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UBC Theses and Dissertations

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

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SYMMETRIC FLOW PAST ORTHOTROPIC BODIES: SINGLE AND CLUSTERS 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 , 1964 M . S c , U n i v e r s i t y o f New B r u n s w i c k , 1966 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e D e p a r t m e n t o f CHEMICAL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA F e b r u a r y , 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o lum b i a , I a g ree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree tha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date j f ^ J 8. !f7Q i ABSTRACT N u m e r i c a l s o l u t i o n o f the N a v i e r - S t o k e s e q u a t i o n was s u c c e s s f u l l y a c c o m p l i s h e d , u s i n g an a d a p t a t i o n o f the r e -l a x a t i o n t e c h n i q u e o f J e n s o n , f o r a x i s y m m e t r i c f l o w p a s t s i n g l e o b l a t e and p r o l a t e s p h e r o i d s at p a r t i c l e Reynolds numbers up t o 100. The a s p e c t r a t i o o f the s p h e r o i d s v a r i e d between 0.999 ( n e a r l y p e r f e c t sphere) and 0.2. For low a s p e c t r a t i o s the s u r f a c e p r e s s u r e and v o r -t i c i t y d i s t r i b u t i o n s showed a marked d i f f e r e n c e from those of a sphere. The appearance o f the wake bubble b e h i n d a s p h e r o i d was found t o be a s t r o n g f u n c t i o n o f the p a r t i c l e shape. N u m e r i c a l 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 f l o w p a s t e l l i p t i c a l c y l i n d e r s , w i t h the f l o w p a r a l l e l t o the major a x i s f o r a s p e c t r a t i o s of 0.995 t o 0.2 at Reynolds numbers up t o 90, and w i t h the f l o w p a r a l l e l t o the minor a x i s f o r an a s p e c t r a t i o o f 0.2 at Reynolds numbers up t o 40. The n u m e r i c a l s o l u t i o n was found t o be l e s s s t a b l e t h a n the c o r r e s p o n d i n g t h r e e -d i m e n s i o n a l a x i s y m m e t r i c case. The v a r i a t i o n o f the t o t a l d r a g c o e f f i c i e n t w i t h Reynolds number f o r the s p h e r o i d s and the e l l i p t i c a l c y -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 not much d i f f e r e n t from t h a t o f a sphere and a c i r c u l a r c y l i n d e r , r e s p e c t i v e l y . The r e s u l t s f o r b o t h the s p h e r o i d s and the e l l i p t i c a l c y -l i n d e r s showed a stea d y t r e n d w i t h Reynolds number from i i S t o k e s a n d / o r Oseen f l o w t o b o u n d a r y l a y e r f l o w . H a p p e l ' s f r e e s u r f a c e c e l l m o d e l and K u w a b a r a ' s z e r o v o r t i c i t y c e l l m o d e l were e m p l o y e d f o r t h e s t u d y o f c r e e p i n g f l o w p a s t swarms o f a l i g n e d s p h e r o i d s a n d c l u s t e r s o f a l i g n e d e l l i p t i c a l c y l i n d e r s . L a r g e d e v i a t i o n s o f t h e K o z e n y c o n s t a n t f r o m i t s commonly assumed v a l u e o f 5 f o r p a c k e d b e d s were f o u n d by b o t h m o d e l s f o r p a r t i c l e s w h i c h d e v i a t e s i g n i f i c a n t l y i n s h a p e f r o m a s p h e r e o r a c i r c u l a r c y l i n d e r . I n g e n e r a l , H a p p e l ' s f r e e s u r f a c e m o d e l p r e d i c t e d l o w e r t o t a l d r a g c o e f f i c i e n t s t h a n d i d K u w a b a r a ' s z e r o v o r t i c i t y m o d e l f o r b o t h t h e swarms o f s p h e r o i d s a nd t h e 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 . C o n t o u r s o f t h e s t r e a m l i n e s , e q u i - v o r t l 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 a r e p r e s e n t e d . i i i TABLE OF CONTENTS Page ABSTRACT i L I S T OF TABLES v i i L I S T OF FIGURES , v i i i L I S T OF DIAGRAMS x v -ACKNOWLEDGEMENTS . . x v i CHAPTER I INTRODUCTION 1 CHAPTER I I REVIEW OF PRIOR WORK 3 1. REVIEW OF ANALYTICAL WORK 3 A. S i n g l e B o d i e s 3 A . l S p h e r o i d s ' 5 A. 2 E l l i p t i c a l C y l i n d e r s .. 17 B. Swarms 23 2. REVIEW OF NUMERICAL WORK 39 CHAPTER I I I FORMULATION OF F I N I T E DIFFERENCE EQUATIONS AND BOUNDARY CONDITIONS < 43 A. S p h e r o i d s 43 A . l F o r m u l a t i o n o f t h e 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 . . 43 A. 2 B o u n d a r y C o n d i t i o n s , 50 B. E l l i p t i c a l C y l i n d e r s 57 B . l F o r m u l a t i o n o f t h e 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 57 B.2 B o u n d a r y C o n d i t i o n s 6l i v Page CHAPTER I V FORMULATION OF DRAG COEFFICIENT EQUATIONS AND THE PRESSURE DISTRIBUTION 67 A. S p h e r o i d s 67 B. E l l i p t i c a l C y l i n d e r s 75 CHAPTER V DISCUSSION AND PRESENTATION OF RESULTS 79 1. A P P L I C A B I L I T Y OF THE NAVIER-STOKES EQUATIONS AND CHOICE OF THE NUMERICAL TECHNIQUE 79 2. RELAXATION PROCEDURE 80 3. COMPUTER PROGRAM 87 4 . V A L I D I T Y OF THE NUMERICAL WORK 88 5 . RESULTS FOR SINGLE SPHEROIDS 101 6. RESULTS FOR SINGLE E L L I P T I C A L CYLINDERS 157 7. RESULTS FOR SWARMS OF PARTICLES I N CREEPING FLOW 212 A. Swarms o f S p h e r o i d s 212 B. 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 222 CHAPTER V I CONCLUSIONS AND RECOMMENDATIONS 2 '35 L I S T OF REFERENCES 237 NOMENCLATURE . 2 4 5 APPENDIX 1 BOUNDARY LAYER THEORY AND POTENTIAL FLOW DEVELOP-MENT 1-1 A. S p h e r o i d 1-1 B. E l l i p t i c a l C y l i n d e r s 1 - 4 V Page APPENDIX I I DATA FOR SINGLE SPHEROIDS, SINGLE ELLIPTICAL CY-LINDERS, SWARMS OF SPHEROIDS AND CLUSTERS OF ELLIP-TICAL CYLINDERS . I I - l Sing l e Oblate Spheroids I I - l Single P r o l a t e Spheroids 11-19 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 ,with O.R.> 1 ....... 11-30 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 with O.R.< 1 11-38 Swarms of Oblate Spheroids, Happel's Model .... II-41 Swarms of Oblate Spheroids, Kuwabara's Model .. 11-49 Swarms of P r o l a t e Spheroids, Happel's Model ... 11-57 Swarms of P r o l a t e Spheroids, Kuwabara's Model . II-63 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 with O.R.> 1, Happel's Model II-69 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 with O.R.