - 1  yields  )  (83)  are determined from the  n o - s l i p c o n d i t i o n s at the s u r f a c e of the i n n e r sphere,  the  zero shear s t r e s s c o n d i t i o n at the o u t e r envelope and  zero  r a d i a l f l u x across i t .  The drag f o r c e i s then g i v e n by  5  P =  4TT  y U a  —  2  y  ,  -2y  where  y  +  +  (84)  3  3Y  - 3Y + 2  i s the r a t i o of the i n n e r sphere r a d i u s t o t h a t of  the o u t e r sphere, and the v o l u m e t r i c c o n c e n t r a t i o n c, i s t h e r e f o r e g i v e n by y  3  •  Equation  84 reduces t o  Stokes'  law f o r y = 0, and has no s i n g u l a r i t y f o r a l l the v a l u e s between 0 and 1.  For s m a l l v a l u e s of y , i . e . d i l u t e  stems, i t reduces to  sy-  ofy  35  P =  6ir  y U a (  1  +  1.5  c 1/3  (85)  I t i s c l e a r t h a t the above e x p r e s s i o n has the same form • as t h a t g i v e n by e q u a t i o n  69.  Happel's f r e e s u r f a c e model  gave e x c e l l e n t agreement i n the d i l u t e range w i t h the p e r i m e n t a l d a t a of McNown and L i n ( 8 8 ) and the expressions  ex-  theoretical  of Smoluchowski (84, 85, 86) and Kawaguchi ( 9 1 )  For the i n t e r m e d i a t e range of c o n c e n t r a t i o n , l e s s f a v o r a b l e agreement was  observed w i t h the e x p e r i m e n t a l  e m p i r i c a l equations  of H a n r a t t y  Hawksley ( m o d i f i e d )  (109),  Adler  (127),  (117),  data  and Bundukwale  Richardson  and  Zaki  Steinour ( 1 1 4 ) , Mertes and Rhodes  Verchoor  (129)  or  (111), . (108),  Wilson  (128),  and Happel and E p s t e i n ( 1 3 0 ) , a n d  t h e o r e t i c a l d e r i v a t i o n of Brinkman  (106).  be mentioned t h a t the e x p e r i m e n t a l  d a t a of the v a r i o u s  the  But i t s h o u l d  workers i n t h i s range are not i n agreement e i t h e r , as i t appears t h a t a unique r e l a t i o n s h i p between r e l a t i v e  velocity  and c o n c e n t r a t i o n does not e x i s t i n the i n t e r m e d i a t e range of c o n c e n t r a t i o n , due p o s s i b l y t o a g g l o m e r a t i o n culation.  (108),  cir-  For h i g h c o n c e n t r a t i o n s good agreement was  o b t a i n e d w i t h the e m p i r i c a l e q u a t i o n s Zaki  and  Hawksley ( m o d i f i e d )  and F a i r and Hatch  (103).  (109),  of R i c h a r d s o n Carman-Kozeny  Subsequently Smith  (131,  again  and (100) 132)  36 used Happel's f r e e s u r f a c e model f o r t h e a n a l y s i s o f t h e d i f f e r e n t i a l s e t t l i n g o f spheres o f d i f f e r e n t  sizes.  Happel (133) a l s o used h i s f r e e s u r f a c e model f o r a n a l y z i n g f l o w p e r p e n d i c u l a r and p a r a l l e l t o a r r a y s of c i r c u l a r c y l i n d e r s . developed  His s o l u t i o n of the f u l l y  laminar flow equation i n c y l i n d r i c a l  coordinates  f o r these two cases a r e , r e s p e c t i v e l y ,  4TT y U  (86)  l n y - 0.5 + [ Y  / ( 1 + Y  ) ]  and P =  _AlULU 4y  2  - Y^  -  3-4  _ ln  (87)  Y  -1 where y  i s the r a t i o of the r a d i u s of the outer  less c y l i n d e r to the inner s o l i d c y l i n d e r . 2 c e n t r a t i o n c i s then g i v e n by Y . t h a t both e q u a t i o n s  friction-  The c o n -  I t i s i n t e r e s t i n g t o note  86 and 87 show zero drag f o r c e p e r u n i t  l e n g t h as Y' -> 0 , thus g i v i n g r i s e t o S t o k e s ' paradox f o r t h e t r a n s l a t i o n of a two-dimensional  body i n an i n f i n i t e medium.  Happel and Brenner (8) made a comparison o f t h e Kozeny cons t a n t as o b t a i n e d from e q u a t i o n s  86 and 87 w i t h t h a t o b t a i n e d  by Emersleben (95) f o r square a r r a y s and by Sparrow and L o e f f e r (119) f o r square and t r i a n g l a r a r r a y s .  Good  agreement was o b t a i n e d f o r t h e d i l u t e range, i n d i c a t i n g t h a t g e o m e t r i c a l arrangement does not e f f e c t t h e p e r m e a b i l i t y s i g n i f i c a n t l y i n t h e d i l u t e range a c c o r d i n g t o t h e v a r i o u s models  37  The  expressions f o r laminar flow p a r a l l e l to c l u s t e r s of  c i r c u l a r c y l i n d e r s a r e a l s o i n g e n e r a l agreement w i t h the e x p e r i m e n t a l d a t a o f Galloway and o f Gunn and D a r l i n g ( 1 4 4 ) . f o r k by Sparrow and L o e f f l e r  ( 1 4 3 ) and E p s t e i n ( 1 4 2 ) The n u m e r i c a l  (119) a t lower  values  fractional  v o i d volumes show s u b s t a n t i a l d e v i a t i o n s b e t w e e n . e q u i l a t e r a l t r i a n g l e and square arrangements, and a l s o from Happel's f r e e s u r f a c e model. At t h e same time as Happel developed s u r f a c e model, Kuwabara (134) suggested  the f r e e  t h a t i n s t e a d o f im-  p o s i n g zero shear s t r e s s a t t h e o u t e r e n v e l o p e ,  one can im-  pose t h e c o n d i t i o n o f zero v o r t i c i t y a t t h i s o u t e r envelope.  F o r t h e case o f c r e e p i n g f l o w p a s t spheres,he o b t a i n e d  f o r t h e drag f o r c e  .. F =  6TT  +  y U a [ 1 + |iy +  Y + §=- Y ] [ 1 - Y ] 7  8  3  | Q Y  2  - J^Y  3  + |Y  4  + |§Y  [ 1 + | Y + |Y + | Y 2  and f o r c r e e p i n g f l o w p e r p e n d i c u l a r t o c i r c u l a r  3  T  2  5  (88)  cylinders,  he o b t a i n e d f o r t h e drag f o r c e p e r u n i t l e n g t h  - I n y - 0.75 + Y  - Y /4  Kuwabara (134) o b t a i n e d t h e drag f o r c e f o r t h e case of spheres  u s i n g t h e energy d i s s i p a t i o n method.  However,  d i f f e r e n t r e s u l t s a r e o b t a i n e d when one uses t h e p r e s s u r e  38  and shear s t r e s s d i s t r i b u t i o n at the sphere s u r f a c e to c a l c u l a t e P.  The  drag f o r c e u s i n g the l a t t e r method  gives  F  =  30ttuU a  5 -  Equation 88.  The  9Y  + 5y  d  -  ( 9 0 )  *  Y  90 g i v e s a s l i g h t l y h i g h e r v a l u e f o r F than  equation  e x p e r i m e n t a l work on f l o w p e r p e n d i c u l a r t o a l i g n e d  c i r c u l a r c y l i n d r i c a l f i b r e s by K i r s c h and Fuchs  (135)  i n d i c a t e s t h a t Kuwabara's model r e p r e s e n t s t h e i r e x p e r i mental d a t a v e r y w e l l , w h i l e Happel's f r e e s u r f a c e model f o r f l o w p e r p e n d i c u l a r to an assemblage of c y l i n d e r s i s a t variance with t h e i r data.  On the o t h e r hand, Happel's model  f o r spheres g i v e s b e t t e r agreement w i t h the a v a i l a b l e e x p e r i mental data on f l o w past c l u s t e r s of spheres (126) Kuwabara's model.  than  I t i s very d i f f i c u l t from t h e o r e t i c a l  c o n s i d e r a t i o n s alone t o d e c i d e which of the two models i s the more r e a l i s t i c , a l t h o u g h i t has been p o i n t e d out  that  Kuwabara's model, u n l i k e H a p p e l ' s , s u f f e r s from an energy imbalance (8).  L e C l a i r and Hamielec(58) extended  Kuwabara's model t o h i g h e r Reynolds numbers. G a l e r k i n ' s method was  used by Snyder and  Stewart  ( l 4 l ) to s o l v e the c r e e p i n g f l o w e q u a t i o n f o r both a and a denser ( o r t h o r h o m b i c )  simple  arrangement of s p h e r e s ,  and  t a i n e d f r i c t i o n f a c t o r s w i t h i n 5% of those o b t a i n e d  ex-  * The r e a s o n f o r t h i s d i s c r e p a n c y  ob-  i n F i s t h a t the energy  d i s s i p a t e d w i t h i n the c e l l i s not due  t o the drag-producing  s t r e s s e s at the s u r f a c e of the sphere alone,as  assumed by  wabara,but a l s o t o the s t r e s s e s on the o u t e r  envelope.  Ku-  39 p e r i m e n t a l l y by M a r t i n et al-(116) f o r s t a c k e d and 9-1  spheres,  % , lower than those p r e d i c t e d by Happel (126). There are s e v e r a l o t h e r f o r m u l a e , many e m p i r i c a l ,  for  p r e d i c t i n g the drag f o r c e , o r more s i m p l y the  de-  v i a t i o n from Stokes f l o w of a p a r t i c l e w i t h c o n c e n t r a t i o n . O l i v e r (140) g i v e s a r e v i e w o f the v a r i o u s 2.  formulae.  Review o f N u m e r i c a l Work I t i s e v i d e n t t h a t the s o l u t i o n o f the N a v i e r -  Stokes e q u a t i o n v i a S t o k e s , Oseen and Proudman and  Pearson  expansions  f a i l s t o d e s c r i b e the f l o w f o r h i g h e r Reynolds  numbers.  Approximate s o l u t i o n s u s i n g the G a l e r k i n method  have been t r i e d .  T h i s method f i r s t assumes t r i a l f u n c t i o n s ,  and an approximate s o l u t i o n i s then o b t a i n e d by d e t e r m i n i n g the t r i a l f u n c t i o n parameters such as t o s a t i s f y Navier-Stokes  the  e q u a t i o n and the boundary c o n d i t i o n s as  c l o s e l y as p o s s i b l e .  Increased accuracy  can be  obtained  by h a v i n g a l a r g e number o f terms i n the t r i a l f u n c t i o n s , as i n d i c a t e d by Snyder (33) (34).  The  and by F l u m e r f e l t and  optimum t r i a l f u n c t i o n s may  Slattery  be determined  by  v a r i a t i o n a l c a l c u l u s o r by e r r o r d i s t r i b u t i o n methods. Kawaguti (35)  used the G a l e r k i n method t o e v a l u a t e unknown  parameters i n h i s assumed stream f u n c t i o n s f o r f l o w p a s t a sphere i n an unbounded f l u i d .  H i s recommendations are t h a t  one s h o u l d use the r e s u l t s from one stream f u n c t i o n at 0 < Re < 10  a n d  10< Re < 70 •  t n e  r e s u l t s from another stream f u n c t i o n a t  Without p r i o r knowledge of the answer, however,  t h i s s l i c i n g c o u l d not be a c c o m p l i s h e d .  Hamielec  and  4o Johnson (37), (hO)  Hamielec et a l , (38)  used t h i s t e c h n i q u e  and. Nakano and  Tien  t o e v a l u a t e the v i s c o u s f l o w  around f l u i d spheres at i n t e r m e d i a t e Reynolds numbers. A f l u i d sphere w i t h a . v i s c o s i t y a p p r o a c h i n g  infinity i s ,  of c o u r s e , a r i g i d  Slattery  (3*0  sphere.  F l u m e r f e l t and  p r e s e n t e d an improvement on the G a l e r k i n method  u s i n g an a u x i l i a r y v a r i a t i o n a l t e c h n i q u e (39).  g i v e n by  Slattery  Such methods are not c o m p l e t e l y r e l i a b l e , however,  as they depend l a r g e l y on the c h o i c e o f the t r i a l f u n c t i o n . Bourot (137,  138)  employed the method o f l e a s t  squares f i t t i n g o f the b i h a r m o n i c  f u n c t i o n t o known  boundary c o n d i t i o n s f o r c r e e p i n g f l o w of a c y l i n d r i c a l envelope.  Bourot (136)  sphere i n a  developed  the method  f o r a s p h e r o i d , but d i d not p e r f o r m the n u m e r i c a l work. T h i s t e c h n i q u e r e q u i r e s the g e n e r a l s o l u t i o n of E %  = 0  and has the d i s t i n c t v i r t u e t h a t the o u t e r boundary  can  take any  shape.  The  technique  has found a wide a p p l i -  c a t i o n f o r f l o w i n ducts and f o r f l o w i n c a v i t i e s  (45,  139). A f i n i t e d i f f e r e n c e method was (42,  43,  hh)  t o .solve the N a v i e r - S t o k e s  past c y l i n d e r s at Re = 10, Kawaguti (35) (46)  equation f o r flow  and h i s method has been used by  f o r f l o w p a s t spheres a t Re = 20 and c y l i n d e r s  at Re = 40.  converted  used by Thorn  T h i s method i s very t e d i o u s and i t was  i n t o a r e l a x a t i o n method by Fox  A l l e n and Dennis (53)  (47,  48,  49),  and by S o u t h w e l l et a l . (50).  f l o w p a s t c y l i n d e r s was  s o l v e d by A l l e n and S o u t h w e l l  by  The (54)  41 f o r Re up t o 1000,  while L i s t e r  (51)  a p p l i e d a modi-  f i c a t i o n o f t h e i r method t o spheres f o r Re up t o 20. Kawaguti (52)  i n d i c a t e d t h a t t h e work o f A l l e n and South-  w e l l was i n e r r o r .  The p r i n c i p l e of t h e r e l a x a t i o n  method i s the a p p r o x i m a t i o n o f t h e d i f f e r e n t i a l e q u a t i o n s by t h e i r e q u i v a l e n t d i f f e r e n c e e q u a t i o n s , which r e l a t e t h e v a l u e s o f the f u n c t i o n and t h e d e r i v a t i v e s i n v o l v e d a t a p o i n t t o those o f n e i g h b o u r i n g l a t t i c e Jenson (55),  points.  i n h i s work on s p h e r e s , s o l v e d t h e  N a v i e r - S t o k e s e q u a t i o n by s p l i t t i n g i t i n t o two s i m u l taneous second o r d e r e q u a t i o n s , u s i n g t h e stream f u n c t i o n and t h e v o r t i c i t y .  Jenson approximated t h e v o r t i c i t y  near t h e s u r f a c e by a t h i r d o r d e r p o l y n o m i a l which c o u l d a l s o be d e r i v e d by u s i n g T a y l o r ' s e x p a n s i o n c o r r e c t t o Re t h i r d o r d e r , and he used an e x p o n e n t i a l t r a n s f o r m a t i o n f o r r a d i a l d i s t a n c e from t h e s u r f a c e .  He o b t a i n e d the  stream f u n c t i o n , v o r t i c i t y , p r e s s u r e v a r i a t i o n a t the. s u r - . f a c e and d r a g c o e f f i c i e n t f o r Re up t o 80 (56) L e C l a i r and Hamielec  (58)  Rhodes  and Hamielec e t a l (59,  (57),  60)  extended Jenson's t e c h n i q u e w i t h t h e a i d o f d i g i t a l comp u t e r s t o i n c l u d e h i g h e r Re.  But t h e i r r e s u l t s f o r an i s o -  l a t e d sphere are not c o n c l u s i v e inasmuch as t h e i r p o r o s i t i e s are i n s u f f i c i e n t l y h i g h t o a c c u r a t e l y r e p r e s e n t an i n f i n i t e medium.  Hamielec and R a a l (6l)  adopted Jenson's t e c h n i q u e  f o r c i r c u l a r c y l i n d e r s f o r Re up t o 500,  assuming t h a t t h e  steady s t a t e form o f t h e N a v i e r - S t o k e s e q u a t i o n s t i l l h o l d s a t t h i s Reynolds number.  They were c a r e f u l t o c o n s i d e r  42 the parameters t h a t a f f e c t the accuracy of the n u m e r i c a l s o l u t i o n and. they have o b t a i n e d the most a c c u r a t e  numerical  v a l u e s o f t h e drag f o r c e t o d a t e . S e v e r a l o t h e r workers (62, 63, 64) have o b t a i n e d ' solutions  o f the unsteady s t a t e  form o f the N a v i e r -  Stokes e q u a t i o n f o r a c i r c u l a r c y l i n d e r .  A numerical  s o l u t i o n f o r f l o w a l o n g a f l a t p l a t e ' h a s been o b t a i n e d by Dennis and Dunwoody  .**  (4l).**  A r e c e n t paper by Rimon and Lugt  a f t e r the p r e s e n t m a n u s c r i p t  was completed,  (164), p u b l i s h e d d e a l s w i t h the  p a r t i c u l a r case o f an o b l a t e s p h e r o i d , p e r f o r m i n g calculations  similar  t o the p r e s e n t work u s i n g a time-dependent  numerical technique.  T h e i r r e s u l t s , which were l i m i t e d t o  Re = 10 and Re = 100, agreed w e l l w i t h t h e c o r r e s p o n d i n g results i n this  thesis.  43 CHAPTER I I I  F o r m u l a t i o n o f F i n i t e D i f f e r e n c e E q u a t i o n s and Boundary C o n d i t i o n s . The  c h o i c e o f s p h e r o i d a l and e l l i p t i c a l c o -  o r d i n a t e s i n t h i s study  f a c i l i t a t e s the formulation of  the boundary c o n d i t i o n s .  I n a d d i t i o n , the e x p o n e n t i a l  p r o p e r t i e s o f these c o o r d i n a t e s g i v e a f i n e l a t t i c e near the s u r f a c e o f t h e p a r t i c l e i n g e n e r a l and near t h e t i p s in particular, surface.  and a coarse  l a t t i c e f a r away from t h e  As t h e i n f l u e n c e o f t h e p a r t i c l e on the. f l o w s  i s mainly manifested  near t h e s u r f a c e ,  especially  around t h e t i p s ' , and as t h i s i n f l u e n c e d e c r e a s e s w i t h d i s t a n c e from t h e s u r f a c e , t h e l a t t i c e thus produced s e r v e d t h e purpose o f g i v i n g a d e s i r e d l a t t i c e g r a d a t i o n . A. A..1  Spheroids Formulation The  as expressed equation  of the F i n i t e D i f f e r e n c e  Navier-Stokes i n orthogonal  equation  f o r steady  curvilinear  state flow  coordinates,  1, reduces t o  VF'V-  ( coshg s i n n r . 5 ( V ,E'V ) c ( s m h C+cos n ) 1  2 E* ifrtcosh g s i n n )" ^ 3 c (sinh S+cos n) 4  and  Equations  2  2  ,C cosh g s i n n ) 9(£>r,) (91)  44  E  12  2  =  =— {  5  —5  c  ( s i n h ^ + c o s n)  2  —TT-  H  tanh £ l - = -  -  2  +—=r  3n  5  - c o t n -|— }  2  9 n  (92)  f o r oblate s p h e r o i d a l c o o r d i n a t e s , the metric coe f f i c i e n t s f o r these c o o r d i n a t e s b e i n g g i v e n by  c { s i n h £+cos  n}  and h, = 3  . . c cosh E, s i n n  (94)  1  c  where c i s a c h a r a c t e r i s t i c l e n g t h o f t h e c o o r d i n a t e system.  The minor s e m i - a x i s , b, and the major  semi-  a x i s , a, of t h e s p h e r o i d a r e g i v e n by  £  Q  a = c cosh E,  (95)  b = c sinh £  (96)  being the value of K at the surface of the spheroid.  