> 1, Kuwabara's Model II-76 •Clusters of E l l i p t i c a l C y l i n d e r s with O.R.< 1, Happel's Model 11-83 Cl 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 with O.R.< 1, Kuwabara's Model II-90 APPENDIX I I I 1. EVALUATION OF THE KOZENY CONSTANT FROM BASIC DEFINITION • • I I I - l A. Spheroids I I I - l B. E l l i p t i c a l C y l i n d e r s . I I I - 3 v i Page 2. EVALUATION OF THE KOZENY CONSTANT USING THE TORTUOSITY METHOD I I I - 4 3. EVALUATION OF A P/L FOR A CLUSTER OF E L L I P -T I C A L CYLINDERS AT CONSTANT Rew • I H - 6 APPENDIX I V COMPUTER PROGRAMS IV-1 1. SPHEROIDS I V - 2 2. E L L I P T I C A L CYLINDERS IV-6 v i i L I S T OF TABLES T a b l e Page 1 V a l u e s o f t h e c o e f f i c i e n t A n 14 2 R e l a x a t i o n f a c t o r s f o r o b l a t e s p h e r o i d s 83 3 R e l a x a t i o n f a c t o r s f o r p r o l a t e s p h e r o i d s 8 4 4 R e l a x a t i o n 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 w i t h O.R. ^ 1.0 85 5 R e l a x a t i o n 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 w i t h O.R. < 1. 86 6 Some s e l e c t e d d a t a f o r s i n g l e s p h e r o i d s 91 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 95 8 E f f e c t o f g r i d s i z e on C D T and a com-p a r i s o n 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 96 v i i i L I S T OF FIGURES Page 1. O b l a t e s p h e r o i d a l mesh s y s t e m . 47 2. O b l a t e s p h e r o i d w i t h i t s o u t e r e n v e l o p e . 51 3. E l l i p t i c a l c y l i n d r i c a l mesh s y s t e m . 60 4. E l l i p t i c a l c y l i n d e r w i t h i t s o u t e r e n v e l o p e . 62 5- V a r i a t i o n o f t o t a l d r a g c o e f f i c i e n t w i t h t h e mean d i a m e t e r . 98 6. 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 w i t h R e y n o l d s number f o r a s p h e r e . 102 7- V a r i a t i o n o f t o t a l d r a g c o e f f i c i e n t w i t h R e y n o l d s 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 8. V a r i a t i o n o f t o t a l d r a g c o e f f i c i e n t w i t h R e y n o l d s 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 9. V a r i a t i o n o f C D S / C D p w i t h a s p e c t r a t i o f o r o b l a t e s p h e r i o d s . 107 10. V a r i a t i o n o f Gpp/Gpg w i t h a s p e c t r a t i o f o r p r o l a t e , s p h e r o i d s . 108 11. V a r i a t i o n o f 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 A.R. f o r o b l a t e s p h e r o i d s . 109 12. V a r i a t i o n o f 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 A.R. f o r p r o l a t e s p h e r o i d s . 110 13. 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 s p h e r e a t Re=10. 112 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 s p h e r e a t h i g h R e y n o l d s number. 113 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 a t Re=1.0. 114 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 a t Re=5-0. 115 17. 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 a t Re=10. 1 1 6 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 a t Re=20. 117 i x 19. Surface pressure d i s t r i b u t i o n f o r oblate spheroids at Re=50. 118 20. Surface pressure d i s t r i b u t i o n f o r oblate spheroids at Re=100. 119 21. 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 at Re=1.0. 121 22. 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 at Re=5.0. 122 23. 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 at Re=10. 123 24. 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 at Re=20. 124 25. 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 at Re=50. • 125 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 spheroid at Re=100. 126 27. 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 an oblate spheroid with A.R. = 0.999 (Sphere). 127 28. 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 an oblate spheroid with A.R. = 0.9. 128 29. 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 an oblate spheroid ; with A.R. = 0.5- 129 30. 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 an oblate spheroid with 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 with A.R. = 0.9. 131 32. 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 ; with A.R. = 0.5. 132 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 of wake length with Re f o r spheroids. 135 35- Location of separation point f o r spheroids. 136 36. Streamlines f o r a nearly s p h e r i c a l oblate spheroid (A.R. = 0.999). 138 37. Streamlines f o r an oblate spheroid with aspect r a t i o 0.9. 139 X 38. 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 r a t i o 0.5. 140 39. 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 r a t i o 0.2. i 4 l 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 s p h e r o i d . (A.R. = 0.999)- 1 4 2 41. 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. 47. 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 a s p e c t r a t i o 0.9-48. 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 a s p e c t r a t i o 0.5. 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 a s p e c t r a t i o 0.2. 143 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 a s p e c t r a t i o 0:5. 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 a s p e c t r a t i o -. hl-0.2. . 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 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. 146 45. S t r e a m 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 a s p e c t r a t i o 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 s p h e r o i d w i t h a s p e c t r a t i o 0.2. 148 149 150 151 50. V e l o c i t y l i n e s f o r s p h e r o i d s a t Re-100. 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 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 ) . 154 52. 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 an o b l a t e 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 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. = 0.2. 156 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 of h i g h a s p e c t r a t i o . -^g 55. 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 of low a s p e c t r a t i o . 159 56. 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 p r o l a t e s p h e r o i d s w t t h A.R. = 0.9. 160 57. 