The f o l l o w i n g d i m e n s i o n l e s s ib = IJJ'/U  a  2  ,  Re  =  q u a n t i t i e s a r e now d e f i n e d :  U/v  2a  and  E  =  2  c  E*  2  2  ,  where Re i s the Reynolds number and U i s t h e v e l o c i t y o f the u n d i s t u r b e d stream.  The N a v i e r - S t o k e s  equation  then  becomes sech E  E (E ^) 2  a  2  =  ge d  t  cosh g s i n n ] | i |_ { £± sinh C+cos n cosh^sin^ C  5  2  cosh  n  C s i nn  }  45 with „2 E =  1  -  d , . 3 • 3 = — { — - t a n h £ ~-=- + — ~ 2  r  3?  sinh ?+co n 2  2  S  3  r  2  3  3 ?  n  . 3 T. c o t n ~— }  2  (98)  3 n  I t i s u s e f u l t o s p l i t e q u a t i o n 97 i n t o two e q u a t i o n s , one p  i n terms of if ^ and  and n and t h e o t h e r i n terms o f E \p, E,  to give  n  . _ „2 / , . >. Re coshg s i n n sech £ E ( t, cosh £ s i n n ) = p— L p p s i n h € +cos n r  2  r  n -  a  r  L  3ifr 3  3g 3n  r  g  1  coshgsinn  3ip 3 :  r  C  3n 3? ^ cosh £ s i n n  J '  i  f  q  q  N  and E \\J = r, cosh £ s i n n s e c h g 2  3  a  (100)  where £ i s t h e d i m e n s i o n l e s s ' v o r t i c i t y d e f i n e d as c, = K £  f  t  a/U,the d i m e n s i o n a l v o r t i c i t y b e i n g = h  3  E'V  For  (101)  a s p h e r o i d which i s almost s p h e r i c a l , cosh 5  becomes v e r y l a r g e and d i f f i c u l t y w i l l be encountered i n the  numerical a n a l y s i s .  For t h i s reason the f o l l o w i n g  functions are introduced:  G = ? cosh g s i n n /cosh £  (102)  P = 5 cosh£  (103)  3,  / cosh £sinri  46  E q u a t i o n s 99 and 100 then become Re sech £ a.r 2  cosh g s i n n 2 s i n h 5+cos n  J L  3^ 9F 3? 3n  3ifr 3F 3n 3C (104)  and  (105)  respectively.  The s o l u t i o n o f the above two  simul-  taneous d i f f e r e n t i a l e q u a t i o n s i n the stream f u n c t i o n and i n t h e m o d i f i e d v o r t i c i t y  G c o u l d be a c h i e v e d by  u s i n g Jenson's r e l a x a t i o n t e c h n i q u e (55).  This tech-  nique i n v o l v e s a p p r o x i m a t i n g the d i f f e r e n t i a l  equations  by f i n i t e d i f f e r e n c e e q u a t i o n s which r e l a t e the v a l u e s of n e i g h b o u r i n g p o i n t s on a f l o w g r i d .  The s o l u t i o n of  these e q u a t i o n s i s f o l l o w e d by the i n t e r p o l a t i o n between the g r i d p o i n t s over the whole f i e l d of i n f l u e n c e . C o n s i d e r i n g l a t t i c e s p a c i n g A i n the  ^-direction  and B i n the n - d i r e c t i o n , as shown i n f i g u r e 1, and u s i n g T a y l o r ' s s e r i e s e x p a n s i o n c o r r e c t t o the second o r d e r , one o b t a i n s f o r a f i e l d f u n c t i o n Q :  3Q 2  =  Q(I,J+1) + Q ( I , J - 1 )  2Q(I,J)  (106)  47  F i g u r e 1. O b l a t e s p h e r o i d a l mesh system.  48  9Q  Q(I+1,J) + Q ( I - 1 , J )  2  9n  =  B  2  9Q _ 9£  _9Q 9n  2Q(I,J) B  2  (  Q(I,J+1) - Q ( I , J - I ) 2 A  0  (  ?  R U  Q(I+1,J) - Q ( I - l . J ) 2 B  =  1  )  2  U  . O  j  (109)  U s i n g e q u a t i o n s 106 - 109, t h e d i f f e r e n t i a l e q u a t i o n 104 and 105 t a k e t h e f o l l o w i n g f i n i t e d i f f e r e n c e forms  G ( I , J ) ( 2 / A + 2/B ) = G(I,J+1) [ (2 - A t a n h 2  G(I J-1){  +  2  2  +  A  }  t  a  n  h  ?  S ( J )  >  +  G(I 1,J){ +  2 A  + G(I-1,J){  r 1  2  +B C  g  t  n ( I )  2 2  } - 5 - sech s  5  2  C(J))/2A ] 2  - B cot n ( I ) 2 B  }  cosh 5 ( J ) s l n n ( I ) ^  [t|»(I,J+l) - i K l , J - l ) ] [ F ( I + l , J ) - F ( I - 1 , J ) ] A.B  L>(l+l,J) - ^(I-1,J)][F(I,J+1) - F(I,J-1) ] }  (  1  1  Q  )  49  and  i K l , J ) (2/A +2/B ) = iJ;(I,J+l) 2  2  2 - A tanh g ( I ) 2 A  + i|>(I,J-l)  + Tp(I-l, J )  2 + A tanh g ( J ) + 2 2 A  2  ip(I+l,J)  2 - B cot n ( D  2 + B cot n ( D 2 2 B (111)  The above f i n i t e d i f f e r e n c e e q u a t i o n s f o r an o b l a t e s p h e r o i d , and t h e i r s o l u t i o n f o r g i v e n boundary c o n d i t i o n s would f u r n i s h v a l u e s o f the stream f u n c t i o n and the. v o r t i c i t y a t each p o i n t o f t h e g r i d i n the f l o w f i e l d .  E q u a t i o n s 110  and 111 reduce t o those o b t a i n e d by Jenson (55) f o r a p e r f e c t sphere when For  £ ->- <»  a p r o l a t e s p h e r o i d , the f i n i t e  difference  e q u a t i o n s c o u l d be o b t a i n e d from those of an o b l a t e s p h e r o i d by r e p l a c i n g each s i n h £ by i coshg  and each coshC  by i s i n h g , or more g e n e r a l l y each c by i c, i n the language of Happel and Brenner ( 8 ) .  From t h e c o m p u t a t i o n a l p o i n t o f  view t h i s i s e x a c t l y e q u i v a l e n t t o changing each s i n h g 2 2 to  coshg , coshg  to sinhg  and cos n  t o - cos n  •  By  t h i s procedure the term a w i l l t h e n r e p r e s e n t t h e l e n g t h o f the  minor s e m i - a x i s and the Reynolds number w i l l be based  on the minor a x i s .  F o r both p r o l a t e and o b l a t e s p h e r o i d s  50  the Reynolds number i s thus based on the e q u a t o r i a l d i a meter. A. 2  Boundary C o n d i t i o n s The n u m e r i c a l a n a l y s i s i s c o n f i n e d between an  i n n e r s u r f a c e r e p r e s e n t e d by the s p h e r o i d i t s e l f and an o u t e r envelope  w h i c h , l i k e the s p h e r o i d a l s u r f a c e ,  c o i n c i d e s w i t h one of the s p h e r o i d a l c o o r d i n a t e g r i d as shown i n f i g u r e  by  lines  2.  As the two f l o w e q u a t i o n s are of second o r d e r , f o u r boundary c o n d i t i o n s are t o be s a t i s f i e d .  For  o b l a t e s p h e r o i d the boundary c o n d i t i o n s f o r for  n = 0  a l o n g BA  , $ = 0  n = ir  a l o n g NO  , ^ = 0  £ = £  a l o n g AMN,  cl  ^ =.0  At the o u t e r boundary, £ = £^  are:  a x i s of symmetry |  s u r f a c e of s p h e r o i d  the f l o w i s assumed to be T  a streaming p a r a l l e l f l o w , g i v i n g  2  ^ = ± sin n 2  for n = 0  a l o n g BA  , r, = 0  n = TT  a l o n g NO  £ = S  a  a l o n g AMN °  , £ = 0 E 2 \p cosh 2 £ , ? = ' sin n  5 = 5  b  a l o n g BRO  , ? = 0  £ = ?  b  a l o n g BRO  , c = C  a x i s of symmetry  a  case I _  0  case I I  2  2  cosh r sech E, , D a  The boundary c o n d i t i o n s f o r r, are :  a  an  F i g u r e 2. Oblate  spheroid with i t s outer  envelope.  The boundary c o n d i t i o n f o r C a t E, - E,  ,  3.  which o r i g i n a t e s from e q u a t i o n 1 0 0 , can be as f o l l o w s : Near t h e s u r f a c e o f t h e s p h e r o i d i t i s assumed t h a t t h e stream f u n c t i o n can be f i t t e d by a c u b i c e q u a t i o n  having  the form  i> = The  K ) + $U 2  a (? -  -  a  ? ) a  (112)  3  above e q u a t i o n s a t i s f i e s t h e n o - s l i p boundary con-  d i t i o n s assuming t h a t t h e c o n s t a n t v a l u e o f \b a t t h e s u r f a c e i s zero f o r any v a l u e o f t h e c o n s t a n t s a and  3 .  98 and 105 combined g i v e  Equations  2 c  G =  o  s  h  £  a  9  2.  2,  3  9  — _ _ { — % . _ tanhCTr + — £ - cot n s i n h ^ + c o s n 8^ 3n 5  2  3 n  .  *  (113)  At t h e s u r f a c e , t h e n o - s l i p conditions noted above r e q u i r e that Sn  3 r )  2  35  and e q u a t i o n 113 t h e r e f o r e reduces t o  G(I,1) =  cosh 5  a 2 ' 2 s i n h £ +•cos n  H  (114) 2  a  Where J = 1 i s t h e v a l u e o f J a t t h e s u r f a c e o f t h e s p h e r o i d , D i f f e r e n t i a t i n g e q u a t i o n 112 t w i c e w i t h r e s p e c t t o E, y i e l d s .2  •2-4 = 2a + 6g ( 5 - ? 35  ) a  (115)  Hence a t £ = £  ,  9,  (116)  2a  as  2  I n t r o d u c i n g e q u a t i o n 116 i n t o 114 y i e l d s  cosh t, G(I,1) =  g ^  2—  -  2  a  (  1  1  7  )  s i n h £ +cos n a  Using the values of  a t J = 2 and J = 3  i n e q u a t i o n 116,  w r i t t e n i n f i n i t e d i f f e r e n c e form a c c o r d i n g t o T a y l o r ' s s e r i e s t o the t h i r d order, gives a  =  8  *  ( I  > > I 4 A  »^>3)  2  •  (118)  and e q u a t i o n 117 becomes cosh £ 2  G(I,1) =  a  2 A  — 2  [ 8 i f ( I , 2 ) - \|»(I,3) 1 , (sinh £ + c o s n ) . a. 2  2  ;  (119)  :  which i s e q u i v a l e n t t o cosh £ 2  ?  (I,1)  =  a  [ 8 i K l , 2 ) - <Kl,3) 1 = :  (  1  2  0  )  ?  2 A ( s i n h La+ c o s n) s i n n  This equation gives the value of £ at the surface of the spheroid  (£ = £ ) and c o u l d a l s o be a r r i v e d a t by exa  panding  directly  as a T a y l o r . s e r i e s expansion"  the t h i r d o r d e r , as s u b s e q u e n t l y  c o r r e c t tc  shown f o r e l l i p t i c a l  cylinders. The boundary c o n d i t i o n f o r . £'at £ = ? which corresponds  b  f o r Case I I  t o Happel's boundary c o n d i t i o n s , can be  d e r i v e d as f o l l o w s : The d i m e n s i o n a l v o r t i c i t y , o b t a i n e d by expanding e q u a t i o n 101, i s g i v e n by  r  '  c  _  i_  y, r3  h r  l  h  3  3^*  - h , h { ^ C—u^2  ? lr > If {  [h  h r  h  1  2  h  3  t—hr  af 3 +  For t h e o b l a t e s p h e r o i d s h  3  =  h  2  3^'  .  3TT  .  ] }  (  1  2  1  and e q u a t i o n 121 becomes  ]+  (122)  or  2 i dh  3^  if'  + h  (123)  The d i m e n s i o n a l shear s t r e s s i s g i v e n (10) by  T  • " h7 Ii <  If'" - * 177 In - '"' ' If' 1 h  For t h e o b l a t e s p h e r o i d s , h = h t  g  h  and e q u a t i o n 124 becomes  ( 1 2 M  55  -2  5n  h  j.o  h  l  3  It-  95  u v,  1  d  ~ , • 9^  ,  1  ,2  9n  9n  2  - h  95  9hr  ;  1  ^  0  1  9  h  3  95 9h  2  ^ /  hi h  95 ~ . » 9^ ,  33  0  ,2  9n  9n  ,  1  3  3  H  »  2  .2. i 9 ib 2  (125)  on  = 0 and a d d i n g e q u a t i o n 125 t o e q u a t i o n 123  Setting yields  ,2,r „ ah, „,« *, 2 . d .ib , „ , 2 = o2 h , h — f + 2 h 9 — 9i|9> ^ +. 2 .h .h 3  9h,  _,f  3n  3n  0  T  5n C  =  3  0  x  x  3  3n 3h,  . . i  95  95  S u b s t i t u t i n g f o r the metric c o e f f i c i e n t s according to equations 93  and  94,  » 5n  T  =  ®  c  _  3  cosh  C  5 s i n n ( s i n h 2 5+cos 2 n)  otn 1^-  + '  9n  c  ...  o  s  n  s  . "\'2L'  l  n  "  3n  2  _3ij/  n  2  s i n h 5 +cos n  9n  s i n h 5 cosh 5 ^ 1 ' s i• n hv 2 5 ^ ,c o s 2 n r  +  95  At 5 = 5^ » t h e stream f u n c t i o n i s  (126)  f  1  2  if)' = 75- U c  2  cosh £  2  b  sin n  (127)  D i f f e r e n t i a t i n g with respect to n yields.  _Oi - U c2 cosh 2 £ s i n n c o s n  (128)  b  :3n  and 2 »  U c2 cosh 2 £  ( cos2 n- s i n2 n )  b  (129)  S u b s t i t u t i n g e q u a t i o n s 128 and 129 i n t o e q u a t i o n 126 and rendering a l l quantities dimensionless y i e l d s 2 cosh  w  0  —  2  - cosh £  2  s i n n ( s i n h g^+cos n)  cosh £  2  cos n s i n n  sinh £  b  + cos n  sinh £  b  cosh g ,  b  + cos n  sinh £  sin n  2  b  2  b  2  ^  b  (130)  and by e q u a t i o n 102, 2 cosh £ G T  Cn  n =  0  2  b  ( s i n h £ +cos n) \ - cosh 2  ^  2  sin n  b  2  +  2  cosh £, cos n s i n n ;—:  2  P_  2  s i n h £ +cos n b  2  sinh  cosh £ 2  2  s i n h g^+cos n  3jjj_ 9?  (131)  57 E q u a t i o n 131 g i v e s t h e modified, v o r t i c i t y  G at the  o u t e r boundary when t h e shear s t r e s s a t t h i s boundary i s equal t o z e r o , i . e . t h e envelope i s f r i c t i o n l e s s .  The  ||r  value o f  can be found by u s i n g Newton's 5=5 b d i f f e r e n t i a t i o n formula. A g a i n the boundary c o n d i t i o n s f o r a p r o l a t e s p h e r o i d c o u l d be o b t a i n e d by r e p l a c i n g s i n h 5 w i t h i cosh 5 and cosh 5 w i t h i s i n h 5 • The boundary c o n d i t i o n f o r 5 = 0  at  5=5  , b  as g i v e n by case I , was used f o r t h e study o f f l o w p a s t isolated  spheroids.  F o r t h e c e l l models used t o r e p r e s e n t  swarms o f s p h e r o i d s , case I r e f e r s t o Kuwabara's zero v o r t i c i t y model, w h i l e c a s e I I r e f e r s t o Happel's f r e e s u r f a c e model.  Both were used f o r t h e study of f l o w  through  swarms o f s p h e r o i d s . B.  E l l i p t i c a l Cylinders  B.l  Formulation of the F i n i t e D i f f e r e n c e The N a v i e r - S t o k e s  as expressed  Equations  e q u a t i o n f o r steady  i n orthogonal c u r v i l i n e a r  state flow  c o o r d i n a t e s , equa-  t i o n 1 ., reduces t o  v.  =—  E'V  '  c  ^ i ^ ) ^ >(  i  ( sinh 5 2  + sin n)  ' ^"  2  d {  1 3 2 )  T ] )  and  E '  2  = - 2  c  \  —  ( s i n h 5 + s i n n)  { —  95  2  + —  9n  2  >  ( 1 3 3 )  58 f o r e l l i p t i c a l c y l i n d r i c a l c o o r d i n a t e s , w i t h t h e und i s t u r b e d f l o w b e i n g p a r a l l e l . t o . t h e x-  or. major a x i s .  The m e t r i c c o e f f i c i e n t s f o r t h i s c o o r d i n a t e system a r e g i v e n by Happel and Brenner (8) as  , hj = h  h  2  =  YPT  p—^ p c [ s i n h ^ S + cos n ] '  (134)  = 1  3  where c i s t h e c h a r a c t e r i s t i c l e n g t h o f t h e c o o r d i n a t e system.  The minor s e m i - a x i s , b, and t h e major semi-  a x i s , a, o f t h e e l l i p t i c a l c y l i n d e r a = c cosh  a r e g i v e n by (135)  a.  b = c sinh E  (136)  Ct  E  0  being the value of £ at the surface o f the c y l i n d e r .  The f o l l o w i n g \|> =  dimensionless  /Ub , E  2  = c  2  E'  2  quantities and  a r e now d e f i n e d :  Re = 2bU / v  ,  where Re i s t h e Reynolds number and U i s t h e v e l o c i t y o f the u n d i s t u r b e d stream a l o n g t h e major a x i s . d i m e n s i o n a l flow: Is; then governed by  2_ Re  \  E fi w  =  :  1 , : 3 i 9 E f i _ 91 . .2 . 2 3£ 3n 3n s i n h E+sm n l  {  r x  The two-  9E ! 9? ' 2  }  ( 1 3 ? ) K ± 5 { )  .  and  E  2  =  2 • '2 p-{ ^-+^-5-} s i n h E+sin n 3? •. 3n •  ;  (138)  59  E q u a t i o n 137 can be s p l i t i n t o two e q u a t i o n s , one i n terms of  y  5 and n and the o t h e r i n terms o f £ , £ and  n  to g i v e  '  sinh  2  2  sin n 2  ? +  3  «  ^  ^  3  *  U  3  9  )  and E TJ, = x, / 2  s  i  n  n  2  5  (140)  a  where z, i s the d l m e n s i o n l e s s v o r t i c i t y d e f i n e d as the d i m e n s i o n a l v o r t i c i t y h The  ? = ? b/U,  being  E'V = E'V  (141)  s o l u t i o n o f the two s i m u l t a n e o u s  i n the stream f u n c t i o n  e q u a t i o n s , 139 and  and i n the v o r t i c i t y z, c o u l d be  a c h i e v e d by u s i n g Jenson's r e l a x a t i o n t e c h n i q u e the case of s p h e r o i d s . the n - d i r e c t i o n and B i n  ( 5 5 ) as f o r  Considering l a t t i c e spacing A i n 5 - d i r e c t i o n , as shown i n f i g u r e  3, and u s i n g T a y l o r ' s s e r i e s e x p a n s i o n c o r r e c t cond o r d e r as g i v e n by e q u a t i o n 106 - 1 0 9 , the  t o the sedifferential  e q u a t i o n s 139 and 140 take the f o l l o w i n g d i f f e r e n c e  G(l,J)(2/A +2/B ) 2  2  forms:  = G(*»J+1)+G(I.J-1) , 0(1+1,J)+0(1-1,J) A B 2  2  - Re \ ^ { I , J + Y ) - t y { I ^ - x ) 2 [ 2 A r  L  _ ^(I+l,J)-iJ;(I-l J) 2 B r  140,  >  J L  i r  J L  G(I+l,J)-G(I-l,J) 2B  n  J  G(I,J+1)-G(I,J-1)-,1 2 A J J  (142)  and ^(I,J)(2/A 2/B ) = 2  2  +  ,  <HI,J+1WI,J-1) A  ^(1+1, J ( 1 - 1 , J )  G ( I , J ) (sinh £(J)+sin n ( I ) 2  2  2  2 sinli E (143)  r e s p e c t i v e l y , where f o r e l l i p t i c a l  cylinders  G = r; .  For f l o w p a r a l l e l t o the minor a x i s , the d i f f e r e n c e equations  cosh E w i t h i s i n h E now  c o u l d be o b t a i n e d from e q u a t i o n s  by r e p l a c i n g each s i n h £ by i cosh E  and 143  finite  .  and  142  each  By t h i s p r o c e d u r e the term b  will  r e p r e s e n t the major s e m i - a x i s and a the minor semi-  a x i s , but the Reynolds number i s a g a i n based on the j e c t e d l e n g t h p e r p e n d i c u l a r t o the f l o w d i r e c t i o n . s o l u t i o n o f the above f i n i t e d i f f e r e n c e e q u a t i o n s  proThe  for  g i v e n boundary c o n d i t i o n s r e s u l t s i n the knowledge o f the flow d e t a i l s .  When E+°°  , equations  142  and  143  i d e n t i c a l t o those o b t a i n e d by Hamielec and R a a l  become  (6l) f o r  a c i r c u l a r c y l i n d e r with the exponential transformation £-5 z a  r - e  e  a s  where r i s the r a d i a l d i s t a n c e f o r the  c y l i n d r i c a l c o o r d i n a t e s and transformed B.2  z the r a d i a l d i s t a n c e i n t h e i r  coordinates.  Boundary The  Conditions computational  field  i n n e r s u r f a c e (the e l l i p t i c a l  i s c o n f i n e d between an  c y l i n d e r ) and an o u t e r  v e l o p e which c o i n c i d e w i t h the i n n e r and o u t e r  en-  elliptical  c o o r d i n a t e g r i d l i n e s , r e s p e c t i v e l y , as shown i n f i g u r e 4.  63  The f o u r boundary c o n d i t i o n s d e f i n i n g t h e f l o w are g i v e n by h a v i n g two boundary c o n d i t i o n s f o r each o f ^ and C.  