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 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 o f low a s p e c t r a t i o . l 6 l x i 58. V a r i a t i o n o f t h e b o u n d a r y 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 number f o r s p h e r o i d s a t n = TT/2. 162 59. D r a g c o e f f i c i e n t s f o r c i r c u l a r c y l i n d e r s . 164 60. V a r i a t i o n o f t o t a l d r a g c o e f f i c i e n t w i t h 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 . 166 61. V a r i a t i o n o f t o t a l d r a g c o e f f i c i e n t w i t h R e y n o l d s 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 as t h e c h a r a c t e r i s t i c l e n g t h . 168 62. V a r i a t i o n o f c n s / c 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 o r i e n t a t i o n r a t i o s o f e l l i p t i c a l c y l i n d e r s . 169 63. 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 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 . 170 64. 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 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). !71 65. 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 n 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. 1 7 2 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 a n 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. 173 67. 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 n e l l i p t i c a l c y l i n d e r w i t h O.R. = 5. 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 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. 71. 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 n 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. 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 a n e l l i p t i c a l c y l i n d e r w i t h O.R. = 5 . 73. 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 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. 174 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 a n 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. !75 69. 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 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). 1 7 7 178 179 180 181 74. 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 e l l i p t i c a l c y l i n d e r . 1 ° 2 75. L o c a t i o n o f s e p a r a t i o n p o i n t f o r e l l i p t i c a l c y l i n d e r s . 183 x i i -76. 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 . 185 77 . " 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 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 . 186 7 8 . 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 e l l i p t i c a l c y l i n d e r s . 187 79. V a r i a t i o n o f t h e b o u n d a r y 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 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 n=ir/2 188 8 0 . F l o w c h a r a c t e r i s t i c s f o r a n 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 . 0 1 , 190 8 1 . 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 . 9 9 5 ) . 191 8 2 . V o r t i c i t y 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 . 0 0 . 5 ) . 192 8 3 a . S t r e a m l i n e s f o r a n 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 R e = l and 5 . 193 8 3 b . Streamlines f o r an e l l i p t i c a l c y l i n d e r with O.R.=10/9 at Re=15 and 40. 194 8 3 c . Streamlines 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 at Re=90. 195 84a. Streamlines 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 at'Re=l and 5. 196 84b. Streamlines f o r an e l l i p t i c a l c y l i n d e r with O.R.= 2 at Re=15 and 50. 197 8 5 a . Streamlines 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.= 5 at Re=l and 5 . 198 85b. Streamlines f o r an e l l i p t i c a l c y l i n d e r with O.R.= 5 at Re=20 and 40. 199 8 6 a . Streamlines f o r an e l l i p t i c a l c y l i n d e r with O.R.= 0.2 at Re=l and 5 . 200 8 6 b . Streamlines 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 0,R.= 0.2 at Re=15 and 40. 201 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 c y l i n d e r w i t h O.R. = 10/9 at Re=l and 5 . 2 0 2 87b. 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 c y l i n d e r w i t h O.R. = 10/9 at. Re=15, 40 and 90. 2 0 3 x i i i 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 c y l i n d e r w i t h O.R. = 2 a t Re=l and 5.' 204 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 c y l i n d e r w i t h O.R. = 2 a t Re=15 and 50. 2°5 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 c y l i n d e r w i t h O.R. = 5 a t Re=l and 5. 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 c y l i n d e r w i t h O.R. = 0.2 a t Re=l and 5. 98. Flow p a t t e r n s f o r an o b l a t e s p h e r o i d a l c e l l model, A.R. = 0.2, c = 0.1832. 206 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 c y l i n d e r w i t h O.R. = 5 a t Re=20 and 40. 2 0 7 208 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 c y l i n d e r w i t h O.R. = o.2 a t Re=l-5 and 40. • 2 °9 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. - 2 1 0 91b. 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. 2 1 1 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. 213 93. 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 : Kuwabara's model, Re=0.01. 214 94. 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 p r o l a t e s p h e r o i d s : Happel's model. Re=0.01. 215 95. 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 p r o l a t e s p h e r o i d s : Kuwabara's model, Re=0.01. 216 96. 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 o b l a t e s p h e r o i d s . 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 s p h e r o i d s . " 221 99. V a r i a t i o n o f the t o t a l d r a g c o e f f i c i e n t w i t h con-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 0,R.>< 1.0: Happel's model, Re=0.01. 223 100. V a r i a t i o n of the t o t a l d r a g c o e f f i c i e n t w i t h con-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: Kuwabara's model, Re=0.01. 224 X I V 101. " 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 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: K u w a b a r a ' s m o d e l , Re=0.01. 225 102. 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 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:' K u w a b a r a ' s m o d e l , Re=0.01. 