F o r f l o w a l o n g t h e major a x i s t h e boundary con-  ditions for  are: a l o n g BA , i|) = 0  for n = 0  a x i s o f symmetry n = ir  a l o n g NO , if = 0  E = E  a l o n g AMN, if = 0 |  &  At t h e o u t e r boundary, £ = 5  surface of c y l i n d e r  , t h e f l o w i s assumed t o be  b  streaming p a r a l l e l f l o w , g i v i n g  ^ =sin n  s i n h E cosech E b  a  The boundary c o n d i t i o n s f o r C a r e : a l o n g BA , z, = 0  for n = 0  a x i s o f symmetry  n = TT  along N  , z, = 0  E = Ea0  a l o n g AMN,  E = Eb  a l o n g BRO, ? = 0  E = Eb  a l o n g BRO, z, = z,  z, = E i f 2  s i n h 2 E a„ case I  T  Q  case I I  En  The boundary c o n d i t i o n f o r z, a t  E = £ can be a  expanded as f o l l o w s : z, i s g i v e n by e q u a t i o n 140, which when com-  The v o r t i c i t y  b i n e d w i t h e q u a t i o n 138 l e a d s t o sinh E„  ~2.  2  sinh  2  a  E+sin  2  n  { 3_1 2 9E  .2  +  9_42  }  (144)  3n  At the s u r f a c e o f t h e c y l i n d e r , if i s a c o n s t a n t f o r a l l .2 values of , leading to = 0, so t h a t e q u a t i o n 144 becomes 3n n  64  sinh E sinh  The as  2  o ^ 2  a  5 +sin a  2  n  stream f u n c t i o n near Taylor's  35  (145) 5 = 5.  2  the surface  s e r i e s expansions  c a n be  to thethird  2  2  d5  5 = 5.  i(;(5 +2A)=^(5 )  + 2A | |  a  a  + 2A  5= 5  At the  146 a n d 147  to eliminate  d  d5  d5  M5  a  +A) - M 5  Eliminating  2  a  ^—!=j-  d5 equation  + 2A) = 2 A  from  5= 5  3  3  |  A  (146)  5=5,  =  m  3  F  d5  J which  0  surface.  13,,..  d ^  5 = 5,  +  2  —|  3  5 = 5.  a  no-slip condition at this  A  2  o f t h e c y l i n d e r , ^ = f|"  the surface  order:  ,  Adj|j 2  expressed  (147)  3  5=5,  satisfies  Combining  equations  yields  5 = 5„  (148)  ^-4 d5 5 = 5.  equation  145 b y u s i n g  t h e above  a  yields sinh 5 2  ?(I D 5  =  2 A  2  a  [ sinh 5  +  2  sin2n(D]  [8  ^(l,2)-^(I,3)]  (149)  3,  The  same r e s u l t  outlined  could  be o b t a i n e d  by u s i n g  t h e method  previously f o r the spheroid. The  boundary  condition for£ at  5 = 5^for  case I I  65 which corresponds  to. Happel's f r e e s u r f a c e model, can  be d e r i v e d as f o l l o w s : E q u a t i n g the shear s t r e s s g i v e n by e q u a t i o n 124 t o zero yields  1 and h = h . I n t r o d u c i n g the v a l u e s o f 9hj h i , - j r — and -^r— o b t a i n e d from e q u a t i o n 134 i n t o t h e on 95  where h dh  3  =  1  2  1  above e q u a t i o n g i v e s 2 2 9 y _ 9 y ~2 „ 2 " 95 ori r  At t h e envelope  2 s i n h 5 cosh 5 ,9i[> . . .2 .2 95 sinh 5+s m n  2 cos n s i n n 9^ . 2 . . 2 9n sinh 5+ s i n n  r x  surface,  5=5  u  U  , the dimensionless  U s i n g e q u a t i o n 152 t o o b t a i n  | 1 and i _ | 9n 9 n  3S  2  _ s r—  then 2  2 s i n h 5 cosh 5 2 2 sinh 5 s i n n b  sinh 5 a  r  ;  (152)  0  s i n n'sinh5^  s  i  stream  f u n c t i o n i s g i v e n by \p = sinrisinh5L cosech ^  2.,. 9 ijj  5  r  +  D  b  95  K  5=^  2 2 cos n. s i n n .sinh 5 + 2 2 " s i n h 5^+sin n The v o r t i c i t y a t 5 = 5 as o b t a i n e d from e q u a t i o n 144 i s K  (  1  5  3  )  b  r  ' ^£ = £ ^ ^b  _  Combination  2 sinh 5 *>2. .2, a r 9_JJ. . 9_1 -, 2 2 2 2 s i n h 5 + s i n n 9n 95 ^ b b  o f e q u a t i o n s 153 and 154 l e a d s t o  (154)  66  'x  r  =  0  2 s i n h E s i n h E, . a . 2 s i n h E^+sin 2r|  cosh E, ^ , ^b s i n h E,. _b 3jp_ . .2- . . 2 3E s i n h E^+sin n ^  cos T) s i n n 2 2 s i n h E^+sin n  The v a l u e o f | 1  5 = 5v  for d i f f e r e n t i a t i o n .  - 2 sin.n  (155) £=£>  i s found by u s i n g Newton's f o r m u l a The above e x p r e s s i o n , e q u a t i o n 155>  g i v e s ' e x p l i c i t l y t h e v o r t i c i t y a t t h e o u t e r boundary when the shear s t r e s s a t t h i s boundary i s z e r o .  Once a g a i n  the boundary c o n d i t i o n s f o r f l o w a l o n g t h e minor a x i s c o u l d be o b t a i n e d by r e p l a c i n g each s i n h E w i t h i coshg and cosh E w i t h i s i n h E • S i n c e t h e two simultaneous  d i f f e r e n t i a l equations  to be s o l v e d a r e o f second o r d e r , f o u r independent boundary c o n d i t i o n s were s u f f i c i e n t t o p r o v i d e a unique s o l u t i o n i n each case.  67 CHAPTER IV  F o r m u l a t i o n o f Drag C o e f f i c i e n t E q u a t i o n s and t h e Pressure D i s t r i b u t i o n The d e r i v a t i o n o f t h e drag c o e f f i c i e n t for  equations  an o b l a t e s p h e r o i d w i l l be g i v e n i n d e t a i l as i t w i l l  serve as an example o f t h e g e n e r a l method. A.  Spheroids G o l d s t e i n (10) g i v e s t h e d i m e n s i o n a l shear  s t r e s s i n g e n e r a l c u r v i l i n e a r c o o r d i n a t e s as  hi T  En  where  = T  n£  =  y  h  L_  (, h  9E  2  V  J  +  71  lll'L-  h  1  3n  ( i h  l  Y  )  ^  (156)  v^ and v^ a r e t h e d i m e n s i o n a l v e l o c i t i e s i n t h e  n - andE- directions respectively. spheroid, 9vl  At t h e s u r f a c e o f t h e  the n o - s l i p c o n d i t i o n s give . . = v- n = Vj. '= 0  9n and e q u a t i o n 156 becomes En  K  =F = v n =E a  9v 1  (157)  3E  The v o r t i c i t y i n g e n e r a l c u r v i l i n e a r c o o r d i n a t e s i s def i n e d as "n  1 = Vxv = h j h j h .  -<J>  h.  h.  h.  3E  9 9n  9_ 9<f>  h.  v h.  v, h.  (158)  68  For axisymmetrie  and t w o - d i m e n s i o n a l  f l o w s a l l de-  t  r i v a t i v e s i n 0 a r e zero and  v^ =0.  The o n l y non-  v a n i s h i n g v o r t i c i t y i s t h a t i n t h e 0 - d i r e c t i o n and i s _»  t  g i v e n by z, = ?  i  n  L  z, , where  i  8?.  n  8Tf  2  ]  5  (159)  and h j = h' f o r both s p h e r o i d a l and e l l i p t i c a l c o ordinates.  Using the n o - s l i p condition at the surface  (5 = 5 a. ) , e q u a t,i o n 159 becomes 8v  c'  =h  ^5=5  (160)  —^  5=5  85  n i  a  S u b s t i t u t i n g e q u a t i o n 157 i n t o e q u a t i o n 160 y i e l d s 5n 5=5  =  y  S  ^a  K  ^a  (161)  £=5  Making a l l terms d l m e n s i o n l e s s t  5n  r  = T'  and  /2pU 2  5n  the d l m e n s i o n l e s s  by p u t t i n g z,  = . s'a/U  >  l o c a l shear s t r e s s a t t h e s u r f a c e i s then  g i v e n by  5n  c _  = 4  ?  r  / Re  (162)  5='5  a  The s u r f a c e of t h e o b l a t e s p h e r o i d f o r z = 0 i s g i v e n by  y with  c  2  x  2  sinh 5  c  a  y = c sinh 5 and  2  2  cosh 5  (163)  cos n  (164)  x = c cosh 5 s i n n a  (165)  cL  69  The s u r f a c e a r e a o f t h e s t r i p on which the shear i s c o n s i d e r e d t o a c t i s 2ir x ( d x • + d y 2  )  2  stress  1 / / 2  Diagram 1. A c t i o n of the normal and t a n g e n t i a l s t r e s s e s on an element of a r e a o f the o b l a t e s p h e r o i d . The drag f o r c e i n the d i r e c t i o n of the s t r e a m i n g f l o w a c t i n g on the whole s u r f a c e due t o shear s t r e s s alone i s then g i v e n by P =  2TT  x ( dx  2  + dy  )  2  1  /  (166)  cos e  2  5 = 5  a  The drag on a body i s u s u a l l y e x p r e s s e d of a d i m e n s i o n l e s s drag  in.terms  coefficient  drag f o r c e i n the f l o w d i r e c t i o n 'D  ( P r o j e c t e d a r e a normal t o t h e ) ( d i r e c t i o n of f l o w ) From e q u a t i1U1o n 166 2TT DS  x  x k i n e t i c pressure  the s k i n drag c o e f f i c i e n t then becomes (dx  (TT C  2  2  + dy )  cosh  2  2  £  1 / 2  T^|g  ) ( |p U  2  =  g )  a  C  Q  S  (167)  70  Now cos 6 =  — ( dx  2 2  dx = c cosh 5  1 / 2  + dy ) 2  1 7 2  cos n dn  ,  a dy = - c s i n h £.a. s i n n dn  ,  and making use o f e q u a t i o n 162 and e q u a t i o n 167 y i e l d s  C  DS  =  I?  t  a  n  h  ?  a  /  s  i  n  The above e q u a t i o n g i v e s  2  n  C  5=C  d  (  n  l  6  8  )  the s k i n ( f r i c t i o n ) drag c o -  e f f i c i e n t f o r an o b l a t e s p h e r o i d .  In the l i m i t i n g  case o f sphere, tanh '£ • i s u n i t y and e q u a t i o n 168 becomes a i d e n t i c a l t o t h a t g i v e n by Jenson ( 5 5 ) sphere.  f o r a perfect  F o r an i n f i n i t e s i m a l l y t h i n c i r c u l a r d i s c , tanh £ a  i s zero and thus e q u a t i o n 168 shows a zero s k i n drag c o e f f i c i e n t , i n c o n f o r m i t y w i t h s i m p l e g e o m e t r i c a l considerations.  For a p r o l a t e spheroid, tanh?  a  i s trans-  formed as u s u a l t o c o t h £ , but i t then becomes d i f f i c u l t a t o p r e d i c t C^g i n t h e l i m i t o f a v e r y l o n g n e e d l e , f o r which C r _ 0 and c o t h B, -* CD . ^ a r  a  The p r e s s u r e d i s t r i b u t i o n around an o b l a t e s p h e r o i d i s o b t a i n e d by c o n s i d e r i n g , t h e i ^ and i ^ of t h e N a v i e r - S t o k e s e q u a t i o n .  The N a v i e r - S t o k e s  as g i v e n by e q u a t i o n 3 can be t r a n s f o r m e d  (8) t o  directions equation  71 -'  v by  x  =  K  i  1  VP  -  -  simple t  e q u a t i o n , V.v  0  =  Let  us  169  equation  vector  x  z  now  h  1  2  h  l  h  dimensional  is -  * p  where V  V x  ~  !h h 2  of  on  side  -v  third  of  1  3  I  h  first  ^  "2  by  3n  equation  9/9E  ;  term  ( V ?  169  h )  second  4.  By  9/9n  9/34)  n  /  h  /  becomes  on  the  h  we  have  3  (171)  2  -direction  n  terms  equation  definition,  1  C  c , the  =  yield  i n /h„2  i n the i  3  and  h  2  two-  left-hand side of  169  I  and  Putting  equation  +  equation  h h |  / 3  t £,/h  the  / h  the  have  (170)  0  =  4 i  and  ±  we  of  V  g'  The  side  h  2  n  t ? /h  -direction  i s defined  ? =  /  component  definition,  V  V,  f l o w s , zl  3P' 3n  h  n  i  3  h ^  right-hand  the c o n t i n u i t y  previously for axisymmetric  term i n the i 169  By  i  r  mentioned  (169)  and  the  term.  ! r, /h  As  £  identities  1  consider  t e r m by  = h  + v  .  V  v  = V x  2  2  P.  using  l - -' + ^ V v  on  the  right-hand  72 The i^-component o f e q u a t i o n 169 thus becomes h  •  -' 2  1  v  _i2 At the s u r f a c e , v =0 , hj = h  i  n  3  (172)  3£  1  = 0 and f o r an o b l a t e s p h e r o i d ,  , so t h a t e q u a t i o n 172 becomes  2  9P  9n  n  5=5  =  y {  95  5=5,  a  35  3  (l/h ) }  (173)  3  I n t r o d u c i n g the d i m e n s i o n l e s s p r e s s u r e 1  t  P - Pst__ .  P =  substituting for h  (173a) and r e n d e r i n g a l l terms d i m e n s i o n l e s s ,  3  then e q u a t i o n 173 becomes Re 9P  T~  9n  _  5=5.  3?  95  5=5 a  +  C _ r  (174)  tanh 5 a  p  4  I n t e g r a t i o n o f e q u a t i o n 174 a l o n g AS o f f i g u r e 2 g i v e s n P  4 " ° " Re P  I  I[  <•  95  H  5=5a  + ? _ r ? _ 5  r  a  tanh 5  Q  ] dn  (175)  a  0 where P  Q  i s the dimensionless f r o n t a l s t a g n a t i o n pressure  (at n = 0) and P i s the d i m e n s i o n l e s s p r e s s u r e a t p o i n t S on t h e s u r f a c e o f the s p h e r o i d . The d i m e n s i o n l e s s s t a g n a t i o n p r e s s u r e P  c  i s ob-  t a i n e d by c o n s i d e r a t i o n o f the i^-component o f e q u a t i o n 169. U s i n g 'equation 170, the term i n the i ^ . - d i r e c t i o n on the l e f t -  73  hand s i d e of e q u a t i o n 169 I s 'h-v^ c'  , s i n c e once  2  again ^  = ^' and ^  =' 5^ = 0 f o r a x l s y m m e t r i c and two-  dimensional flows.  The f i r s t and second terms on the  r i g h t - h a n d s i d e o f e q u a t i o n 169 y i e l d f o r the i ^ - d l r e c t i o n h  '  ~  -' 2  dK  2  H  1  The l a s t term o f e q u a t i o n 169 g i v e s h h 2  3  7^-( ? / h ) . 3  The i ^.-component of e q u a t i o n 169 thus becomes h  *V  rlr  ,B  I i H'  + v h  h  +  * 3 In ^'/h, ) (176) h  C o n s i d e r i n g f l o w a l o n g the a x i s BA-shown i n f i g u r e 2 g i v e s v  r  n t  v. Hence  =0  (no-cross flow)  =0  (axial  _«2_.2 v  =v  _,2  + v  c  5  symmetry) _.2  + v,  n  _ i  = v  ,  r  <p  2  5  and e q u a t i o n 176 a l o n g n = 0 becomes 3 v  £  ,  3P  (177)  where a l l q u a n t i t i e s have been r e n d e r e d d l m e n s i o n l e s s and the v a l u e s o f h , ,  h  2  and h  3  f o r an o b l a t e s p h e r o i d have  been s u b s t i t u t e d . I n t e g r a t i n g a l o n g BA y i e l d s B dC n=0  (178)  74 As p o i n t B l i e s on t h e o u t e r boundary and p o i n t A on the s u r f a c e o f t h e s p h e r o i d respectively that P  =0,  B  v  (stagnation p o i n t ) , i t follows B  = 1 and P  A  = P , v D  A  = 0.  E q u a t i o n 178 t h e r e f o r e becomes E, p  °  =  1  J / U  Si"  +  cotn  c o t  U  +  n+ In" f^- ) )  (179)  d£  ^a Along t h e a x i s  n  = 0, c o t n = °° and t, = 0, u s i n g L* H o s p i t a l ' s  r u l e , e q u a t i o n 179 becomes  P F  ! + §_  =  °  EVj  fn  Re J ~r\  1  (180)  d£  n=o  E q u a t i o n 180 g i v e s t h e d i m e n s i o n l e s s  f r o n t a l stagnation  p r e s s u r e and e q u a t i o n 175 g i v e s t h e d i m e n s i o n l e s s  pressure  d i s t r i b u t i o n around t h e s u r f a c e o f t h e o b l a t e s p h e r o i d . I t i s i n t e r e s t i n g t o note t h a t P becomes a n t i - s y m m e t r i c a l about  n  =TT/2  as Re + o and t h a t t h e u n i t y t e r m i n e q u a t i o n  180, which a r i s e s from t h e i n e r t i a l term o f the. N a v i e r Stokes  e q u a t i o n , does not appear when t h e i d e n t i c a l  a n a l y s i s i s made on the c r e e p i n g f l o w e q u a t i o n . U s i n g diagram 1, t h e form ( p r e s s u r e ) drag c o e f f i c i e n t i s g i v e n by  c  D p  =  - L  x ( dx + dy -jTT c cosh E a  2TT  2  ) '  sin 6 P .  (181)  75  where  s i n 0 = dx / ( dx  and at the  2  + dy  2  )  1 / 2  Surface, sin n  x = c cosh £ £1  dx = c cosh £  3.  cos n dn ,  rendering equation l 8 l  TT  C  D p  JP  =  sin  dn  2 n  ( 1 8 2 )  0  The  t o t a l drag c o e f f i c i e n t i s o b t a i n e d  adding the s k i n and form drag c o e f f i c i e n t s . ponding e q u a t i o n s  The  by corres-  f o r a p r o l a t e s p h e r o i d c o u l d be d e r i v e d  by the u s u a l t r a n s f o r m a t i o n . B. E l l i p t i c a l  Cylinders  F o l l o w i n g the a n a l y s i s g i v e n f o r an o b l a t e s p h e r o i d , e q u a t i o n 1 6 2 i s a l s o a p p l i c a b l e t o an e l l i p t i c a l for  cylinder,  which the shear s t r e s s at the s u r f a c e i s t h e r e f o r e  T  =  £n  The  4  ?  /Re  ( 1 6 2 )  c y l i n d e r at z = 0 i s g i v e n by  s u r f a c e of the e l l i p t i c a l 2  2 +  y  c  2  2  sinh £  a  c  cosh E  =  a  1  d  8  3 )  76  with y = c sinh £  sinn  (184)  x = c cosh £ a  cos n  (185)  cl  and  The s u r f a c e  a r e a o f t h e s t r i p on which t h e shear s t r e s s i s  acting i s £ ( dx  2  + dy  )  2  1  /  2  ,  dy  U  U  Diagram 2. A c t i o n o f t h e normal and t a n g e n t i a l s t r e s s e s an element of a r e a o f the e l l i p t i c a l  on  cylinder.  where I i s the l e n g t h of the s t r i p i n t h e z - d i r e c t i o n . The s k i n drag c o e f f i c i e n t i s then g i v e n by 2 J 'DS where cos 0=  ft  T  l  R  ( dx  2  + dy  )  2  5n  /  2  cos 6 (186)  ( \ pU  2  ) 2 c sinh £  dx ( dx  1  2  dy ) / 2  +  1  2  I  77  dx = - c cosh E  s i n ndn  and u s i n g e q u a t i o n 162,  ,  e q u a t i o n 186  becomes  TT C DS  Re  coth  5=5  0  sin n  dn  (187)  a  The above e x p r e s s i o n g i v e s the s k i n drag for  an e l l i p t i c a l  to  i t s major a x i s .  coefficient  c y l i n d e r w i t h the main f l o w p a r a l l e l For an e l l i p t i c a l  c y l i n d e r w i t h the  main f l o w p a r a l l e l to i t s minor a x i s , c o t h £  becomes  a and hence i n the l i m i t i n g case o f a f l a t  tanh E  plate  CL  set  p e r p e n d i c u l a r t o the f l o w d i r e c t i o n , C^g  -* 0 as  tanh £ a ->• 0, which i s i n t u i t i v e l y c o r r e c t . The d l m e n s i o n l e s s s u r f a c e p r e s s u r e , P, can  be  deduced from the a n a l y s i s of t h e o b l a t e s p h e r o i d by approp r i a t e l y m o d i f y i n g e q u a t i o n 173  and 175,  the r e s u l t  being  n (188)  s i n c e the l a s t term o f b o t h e q u a t i o n s zero because h The  3  =1  for e l l i p t i c a l  stagnation pressure P  equations  176  and 178,  G  and 175  cylindrical  becomes coodinates.  can be s i m i l a r l y o b t a i n e d from  where the l a s t term of e q u a t i o n  a g a i n reduces t o zero f o r h e q u a t i o n 179  173  i s modified to  3  176  = 1, so t h a t the r e s u l t i n g  78  ^b P  o  =  1  k/In-  +  U s i n g diagram  I  d ?  (189)  n  ,n=o  2, the form  ( p r e s s u r e ) drag c o e f f i c i e n t i s  g i v e n by 2 ^  PP ssiinn 6 H ( dx^ + dy  J  C  2  =  I  u r  .) •  (190)  2 c sinh £ a  C  DF  =  JP  cos n dn  (19D  o where  sin9 =  ~ — (dx  2  —  0  + dy  >  N  )  2  0  1  /  2  and dy = c s i n h E  cl  cos n  dn  The e x p r e s s i o n g i v e n by e q u a t i o n 191 i s i d e n t i c a l t o t h a t o b t a i n e d by Hamielec  and R a a l (.61) f o r c i r c u l a r c y l i n d e r s .  The t o t a l drag c o e f f i c i e n t , t a i n e d by adding C  D S  and  C . Dp  c D  rp>  i s again  ob-  79  CHAPTER V  D i s c u s s i o n and P r e s e n t a t i o n o f R e s u l t s 1. A p p l i c a b i l i t y o f the N a v i e r - S t o k e s E q u a t i o n s and of the N u m e r i c a l  Choice  Technique  The steady s t a t e form of the N a v i e r - S t o k e s e q u a t i o n i s a s p e c i a l case of the g e n e r a l momentum e q u a t i o n g o v e r n i n g the f l o w f i e l d where both the v i s c o s i t y and d e n s i t y o f the f l u i d are c o n s t a n t .  The momentum e q u a t i o n i s d e r i v e d f o r a  continuous f l u i d , even though r e a l f l u i d s are composed o f f i n i t e elementary p a r t i c l e s  i n c o n t i n u o u s motion i n a r e -  l a t i v e l y l a r g e expanse o f space.  As the f l u i d under study  i s c o n t i n u o u s even t o the i n f i n i t e s i m a l l i m i t , t i e s such as p r e s s u r e , v e l o c i t y  i t s proper-  and temperature can be r e g a r d e d  as c o n t i n u o u s and can be m a t h e m a t i c a l l y d e f i n e d .  When a body  i s immersed w i t h i n a r e a l f l u i d and i t s dimensions are much l a r g e r than the mean f r e e p a t h of the f l u i d m o l e c u l e s , t h e n the momentum e q u a t i o n i s adequate field.  i n d e s c r i b i n g the f l o w  However, f o r s m a l l body d i m e n s i o n s , the  Brownian  Movement of the f l u i d m o l e c u l e s c o n t r i b u t e s t o the f l o w f i e l d . Lapple (148) has shown t h a t the c o n t r i b u t i o n t o the drag f o r c e e x p e r i e n c e d by a 10y i s about 0.5%, 130$,  diameter sphere from Brownian Movement  whereas f o r a l u  of the g r a v i t a t i o n a l l y  d i a m e t e r sphere i t .is over  induced drag.  This indicates  t h a t the a n a l y s i s i n the p r e s e n t work i s o n l y v a l i d where the p a r t i c l e (s)  under c o n s i d e r a t i o n i s l a r g e enough t h a t the de-  t a i l s of i t s shape are "seen" by the f l u i d . -  The  80  axisymmetric  steady s t a t e form o f t h e N a v i e r - S t o k e s  i s used, which means t h a t t h e r e i s complete "top"  and "bottom" o f t h e f l o w f i e l d  equation  symmetry between  and hence t h a t any  asymmetric phenomena such as v o r t e x shedding and wake i n s t a b i l i t y c o u l d not be d e t e c t e d . The main o b j e c t i v e o f t h i s work i s t o f i n d t h e f l o w c h a r a c t e r i s t i c s f o r i s o l a t e d s p h e r o i d s and e l l i p t i c a l c y l i n d e r s and f o r t h e i r r e s p e c t i v e c l u s t e r s u s i n g t h e c e l l models, r a t h e r than t o i n v e s t i g a t e and experiment various numerical techniques. has been found  Jenson's r e l a x a t i o n  c y l i n d e r s and spheres.  2.  technique  (58-61) t o be v e r y adequate and s u c c e s s f u l ' i n  accurately d e s c r i b i n g the flow f i e l d  elliptical  with  f o r both  circular  I t s a d a p t a t i o n t o s p h e r o i d s and  cylinders follows quite readily.  R e l a x a t i o n Procedure The r e l a x a t i o n t e c h n i q u e used  involved calculating  new v a l u e s f o r if.(I,J) and G ( I , J ) from t h e f o u r a d j a c e n t p o i n t s l y i n g on t h e o r t h o g o n a l l i n e s c r o s s i n g a t ( I , J ) , a s shown i n f i g u r e 1 f o r s p h e r o i d s and f i g u r e 3 f o r e l l i p t i c a l The new v a l u e s o f  cylinders.  and G a r e c a l c u l a t e d by means o f e q u a t i o n s  110 and 111 f o r t h e s p h e r o i d s and 142 and 143 f o r t h e elliptical  cylinders.  As t h e r e l a x a t i o n procedure  :  i s the  same f o r t h e s p h e r o i d s and t h e c y l i n d e r s , t h e t e c h n i q u e f o r the former o n l y i s d i s c u s s e d i n d e t a i l .  I n the. i t e r a t i v e p r o -  cedure f o r s o l v i n g t h e d i f f e r e n c e e q u a t i o n s , r e l a x a t i o n parameters  a  g  and ag  p u t a t i o n s as f o l l o w s :  were i n t r o d u c e d t o s t a b i l i z e t h e com-  81  1^(1,J) = ^ _ ( I , J ) + n  G  n  ( I  '  J )  =  G  n-l  a [ i[»*(I,J) - * _ ( I , J )  1  (  I  '  J  )  +  s  a  g  [  G  n  n  ( I  > ) ~ n-l J  where t h e s u b s c r i p t n denotes t h e  G  1  (  I  '  J  )  ]  ]  (192)  ( 1 9 3  >  n t h i t e r a t i o n and s u p e r -  s c r i p t * s i g n i f i e s t h e u n m o d i f i e d r e s u l t o f the n t h i t e r a t i o n . The p r o p e r c h o i c e of a  and a s  were v e r y i m p o r t a n t f o r cong  t r o l l i n g t h e r a t e of convergence.  A Gauss-Seidel. s o l u t i o n .  uses a r e l a x a t i o n parameter o f u n i t y , i . e . t h e new v a l u e c a l c u l a t e d f o r p o i n t ( I , J ) i s used u n a l t e r e d i n t h e next iteration.  I t has been shown by P o r s y t h e and Wasow (152)  that f o r l i n e a r equations, successive o v e r - r e l a x a t i o n using Ka<2 suggest  g r e a t l y improves t h e r a t e o f convergence.  A l s o they  t h a t i t i s b e t t e r t o be s l i g h t l y on t h e h i g h s i d e o f  the optimum a than on t h e low s i d e f o r h i g h e r r a t e o f convergence.  I n the p r e s e n t problems e q u a t i o n s 111 and 143  are l i n e a r , and i n g e n e r a l a v a l u e o f 1.8 was a s s i g n e d t o a  f o r s p h e r o i d s and 1.8 - 1.6 f o r e l l i p t i c a l c y l i n d e r s . .  On t h e o t h e r hand, e q u a t i o n s 110 and 142 a r e l i n e a r o n l y when Re i s e q u a l t o z e r o , and t h e i r " n o n l i n e a r i t y " becomes more pronounced as Re i n c r e a s e s . This n e c e s s i t a t e d a c o n t i n u a l r e d u c t i o n i n a as Re was i n c r e a s e d f o r t h e s i n g l e b o d i e s , i n g o r d e r t o o b t a i n convergence. the comments made by B u r g g r a f  The p r e s e n t work s u b s t a n t i a t e s (153) t h a t the r e l a x a t i o n  parameter a a p p e a r s t o be f a i r l y i n s e n s i t i v e t o mesh s i z e , but s t r o n g l y dependent on the p r e c i s e method o f i t e r a t i o n used. In t h i s work, f o r each a l t e r n a t i v e c y c l e o f these  calculations,  p o i n t s were c o n s i d e r e d from the s u r f a c e o f the s p h e r o i d t o t h e  82  o u t e r boundary, s t a r t i n g w i t h n=0 a t t h e f r o n t t o n=tr a t t h e rear.  Once a c y c l e was completed,  t h e next c y c l e was  started  from n = TT p r o c e e d i n g down t o n=G,points b e i n g t a k e n from t h e o u t e r boundary t o t h e s o l i d s u r f a c e .  C o r r e c t e d v a l u e s o f 4*  and G were used as soon as they became a v a i l a b l e .  However, i f  the c o r r e c t e d v a l u e s o b t a i n e d d u r i n g a g i v e n c y c l e were not u t i l i z e d once a v a i l a b l e , i t was found t h a t t h e optimum v a l u e of a  g  would be l e s s than t h a t o f the former method, and con-  s e q u e n t l y t h e number of i t e r a t i o n s would be i n c r e a s e d f o r proper convergence.  Rhodes (57) used t h e l a t t e r method and  found t h a t he r e q u i r e d an e x c e s s i v e number Of i t e r a t i o n s , f o r example 10,000 i t e r a t i o n s f o r convergence i n the case o f f l o w p a s t a.sphere a t Re = 100.  The number o f i t e r a t i o n s f o r the  l i m i t i n g case o f a sphere u s i n g the, former method was 1000 as o b t a i n e d i n t h e p r e s e n t work. D u r i n g t h e course o f t h e work i t was found t h a t large. values of a  i n t r o d u c e d an o s c i l l a t i o n o f t h e v o r t i c i t y a t  the s u r f a c e , and t h a t such an o s c i l l a t i o n sometimes l e d t o divergence.  F o r t h i s reason a s e p a r a t e r e l a x a t i o n parameter,  a  ,was i n t r o d u c e d t o t h e v o r t i c i t y a t t h e s u r f a c e . This gs made i t p o s s i b l e t o a s s i g n a h i g h e r v a l u e t o a than was p r e viously possible, a and a  gs  b e i n g s m a l l e r than a..  Values of a  f o r v a r i o u s v a l u e s o f Re and aspect r a t i o , A.R., are  g i v e n i n t a b l e s 2 and 3 f o r s p h e r o i d s and t a b l e s 4 and 5 f o r ' elliptical  cylinders.  C a l c u l a t i o n s were begun by f i r s t a s s i g n i n g approximate v a l u e s t o if and G on the g r i d from the c r e e p i n g f l o w e q u a t i o n for  a sphere i n the case o f t h e s p h e r o i d s and from the  83 Table 2 Relaxation factors f o r oblate  spheroids No. i t e r a t i o n s f o r convergence  Re  A.R.  ag  a gs  1 1 1 1  0.999 0.9 •0.5 0.2  0.7 0.7 0.7 0.7  0.5 0.5 0.5 0.4  400 450 500 600  5 5 5 5  0.999 0.9 0.5 0.2  0.5 0.5 0.5 0.5  0.3 0.3 0.3 0.3  400 400 450 450  10 10 10 10  0.999 0.9 0.5 0.2  0.3 0.3 0.3 0.3  0.15 0.15 0.15 0.15  1750 2100 2200 2300  20 20 20 20  0.999 0.9 0.5 0.2  0.25 0.25 0.25 0.20  0.13 0.13 0.13 0.10  1100 1050 1200 1300  50 50 50 50  0.999 0.9 0.5 0.2  0.2 0.2 0.2 0.15  0.1 0.1 0.1 0.08  900 1000 1000 . 1500  100 100 100 100  0.999 0.9 0.5 0.2  0.08 0.08 0.08 0.08  0.05 0.05 0.05 0.05  1000 1150 1250 1800  0.01  9-0.2  0.8  0.6  250 - 400 swarms  84 Table  3  Relaxation f a c t o r s f o r prolate spheroids A.R.  R e  a  g  a  gs  No. i t e r a t i o n s f o r convergence  1 1 1  0.9 0.5 0.2  0.70 0.70  0.60  0.55 0.55 0.50  450 500 400  5 5 5  0.9 0.5 0.2  0.5 0.5 0.45  0.3 0.3 0.3  550 550 600  10 10 10  0.9 0.5 0.2  0.25 0.25 0.25  0.15 0.15 0.15  700 550 900  20 20 20  0.9 0.5 0.2  0.25 0.25 0.40  0.15 0.15 0.30  750  50 50 50  0.9 0.5 0.2  0.20 0.20 0.15  0.10 0.10 0.10  850 850 650  100 100 100  0.9 0.5 0.2  0.08 0.08 0.06  0.05 0.05  1000 1000 650  0.01 0.01  0.9-0.5 0.2  0.6 0.5  0.50 0.40  0.04  800  600  400 swarms 800 swarms  85 Table 4 Relaxation  f a c t o r s f o r e l l i p t i c a l c y l i n d e r s w i t h O.R. > 1  Re  O.R.  a g  oi gs  1.0 5.0  %1.0 <vi.O  0.6 0.2  0.45 0.15  1900 2500  1.0 5.0 15.0 40.0 90,0  10/9 10/9 10/9 10/9 10/9  0.4 0.2 0.08 0.03 0.005  0.3 0.15 0.05 0.02 0.003  2150 2950 2250 2500 3500  .5 1.0 5.0 15.0 50.0  2 2 2 2 2  0.65 0.40 0.2 0.06 0.02  0.55 0.3 0,15 0.03 ... 0.01  1150 1750 1800 2000 2150  2 1.0 5-0 10.0 20.0 40.0  5 5 5 5 5 5  0.6 0.5 0.12 0.05 0.03 0.02  0.5 0.4 0.10 0.035 0.015 0.01  1150 1500 2150 1800 1950 1500  0.7  0.5  v  0.01  10/9-5  No. i t e r a t i o n s f o r convergence  450 - 600 swarms  86  Table 5 Relaxation factor f o r e l l i p t i c a l cylinders Re 1.0 5.0 15.0 40.0  0.01  O.R.  a •  0.2 0.2 0.2 0.2  0.4 0.3 0.06 0.018  0.3 0.15 0.03 0.012  0.7  0.5  0.9-0.2  a  g  g  w i t h O.R. < 1 No. i t e r a t i o n s f o r convergence 1900 2500 5500 3500  450 - 600 swarms  87  n u m e r i c a l s o l u t i o n f o r a c i r c u l a r c y l i n d e r i n case o f the elliptical  cylinders.  s p h e r o i d s a new  Using equation  v a l u e o f if was  by e q u a t i o n 192.  The new  m  ,  f o r oblate  c a l c u l a t e d and then  v a l u e o f if was  modified  used i n e q u a t i o n  110 t o f i n d an improved v a l u e o f G, which was  subsequently  m o d i f i e d by e q u a t i o n 193.  repeated  no f u r t h e r change was  T h i s procedure was  apparent i n the d e s i r e d s i g n i f i c a n t ,  f i g u r e o f any f u n c t i o n between s u c c e s s i v e 3.  until  iterations.  Computer Programs The n u m e r i c a l a n a l y s i s o f f l o w p a s t o b l a t e s p h e r o i d s ,  i n c l u d i n g the c e l l models, were c a r r i e d out u s i n g an IBM  7044  d i g i t a l computer, but the f u r t h e r work on p r o l a t e s p h e r o i d s and e l l i p t i c a l c y l i n d e r s was  c a r r i e d out on an IBM  i n c o n j u n c t i o n w i t h M i c h i g a n T e r m i n a l Systems.  360/67  Two  programs were w r i t t e n , one f o r the s p h e r o i d s and the for  the e l l i p t i c a l  cylinders.  Each program was  i n c o r p o r a t e a l l p o s s i b l e v a r i a b l e s t o be s t u d i e d :  general other  w r i t t e n to o b l a t e or  p r o l a t e i n the case o f the s p h e r o i d s , f l o w p a s t major or minor a x i s i n the case o f the e l l i p t i c a l  c y l i n d e r s , v o l u m e t r i c con-  c e n t r a t i o n , Reynolds number, g r i d s i z e , r e l a x a t i o n parameters and number of i t e r a t i o n s per r u n .  Each program e v a l u a t e d  the s k i n and drag c o e f f i c i e n t s and the d i m e n s i o n l e s s  surface  p r e s s u r e , and i t i n i t i a t e d the c o n t o u r i n g o f s t r e a m l i n e s , e q u i - v o r t i c i t y l i n e s and e q u i - v e l o c i t y l i n e s * . * The  c o n t o u r i n g program was  The  listing  w r i t t e n by the Computing  Centre S t a f f of the U n i v e r s i t y o f B r i t i s h Columbia.  88 of the two programs i s g i v e n i n Appendix  IV..  The p o i n t v a l u e s of the stream f u n c t i o n and  vorticity  for  a l l the cases are s t o r e d i n a master tape i n d u p l i c a t e .  The  computer times t o produce a new  v a l u e f o r one g r i d p o i n t  are 0,000441 and 0.0002 61 second p e r i t e r a t i o n f o r the  spheroids  and the e l l i p t i c a l c y l i n d e r s , r e s p e c t i v e l y , based on the  IBM  360/67. 4.  V a l i d i t y of the n u m e r i c a l work In the r e l a x a t i o n p r o c e d u r e , o r i n any o t h e r  technique  i n which o n l y p o i n t v a l u e s o f the f i e l d f u n c t i o n under cons i d e r a t i o n are known, m i s l e a d i n g r e s u l t s c o u l d be  obtained  from the v a l u e s o f the f i e l d f u n c t i o n s at the l a t t i c e p o i n t s u n l e s s g r e a t care were t a k e n by c o n s i d e r i n g the v a r i o u s comp u t a t i o n a l parameters t h a t c o u l d i n f l u e n c e the f l o w f i e l d . These parameters a r e : (a) the number o f T a y l o r expansion terms used,  (b) Convergence t o l e r a n c e , (c) mesh s i z e , (d) p o s i t i o n  of the o u t e r boundary f o r s i n g l e b o d i e s and i a t i o n technique. (a)  The  (e) the d i f f e r e n t -  Comments on each parameter f o l l o w :  f l o w e q u a t i o n s were transformed t o  finite  d i f f e r e n c e ' e q u a t i o n s by u s i n g second o r d e r T a y l o r expansions.  Jenson (55), i n h i s work on  used b o t h second and f o u r t h o r d e r T a y l o r for  spheres, expansions  Re = 5, and found no d i f f e r e n c e i n the  accuracy  of the f i n a l v a l u e s of the d r a g c o e f f i c i e n t s f o r a l a t t i c e s p a c i n g o f 12 degrees i n the a n g u l a r , and 0.2  u n i t s of I n r (where r = d i m e n s i o n l e s s  c o o r d i n a t e ) i n the r a d i a l , d i r e c t i o n .  radial  Although  a  s i m i l a r i n s e n s i t i v i t y t o the order of the T a y l o r e x p a n s i o n would be l e s s p r o b a b l e f o r a s p h e r o i d o r e l l i p t i c a l c y l i n d e r of small aspect r a t i o , the second o r d e r e x p a n s i o n was n e v e r t h e l e s s used throughout t h i s a n a l y s i s i n o r d e r t o keep t h e comp u t e r time w i t h i n r e a s o n a b l e bounds. (b)  Convergence  o f t h e computed v a l u e s f o r t h e stream  f u n c t i o n and v o r t i c i t y was assumed when no f u r t h e r change was apparent I n t h e f o u r t h s i g n i f i c a n t f i g u r e o f any f u n c t i o n between s u c c e s s i v e iterations.  With such a t o l e r a n c e t h e d r a g  c o e f f i c i e n t s and s u r f a c e p r e s s u r e d i s t r i b u t i o n s would not change by more than 0.2% and 0.5%,  res-  p e c t i v e l y , f o r t h e case o f a s p h e r o i d and 0.5% and 1.0$ r e s p e c t i v e l y , f o r t h e case o f an e l l i p t i c a l c y l i n d e r , f o r a f u r t h e r 50 i t e r a t i o n s .  I n some  cases o f low Re and l a r g e mean d i a m e t e r f o r t h e e l l i p t i c a l c y l i n d e r s , t h e convergence  c r i t e r i o n was  t a k e n when no f u r t h e r change o c c u r r e d i n t h e f i f t h significant  f i g u r e o f any f u n c t i o n between s u c c e s s i v e  iterations.  D u r i n g t h e course o f t h i s work i t was  found t h a t w h i l e t h e drag c o e f f i c i e n t s were not v e r y s e n s i t i v e t o t h e degree o f convergence points,P  of the l a t t i c e  was v e r y s e n s i t i v e t o t h e degree o f con-  vergence. (c)  The e f f e c t o f t h e mesh s i z e f o r t h e o b l a t e s p h e r o i d s at Re: = 50 and Re = 100 was e v a l u a t e d by changing t h e  90  g r i d s i z e f o r t h e angles from ^6° t o ^ 3 ° , and the r a d i a l s t e p from ^0.06 t o ^0.03.  The r e s u l t i n g  changes i n t h e t o t a l drag c o e f f i c i e n t s f o r t h e two Reynolds numbers a t any aspect r a t i o were w i t h i n t h e n u m e r i c a l e r r o r (e.g. a p p r o x i m a t e l y  2%).  