226 103. V a r i a t i o n o f K o z e n y 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 w i t h O.R. .< 1.0 a t Re = 0.01. ' 227 104. V a r i a t i o n o f K o z e n y 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 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. 229 106. S q u a r e c e l l m o d e l f o r e l l i p t i c a l c y l i n d e r s . 231 107. C o m p a r i s o n b e t w e e n t h e t o t a l d r a g c o e f f i c i e n t o b t a i n e d by K u w a b a r a ' s z e r o v o r t i c i t y c e l l m o d e l and f l o w p a s t a s i n g l e row o f e l l i p t i c a l c y l i n d e r s , a l s o by K u w a b a r a , e x t e n d e d t o a s q u a r e c e l l m o d e l : Re=0.01. 232 108. E q u i - v o r t i c i t y l i n e s f o r e l l i p t i c a l c e l l m o d e l s , O.R. = 5, c =-0.200. 233 XV LIST OF DIAGRAMS Diagram Page 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 area of the oblate spheroid 69 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 area of the e l l i p t i c a l c y l i n d e r 76 Appendix I I I 111.1 Oblate spheroid c e l l model I I I - l 111.2 E l l i p t i c a l cell.model I I I - 3 111.3 Main and average flow d i r e c t i o n s f o r packed beds I I I - 4 111.4 E v a l u a t i o n of angle 6 I I I - 5 x v i ACKNOWLEDGEMENT I g r a t e f u l l y a c k n o w l e d g e t h e h e l p f u l g u i d a n c e a nd e n c o u r a g e m e n t f r o m D r . Norman E p s t e i n , u n d e r whose d i r e c t i o n t h i s i n v e s t i g a t i o n was c o n d u c t e d . T h a n k s a r e e x t e n d e d t o Dr.. Zeev Rotem f o r h i s c o n t i n -uous i n t e r e s t i n t h i s w o r k . I am i n d e b t e d t o t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a a nd t o t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r f i n a n c i a l s u p -p o r t . T h a n k s a r e a l s o due t o t h e members o f t h e U.B.C. C o m p u t i n g C e n t r e f o r t h e i r c o o p e r a t i o n a n d f o r b e a r a n c e . I am a l s o i n d e b t e d t o my w i f e E s t h e r f o r h e r c o n t i n u a l s u p p o r t t h r o u g h o u t t h i s w o r k . CHAPTER I I n t r o d u c t i o n The w i d e r a n g e o f a p p l i c a t i o n s i n c h e m i c a l and p r o c e s s e n g i n e e r i n g o f s y s t e m s i n w h i c h m o t i o n t a k e s p l a c e b e t w e e n a f l u i d a n d s u s p e n d e d s o l i d p a r t i c l e s h a s l e d t o t h e n e e d f o r a g r e a t e r u n d e r s t a n d i n g o f t h e i r u n d e r l y i n g c h a r a c t e r i s t i c s . The most s t r a i g h t f o r w a r d a p p l i c a t i o n s o f p a r t i c l e d y n a m i c s o c c u r i n s i t u a t i o n s where m u t u a l i n t e r a c t i o n o f p a r t i c l e s c a n be t a k e n as n e g l i g i b l e . D u s t and m i s t c o l l e c t i o n f r o m d i l u t e s u s p e n s i o n s o f f i n e s o l i d p a r t i c l e s a n d f r o m l i q u i d d r o p s i n g a s e s a r e s i m p l e p r a c t i c a l e x a m p l e s o f s u c h i d e a l i z a -t i o n . One common a p p l i c a t i o n i s i n t h e e l i m i n a t i o n o f a t m o s -p h e r i c p o l l u t i o n , 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 a i r . The r e c o v e r y o f v a l u a b l e b y - p r o d u c t s f r o m d u s t s l e a v i n g d r y e r s and s m e l t e r s r e p r e s e n t s a n o t h e r i m p o r t a n t a p p l i c a t i o n . S e epage o f u n d e r g r o u n d w a t e r , f l o w o f o i l i n w e l l s and s e d i m e n t a t i o n a r e a few more common e x a m p l e s o f s o l i d - f l u i d i n t e r a c t i o n s . F l u i d i z a t i o n h a s become an i m p o r t a n t u n i t o p e r a t i o n i n t h e c h e m i c a l i n d u s t r i e s , where f l u i d i z e d bed r e a c t o r s a r e d e s i g n e d i n w h i c h n o t o n l y c h e m i c a l r e a c t i o n , b u t a l s o h e a t t r a n s f e r and mass t r a n s f e r , c a n t a k e p l a c e i n t h e b e d . R a t e s a t w h i c h t h e s e t r a n s f e r p r o c e s s e s o c c u r a r e s t r o n g l y i n f l u e n c e d by t h e m e c h a n i c s o f t h e f l u i d f l o w , k n o w l e d g e o f w h i c h i s b a s i c 2 i n t h e f u n d a m e n t a l s t u d y o f t h e t r a n s p o r t phenomena and t h e r e l a t i o n s h i p b e t w e e n them ( 1 , 2, 4, 8 ) . G r a v i t a t i o n a l and c e n t r i f u g a l s e p a r a t i o n p r o c e s s e s a r e a l s o w i d e l y e m p l o y e d i n t h e c h e m i c a l i n d u s t r y and h a v e r e c e i v e d t h e a t t e n t i o n o f c h e m i c a l and m i n i n g e n g i n e e r s o v e r a l o n g p e r i o d . However, s e d i m e n t a t i o n and t h i c k e n i n g d e v i c e s and t h e c l a s s i f i c a t i o n o f p a r t i c l e s by h i n d e r e d s e t t l i n g a r e u s u a l l y s t i l l d e s i g n e d on a f a i r l y e m p i r i c a l b a s i s . More t h e o r e t i c a l work w i l l g i v e a b e t t e r u n d e r s t a n d i n g o f p a r t i c l e - f l u i d i n t e r a c t i o n s , and i t seems r e a s o n a b l e t o s u p p o s e 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 i n v o l v e s s u c h i n t e r -a c t i o n s c o u l d be made more r a t i o n a l as one a c q u i r e s more know-l e d g e o f t h e l a w s g o v e r n i n g them. 3 CHAPTER' I I R e v i e w o f P r i o r Work . . The s t u d y o f t h e s t e a d y i s o t h e r m a l l a m i n a r f l o w o f an i n c o m p r e s s i b l e v i s c o u s N e w t o n i a n f l u i d a b o u t an i m p e r m e a b l e r i g i d body i m m e r s e d t h e r e i n r e q u i r e s t h e s i m u l t a n e o u s s o l u t i o n o f t h e N a v i e r - S t o k e s and t h e c o n t i n u i t y e q u a t i o n s , s u b j e c t t o t h e p r e v a i l i n g b o u n d a r y c o n d i t i o n s . The n o n - l i n e a r i t y 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 r e n d e r s t h e i r s o l u t i o n v e r y d i f f i c u l t , and o n l y a r e l a t i v e l y s m a l l number o f e x a c t s o l u t i o n s o f r a t h e r s p e c i a l i z e d c h a r a c t e r a r e known: Lamb (5), D r y d e n ( 6 ) , and Payne and P e l l ( 7 ) . I n many c a s e s , h o w e v e r , t h e f l o w i s s u c h t h a t r e a s o n a b l e s i m p l i f y i n g a s s u m p t i o n s r e g a r d i n g i t s c h a r a c t e r c a n be made w h i c h r e s u l t i n a l e s s c o m p l i c a t e d m a t h e m a t i c a l p r o b l e m . I n d e e d , a l a r g e p a r t o f t h e t h e o r y o f v i s c o u s f l o w s c o n s i s t s o f p r o b l e m s o b t a i n e d i n t h i s manner, c . f . H a p p e l and B r e n n e r (8) and B r e n n e r ( 9 ) . 1. R e v i e w o f A n a l y t i c a l Work A. S i n g l e B o d i e s The m a t h e m a t i c a l a n a l y s i s o f t h e f l o w p a s t i s o l a t e d b o d i e s i s t h e f i r s t s t a g e i n t h e u n d e r s t a n d i n g o f p a r t i c l e -f l u i d i n t e r a c t i o n s . I n t h e p r e s e n t s t u d y ' o f s y m m e t r i c f l o w p a s t o r t h o t r o p i c b o d i e s , two c l a s s e s o f b o d i e s were c o n s i d e r e d , n a m e l y , 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 s p h e r e and a c i r c u l a r c y l i n d e r a r e e a c h a s p e c i a l c a s e o f a s p h e r o i d and an e l l i p t i c a l c y l i n d e r , r e s p e c t i v e l y . An o r t h o t r o p i c p a r t i c l e i s d e f i n e d as a body w h i c h p o s s e s s e s t h r e e m u t u a l l y p e r p e n d i c u l a r symmetry p l a n e s . I t i n c l u d e s e l l i p s o i d s ( s p h e r o i d s b e i n g a s p e c i a l c a s e o f an e l l i p s o i d ) , r i g h t e l l i p t i c a l c y l i n d e r s and r e c t a n g u l a r p a r a l l e l e p i p e d s (8). The N a v i e r - S t o k e s e q u a t i o n f o r s t e a d y s t a t e a x i -s y m m e t r i c o r t w o - d i m e n s i o n a l f l o w , e x p r e s s e d i n o r t h o g o n 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 n t e r m s o f t h e d i m e n s i o n a l s t r e a m t f u n c t i o n ^ , i s , Zj 9 (ijAE ) ,2- 2 • 3 (^'> h 3 ) v E r = h 1 h 2 h 3 a ( g > 0 - 2E <M H 2 * ; _ T F - R R (1) where h : , h 2 and h 3 a r e t h e d i m e n s i o n a l m e t r i c c o e f f i c i e n t s , v i s t h e k i n e m a t i c v i s c o s i t y and , 2 h i h o E = - L h 3 J3 X12 3 . h 3 h j 3 . 3 L h 2 ' 3n hi 8n J ( 2 ) (£,n) b e i n g t h e c u r v i l i n e a r c o o r d i n a t e s . The e l l i p t i c a l , s p h e r i c a l and s p h e r o i d a l c o o r d i n a t e s y s t e m s a r e d i s c u s s e d w i t h e x c e l l e n t c l a r i t y by H a p p e l and B r e n n e r (8) and by G o l d s t e i n (10). E q u a t i o n 1 i s o f t h e f o u r t h o r d e r i n V and i t i s n o t l i n e a r b e c a u s e o f t h e p r e s e n c e o f t h e t e r m s on t h e r i g h t -h and s i d e , c o n t r i b u t e d by t h e i n e r t i a l e f f e c t s . A n o t h e r f o r m 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 i s g i v e n by p v' . V v' -y .V 2v' = - V P * ( 3 ) where p i s t h e d e n s i t y , y i s t h e v i s c o s i t y , p» i s t h e d i m e n -s i o n a l d y n a m i c p r e s s u r e and v' i s t h e d i m e n s i o n a l v e l o c i t y — — 2 v e c t o r . The o p e r a t o r s V and V a r e g i v e n by 5 (4) and V 2 = hi h 2 h 3 r 3 + l 3qj h 2 h 3 9Qi 3 q 2 h 3 hi 3q 2 3 h 3 3 + (5) 3q h i h 2 -I r e s p e c t i v e l y , ( q 1 } q 2 >Q 3 ) b e i n g t h e c u r v i l i n e a r c o o r d i n a t e s and ( i x , i 2 , i 3 ) t h e u n i t v e c t o r s . A . l S p h e r o i d s The o l d e s t f o r m u l a t i o n o f f l o w p a s t an o b s t a c l e i s t h e s o - c a l l e d " S t o k e s f l o w " . I t i s d e f i n e d by t h e a s s u m p t i o n t h a t t h e i n e r t i a l e f f e c t s a r e n e g l i g i b l e i n c o m p a r i s o n w i t h t h o s e o f v i s c o s i t y , o r more p r e c i s e l y , t h a t t h e R e y n o l d s number o f t h e f l o w i s v e r y s m a l l . 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 c h a r a c t e r i s t i c f l o w v e l o c i t y a n d / o r t h e body d i m e n s i o n a p p e a r -i n g i n t h e R e y n o l d s number a r e s u f f i c i e n t l y s m a l l , a n d / o r t h e k i n e m a t i c v i s c o s i t y i s l a r g e ( 8 ) . S t o k e s ( 3 ) j i n 1851, w h i l e c o n s i d e r i n g t h e m o t i o n o f a s p h e r e i n a v i s c o u s f l u i d , seems t o h a v e b e e n t h e f i r s t t o o m i t t h e i n e r t i a l t e r m s 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 , w h i c h t h e n r e d u c e s t o Su c h 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 i s v e r y s u b -s t a n t i a l , f o r t h e e q u a t i o n o f m o t i o n t h u s becomes l i n e a r . = 0 (6) 6 T h i s means, f o r e x a m p l e , t h a t i f i j / and t|/2 e a c h s e p a r a t e l y s a t i s f y t h e a f o r e m e n t i o n e d e q u a t i o n , t h e n so does ty\ + ty\ . E x a c t s o l u t i o n s a r e t h e n f e a s i b l e e v e n f o r r e l a t i v e l y c o m p l e x g e o m e t r i e s , v i a c l a s s i c a l s u p e r p o s i t i o n t e c h n i q u e s a p p l i c a b l e t o l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . A g e n e r a l s o l u t i o n i n s p h e r i c a l c o o r d i n a t e s h a s b e e n d e v e l o p e d by Sampson (11) u s i n g t h e s e p a r a t i o n o f v a r i a b l e s t e c h n i q u e . I t h a s a l s o b e e n g i v e n i n d e p e n d e n t l y by S a v i c (12) and by Haberman and S a y r e (13), and has b e e n r e v i e w e d by H a p p e l and B r e n n e r (8). The c o m p l e t e s o l u t i o n o f e q u a t i o n 6 f o r t h e s t r e a m f u n c t i o n s i s g i v e n by i M r , 6 ) = X (An r " + B n r n + 1 _ + C n r n + 2 + D n r n + 3 ) I n (O + | ( A n r n + B n r _ n + 1 + C n r n + 2 + D n r _ n + 3 ) H n ( ? ) (7) where (r,0) a r e t h e s p h e r i c a l c o o r d i n a t e s , w h i l e A n , Bn, Cn and Dn and t h e c o r r e s p o n d i n g p r i m e d t e r m s a r e c o n s t a n t s . The t e r m 5 i s d e f i n e d by t, = c o s 8. I n ( C ) and H. n(0- a r e G e g e n b a u e r f u n c t i o n s o f o r d e r n a n d d e g r e e -1/2 o f t h e f i r s t a n d s e c o n d t y p e , r e s p e c t i v e l y . T h e c o n s t a n t s a r e e v a l u a t e d f r o m t h e a p p l i c a b l e b o u n d a r y c o n d i t i o n s . F o r t h e n o - s l i p c o n d i t i o n a t t h e s p h e r e and t h e u n i f o r m s t r e a m i n g c o n d