No a p p r e c i a b l e change i n the d i m e n s i o n l e s s  frontal  s t a g n a t i o n p r e s s u r e was d e t e c t e d , but t h e r e a r s t a g n a t i o n p r e s s u r e was a l t e r e d by about 30$ w h i c h , however, r e p r e s e n t e d o n l y a 5% change r e l a t i v e t o the f r o n t a l s t a g n a t i o n p r e s s u r e .  F o r lower Re, t h e  l a r g e r step s i z e was used because t h e r a t e o f change o f t h e f l o w f u n c t i o n s w i t h s t e p s i z e was not as r a p i d as t h a t f o r h i g h e r v a l u e s o f Re.  The  shape o f the wake was found t o be almost t h e same f o r t h e two mesh s i z e s used except  f o r A.R. o f 0.2,  where the wake l e n g t h was about 10% s m a l l e r f o r t h e finer grid size.  F o r the p r o l a t e s p h e r o i d s , t h e  step s i z e was found t o be more c r i t i c a l , and t h e r e f o r e a s m a l l e r step s i z e was used throughout.  A change  i n s t e p s i z e o f ^6° i n the a n g u l a r and ^0.04 i n the r a d i a l d i r e c t i o n t o ^3° and^0.02°  respectively,  f o r Re = 100 and A.R.= 0.5, gave a H% d i f f e r e n c e i n the t o t a l drag c o e f f i c i e n t , u s i n g t h e same d i f f e r e n t i a t i o n and i n t e g r a t i o n  formulae.  t a n e o u s l y changed by 1% and P to P . 0  n = T r  P  Q  was s i m u l -  by about 6% r e l a t i v e  Some s e l e c t e d v a l u e s f o r t h e drag c o -  e f f i c i e n t s and P . a r e g i v e n i n Table 6 . 0  The angle  Table 6 Some s e l e c t e d d a t a f o r s i n g l e Mark Type o f spheroid  A.R.  Mean IMxJM diameter m  Re  C  D S  d  266 137 234 113 165 97 96 128 224 111 164 59 84 126 217 126 217 112 161 61 8 106 150 90 283 58  Oblate " . "  " "  " " "  " " • »  0 .2 0 .2 0 .2 0 .2 0 .2 0 .2 0 .2 0 .5 0 .5 0 .5 0 .5 0 .5 0 .5 0• 9 0 .9 0 .9 0 .9 0• 9 0• 9 0• 9 0 .9 0 • 999 0 .999 0 • 999 0 .999 0 .999  7. 0 17. 11 17. 11 17. 11 17. 11 17. 11 29. 90 17. 13 17. 13 17. 13 17. 13 17. 13 29- 95 17. 10 17- 10 17. 10 17- 10 17. 10 17- 10 17. 10 29. 63 17. 10 17. 10 17. 10 17. 10 17. 10  41x65 33x59 61x105 33x59 61x105 33x59 33x69 33x59 61x105 33x59 61x105 33x59 33x73 33x59 61x105 33x59 61x105 33x59 61x105 33x59 33x65 33x59 61x105 33x59 61x65 33x59  100 100 100 50 50 10 10 100 100 50 50 10 10 100 100 100 100 50 50 10 10 50 50' 20 20 10  0 .1947 0 .1755 0 .1666 0 .2928 0 .2855 0 .9186 0 .9055 3 .747 3 .627 0 .601 0 .586 1 .818 1 .802 0 .576 0 .554 0 • 576 0 • 554 0 .890 0 .868 2.665 2 . 650 0 .949 0•925 1 .740 1 • 715 2 .844  C 3 points 1 .155 1 .157 1 .123 1 .378 1 .368. 2 .985 2 .940 0 .812 0 .769 0 .9996 0 .981 2 .162 2 .169 0 .558 0 .548 0 .558 0 .548 0 .711 0 .703 1 .626 1 .628 0 .662 0 .656 1 .025 1 .018 1 .534  spheroids D p  C 3 5 points' points  P 3 points  1 • 350 1 .333 1 .289 1 .671 1 .653 3 .904 3 .846 1 .187 1 .132 1 .600 1 .567 3 .971 3 .970 1 .134 1 .102 1 .134 1 .102 1 .601 1 .571 4 .291 4 .278 1 .611 1 .580 2 .765 2 .734 4 • 378  1 .060 1 .057 1 .056 1 .116 1 .115 1 .559 1 .554 1 .077 1 .076 1 .151 1 .149 1 .685 1 .682 1 .103 1 .100 1 .103 1 .100 1 .196 1 .192 1 .847 1 .845 1 .207 1 .202 1 .477 1 .467 1 .887  D T  5 points 1 .145 1 .146 1 .118 1 • 371 1 .366 2 .982 2 .936 0 .809 0 .774 1 .006 0 .984 2 .173 0 .557 0 .562 0 .557 0 .562 0 .728 0 .713. 1 .638 1 .641 0 .681 0 .666 1 .040 1 .032 1 .548  1 .340 1 .321 1 .285 1 .664 1 .651 3 .900 3 .842 1 .184 1 .137 1 .607 1 .570 3 • 974 1 .133 1 .116 1 .133 1 .116 1 .617 1 .580 4 .303 4 .290 1 .630 1 .591 2 .780 2 .748 4 .392  D  cont.  5 points 1 .061 1 .058 1 .057 1 .117 1 .115 1 .564 1 .560 1 .078 1 .076 1 • 153 1 .150 1 .687 1 .104 1 .100 1 .104 1 .100 1 .196 1 .192 1 .848 1 .846 1 .207 1 .202 1 .476 1 .467 1 .886  Table 6 ( cont'd Mark Type o f spheroid  73 249 124 218 588 580 613 578 572 590 555 549 537 530 566 558 550 534 531  Oblate It  11 tt  Prolate t! tt It It tl It II tt It tt II tt It tt  A.R.  0 .999 0 .999 0 • 999 0 .999 0 .2 0 .2 0 .2 0 .2 0 .2 0 .5 0 .5 0 .5 0 .5 0 .5 0 .9 0 .9 0 .9 0 .9 0 .9  Mean IMxJM .diameter  ;  33x69 29 • 96 7 .00 . 41x65 17 .10 ' 33x59 17 .10 6lxi05 14 .9 65x105 14 .9 33x95 14 • 9 65x105 14 • 9 33x95 14 • 94 33x95 14 .95 65x105 14 .95 33x65 14 .95 33x65 14 .95 33x65 33x65 29 • 67 15 .08 65x95 15 .08: 33x65: 33x65 15 .08 15 .08 33x65 29 • 93 33x65  Re  C  n Q  10 2 .833 100 •0 .611 100 0 .617 100 0 .591 100 1 .481 100 1 • 551 50 2 .361 50 2 .40 10 7 .49 100 0 .895 100 0 .935 50 1 .408 10 4 .316 10 4 .292 100 0 .626 100 0 .655 50 .999 10 3 .085 10 3 .023  )  C _,  C  n  nrT1  .  P  Q  ^ points  5 points  3 points  5 points  3 points  5 points  1 .525 0 .533 0 .516 0 .508 0 .104 0 .124 0 .1508 0 .160 0 .430 0 .279 0 .278 0 .374 0 .987 0 .954 0 .447 0 .473 .607 1 .674 1 .447  1 .539 0 .551 0 .515 0 .524 1 .50 0 .175 0 .190 0 .205 0 • 507 0 .303 0 .293 0 .404 1 .018 1 .001 0 .466 0 .487 0 .626 1 .685 1 .465  4 .357 1 .144 1 .133 1 .099 1 .585 1 .675 2 .511 2 .560 7 .920 1 .174 1 .213 1 .782 5 .303 5 .246 1 .074 1 .128 1 .606 4 .759 4 .470  4.371 1.162 1.132 1.115 1.631 1.726 2.551 2 .606 7.997 1.198 1.228 1.813 5.334 5.293 1.093 1.142 1.625 4.77 4.489  1 .885 1 .108 1 .109 1 .106 1 .332 1 .350 1 .599 1 .630 3 .389 1 .162 1 .172 .1 .317 2 .285 2 .284 1 .112 1 .115 1 .218 1 .923 1 .929  1 .885 1 .108 1 .110 1 .106 1 .481 1 .372 1 .573 1 .661 3 .461 1 ".16 1 .167 1 .408 2 .249 2 .248 1 .112 1 .115 1 .217 1 .920 1 .926  cont. . . .  Table 6 ( cont'd Mark  Angle of Separation  ) w' ^  H 266 137 234 113 165 97 96 128 224 111 164 59 84 126 217 i 126 | 217 112 161 61 8 106 150 90 283 58 73 " 249 124 218 588 580 613 • 578 • 572 590 555 549 537 530 566 558 550 534 531  99.3 100. 3 99 .7 104.3 103.4 122.4 122.3 110.9 110.1 118.8 118.3  2.3 3.9 3.6 2.3 2.3 0.52 0.52 2.8 2.7 1.7 1.6  161.4 125.1 124.5 125.1 124.5 . 136.1 135.1  0.08 1.9 1.8 1.9 1.8 1.0 1.0  —  _  _  139.9 138.6 173-9 175.5  -  128.3 128.4 127.7  — — —  151.8  155.4 172.2  -—  129.4 132.0 143.4  -  0.85 0.95  —  — 1.65 1.7 1.7  — _ — — 0.65 0.55  — — _  i:55 1.50 0.75  -  94  of s e p a r a t i o n i s seen t o be s e n s i t i v e t o t h e s t e p s i z e chosen, f o r t h e p r o l a t e For  spheroids.  single e l l i p t i c a l cylinders  less  extensive  t e s t s were made and, i n g e n e r a l , s m a l l e r g r i d were used.  The g r i d s i z e s used v a r i e d  t o %3° i n t h e a n g u l a r d i r e c t i o n and 0.053 i n t h e r a d i a l d i r e c t i o n .  sizes  from ^6°  M).044  t o about  The d e t a i l s a r e  g i v e n i n T a b l e 7 and Appendix I I . For b o t h t h e s p h e r o i d s and e l l i p t i c a l c y l i n d e r s a t Re  = 0.01, u s i n g e i t h e r Happel's f r e e s u r f a c e  cell  model o r Kuwabara'szero v o r t i c i t y c e l l model, .where the  o u t e r envelope i s r e l a t i v e l y c l o s e t o t h e i n n e r  s u r f a c e o f t h e body a r e l a t i v e l y s m a l l g r i d i n the r a d i a l d i r e c t i o n i s possible the  even though  t o t a l number o f p o i n t s i s not l a r g e .  general the g r i d s i z e varied  size  In  from 6° t o 4.5°  i n t h e a n g u l a r d i r e c t i o n and from 0.08 t o 0.003 i n the r a d i a l d i r e c t i o n .  T a b l e 8 shows t h a t t h e g r i d  s i z e s used i n t h i s work were adequate, as they gave very close r e s u l t s t o the a v a i l a b l e solutions,  analytical  t h e d e t a i l s o f which a r e once a g a i n g i v e n  i n Appendix I I . For n u m e r i c a l a n a l y s i s  purposes t h e f i e l d o f com-  p u t a t i o n f o r s i n g l e b o d i e s must be r e s t r i c t e d  within  an o u t e r envelope which i s made t o c o i n c i d e w i t h one  o f t h e g r i d l i n e s used.  The p o s i t i o n o f such an  envelope i s an i m p o r t a n t f a c t o r i n t h e n u m e r i c a l  Table 7 Some s e l e c t e d d a t a f o r s i n g l e e l l i p t i c a l c y l i n d e r s Mark  O.R.  Mean I xJM diameter d ^ m M  Re  C D  b  C , * 3 points D  _  _  C D  i  5 points  3 points  nrp  _  5 points  P 0  3 points  0  Angle „' of 1 5 sepa- w' P°l ' ation  _  n  bs  r  1254  0.2  133.04  65x105 40  0.1725 1.656  1.642  1.829  1.814  1.086  1.086  103.2  6.9  1143  0.2  133.04  33x93  15  0.3097 2.283  2.276  2.597  2.586  1.220  1.222  108.8  2.95  1112  0.2  133.04  33x93  1  1.7795 9.038  9.030  10.817  10.810  3.642  3.658  1139  5  133-04  33x93  40  1.138  0.573  0.594  1.711  1.732  1.438  1.630  1117  5  133.04  65x105 40  1.100  O.631  0.652  1.730  1.752  1.342  1.324  1021  5  59.78  33x69  1  11.577  2.436  2.437  14.013  14.014  6.264  6.239  1023  5  89.18  33x85  1  11.247  2.408  2.417  13.655  13.665  1028  5  133.04  33x93  1  11.236  2.370  2.370  13.606  13.607  6.118  6.092  1096  2  111.05  33x93  50  0.7163 0.744  0.752  1.460  1.469  1.185  1.183  150.6  1.14  125110/9  133.79  65x105 90.  0.3156 0.8014  0.8017  1.117  1.117  1.070  1.067  118.1  7.1  1266 10/9  133.79  65x105 40  0.5587 0.861  0.861  1.420  1.420  1.151  1.151  128.4  3-45  1066 ^1.0  121.5  33x93  5  1.891  2.161  4.049  4.052  1.86  I.856  -  -  1045 ^1.0  257.23  33x93  1  5.3144 5.461  5.464  1.077  1.078  4.001  3-989  -  -  2.158  .6.150 , 6.124  S  96 Table 8 E f f e c t o f g r i d s i z e on C  D T  and a comparison  w i t h a n a l y t i c a l s o l u t i o n s f o r Re = 0.01 Mark  A.R. O.R. (spheroids) ( c y l i n d e r s )  Concen- IMxJM tration c 0.4025  33x33  Model  'DT  0.265xl0 0.266xl0  Happel Happel  1525 Analytical  1000/995 1.0  0.4025  1544  1000/999 999/1000 1.0  0.01005 33x33 0.01005 33x33 0.01005  0.140x10  0.4025  0.373xl0 ,5 0.372x10""  Kuwabara  0.01005 33x33 0.01005 33x33 0.01005 -  0.161x10  Kuwabara Kuwabara Kuwabara  1535  0.l40xl0 0.l40xl0  5  5  Happel Happel Happel  i  J  Analytical 1000/995  1526 Analytical  1.0  1272  1000/995 999/1000 1.0  1276 Analytical  33x33  0.4025  5  0.l6lxl0  J  0.l62xlO  J  5517 5516  0.2, p r o l a t e 0.2, p r o l a t e  0.5959  65x21  0.505x10  0.5959  33x21  5138  o.2, o b l a t e  0.4011  5117  0.2, o b l a t e  5150  0.999,oblate  A n a l y t i c a l 1.0 sphere 5051  0.999,oblate  A n a l y t i c a l 1.0, sphere 272  0.999,oblate  A n a l y t i c a l 1.0, sphere  Kuwabara  Happel  0.505xl0  6  Happel  41x33  0.336xl0  5  Happel  0.4011  61x33  0.336xl0  5  Happel-  0.4066 0.4066  33x33 -  0.474xl0 0.475xl0  5  Happel Happel.  0.0498 0.0498  33x33  0.529x10  Happel  0.529x10  Happel  0.0498 0.0498  33x33  0.619x10  Kuwabara  0.620x10  Kuwabara  5  97 results.  The a n a l y t i c a l s o l u t i o n of the  c r e e p i n g flow equations f o r a sphere  surrounded  by an outer s p h e r i c a l envelope under the boundary c o n d i t i o n s of zero v o r t i c i t y at the outer envelope i s g i v e n by equation 90 : p = —3_PjT_uJJ__a 5 -  (90)-  + 5Y - Y  9Y  where y i s the diameter r a t i o of s o l i d sphere t o spherical  envelope  l\y ' I =0 = 0  ,then =  0 1  1.018  '  :  P  Y  This i n d i c a t e s that even when the outer envelope i s l o c a t e d at 100 diameters from the sphere, i t s i n f l u e n c e on the drag f o r c e or drag c o e f f i c i e n t i s s t i l l perceptible. Re =1.0  The drag c o e f f i c i e n t s at  f o r both o b l a t e and p r o l a t e spheroids were  e x t r a p o l a t e d to i n f i n i t e envelope volume from a plot of C  D T  meter,, d  a g a i n s t the r e c i p r o c a l of the mean d i -  (=Y~ =C~ 1  1//3  ),  d e f i n e d as the t o t a l volume  of the outer envelope d i v i d e d by the volume of the s p h e r o i d , a l l r a i s e d to the power o n e - t h i r d . range of d  m  i n f i g u r e 5.  The  a c t u a l l y computed was 17-102, as shown At Re.= 10 the d i f f e r e n c e between  the computed v a l u e s of the drag c o e f f i c i e n t s f o r d^ of 17 and 30 was n e g l i g i b l e f o r both the o b l a t e and prolate spheroids. For h i g h e r Re the values of d used to r e p r e s e n t an i n f i n i t e medium were 15 and m ^  99 17 f o r p r o l a t e and It  oblate spheroids, r e s p e c t i v e l y .  i s i n t e r e s t i n g to note t h a t a d of 7.0 • m = 0.2  an o b l a t e spheroid with A.R. s m a l l e r wake than d^ = 17, p e c t i v e values  of C  D T  gave a much  even though the r e s -  were w i t h i n 2% of each other.  = 0.999 the wake shape was  However, f o r A.R.  not  7.0.  a f f e c t e d by d e c r e a s i n g d to • m For e l l i p t i c a l  for  c y l i n d e r s no a n a l y t i c a l s o l u t i o n  of the c r e e p i n g flow equation  i s a v a i l a b l e and-  hence no d i r e c t comparisons could be made as i n the case of s p h e r o i d s . c y l i n d e r of O.R.  R e s u l t s f o r an  = 5 at Re  = 1.0  with d  elliptical m  = 60-133  i n d i c a t e d that there i s no a p p r e c i a b l e d i f f e r e n c e = 89 and d  i n r e s u l t s between d m  = 133.  For  m  higher Reynolds numbers the values of d^ used to represent  111 -  :  an i n f i n i t e medium were i n the range  133.  For the c l u s t e r s , the p o s i t i o n of the outer velope was  used to evaluate  en-  the c o n c e n t r a t i o n  of  the swarm i n the c e l l model. The  e v a l u a t i o n of the pressure 175  given by equations d i f f e r e n t i a t i o n and no d i f f i c u l t y  and  and  180  as  i n v o l v e d both  integration.  The  l a t t e r posed  i n i t i a l t e s t s i n d i c a t e d that  Simpson's r u l e (3 p o i n t s ) and gave almost i d e n t i c a l r e s u l t s . numerical  distribution  Boole's r u l e (5  points)  On the other hand,  d i f f e r e n t i a t i o n could give an u n c e r t a i n t y  100  i n the computed v a l u e s of P, and i n the form drag c o e f f i c i e n t . the s t a b i l i t y  consequently I n o r d e r t o examine  of t h e d i f f e r e n t i a t i o n , Lagrange f o r 3 and 5 p o i n t s were  d i f f e r e n t i a t i o n formulae  used ( i n i t i a l t e s t s were made u s i n g 3 4 and 5 S  points).  I n g e n e r a l f o r an o b l a t e s p h e r o i d w i t h ,  s m a l l A.R. the v a r i a t i o n i n t h e computed v a l u e s of C  D p  0.5%.  was w i t h i n  For a p r o l a t e spheroid,  however, d i f f i c u l t y was e n c o u n t e r e d , as much as 20% f o r s m a l l A.R.  C  D p  varying  An e x a m i n a t i o n o f  e q u a t i o n 175 and i t s n u m e r i c a l e v a l u t i o n i n d i c a t e s t h a t the v a l u e o f  a t  -j^\E =E,  l o w  a s  j  P  e c t  r a t i o Is  £1  n e g a t i v e and n e a r l y e q u a l t o C  a  tanh. K ,  except  a  near the t i p s o f t h e s p h e r o i d , r e s u l t i n g i n the f l a t n e s s o f the s u r f a c e p r e s s u r e d i s t r i b u t i o n f o r A.R. = 0 . 2  shown i n f i g u r e s 23 - . 2 6 .  T h i s gave  r i s e t o a l a r g e e r r o r i n the computed v a l u e o f P and hence C . Dp  F o r t u n a t e l y , f o r s m a l l aspect  r a t i o t h e major c o n t r i b u t i o n t o C  D T  i s from the.  s k i n drag c o e f f i c i e n t , which c o u l d be o b t a i n e d w i t h good a c c u r a c y .  For Re = 1,  and A.R.  =0.2.  and 0 . 5 , the Lagrange 5 - p o i n t d i f f e r e n t i a t i o n  formula  as compared t o the 3 - p o i n t f o r m u l a gave a c l o s e r v a l u e o f Opp/Cpg t o t h a t given.by  equation 34.  For h i g h e r Re the 5 - p o i n t f o r m u l a was used. For e l l i p t i c a l c y l i n d e r s , C^p i s g i v e n by one term o n l y and hence the e r r o r i n u s i n g d i f f e r e n t  different-  101 l a t i o n formulae was  not as severe as f o r the  of p r o l a t e s p h e r o i d s .  The worst case was  e l l i p t i c a l c y l i n d e r of A.R. for  = 5 at Re =  case  the  40,  which the v a r i a t i o n i n C  It DF was about 3%. i s u n f o r t u n a t e t h a t most p u b l i c a t i o n s i n t h i s f i e l d d i s c u s s i n d e t a i l the s e v e r a l methods o f  o b t a i n i n g converged and s t a b l e l o c a l f l o w f u n c t i o n s , but f a i l t o d i s c u s s the method used i n o b t a i n i n g the dependent c u m u l a t i v e f u n c t i o n s , which u s u a l l y r e q u i r e n u m e r i c a l i n t e g r a t i o n and 5.  differentiation.  R e s u l t s f o r S i n g l e Spheroids The boundary c o n d i t i o n s used f o r the  s p h e r o i d s are g i v e n i n Chapter  III.  isolated  At the o u t e r  the zero v o r t i c i t y model as g i v e n by Case I was  envelope  used. •  For an aspect r a t i o of u n i t y , the s p h e r o i d comes a p e r f e c t sphere. w i t h A.R.  of 0.999 was  I n t h i s study an o b l a t e s p h e r o i d c o n s i d e r e d t o be a sphere.  v a r i a t i o n of the t o t a l drag c o e f f i c i e n t w i t h Re for  be-  The  obtained  the l i m i t i n g case of a sphere i s shown i n f i g u r e  6.  The p r e s e n t computed r e s u l t s agree w e l l w i t h the e x p e r i - ' mental v a l u e s summarized by L a p p l e and Shepherd (154). reason t h a t the computed v a l u e s of C ^  The  I n g e n e r a l are above  the e x p e r i m e n t a l curve seems m a i n l y t o be t h a t the o u t e r envelope was negligible.  