i t i o n a t i n f i n i t y , t h e S t o k e s s t r e a m f u n c t i o n as g i v e n by e q u a t i o n 7 becomes 2 i j / = % U a 2 (| - ^ + 2 ^ ) s i n 2 e (8) where a i s the r a d i u s o f the sphere and U i s the v e l o c i t y o f the u n d i s t u r b e d stream, the l i n e 6 = 0 b e i n g i n the d i r e c t i o n o f the u n d i s t u r b e d stream. The d i m e n s i o n a l s u r f a c e p r e s s u r e i s th e n g i v e n by P* = I p U 2 (1 + p | cose ) (9) where Re i s the Reynolds number d e f i n e d as Re = ^ (10) The drag f o r c e e x p e r i e n c e d by the sphere i s g i v e n by F = 4ITU D 2 (11) where D 2 i s a c o n s t a n t from e q u a t i o n 7 which i s e q u a l t o 75- a U . Hence F = 6Try aU (12) T h i s i s the 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 the 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 = C0 + D j T + ^ (C ) { B 2T + C2 X, (T) + D 2 H, ( T ) + E*.H ( T )} + I 3 ) { B3 + D3 H3 (x ) + B* Hg (T ) } + \ (C ) { B* H 2 (T ) + D , H, (T ) + B* H, (T ) } (13) where x = cosh 4 (14) 8 and z, = c o s n F o r a p r o l a t e s p h e r o i d one o b t a i n s ip' = I 2 ( C ) { B 2T + C 2 I 2. (T ) + D 2 H 2 (x ) } (15) (16) F o r an o b l a t e s p h e r o i d t h e s t r e a m f u n c t i o n i s g i v e n by s i m p l y r e p l a c i n g x i n e q u a t i o n 16 by IX, where X i s sinh£ . F o r t h e n o - s l i p c o n d i t i o n a t t h e s p h e r o i d s u r f a c e and a u n i f o r m s t r e a m c o n d i t i o n a t i n f i n i t y , t h e S t o k e s s t r e a m f u n c t i o n s g i v e n by e q u a t i o n 16 f o r a p r o l a t e s p h e r o i d and by i t s t r a n s f o r m e d f o r m f o r an o b l a t e s p h e r o i d become r e s p e c t i v e l y ib' = 1/2 U c 2 ( x 2 - l ) ( l - ? 2 ) 1 -{(T 2+1)/(T 2-1)} C O t h " ^ - { T / ( T 2 -1)} a. d. {(X 2+1)/(T 2-1)} C O t h _ 1 T - { T a / ( T 2 -1)} a a g_ d. d. and ip'= 1/2 U c 2 ( A 2 + D ( l - C 2 ) 1 -{ X / ( X 2 + 1 ) } - { ( X 2 - D / ( A 2 +1)} c o t _ 1 A ci d, {X / ( X 2 + 1 ) } - { ( X 2 - D / ( A 2 +1)} c o t _ 1 X , a. a. a. ci <• (17) (18) where s u b s c r i p t a d e n o t e s t h e s u r f a c e and c t h e f o c a l l e n g t h o f t h e s p h e r o i d . Payne and P e l l (7) a r r i v e d a t t h e same e x p r e s s i o n s by u s i n g W e i n s t e i n ' s g e n e r a l i z e d a x i a l l y s y m m e t r i c p o t e n t i a l t h e o r y . The d r a g f o r c e (7*8) e x p e r i e n c e d by t h e s p h e r o i d i s t h e n g i v e n by F = 6 TTU Ua„K (19) 9 where aQ i s the e q u a t o r i a l radius of the spheroid and K = — — — (20) oblate { A'. - (A -1) cot xx } a a a while K , . = 1 (21) p r o l a t e ZZZZZZ | Al -1 { ( T 2 +1) C O t h ' 1 x a - T a> A comparison of the drag force experienced by a spheroid as given by equation 19 with that f o r a sphere, aquation 12, shows that the r a t i o of the two drag forces Is given by the f a c t o r K. Despite a m i s p r i n t i n t h e i r expression f o r ^ •Q-^j.ate' ^ a P P e l and Brenner (8) provide an accurate t a b u l a t i o n of K as a fu n c t i o n of aspect r a t i o A.R. f o r both oblate and p r o l a t e spheroids, A.R. being defined as the r a t i o of minor a x i s to major a x i s of the spheroid. For a t h i n c i r c u l a r d i s c (A.R. = 0), the ta b u l a t e d value of K i s 0.84883- A c i r c u l a r d i s c ; . i s , of course, the l i m i t i n g case of an oblate spheroid whose minor axis i s zero. The drag f o r c e experienced by a d i s c can be obtained from equations 19 and 20 as \ -y 0, g i v i n g a F = 16 y a e U (22) S i m i l a r expressions f o r the drag force are given by Sampson (11), Ray (14), Roscoe (15) and Gupta (16). The dimensionless surface pressure d i s t r i b u t i o n f o r Stokes flow past an oblate spheroid i s p = 1 + 6 K o b l a t e / + 1 Re 1 x2* +cos 2n / * a ' 2 , -, \ cos n (22a) 10 and f o r a p r o l a t e spheroid, 6K P = 1 + p r o l a t e 1 ^ \ a ! -f p-)cos n C 22b) Re \ X +sin n ./ a Once again the u n i t y term of equations 22a and 22b comes from c o n s i d e r a t i o n of the i n e r t i a l terms i n the Navier-Stokes equations, s i n c e - f o r s t r i c t l y creeping flow i t i s absent. I t should a l s o be mentioned that Oberbeck (17) i n 1876 solved the problem of the steady t r a n s l a t i o n of an e l l i p s o i d i n a viscous l i q u i d without the use of the stream f u n c t i o n (due to lack of symmetry), using i n s t e a d the v e l o c i t y compo-nents expressed i n terms of the g r a v i t a t i o n a l p o t e n t i a l of the e l l i p s o i d . For the e l l i p s o i d s of r e v o l u t i o n one obtains the same r e s u l t s as given by equations 17 and 18. Lamb (5) d i s -cusses Oberbeck's technique i n d e t a i l . The case when the i n e r t i a l e f f e c t s of the flow are small but 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 to extend Stokes' o r i g i n a l s o l u t i o n f o r a t r a n s l a t i n g sphere to higher Reynolds numbers using a simple p e r t u r b a t i o n technique. This technique s t a r t s with the Stokes' s o l u t i o n of the creeping flow equation, V V2v„ - v Pj = 0 ( 2 3 ) and the c o n t i n u i t y equation, V . v0' = 0 ( 2 l j ) f o r the boundary co n d i t i o n s of no, s l i p at the surface 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 0 , P 0 ) , i s then used f o r the i n e r t i a l term of the Navier-Stokes equation as f o l l o w s : y V 2 v[ - V p j =p VQ.V VQ (25) * • f i = o ( 2 6 ) v x being the improved v e l o c i t y vector of the flow f i e l d . Even though equation 25 i s now l i n e a r and hence e a s i e r to solve than the complete Navier-Stokes equation, there e x i s t s u n f o r t u n a t e l y no e x p o n e n t i a l l y decaying s o l u t i o n that can s a t i s f y the boun-dary c o n d i t i o n s , v, = 0 at the surface and v = U f a r away from the surface. Moreover, as Happel and Brenner (8) point out, the next approximation f o r v e l o c i t y , v 2 , becomes i n f i n i t e at i n f i n i t y . The r e s u l t s given by such p e r t u r b a t i o n are r e -f e r r e d to as Whitehead's paradox. Oseen (19, 20) showed that Stokes' s o l u t i o n of the l i n e a r -i z e d Navier-Stokes"equation i s of the. form v = U + Ua 0(l/v). Hence