not t a k e n f a r enough f o r i t s i n f l u e n c e t o be The  same d i r e c t i o n of i n f l u e n c e would be  p e c t e d from the presence  ex-  o f w a l l s c o n f i n i n g the f l o w f i e l d .  Re F i g u r e 6. V a r i a t i o n of the t o t a l drag c o e f f i c i e n t w i t h Reynolds number f o r a sphere.  103 The v a r i a t i o n o f t h e t o t a l drag c o e f f i c i e n t w i t h Reynolds number o f  an o b l a t e s p h e r o i d h a v i n g v a r i o u s  aspect r a t i o s i s g i v e n i n f i g u r e 1.  At low Re, an o b l a t e  s p h e r o i d w i t h h i g h e r A.R. g i v e s a h i g h e r drag  coefficient,  i . e . , C , j f o r a sphere i s h i g h e r t h a n t h a t f o r a c i r c u l a r D  disc.  F o r h i g h e r Re, t h i s tendency i s e v e n t u a l l y r e v e r s e d  and a s p h e r o i d w i t h lower A.R. g i v e s a h i g h e r C ^ . low Re, f o r which the c o n t r i b u t i o n o f C  D S  to C  D T  At -  i s large,  t h i s r e s u l t appears t o be q u i t e r e a s o n a b l e , s i n c e a t a f i x e d v a l u e o f Re a l l the s p h e r o i d s may be c o n s i d e r e d t o have t h e same e q u a t o r i a l diameter  ( f o r c o n s t a n t U and v ) , and con-  s e q u e n t l y the s u r f a c e a r e a and hence t h e s k i n drag c o e f f i c i e n t i n c r e a s e s w i t h A.R.  F o r h i g h e r v a l u e s o f Re .  the wake bubble becomes l a r g e r f o r an o b l a t e s p h e r o i d as A.R. decreases  and s i n c e , n o r m a l l y , h i g h form drag c o -  e f f i c i e n t s a r e a s s o c i a t e d w i t h l a r g e wakes, and i n c r e a s i n g Re i n c r e a s e s the r e l a t i v e c o n t r i b u t i o n o f C r e s u l t i s t h a t a t s u f f i c i e n t l y h i g h Re, r a t i o i s h i g h e r than t h a t f o r a sphere. on d i s c s by Schmiedel  D p  t o C^rp, the  f o r a low aspect E x p e r i m e n t a l work  (155) i n d i c a t e s t h a t the  - Re  curve f o r a d i s c ( c r o s s e s t h a t o f a sphere at Re = 50. d i s c may be thought zero aspect r a t i o .  A  o f as the l i m i t i n g o b l a t e s p h e r o i d o f F i g u r e 8 shows the v a r i a t i o n o f  w i t h Re f o r various" aspect r a t i o s o f a p r o l a t e s p h e r o i d . The comparison at a f i x e d v a l u e o f Re may once a g a i n be cons i d e r e d as based on s p h e r o i d s h a v i n g the same e q u a t o r i a l diameter f o r f i x e d U and v .  106  The  r a t i o of s k i n t o form drag c o e f f i c i e n t f o r  oblate spheroids  at d i f f e r e n t Reynolds numbers i s shown i n  f i g u r e 9 as a f u n c t i o n of aspect r a t i o . expression  as g i v e n by e q u a t i o n  w i t h the p r e s e n t n u m e r i c a l  , Aoi's  35 i s i n good agreement  study.  The  r e l a t i v e con-  t r i b u t i o n of C  D S  ratios.  same i s observed f o r p r o l a t e s p h e r o i d s  The  to C  At Re = 1.0  D T  d e c r e a s e s w i t h Re f o r a l l aspect  g i v e n by the p l o t of Cjjp/^DS v s . Aoi's expression, equation at Re = 1.0. to C  D T  34,  A.R.  as  i n f i g u r e 10.  i s a g a i n i n good agreement  However, t h e r e l a t i v e c o n t r i b u t i o n of  i n c r e a s e s w i t h A.R.  C^g  f o r an o b l a t e , but d e c r e a s e s f o r  a p r o l a t e , spheroid. The  v a r i a t i o n of the d i m e n s i o n l e s s  f r o n t a l stag-  n a t i o n p r e s s u r e w i t h Re f o r an o b l a t e and a p r o l a t e s p h e r o i d i s shown i n f i g u r e s 11 and 12, r e s p e c t i v e l y . e s t i n g t h a t the f r o n t a l s t a g n a t i o n p r e s s u r e  It is inter-  f o r the v a r i o u s  aspect r a t i o s of both o b l a t e and p r o l a t e s p h e r o i d s c o n s i d e r a b l y from each o t h e r and from u n i t y , P s p h e r o i d w i t h A.R. Re.  = 0.2  c  differ  f o r an  oblate  b e i n g the c l o s e s t t o u n i t y at each  Moreover the d e v i a t i o n of P  d e c r e a s i n g Reynolds number.  Q  from u n i t y i n c r e a s e s w i t h  This.seems t o c o n f i r m two w e l l  observed c h a r a c t e r i s t i c s of s t a t i c p i t o t t u b e s , namely t h a t the c o r r e c t i o n . f a c t o r i s l a r g e f o r low Re and t h a t i t v a r i e s c o n s i d e r a b l y f o r d i f f e r e n t t i p shapes (156). A comparison of the s u r f a c e p r e s s u r e  distribution  o b t a i n e d by s e v e r a l workers (25a,55,57,163) f o r a sphere a t Re = 10 i s made w i t h the p r e s e n t  l i m i t i n g c a s e of a sphere i n  107  A.R. F i g u r e 9. V a r i a t i o n of C / C oblate spheroids. D S  D p  w i t h aspect r a t i o f o r  109  110  F i g u r e 12. V a r i a t i o n of f r o n t a l s t a g n a t i o n p r e s s u r e A.R. f o r p r o l a t e s p h e r o i d s .  with  Ill f i g u r e 13, w h i c h shows good agreement between the  numerical  work and t h e a n a l y t i c a l s o l u t i o n o f Oseen's e q u a t i o n s by P i e r c e y and McHugh (25a).  as g i v e n  F i g u r e 14 shows t h e comparison  of P f o r a sphere a t Re=100 and Re+°°,Stewartson (25b,equation 32). The  dlmensionless  pressure d i s t r i b u t i o n at  the s u r f a c e o f an o b l a t e s p h e r o i d f o r d i f f e r e n t A.R. i s shown i n f i g u r e 15 f o r Re = 1.0.  The Stokes p r e s s u r e  dis-  t r i b u t i o n f o r b o t h a sphere and an o b l a t e s p h e r o i d w i t h > A.R.  = 0.2 a r e i n good agreement w i t h t h e computed d i s -  t r i b u t i o n , c o n s i d e r i n g , t h a t even f o r a sphere t h e Stokes regime i s s t r i c t l y v a l i d o n l y f o r Re ^ -0.1.  I tis inter-  e s t i n g t o note sharp maxima and minima f o r A.R. = 0.2, l e s s pronounced ones f o r A.R. = 0.5 and none a t a l l f o r A.R.  o f 0.9.  The exact v a l u e o f A.R. where maxima and  minima would be o b l i t e r a t e d f o r Stokes f l o w c o u l d be found 22a.  from e q u a t i o n  Such s u r f a c e p r e s s u r e maxima are  still  e x h i b i t e d f o r A.R. = 0.2 up t o Re = 10, as shown i n f i g u r e s 16 and 17.  The s u r f a c e p r e s s u r e d i s t r i b u t i o n on an o b l a t e  s p h e r o i d f o r s t i l l h i g h e r Reynolds numbers a r e shown i n . figures  18-20. For an o b l a t e s p h e r o i d w i t h A.R. = 0.2, f i g u r e s 18,  19 and 20 show t h a t t h e p r e s s u r e  at the surface  increases  r a p i d l y j u s t b e f o r e t h e boundary l a y e r s e p a r a t i o n p o i n t , whereas f o r a p r o l a t e  s p h e r o i d w i t h t h e same A.R. t h e s u r f a c e  pressure at corresponding or not a t a l l ,  Reynolds numbers i n c r e a s e s s l o w l y  and no s e p a r a t i o n I s o b s e r v e d , a c c o r d i n g t o  f i g u r e s 24, 25 and 26.  The d i m e n s i o n l e s s  surface  pressure  d i s t r i b u t i o n on a p r o l a t e s p h e r o i d w i t h d i f f e r e n t A.R. i s  1 1 2  F i g u r e 1 3 . S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r a sphere at Re=10.  -Figure.14.  S u r f a c e p r e s s u r e d i s t r i b u t i o n o f a sphere a t h i g h Reynolds number.  114  F i g u r e 15. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r o b l a t e s p h e r o i d s at Re=1.0.  115  F i g u r e 16. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r o b l a t e s p h e r o i d s at  Re-5.0.  116  P  Figure 17.  Surface pressure d i s t r i b u t i o n f o r oblate spheriods  at  Re=10.  117  F i g u r e 18. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r o b l a t e s p h e r o i d s at Re=20.  118  F i g u r e 19. S u r f a c e p r e s s u r e at Re=50.  d i s t r i b u t i o n f o r oblate  spheroid  119  g i v e n i n f i g u r e 21 f o r Re = 1.0,  A comparison w i t h t h e  Stokes p r e s s u r e d i s t r i b u t i o n f o r A.R. = 0.2 shows g e n e r a l agreement i n t h e shape o f t h e c u r v e , but a c o n s i d e r a b l e d i f f e r e n c e i n the f r o n t a l s t a g n a t i o n pressure.  Figures  22, 23, 24, 25 and 26 show t h e s u r f a c e p r e s s u r e  distribution  on a p r o l a t e s p h e r o i d f o r v a r i o u s A.R. a t Re = 5, 10, 20, 50 and 100, r e s p e c t i v e l y . The d l m e n s i o n l e s s  surface v o r t i c i t y i s a d i r e c t  measure o f t h e shear s t r e s s a t t h e s u r f a c e o f t h e s p h e r o i d , and i t s change i n s i g n i n d i c a t e s t h e onset  of s e p a r a t i o n ,  i . e . , a r e v e r s a l i n t h e f l o w d i r e c t i o n i n t h e immediate v i c i n i t y o f t h e body.  The d i m e n s i o n l e s s  surface  vorticity  i s p l o t t e d i n f i g u r e s 27 - 30 f o r an o b l a t e s p h e r o i d i n • f i g u r e s 31 - 33 f o r a p r o l a t e s p h e r o i d , a t t h e v a r i o u s A.R. and Re.  F o r t h e o b l a t e s p h e r o i d a maximum i s ob-  s e r v e d , which s h i f t s toward t h e f r o n t a l s t a g n a t i o n p o i n t ; w i t h i n c r e a s e o f Re, thus r e n d e r i n g t h e curve asymmetric about n= TT/2 .  more  For the p r o l a t e spheroid w i t h  A.R. = 0.2 a t Re = 1.0, f i g u r e 33 shows t h a t t h e r e a r e two maxima, the one n e a r e r t h e r e a r o f t h e s p h e r o i d i n g w i t h i n c r e a s e o f Reynolds number.  The  disappear-  corresponding  asymmetry o f the f l o w i s a r e s u l t o f i n c r e a s i n g l y dominant inertial effects.  The maximum near t h e f r o n t o f t h e p r o -  l a t e s p h e r o i d s w i t h A.R. o f 0.2  ( f i g u r e 33) and A.R. o f  0.5 ( f i g u r e 32) seems t o be r a t h e r f i x e d f o r t h e v a r i o u s Re at 11° and.28°, r e s p e c t i v e l y .  121  F i g u r e 21. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r p r o l a t e s p h e r o i d s a t Re'=1.0.  122  F i g u r e 22. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r p r o l a t e s p h e r o i d s a t Re=5.0.  1 2 3  Figure 2 3 . Surface pressure d i s t r i b u t i o n f o r p r o l a t e at  Re=10.  spheroids  12 k  F i g u r e 2k.  Surface pressure d i s t r i b u t i o n f o r p r o l a t e  at  Re=20.  spheroids  126  Figure 26.  Surface pressure d i s t r i b u t i o n f o r a p r o l a t e s p h e r o i d at Re=100.  127  F i g u r e 27.  S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an o b l a t e w i t h A.R. = 0.999 ( s p h e r e ) .  spheroid  128  129  F i g u r e 29. S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an s p h e r o i d With A.R. = .0.5.  oblate  130  F i g u r e 30. S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an s p h e r o i d w i t h A.R, = 0.2.  oblate  131  F i g u r e 31. S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r a p r o l a t e w i t h A.R. = 0.9. •  spheroid  132  Figure 3 2 .  Surface v o r t i c i t y d i s t r i b u t i o n f o r a p r o l a t e w i t h A.R. = 0.5.  spheroid  133  Figure.33. Surface v o r t i c i t y d i s t r i b u t i o n f o r a p r o l a t e spheroid w i t h A.R. = 0.2.  134  The  appearance o f t h e wake bubble behind a  s p h e r o i d i s c l e a r l y a f u n c t i o n o f t h e s o l i d body's shape and o f t h e Reynolds number.  I t was found t h a t a wake  bubble d e v e l o p s a t Re as low as 20, 10 and 5 f o r an o b l a t e s p h e r o i d w i t h A.R. o f 0.9, 0.5 and 0.2 r e s p e c t i v e l y . an almost s p h e r i c a l o b l a t e - s p h e r o i d  (A.R.  For  = 0.99 ) , however,  the p r e s e n t work shows no marked v o r t e x development a t Re=20, though a s e p a r a t i o n was i n d i c a t e d a t t h i s Reynolds number, by a n e g a t i v e v o r t i c i t y a t t h e s u r f a c e . e x t r a p o l a t i o n of h i s experimental  Taneda's (160)  work gave t h e onset o f  s e p a r a t i o n a t Re = 24, w h i l e Jenson's work (55) i n d i c a t e d the Re a t s e p a r a t i o n t o be 17. P o r t e r and N i s i  The c o r r e c t e d v a l u e o f  (157) i s Re = 17 (158)..  For a prolate  s p h e r o i d o f A.R. - 0.2, no v o r t e x was observed even f o r Re o f 100.  The v a r i a t i o n o f t h e d i m e n s i o n l e s s  w i t h Re i s shown i n f i g u r e 34.  wake l e n g t h  I t i s i n t e r e s t i n g that  Van Dyke's (159)-equation, d e r i v e d from Oseen's a p p r o x i mation f o r t h e wake l e n g t h behind  a sphere i s i n e x c e l l e n t  agreement w i t h t h e v a l u e s o f wake l e n g t h computed f o r a nearly spherical oblate spheroid  (A.R.  = 0.999).  Figure  35 g i v e s t h e p o i n t o f s e p a r a t i o n f o r t h e v a r i o u s shapes and Reynolds numbers.  Taneda's (160) e x p e r i m e n t a l  sphere a r e i n e x c e l l e n t agreement w i t h those  data f o r a  computed f o r t h e  o b l a t e s p h e r o i d o f A.R. = 0.999, but Van Dyke's t h e o r e t i c a l equation f o r the separation point i s a t variance with results.  these  135  F i g u r e 34. V a r i a t i o n of wake l e n g t h w i t h Re f o r s p h e r o i d s Oblate s p h e r o i d s : A.R. = 0.2; v , A.R. = 0.5; A , A.R. = 0.9;o , A.R. = 0.999 ( s p h e r e ) . P r o l a t e . s p h e r o i d s : • , A.R. = 0.9; CD , A.R. = 0.5. • Sphere: e , (Hamielec et a l ) ; , Van Dyke j (theoretical). I  Re F i g u r e 35. L o c a t i o n of s e p a r a t i o n p o i n t f o r s p h e r o i d s .  Streamlines  and e q u i - v o r t i c i t y l i n e s at v a r i o u s  Reynolds numbers and aspect r a t i o s f o r o b l a t e and p r o l a t e spheroids  are shown i n f i g u r e s 36 - 49.  It i s interesting  t o note t h a t the s t r e a m l i n e of magnitude 4.0 l e s s curved w i t h i n c r e a s i n g Re,  tends t o be  i n d i c a t i n g t h a t the flow,  becomes u n d i s t u r b e d a t a s h o r t e r d i s t a n c e from the as Re i n c r e a s e s .  spheroid  T h i s r e s u l t c o r r e l a t e s w i t h the f a c t t h a t  the w a l l e f f e c t on the drag c o e f f i c i e n t d e c r e a s e s w i t h i n c r e a s i n g Re, as e x p e r i m e n t a l l y shown by McNown et a l . ( l 6 l ) . The i n which  d e v i a t i o n of the f l o w from the Stokes r e g i m e ,  symmetry p r e v a i l s , i s b e s t observed by  asymmetry of the v o r t i c i t y l i n e s .  The  the  v o r t i c i t y i s generated  upstream and i s c a r r i e d by the f l u i d around the s p h e r o i d : t o c o n s i d e r a b l e d i s t a n c e s downstream, these d i s t a n c e s i n c r e a s i n g w i t h Reynolds number.  The  h e a t - f l o w analogy i n -  d i c a t e s t h a t i n time t v o r t i c i t y would d i f f u s e outwards a  •, .  1/2  d i s t a n c e d=(ty/p) a d i s t a n c e s'= 1/2  S d  Ut.  w h i l e i t i s b e i n g c o n v e c t e d downstream f o r E l i m i n a t i n g t y i e l d s the  result  1/2 a  Re  > where S and d have been rendered  dimensionless  by use of the e q u a t o r i a l r a d i u s of the s p h e r o i d .  The  v o r t i c i t y l i n e s f o r a g i v e n s p h e r o i d c l e a r l y support r e s u l t i n t h a t f o r Re = 1.0  the above  the c o n v e c t i o n of v o r t i c i t y i s  much l e s s than t h a t a t Re = 100  47 -  equi-  (see f i g u r e s 40 - 42 and .  49). The  e q u i - v e l o c i t y l i n e s at Re = 100  f o r b o t h an  o b l a t e and a p r o l a t e s p h e r o i d h a v i n g an aspect r a t i o of and f o r an almost s p h e r i c a l o b l a t e s p h e r o i d , are shown i n  0.2,  F i g u r e 36,  Streamlines (A.R.  f o r a nearly spherical oblate  = 0.999).  spheroid  Re  F i g u r e 37.  Streamlines  =  100.  139  f o r an o b l a t e s p h e r o i d w i t h aspect r a t i o  0.9.  140  Figure 3 8 .  Streamlines  f o r an o b l a t e s p h e r o i d w i t h aspect r a t i o 0 . 5  F i g u r e 40. V o r t i c i t y l i n e s f o r a n e a r l y s p h e r i c a l o b l a t e (A.R. = 0.999).  spheroid  143  F i g u r e i l l . V o r t i c i t y l i n e s f o r an o b l a t e s p h e r o i d w i t h a s p e c t r a t i o 0.9.  144 Re =  100.  F i g u r e 42. V o r t i c i t y l i n e s f o r an o b l a t e s p h e r o i d w i t h aspect r a t i o 0.5.  Re =  100.  F i g u r e 43. V o r t i c i t y l i n e s f o r an o b l a t e s p h e r o i d w i t h aspect r a t i o 0.2.  146 Re = 100.  Re = 100.  147  148  150  Re a  100.  F i g u r e 48.' . V o r t i c i t y l i n e s f o r a p r o l a t e , s p h e r o i d w i t h r a t i o 0.5.  aspect  151  Re =  50.  .1  Re = 1.  . .1  (d) F i g u r e 49. V o r t i c i t y l i n e s f o r a p r o l a t e s p h e r o i d w i t h r a t i o 0.2.  aspect  152 f i g u r e 50.  The  modulus of the v e l o c i t y v e c t o r was  u s i n g the Lagrange 3-point lattice.  