at large distances from the surface of the sphere the i n -- ' - - ' 2 i e r t i a l term p v„.V v 0 becomes p U a0( 1 / r 2) and the viscous term -2 -' i y V v 0 becomes y Ua 0 ( ± / r 3) . The r a t i o of the i n e r t i a l to the viscous term then becomes of the order This r a t i o i n -P creases i n d e f i n i t e l y with r however small U/v may be. For t h i s reason Stokes' creeping flow equation cannot be considered as ; v a l i d at points f a r away from the sphere unless Re+0. At points near the sphere the i n e r t i a l forces tend to vanish while the viscous forces are of the order ^f^-. This means that Stokes' approximation f o r the flow f i e l d i s not uniformly v a l i d throughout but breaks down at a large distance from the sphere, and hence v Q does not give an accurate estimate of the i n e r t i a l term at such la r g e d i s t a n c e s . Consequently i t cannot be used f o r the f i r s t approximation of the i n e r t i a l terms as proposed by Whitehead. _ i _ i Oseen suggested that the i n e r t i a l term p v .V v be _ _ _ i replaced by pU .V v , g i v i n g y v 2 v' - VP* = p U • vv' (27) V.v' = 0 (28) The i n e r t i a l terms are thus to some extent taken i n t o account, the previous approximation being much improved at i n f i n i t y but somewhat impaired near the surface. The drag force experienced by a sphere i s then given by P = 6TTu a U (1 + (3/16) Re) (29) where a i s the radius of the sphere and Re i s the Reynolds number based on the sphere diameter. Oseen's o r i g i n a l s o l u t i o n of equations 27 and 28, a l s o developed independently by Burgers et a l . (21) i n 1916, i s only an approximate s o l u t i o n . The complete a n a l y t i c a l s o l u t i o n of Oseen's equations was f i r s t derived by G o l d s t e i n (22, 23), g i v i n g the drag force as a s e r i e s i n Reynolds number: f 3 19 Re 71 Re3 30179 Re F = 6 iryU 1 + jj-Re - - y ^ - +. - 344 06400 122519 R e 5  + 560742400 " (30) The f i r s t term of the above equation i s the value given by 13 Stokes, while the second term i s that given by Oseen and by Burgers et a l . For values of Re up to and i n c l u d i n g 2, the drag can be c a l c u l a t e d from the s e r i e s as given by equation 30. For Re up to 20, G o l d s t e i n presented a t a b u l a t i o n of F against Re obtained by s o l v i n g one of the intermediate steps of Oseen*s equations n u m e r i c a l l y . Tomotika and Aoi (24) considered Oseen's equations and obtained the dimensional stream f u n c t i o n as </ = _ u a Y | - 1 6 + 2 3 R e (4 " • 2 2 + - ^ cos 9} s i n 2 6 (3D ^ a r In the l i m i t when Re-»- 0 the above equation degenerates i n t o the w e l l known Stokes stream f u n c t i o n as given by equation 8. t Based on the above expression f o r \p , Tomotika and Aoi claimed that a small vortex i s formed behirid a sphere even f o r Re as low as 1.0. Pearcey and MoHugh (25a) , however, i n t h e i r d e t a i l e d computation of Oseen's equations, pointed out that no vortex i s attached to the rear of the sphere even at Re = 10, and that equation 31 Is only very approximate as i t was obtained by an e a r l y t r u n c a t i o n of the s e r i e s f o r one of the constants i n the s o l u t i o n . The dimensionless surface pressure d i s t r i b u t i o n , P, given by Pearcey and McHugh (25a) ' could be put i n the form P = , F = 2 f A P n + i cos 6 (32) 1/2 P U2 * n where P n +]_ • i s a Legendre polynomial of the f i r s t k i n d . A 14 t a b u l a t i o n of the c o e f f i c i e n t An. i s given i n Table 1. Table 1. Values of the c o e f f i c i e n t Re n 1 Ref (25a) 4 Ref (25a) 10 Ref (25a) Ref (25b) 0 3.52627 1.21748 0.7169 0.13225 1 0.45078 0.38926 0.32761 0.23725 2 0.2216 x 10" •2 0.15192 x 1 0 - 1 0.3363 x 10" 3 0.1600 3 -0.7004 x 10" •4 -0.1323 x 1 0 _ 1 -0.402 x 10" 2 -0.0055 4 0.1726 x 10" •5 0.861 x 10 _ i | 0.22 x 10" 3 -0.05 5 -0.3496 x 10" •7 -0.421 x 10~5 -4 -0.1 x 10 0.003 6 0.6 x 10~9 0.14 x 10~6 0.0270 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 to note that Tomotika and Aoi (24) observed Oseen's s o l u t i o n to give a constant r a t i o of the form (pressure) to f r i c t i o n (skin) drag c o e f f i c i e n t whatever the value of the Reynolds number. The s o l u t i o n of Oseen's equations f o r the case of a spheroid was e f f e c t e d by an approximate method by Oseen himself (26) . Aoi (27) j using an exact a n a l y t i c a l technique as o u t l i n e d by Gol d s t e i n (22, 23) , obtained i d e n t i c a l expressions to Oseen 's fo r both the form and.. the ..skin drag forces... In order to maintain conformity with the corresponding creeping flow s o l u t i o n s , the expression f o r the t o t a l drag f o r c e becomes 15 F = 6 TT u U a K { 1 + 3 I* R e } (33) e l b where K i s given by equation 20 f o r an oblate spheroid and by equation 21 f o r a p r o l a t e spheroid, while Re i s based on the e q u a t o r i a l diameter of the spheroid. The r a t i o of the form drag to the s k i n drag f o r a p r o l a t e and an oblate spheroid as developed by Aoi (27) i s given by 2 * S 1 - (Ta - 1) ( Ta coth 1 T a - 1) and P p / F q - J ^ + » U - *a o o f 1 P S 1 - + 1) (1 - X a o o t - ^ ^ ) f o r a p r o l a t e and an oblate spheroid, r e s p e c t i v e l y . Accord-ing to the above expressions, the r a t i o of the drag forces 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 to the true i n e r t i a l term i n the Navier-Stokes equation at great distances from the body, controversy has been aroused by the f a c t that i t appears to be a poor approximation i n the neighbourhood of the body. The u n s a t i s f a c t o r y status of the Oseen equations p e r s i s t e d u n t i l the work of Lagerstrom and Cole (28) , Kaplun (29) , Proudman and Pearson (30) and Kaplun and Lagerstrom (32) , who suggested that one should abandon the attempt to obtain p e r t u r b a t i o n f i e l d s which are uniformly v a l i d throughout the flow f i e l d and seek i n s t e a d to f i n d separate asymptotic 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 separate regions near t o , and f a r from, the body. These "inner" and "outer" s o l u t i o n s are each deter-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 . The expression of Proudman and Pearson (30) f o r