f o r m u l a on the stream f u n c t i o n  I t i s i n t e r e s t i n g t o note t h a t the  e n c l o s i n g the wake f o r the 0.2 ( f i g u r e 39a)  found by  has a p p r o x i m a t e l y  streamline  aspect r a t i o o b l a t e  spheroid  the same c u r v a t u r e over the  wake r e g i o n as the e q u i - v e l o c i t y l i n e of u n i t y ( f i g u r e and i t may  be a c o n j e c t u r e t h a t b o t h l i n e s c o i n c i d e w i t h each  o t h e r when the Reynolds number I s i n c r e a s e d  indefinitely;.  A l s o i t can be seen t h a t the c e n t r e of the v o r t e x 39a)  50a),  l i e s i n a r e g i o n of low v e l o c i t y  (figure  (figure  50a).  I n p o t e n t i a l f l o w , i . e . when an i d e a l f l u i d  flows  p a s t a s p h e r o i d , symmetry e x i s t s between the upstream f l o w and the downstream flow.. energy l o s s e s due dimensionless flow past a  T h i s i s a r e s u l t of no  frlctional  t o the absence of v i s c o u s f o r c e s .  surface pressure  The  distribution for potential  s p h e r e , as g i v e n i n Appendix I , i s p l o t t e d i n  f i g u r e 51 t o g e t h e r w i t h the n u m e r i c a l r e s u l t s at Re = 50 and Re =100  f o r a s p h e r o i d h a v i n g A.R.  = 0.999-  f o r the h i g h e r Re shows c l o s e r agreement w i t h the  The  curve  pressure  d i s t r i b u t i o n f o r p o t e n t i a l f l o w than t h a t f o r the lower  Re.  T h i s agreement i s b e s t f o r p o i n t s near the f r o n t end of the spheroid.  S i m i l a r p l o t s f o r A.R.  prolate spheroid  = 0,2  of an o b l a t e and  are g i v e n i n f i g u r e s 52 and  For the p r o l a t e s p h e r o i d a t Re = 100  a  53 r e s p e c t i v e l y .  the agreement w i t h  the  p o t e n t i a l f l o w extends t o about 3TT/2, which i s undoubtedly r e l a t e d to the absence of s e p a r a t i o n f o r the r e a l f l o w at Re =  100.  153  (c)  F i g u r e 51. S u r f a c e p r e s s u r e d i s t r i b u t i o n at h i g h Re f o r an o b l a t e s p h e r o i d w i t h A.R. = 0.999 ( s p h e r e ) .  1 5 5  F i g u r e 53. S u r f a c e p r e s s u r e d i s t r i b u t i o n at h i g h Re f o r p r o l a t e s p h e r o i d w i t h A.R. = 0.2.  157  F i g u r e 54a  shows the e x c e l l e n t agreement of  Homann's r e s u l t s w i t h those o b t a i n e d i n the work on a . n e a r l y p e r f e c t sphere and by o t h e r n u m e r i c a l and work.  analytical  F i g u r e s 54 - 57 compare the d i m e n s i o n l e s s  frontal-  s t a g n a t i o n p r e s s u r e f o r v a r i o u s aspect r a t i o s o f o b l a t e  and:  p r o l a t e s p h e r o i d s w i t h the s o l u t i o n d e r i v e d i n Appendix I u s i n g boundary l a y e r t h e o r y and the p o t e n t i a l f l o w s o l u t i o n for  spheroids.  I n a l l cases the agreement i s much c l o s e r  for  h i g h than low Reynolds number which i s not  s i n c e the boundary l a y e r t h e o r y i s s t r i c t l y the h i g h e r Re.  surprising  applicable for  The m o d i f i e d Stokes s o l u t i o n (which adds  the f i r s t term o f u n i t y ) t o the u n m o d i f i e d  s o l u t i o n gives  s u r p r i s i n g l y c l o s e r e s u l t s t o the n u m e r i c a l v a l u e s  obtained  i n t h i s work. The modulus of the v e l o c i t y v e c t o r i n the d i r e c t i o n was along  determined  at n=ir/2  , i . e . |v | was  evaluated  n = TT/2 and the p o s i t i o n o f i t s maximum v a l u e  t a k e n as the edge o f the boundary l a y e r .  n- .  was;  This p o s i t i o n  was  n o n - d i m e n s i o n a l i z e d by the use of the e q u a t o r i a l r a d i u s o f the s p h e r o i d , and the d i m e n s i o n l e s s boundary t h i c k n e s s , # was p l o t t e d a g a i n s t the Reynolds number i n f i g u r e 5 8 . p l o t gave s t r a i g h t l i n e s h a v i n g s l o p e s o f about i n agreement w i t h boundary l a y e r t h e o r y 6.  -1/2  , The  ,  (162).  Results f o r Single E l l i p t i c a l Cylinders The boundary c o n d i t i o n s used f o r the  elliptical  c y l i n d e r s are g i v e n i n Chapter  isolated  I I I as Case I , i n  which the v o r t i c i t y i s t a k e n as zero at the o u t e r boundary.  158  Re  F i g u r e 54. V a r i a t i o n o f the f r o n t a l s t a g n a t i o n p r e s s u r e w i t h Reynolds number f o r o b l a t e s p h e r o i d s o f h i g h aspect r a t i o r B.L.T., Boundary l a y e r t h e o r y ; o , t h i s work; • , J e n s o n , A , Rhodes; V , Hamielec e t a l ; ? , Pearcey and McHugh.  159  T  3  5  10  30  50  Re Figure  55. V a r i a t i o n o f t h e f r o n t a l s t a g n a t i o n p r e s s u r e with R e y n o l d s number f o r o b l a t e s p h e r o i d s o f l o w a s p e c t • ratio: B.L.T.., B o u n d a r y l a y e r t h e o r y ; o , t h i s work.  100  10.  1  i  1  i  1  1  r  2  3  5  10  2  3  5  100  Re F i g u r e 5 6 . V a r i a t i o n o f t h e f r o n t a l s t a g n a t i o n p r e s s u r e w i t h Reynolds number f o r p r o l a t e s p h e r o i d s w i t h A.R. .= 0.9: B.L.T., Boundary l a y e r t h e o r y ; o , t h i s work.  161  I  2  3  5  10  20  30  50  100  Re F i g u r e 57. V a r i a t i o n o f t h e f r o n t a l s t a g n a t i o n p r e s s u r e -with Reynolds number f o r p r o l a t e s p h e r o i d s o f low aspect r a t i o * . B..-L.T., Boundary l a y e r t h e o r y ; o , t h i s work.  162  SLOPE = -1/2  SOC Re-1/2 02  10  20  SLOPE = - 1/2  Re  50  100  F i g u r e 58. V a r i a t i o n o f t h e boundary l a y e r t h i c k n e s s w i t h Reynolds number f o r s p h e r o i d s a t n = TT/2 O b l a t e s p h e r o i d s : 6, A.R. = 0.999 ( s p h e r e ) ; v , A.R. = 0.2. P r o l a t e s p h e r o i d s : e , A.R. = 0 . 5 ; A , A.R. = 0.2. Sphere: • , Jenson.  163 The n u m e r i c a l work on c i r c u l a r c y l i n d e r s has been w e l l covered by p r e v i o u s workers. a nearly c i r c u l a r e l l i p t i c a l computer program.  I n i t i a l work on  c y l i n d e r was done t o t e s t t h e  An aspect r a t i o of 0.995 was used.  F i g u r e 59 shows t h e e x p e r i m e n t a l d a t a o f v a r i o u s workers t o g e t h e r w i t h t h e s o l u t i o n s o f Lamb (5) and Kaplun w e l l as the p r e s e n t work. for  O.R.  The C  (29) as  r e s u l t s a t Re = 0.01 .  D T  = 0.995 and a t Re - 1 and 5 f o r O.R.  = 1/0.995  agree v e r y w e l l w i t h t h e e x p e r i m e n t a l d a t a of the v a r i o u s workers (145, 146, 147).  I t i s i n t e r e s t i n g t h a t Lamb's,  s o l u t i o n i s q u i t e accurate i n e s t i m a t i n g the t o t a l c o e f f i c i e n t up t o Re = 1.0.  drag  The r e c e n t e x p e r i m e n t a l work  of Jayaweera and Mason (145) a t Re - 0.02 shows t h a t Lamb's s o l u t i o n s t i l l h o l d s a t t h i s v e r y low Reynolds number. I t s h o u l d be borne i n mind t h a t t h e r e i s no v a l i d  creeping  f l o w s o l u t i o n f o r the c i r c u l a r c y l i n d e r , s i n c e Stokes s o l u t i o n g i v e s zero drag c o e f f i c i e n t when t h e o u t e r envelope i s at i n f i n i t y .  At Re = 0.01 t h e presence  of the  o u t e r boundary i s very i n f l u e n t i a l i n d e t e r m i n i n g t h e exact v a l u e o f t h e t o t a l drag c o e f f i c i e n t . was found t h a t  For d  m  = 257, i t  o b t a i n e d by the n u m e r i c a l s o l u t i o n o f  the complete N a v i e r - Stokes e q u a t i o n agrees w i t h t h a t obt a i n e d by the c r e e p i n g f l o w s o l u t i o n as g i v e n by e q u a t i o n 89. However, t h i s  v a l u e i s about 30% h i g h e r than the ex-  p e r i m e n t a l v a l u e (see Appendix I I , Mark 1524).  At d  m  = 1808  the n u m e r i c a l l y computed v a l u e of C^ ' i s v e r y c l o s e t o t h e T  experimental data, i n d i c a t i n g that the I n f l u e n c e of the outer  165  envelope  i s n e g l i g i b l e , w h i l e the c r e e p i n g • f l o w s o l u t i o n  g i v e s a lower v a l u e than the e x p e r i m e n t a l d a t a . s p h e r o i d s , the i n f l u e n c e o f the o u t e r envelope  For becomes  n e g l i g i b l e a t much s h o r t e r d i s t a n c e s than f o r c y l i n d e r s . T h i s c o u l d be due t o the f a c t t h a t a t w o - d i m e n s i o n a l c r e a t e s a l a r g e r d i s t u r b a n c e than a t h r e e body.  body  dimensional  Lamb's s o l u t i o n o f Oseen's e q u a t i o n s and  Kaplun's  i n n e r and o u t e r expansion  s o l u t i o n f o r low Reynolds number  g i v e lower v a l u e s  on a c y l i n d e r than the e x p e r i -  of C  D T  mental d a t a , whereas the c o r r e s p o n d i n g s o l u t i o n s f o r a s p h e r o i d f a l l h i g h e r than the e x p e r i m e n t a l curve C  D T  vs.  of  Re. The t o t a l drag c o e f f i c i e n t based on the l e n g t h  of the e l l i p t i c a l  c y l i n d e r (axis p e r p e n d i c u l a r t o the f l o w  d i r e c t i o n ) i s shown i n f i g u r e 60, where i t i s seen t h a t does not vary much w i t h O.R.  = 0.2  shows v a l u e s of C  exceed a l l the o t h e r s .  D T  (O.R.=o )•  v s . Re f o r O.R.  D T  However, f o r  = 0.2  crosses  = 0) g i v e s v a l u e s of  much h i g h e r even than those computed f o r O.R.  C  ob-  and  The work of M i y a g i (8la) f o r flow:  p e r p e n d i c u l a r to f l a t p l a t e s (O.R.  h i g h e r Re.  cylinder  lower than those  D T  t a i n e d f o r f l o w a l o n g the major a x i s h i g h e r Re, the curve of C  D T  I t i s i n t e r e s t i n g t h a t at  Re < 6, f l o w a l o n g the minor a x i s of an e l l i p t i c a l w i t h O.R.  C  = 0.2  C  D T  at  The reason f o r the s l o w e r r a t e of decrease  w i t h Re f o r the low o r i e n t a t i o n r a t i o s i s p r o b a b l y  In the  e a r l y f o r m a t i o n of a wake bubble when the f l o w i s a l o n g the minor a x i s a t low aspect  ratios.  167 In o r d e r t o compare the t o t a l drag c o e f f i c i e n t w i t h t h a t of a f l a t p l a t e , jjrp$ i s p l o t t e d a g a i n s t Rew c  61, the s u b s c r i p t w s i g n i f y i n g t h a t the  i n figure  c o o r d i n a t e s are  based on the w i d t h of the e l l i p t i c a l c y l i n d e r ( a x i s the f l o w d i r e c t i o n ) . •and 10/9  l i e between those of a c i r c u l a r c y l i n d e r (O.R. =  1.0)  =  ), as computed numeri-  00  c a l l y by Dennis and Dunwoody ( 4 l ) . (82)  along  I t i s seen t h a t the r e s u l t s f o r O.R.=5,2  and those o f a f l a t p l a t e (O.R.  Kuo  both  The  B l a s i u s (83)  and  s o l u t i o n s f o r a f l a t p l a t e , u s i n g boundary l a y e r  t h e o r y , f a l l below the  latter.  F i g u r e 62 shows t h a t as Re i n c r e a s e s the CpS t o  ratio of :  d e c r e a s e s , t h a t t h i s r a t i o approaches t h a t o f  Oseen's s o l u t i o n a t low Reynolds number, and t h a t i t i n creases w i t h o r i e n t a t i o n r a t i o .  From s i m p l e  geometrical =0,  c o n s i d e r a t i o n s the s k i n drag c o e f f i c i e n t i s zero f o r O.R. whereas the form drag c o e f f i c i e n t i s zero f o r O.R. The v a r i a t i o n of the d i m e n s i o n l e s s stagnation pressure, P , 0  00  .  frontal  f o r the e l l i p t i c a l c y l i n d e r s i s :  q u i t e s i m i l a r t o t h a t of the s p h e r o i d s . d e v i a t i o n from u n i t y i s l a r g e and P O.R.,  =  as shown i n f i g u r e 63.  from u n i t y i s s m a l l e r and P  0  0  For low Re,  the  increase rapidly with  For h i g h e r Re the d e v i a t i o n does not i n c r e a s e much w i t h  The v a r i a t i o n of the d i m e n s i o n l e s s  pressure  O.R.  dis-  t r i b u t i o n on the s u r f a c e of a n e a r l y c i r c u l a r c y l i n d e r i s shown i n f i g u r e 64. ri = TT/2  Any tendency towards antisymmetry about  a t Re = 1 i s no l o n g e r p r e s e n t a t Re = 5.  f i g u r e s 65,  66,  67 and 68 the s u r f a c e p r e s s u r e  In  distribution  0  I  2 |  I  2  ,  5  1  10  I  20  ;  ,  50  |  |  100  Rew F i g u r e 6 1 . . V a r i a t i o n o f t o t a l drag c o e f f i c i e n t w i t h Reynolds number f o r f l o w a l o n g the major a x i s , which i s t a k e n as the c h a r a c t e r i s t i c l e n g t h .  F i g u r e 62. V a r i a t i o n of C / C n q  of e l l i p t i c a l  n i T  w i t h Reynolds number f o r v a r i o u s o r i e n t a t i o n r a t i o s  cylinders. VO  172  173  P  0  7T/2 17  F i g u r e 66.  S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r an e l l i p t i c a l c y l i n d e r w i t h O.R. = 2.  1Y5  1  F i g u r e 68. S u r f a c e p r e s s u r e d i s t r i b u t i o n f o r an t i c a l c y l i n d e r w i t h O.R. = 0.2.  ellip-  176 i s g i v e n a t v a r i o u s Re f o r O.R. respectively.  At O.R.  d i s t r i b u t i o n was except  = .10/9, 2,5,  = 5 ( f i g u r e 65)  and  a flat  o b t a i n e d on the e l l i p t i c a l  0.2,  pressure  cylinder,  f o r the l e a d i n g and t r a i l i n g edges at low Re =0.2  and 68)  the l e a d i n g edge a t h i g h Re.  At O.R.  a maximum was  as i n the case of t h e o b l a t e  observed  s p h e r o i d w i t h A.R.  for P  = 0.2  ( f i g u r e 17).  (figure  Such a maximum d i d  n o t , however, appear f o r Re h i g h e r than about The  dimensionless  surface v o r t i c i t y  5. distribution  f o r the v a r i o u s o r i e n t a t i o n r a t i o s and Reynolds numbers are g i v e n i n f i g u r e s 69 - 73.  As i n d i c a t e d p r e v i o u s l y ,  the change i n the s i g n of the s u r f a c e v o r t i c i t y i s an i n d i c a t i o n o f a r e v e r s a l i n the f l o w d i r e c t i o n and of s e p a r a t i o n .  Once a g a i n the maximum o f the 5 - n  curve s h i f t s s l i g h t l y towards n = 0 w i t h of Re.  At O.R.  Re = 1 . 0 .  = 5  ( f i g u r e 72)  increase  t h e r e e x i s t s two maxima f o r  The maximum n e a r e r the t r a i l i n g edge  f o r Re > 5.  hence  At O.R,  = 0.2  ( f i g u r e 73)  the  disappears  surface  v o r t i c i t y changes s i g n s v e r y r a p i d l y j u s t beyond n=Tr/2 f o r the h i g h e r The  Re. v a r i a t i o n of the d i m e n s i o n l e s s  wake l e n g t h  w i t h Re f o r the e l l i p t i c a l c y l i n d e r s i s shown i n f i g u r e I t i s c l e a r t h a t t h e wake l e n g t h s are l o n g e r 'than produced by the c o r r e s p o n d i n g two-dimensional u s u a l l y occurs.  spheroids.  those  However, such l o n g  wakes are i n g e n e r a l not s t a b l e , and The  e x p e r i m e n t a l d a t a o f Taneda  on c y l i n d e r s are p l o t t e d f o r comparison.  74.  shedding  (160)  F i g u r e 75 shows  177  178  0  TT/2  TT  V F i g u r e 70. S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an t i c a l c y l i n d e r w i t h O.R. = 10/9.  ellip-  179  5i  0  n——  1  7T/2  TT  V F i g u r e 72. S u r f a c e v o r t i c i t y d i s t r i b u t i o n f o r an t i c a l c y l i n d e r w i t h O.R. = 5 .  ellip-  180  184 the angle o f s e p a r a t i o n f o r O.R.  =0.2,  For a n e a r l y c i r c u l a r c y l i n d e r (O.R. s e p a r a t i o n was o b s e r v e d ,  10/9 and 2.  =; 1.0) a t Re = 5 no  and f o r O.R.  = 5 no f l o w r e -  v e r s a l was noted a t Re as h i g h as 40. The s u r f a c e p r e s s u r e d i s t r i b u t i o n s a t Re = 40 for  A.R.  = 0.2 w i t h t h e f l o w a l o n g both t h e minor a x i s  (O.R. = 0.2) and t h e major a x i s (O.R. = 5.0) a r e compared w i t h the c o r r e s p o n d i n g p o t e n t i a l f l o w d i s t r i b u t i o n s i n f i g u r e 76.  As e x p e c t e d ,  a x i s t h e r e was f a i r l y  f o r t h e f l o w a l o n g t h e major  c l o s e agreement t o n g r e a t e r  TT/2 due t o the absence o f s e p a r a t i o n .  For t h e flow along  the minor a x i s t h e agreement was r e a s o n a b l e f r o n t o f the e l l i p t i c a l  cylinder.  o n l y f o r the  F i g u r e s 77 and 78  show t h e v a r i a t i o n o f t h e d i m e n s i o n l e s s for  than  p r e s s u r e w i t h Re  the f r o n t a l s t a g n a t i o n p o i n t , compared w i t h t h e f i r s t  approximation  o f t h e boundary l a y e r t h e o r y where t h e o u t -  s i d e f l o w i s t a k e n t o be a p o t e n t i a l f l o w .  The  agreement  i s v e r y good f o r the h i g h e r Reynolds numbers, as would be expected.  The d e r i v a t i o n of t h e e x p r e s s i o n used f o r P -• Q  i s g i v e n i n Appendix I . The boundary l a y e r t h i c k n e s s a t n = TT/2 v a r i o u s e l l i p t i c a l c y l i n d e r s i s " shown i n f i g u r e 79. edge o f the boundary l a y e r I s a g a i n t a k e n a t  3  %  f o r the The = 0  .  From t h e p l o t s of f i g u r e 79 i t i s seen t h a t the boundary l a y e -1/2 t h i c k n e s s v a r i e s as Re theory. after  , as p r e d i c t e d by boundary l a y e r  S e p a r a t i o n ' f o r t h e cases c o n s i d e r e d o c c u r s w e l l  n = TT/2  185  1  1  P O T E N T I A L F L O W  - 11  \  •  /  >  P  0  Q R.= 5 Re =40*  1  P O T E N T I A L F L O W  Re= 40  —  QR.-02' —  i  0  1  7T/2  V  F i g u r e 76. Comparison o f s u r f a c e p r e s s u r e d i s t r i b u t i o n on e l l i p t i c a l c y l i n d e r s a t Re=40 w i t h p o t e n t i a l f l o w .  186  gure 77. V a r i a t i o n o f the f r o n t a l s t a g n a t i o n p r e s s u r e w i t h Reynolds number f o r e l l i p t i c a l c y l i n d e r o f A.R. = 0.2: , Boundary l a y e r t h e o r y ; o, t h i s work.  F i g u r e 78. V a r i a t i o n o f the f r o n t a l s t a g n a t i o n p r e s s u r e w i t h Reynolds number f o r e l l i p t i c a l c y l i n d e r s : Boundary l a y e r t h e o r y ; o, t h i s work.  ,  188  F i g u r e 79. V a r i a t i o n of the Boundary l a y e r t h i c k n e s s w i t h Reynolds number f o r e l l i p t i c a l c y l i n d e r s a t TI=TT/2.  189  The  c o n t o u r i n g o f the s t r e a m l i n e s and  v o r t i c i t y l i n e s f o r the e l l i p t i c a l f i g u r e s 86 - 9 0 .  = 2.0  For O.R.  equi-  c y l i n d e r s i s shown i n  a t Re = 0 . 0 1 ,  the con-  t o u r s of stream f u n c t i o n , v o r t i c i t y and v e l o c i t y i n d i c a t e d t h a t symmetry p r e v a i l e d between the upstream and r e g i o n s of the f l o w .  downstream  At t h i s low Reynolds number, as  opposed t o those p l o t t e d i n f i g u r e s 8 l - 90 the  diffusion  of v o r t i c i t y i s dominant and t h e r e i s no apparent conv e c t i o n o f v o r t i c i t y i n the downstream r e g i o n .  It i s well  known t h a t the. d i s t u r b a n c e due t o a t w o - d i m e n s i o n a l  object  (159).  i s more i n t e n s e t h a n f o r a t h r e e - d i m e n s i o n a l one  T h i s i s c o n f i r m e d i n the p r e s e n t work, i n t h a t a t a g i v e n d i s t a n c e from the e l l i p t i c a l  c y l i n d e r the s t r e a m l i n e i s ,  more c u r v e d than t h a t f o r the c o r r e s p o n d i n g A.R.;  also.O.R. < 1 f o r e l l i p t i c a l  o b l a t e s p h e r o i d and O.R.  s p h e r o i d (same  c y l i n d e r corresponds  > 1 to p r o l a t e spheroid).  o v e r , even f o r the h i g h e r Re; the if = 4 . 0  to  More-  line i s s t i l l  curved f o r the c y l i n d e r s (as opposed t o the s p h e r o i d s ) , i n d i c a t i n g t h a t the e f f e c t o f a w a l l would be more apprec i a b l e f o r the c y l i n d e r s than f o r the spheroids.  corresponding  For a g i v e n Re, the wakes f o r the  c y l i n d e r s are much l o n g e r than those of the spheroids.  The b e h a v i o u r  elliptical  corresponding  o f the v o r t i c i t y f o r the cy- .,  l i n d e r s i s s i m i l a r t o t h a t o f the s p h e r o i d s .  The  contour i n f i g u r e 91 I n d i c a t e s t h a t the v e l o c i t y the wake i s very s m a l l .  velocity inside  Re  F i g u r e 80.  =  0.01  190  Flow c h a r a c t e r i s t i c s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R. = 2 a t Re = 0.01: a) v e l o c i t y l i n e s , b) v o r t i c i t y l i n e s , c) s t r e a m l i n e s .  191  Re = 1.  F i g u r e 81. S t r e a m l i n e s f o r a n e a r l y c i r c u l a r c y l i n d e r  (O.R.=1/0.995).  192  193  F i g u r e 83a.  S t r e a m l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R. = 10/9 a t Re=l and 5-  194  Figure 8 3 b .  Streamlines 0...R. = 10/9  f o r an e l l i p t i c a l c y l i n d e r w i t h at Re=15 and 40.  Re = 90.  196 Re *  F i g u r e 84a.  1.  S t r e a m l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R.=2 Re=l and 5 .  197  F i g u r e 84b.  Streamlines  Re=15 and  f o r an e l l i p t i c a l c y l i n d e r w i t h O.R.  50.  = 2  1  F i g u r e 85a. S t r e a m l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R. at Re=l and 5-  =  199  Re *  ho.  x02  F i g u r e 85b.  S t r e a m l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R at Re = 20 and 40.  = 5  200  Re  F i g u r e 86a.  =  1.  S t r e a m l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h at Re=l and 5.  O.R. =  0.2  Re =  1*0.  o F i g u r e 86b.  S t r e a m l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R. a t Re = 15 and 40.  =  0.2  202  F i g u r e 87a.  V o r t i c i t y l i n e s f o r an e l l i p t i c a l w i t h O.R. = 10/9 at Re=l and 5.  cylinder  203  204  F i g u r e 88a.  V o r t i c i t y l i n e s f o r an e l l i p t i c a l w i t h O.R. = 2 at Re=l and 5.  cylinder  F i g u r e 88b.  V o r t i c i t y l i n e s f o r an e l l i p t i c a l w i t h O.R. = 2 at Re=15 and 50.  cylinder  206  F i g u r e 89a.  V o r t i c i t y l i n e s f o r an e l l i p t i c a l w i t h O.R. = 5 at Re=l and 5.  cylinder  F i g u r e 89b.  V o r t i c i t y l i n e s f o r an e l l i p t i c a l w i t h O.R. = 5 at Re=20 and 40.  cylinder  208  F i g u r e 90a.  V o r t i c i t y l i n e s f o r an e l l i p t i c a l w i t h O.R. = 0.2 at Re=l and 5-  cylinder  2Q9.  Re  =  1*0.  .1  .1  F i g u r e 90b.  V o r t i c i t y l i n e s f o r an e l l i p t i c a l w i t h O.R. = 0.2 at Re=15 and 40.  cylinder  F i g u r e 91a. V e l o c i t y l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R.  =  0.2. o  Re  A.R.  F i g u r e 91b.  B  90.  =  .9  V e l o c i t y l i n e s f o r an e l l i p t i c a l c y l i n d e r w i t h O.R.  =  10/9.  7.  R e s u l t s f o r Swarms o f P a r t i c l e s i n C r e e p i n g Flow The Reynolds number used throughout was 0.01,  based on the e q u a t o r i a l d i a m e t e r and on t h e a x i s p e r p e n d i c u l a r (except i n f i g u r e 105) t o t h e main -flow d i r e c t i o n , i n t h e case o f s p h e r o i d s and e l l i p t i c a l respectively.  cylinders  The c o n c e n t r a t i o n o f t h e swarms was t a k e n  up t o about 0.6 and t h e r a t i o o f minor t o major a x i s  con-  s i d e r e d ranged from 0.999 t o 0.2. A. Swarms of S p h e r o i d s F i g u r e s 92 and 93 show t h e v a r i a t i o n o f V/V  with  0  c o n c e n t r a t i o n , c, f o r v a r i o u s a s p e c t r a t i o s o f o b l a t e s p h e r o i d s u s i n g t h e Happel f r e e s u r f a c e and Kuwabara zero v o r t i c i t y models, r e s p e c t i v e l y . d i c t t h a t t h e v e l o c i t y r a t i o V/V A.R. = 0.2 t h a n f o r s p h e r e s .  Q  Both models p r e -  i s much lower f o r  This i n d i c a t e s that the drag  f o r c e s on a swarm o f d i s c - s h a p e d s p h e r o i d s a l i g n e d p e r p e n d i c u l a r t o t h e main f l o w a r e much l a r g e r than those exp e r i e n c e d by spheres h a v i n g the same e q u a t o r i a l d i a m e t e r . Happel's model p r e d i c t s a lower v a l u e o f V/V  0  than  Kuwabara's f o r a g i v e n A.R. and c. The v a r i a t i o n o f V/V  G  with concentration f o r pro-  l a t e s p h e r o i d s i s shown i n f i g u r e 94 and 95 f o r t h e Happel and Kuwabara models, r e s p e c t i v e l y . vs.  The curves o f V/V  0  c f o r t h e d i f f e r e n t A.R. a r e c l o s e t o each o t h e r , but i t  i s i n t e r e s t i n g t o note t h a t t h e curve f o r A.R. = 0.2 ( n e e d l e shaped s p h e r o i d s ) f a l l s below t h e o t h e r s .  T h i s can be  213  F i g u r e 92. V a r i a t i o n o f v e l o c i t y r a t i o w i t h c o n c e n t r a t i o n f o r o b l a t e s p h e r o i d s : Happel's model, Re=0.-01.  214  .Figure 93- V a r i a t i o n of v e l o c i t y r a t i o w i t h c o n c e n t r a t i o n f o r o b l a t e s p h e r o i d s : Kuwabara's model, Re=0.01.  215  1  -l  10  V/Vo 5  -  \V  A.R.  \\._0-9  VV-io  -2  10  \  —  1  0  1  02  1  0-4  0-5 \—0-2 1 06  -  F i g u r e 9 4 . V a r i a t i o n of v e l o c i t y r a t i o w i t h c o n c e n t r a t i o n f o r p r o l a t e s p h e r o i d s : Happel's model, R e = 0 ; 0 1 .  216  Figure 95.  V a r i a t i o n of v e l o c i t y r a t i o w i t h c o n c e n t r a t i o n f o r p r o l a t e s p h e r o i d s : Kuwabara's model, Re=0.01.  217 a t t r i b u t e d t o t h e f a c t t h a t a t t h i s low a s p e c t r a t i o t h e surface a r e a i s l a r g e f o r a g i v e n e q u a t o r i a l d i a m e t e r , and consequently  a h i g h s k i n drag  predominates.  For A.R. = 0.9, both o b l a t e and p r o l a t e s p h e r o i d s showed v a l u e s o f V/V v e r y c l o s e t o t h a t o f a p e r f e c t 0  sphere, i n d i c a t i n g t h a t l a r g e v a r i a t i o n s i n t h e e x p e r i mental r e s u l t s f o r swarms o f " s p h e r e s " r e c o r d e d i n t h e literature  (126) cannot be a t t r i b u t e d t o t h e use o f im- ,  perfectly spherical particles. The v a r i a t i o n o f t h e Kozeny c o n s t a n t , k, w i t h  con-  c e n t r a t i o n , e v a l u a t e d from the drag c o e f f i c i e n t s f o r o b l a t e and p r o l a t e s p h e r o i d s , i s shown i n f i g u r e s 96 and 97, respectively.  F o r o b l a t e s p h e r o i d s w i t h A.R. = 0.2,  minimum v a l u e s o f k are observed  a t c = 0.15 f o r t h e Happel  model and 0.1 f o r t h e Kuwabara model.  Thereafter k i n - :  c r e a s e s w i t h c o n c e n t r a t i o n t o v a l u e s o f 48 and 70 a t c = 0.6, f o r t h e Happel and Kuwabara models, r e s p e c t i v e l y . A.R.  For  =0.5 - 0.999, k showed o n l y a s l i g h t v a r i a t i o n w i t h •  c o n c e n t r a t i o n i n t h e range c = 0.3 - 0.6. For p r o l a t e spheroids- ( f i g u r e 97) no minimum i n k'was observed.  The v a l u e o f k a t c = 0.6 f o r A.R. = 0.2  i s 3.7» which compares f a v o u r a b l y w i t h t h e v a l u e o f 3 - 4 4 ; f o r f l o w p a r a l l e l t o c y l i n d e r s (133).  T h i s agreement i s not  s u r p r i s i n g I n view o f t h e f a c t t h a t t h e p r o l a t e s p h e r o i d s w i t h A.R. = 0.2 r e p r e s e n t n e e d l e - l i k e p a r t i c l e s a l i g n e d w i t h the f l o w . The broken l i n e s i n f i g u r e s 96 and 97 r e p r e s e n t the Kozeny c o n s t a n t e v a l u a t e d by t h e t o r t u o s i t y method, where  A"R  I  KUWABARA 0-2 50 H A P P E L 0-2  30 20  KUWABARA 0-5 10  H A P P E L 0-5 KUWABARAO-9 KUWABARA 10 H A P P E L 0-9 H A P P E L 10  0 2  0 4  1  06  Figure. 9.6. V a r i a t i o n o f t h e Kozeny c o n s t a n t w i t h c o n c e n t r a t i o n f o r o b l a t e spheroids: , t o r t u o s i t y method.  oo  20  10 A.  KUWABARA HAPPEL KUWABARA HAPPEL  R  0-9 0-9 05 0-5  BOTH MODELS 0 2  0  0-2  0-4  0-6  F i g u r e 97- . V a r i a t i o n o f the. Kozeny c o n s t a n t w i t h c o n c e n t r a t i o n f o r p r o l a t e spheroids: , t o r t u o s i t y method.  220  o n l y the shape of the p a r t i c l e i s t a k e n i n t o a c c o u n t . Even though the method i s v e r y a p p r o x i m a t e , i t a p p a r e n t l y g i v e s a good e s t i m a t e of t h e o r d e r of the Kozeny  constant  as o b t a i n e d from the computed drag c o e f f i c i e n t s , f o r the higher concentrations.  The methods of e v a l u a t i n g k are  g i v e n i n Appendix I I I . The  l a r g e d e v i a t i o n s from the commonly  accepted  v a l u e of 5 c l e a r l y i n d i c a t e t h a t t h i s v a l u e i s q u i t e i n a p p l i c a b l e t o a l i g n e d p a r t i c l e s which d e v i a t e  significantly  i n shape from a sphere. A comparison between the s t r e a m l i n e s , e q u i v o r t i c i t y l i n e s and e q u i - v e l o c i t y l i n e s f o r o b l a t e w i t h A.R.  = 0.2  f i g u r e 98.  spheroids  and c = 0.1832, of b o t h models, i s shown i n  I n a l l cases complete symmetry e x i s t s between  upstream and downstream c o n d i t i o n s , because of c r e e p i n g The  flow.  s t r e a m l i n e s and e q u i - v e l o c i t y l i n e s appear t o be r e s -  p e c t i v e l y very s i m i l a r f o r the two models, but the e q u i v o r t i c i t y l i n e s d i f f e r from each o t h e r .  For Happel's model  some of the e q u i - v o r t i c i t y l i n e s l e a v i n g the s u r f a c e of the s p h e r o i d meet the o u t e r c e l l e n v e l o p e , w h e r e a s , f o r Kuwabara's model none of the l i n e s l e a v i n g the s u r f a c e meet the o u t e r . envelope.  The  d i f f e r e n c e a r i s e s from the f a c t t h a t f o r .  Happel's model the v a l u e of v o r t i c i t y a t the o u t e r envelope i s non-zero except at the a x i s of r e v o l u t i o n of the whereas f o r Kuwabara's i t i s always zero by  1.  \  cell',  definition.  Streamlines  222 B.  C l u s t e r s of E l l i p t i c a l C y l i n d e r s The v a r i a t i o n of the t o t a l drag n  w i t h c o n c e n t r a t i o n f o r O.R.  coefficient  i s given i n figures  99  and 100 f o r the Happel and Kuwabara models, r e s p e c t i v e l y . Both f i g u r e s show t h a t C  D T  f o r O.R.  = 0.2  than t h a t f o r c i r c u l a r c y l i n d e r s (O.R. O.R.  ^1.0,  the v a r i a t i o n of  i s much h i g h e r  = 1.0).  For  with concentration i s  g i v e n by f i g u r e s 101 and 102, f o r the Happel and Kuwabara models, r e s p e c t i v e l y . curves o f C  U n l i k e the curves f o r O.R.<: 1,  v s . c f o r 1.0 « O.R.  D T  o t h e r a c c o r d i n g t o both models.  < 5.0. are c l o s e t o each .  The v a r i a t i o n o f the Kozeny c o n s t a n t w i t h ,c for  O.R.  < 1.0,  shown i n f i g u r e 103, i s v e r y s i m i l a r t o  t h a t o f the o b l a t e s p h e r o i d s ( f i g u r e 96) i n t h a t a m i n i mum  p o i n t i s a g a i n e x h i b i t e d f o r . O.R.  = 0.2.  The  results  by both models are c l o s e t o each o t h e r f o r both O.R. and O.R.  = 0.5-  However, f o r O.R.  = 0.9  and 1.0,  =  0.2  Happel's  model g i v e s h i g h e r v a l u e s of k than Kuwabara's model.  For  O.R.  104,  > 1.0,  the v a r i a t i o n o f k w i t h c, g i v e n i n f i g u r e  i s s i m i l a r t o t h a t of p r o l a t e s p h e r o i d s for  elliptical  ( f i g u r e 97).  Thus  c y l i n d e r s the d e v i a t i o n from k - 5 i s a g a i n  l a r g e f o r shapes t h a t are s i g n i f i c a n t l y n o n - c i r c u l a r .  The  Kozeny c o n s t a n t as found by the t o r t u o s i t y method once again gives a reasonable estimate of k f o r high c o n c e n t r a t i o n s . The v a r i a t i o n o f n o n - d i m e n s i o n a l w i t h O.R. Rew  i s g i v e n i n f i g u r e 105 f o r Rew  pressure gradient = 0.01,  i s the Reynolds number based on the a x i s , 2b  where , p a r a l l e l to  2 2 3  F i g u r e 99.  V a r i a t i o n of the t o t a l drag c o e f f i c i e n t w i t h concentration for e l l i p t i c a l cylinders with O.R. ^ 1.0: Happel's model, Re=0.01.  224  F i g u r e 100.  V a r i a t i o n of the t o t a l drag c o e f f i c i e n t w i t h concentration for e l l i p t i c a l cylinders with •O.R... 4. -1.0: Kuwabara's model, Re=0.01.  225  F i g u r e 101. V a r i a t i o n of the t o t a l drag c o e f f i c i e n t w i t h concentration f o r e l l i p t i c a l cylinders with O.R. »1.0: Happel's model, Re=0.01.  226  F i g u r e 102. V a r i a t i o n of the t o t a l drag c o e f f i c i e n t w i t h concentration f o r e l l i p t i c a l cylinders with O.R.^1.0: Kuwabara's model, Re=0.01.  2-  0. R. HAPPEL  2  n  p  KUWABARA -1  10  KUWABARA 05 HAPPEL 10 KUWABARA 0-9 KUWABARA10  ^HAPPFI  0 9  HAPPEL 10 0 2  04  06  gure 103. V a r i a t i o n o f Kozeny c o n s t a n t f o r e l l i p t i c a l c y l i n d e r s O.R. < 1.0 a t Re=0.01: , t o r t u o s i t y method.  with  3 0 20  10 Q  R  KUWABARA10/9 H A P P E L 10/9 KUWABARA 2 HAPPEL 2 N<UWABARA5 HAPPEL 0-4  0-2  5  0-6  C F i g u r e 104. V a r i a t i o n o f Kozeny constant w i t h c o n c e n t r a t i o n f o r e l l i p t i c a l .with O.R. > l.O.at Re=0.....01: t o r t u o s i t y method.  cylinders ro  CO  229  0. R. F i g u r e 105. V a r i a t i o n of AP/L w i t h o r i e n t a t i o n r a t i o f o r e l l i p t i c a l c y l i n d e r s , Rew = 0.01.  230 the main f l o w d i r e c t i o n .  For a g i v e n f l u i d and a f i x e d  l e n g t h of the a x i s , 2b , f i g u r e 105 A.R. to  = 0.2  at c = 0.6,  the major a x i s (O.R.  for  AP/L  shows t h a t f o r  f o r the f l o w p e r p e n d i c u l a r  = 0.2)  i s as much as 1000  the f l o w p a r a l l e l t o t h i s a x i s (O.R.  times  = 5.0).  d e r i v a t i o n of the e x p r e s s i o n used f o r AP/L  that  The  i s g i v e n i n Appendix  III. When the c o n c e n t r a t i o n of a bank of c y l i n d e r s i s > v e r y low  ( h i g h c l e a r a n c e ) , then the p r e s s u r e drop a c r o s s  s e v e r a l rows of c y l i n d e r s can be taken as e q u a l t o the of p r e s s u r e drops a c r o s s i s o l a t e d rows.  Kuwabara  e v a l u a t e d the t o t a l drag c o e f f i c i e n t .on a row of cylinders.  sum  (134)  elliptical  Here the i s o l a t e d row has been extended t o a  square c e l