the drag force experienced by a sphere placed i n a uniform stream-ing flow i s given by P = 6 ir y D a (1 + ^  .Re + Re 2 In- (Re/2)) (36) The f i r s t two terms are i d e n t i c a l to Oseen's expression, equation 29. The above expression gives b e t t e r agreement with experimental data than Oseen's expression, up to Re of 2. Unfortunately, f o r Re> 2, the l o g a r i t h m i c term of equation 36 becomes large and thus the equation diverges and can no longer be used to evaluate the drag f o r c e . Breach (31b) extended the technique of Proudman and Pearson to incl u d e some other terms, g i v i n g F = 6 T P U a (1 + l ! T R e + 1^ 0 ~ R e 2 l n ^ ' R e / 2 ) Re2•, 9^ 3 323 27 3 Re , + — (4u- + X l n 2 - T 6 W - ) + W - R e l n 2 " ) + ' ' - ( 3 7 ) p where y = 0.5772... i s Euler's constant. The terms i n Re 3 Re and Re l n — ^ ~ a r e new. The new drag equation does not provide any improvement on the previous 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 the above r e s u l t s to render them a p p l i c a b l e to spheroids. These s o l u t i o n s are again l i m i t e d by the nature of the approximations and prove to be inadequate f o r Re > 2. The expression f o r the drag force on spheroids can be r e w r i t t e n i n the form !7 2 P = 6 i T y U a e K { l + 1|-ReK + -9-^ |Q Re 2 In~r } 08) Once again the f i r s t term i n the above equation i s f o r creep-ing flow, the second being Oseen's extension while the t h i r d i s new. For a sphere, K = 1 and equation 38 becomes i d e n t i c a l to that of Proudman and Pearson. A.2 E l l i p t i c a l C y l i n d e r s For the two-dimensional 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 creeping flow equation 6 has no v a l i d s o l u t i o n that can s a t i s f y both the n o - s l i p c o n d i t i o n at the surface of the c y l i n d e r and uniform streaming flow at i n f i n i t y . This s i t u a t i o n was discovered by Stokes (65) and i s u s u a l l y r e f e r r e d to as "Stokes Paradox". Krakowski and Charnes (66) 'generalized Stokes' Paradox to include any two-dimensional body placed i n a medium which i s i n f i n i t e i n a l l d i r e c t i o n s . I n v a l i d s o l u t i o n s which defy Stokes' paradox f o r the streaming motion perpendicular to the a x i s of a c i r c u l a r c y l i n d e r are given by Berry and Swain (67) , Wilton (68) and Harrison (69) . Lamb (5) a p p l i e d Oseen's equations to the case of flow 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 co n d i t i o n s disappeared. His approximate s o l u t i o n to Oseen's equations gives the drag f o r c e per u n i t length of the c y l i n d e r as 18 The v a l i d i t y of Lamb's equation i s w e l l e s t a b l i s h e d by the experimental r e s u l t s of Wieselsberger (147), Finn (146), and Jayaweera and Mason (145). The exact s o l u t i o n of Oseen's equations was obtained by Tomotika and Aoi (24). Their expression f o r the dimensional stream f u n c t i o n i s ij/= a U {A ( - - - ) - B - l n ( r / a ) } s i n e + a r a 2 2 2 { C ^2 ~ § 2 ) + D l n ( r / a ) } s l n 2 9 ( 4 0 ) a r a where A = f B = 2 ( l n Re - 2.0022) ' C 5 f > D = - B C , and a i s the c y l i n d e r r a d i u s . As f o r the case of Sphere, the authors (24) cl a i m that the above expression p r e d i c t s a vortex at the rear of t h e . c y l i n d e r . I t i s very l i k e l y that t h i s expression i s too approximate i n i t s d e r i v a t i o n to put any weight on the streamlines given by i t . The for c e per u n i t length i s given by F = 2uyU Z B m , (41) o and the e v a l u a t i o n of the constants B^, B r and B 2were done num e r i c a l l y . In a f u r t h e r paper Tomotika and Aoi (70) expressed the drag f o r c e per u n i t length i n s e r i e s terms of S and Re: 19 P 47rUy S 102 (42) where S 0.5 -Y - In (Re/8) (43) Y being Euler's constant. The f i r s t term i n equation 42 i s that obtained by Lamb (5). For Re < 0.5, Lamb's expression i s qu i t e adequate. The drag f o r c e per u n i t length obtained by equation 42 agrees q u i t e w e l l w i t h the numerical s o l u t i o n of equation 4 l up to Re = 2.0. Using Oseen's equation, Tomotika and Aoi found that the r e -l a t i v e c o n t r i b u t i o n s of the form drag and the s k i n drag to the t o t a l drag were the same and independent of Reynolds number. Kaplun (29) a p p l i e d the inner and outer ex-pansions f o r the flow past a c i r c u l a r c y l i n d e r and ob-tained the for c e per u n i t length as where S i s given by equation 43. The flow past an e l l i p t i c a l c y l i n d e r i n an un-bounded f l u i d f a l l s i n the realm of two-dimensional flow and thus no v a l i d s o l u t i o n could be obtained f o r the creeping flow equation. I n v a l i d s o l u t i o n s which contravene Stokes paradox are discussed by Berry and Swain (67). 20 Oseen's equations were solved approximately by Harrison (69) and by Bairstow et a l . (71). The drag f o r c e per u n i t length on the e l l i p t i c a l 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 to the stream i s given by 4rruU ; ( 4 5 ) F = —; ± (1 + a) - y - In Re/8 (1 + a) 1 - b/a , D 2pUa w h e r e 0 = l + b/a a n d R e = ~ The drag f o r c e per u n i t length experienced 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 to the stream i s given by p = — i ^ l U _ (46) _ ( 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 equations 45 and 46 c o l l a p s e i n t o equation 39, given by Lamb. When a=1 equations 45 and 46 give the drag f o r c e per u n i t length on a f l a t p l a t e f o r flow p a r a l l e l and perpendicular to the p l a t e , r e s p e c t i v e l y . Tomotika and Aoi (73), using an exact s o l u t i o n of Oseen's equation, obtained the drag force per u n i t length f o r the flow along and perpendicular to the major a x i s of an e l l i p t i c a l c y l i n d e r as 21 4TTU I , Re 2 F = -j^- 1 — - x L \ 3 2[l+a] 2L {L 2-|[l+ 2 a-a 2 ]L+^[15-4a- l8a 2 -12a 3 - a 4 ]} -— 0 R E M 0 { L 4 - k - [31a+10a 2 -6a 3 +10a 4+a 5]L 2 ~[4-3a 2 -2a 3]L 3+ 3 2 2 [ l + a J L 2 4 5 1 2 [280+ll85a-987a2-2065a3-90a2,+525a5+85a6+3a7] L -r i r r [ 225- 12Oa-524a 2-2l6a 3+39OaV44Oa 5+l80a 6+24a 7+a 8 ]} 2304 '(47) where L = | ( l + a ) - Y - l n Re/8(1 + a) and v - ^ 1 _ R e 2 32[l+a] 2 L {L 2-|[l -2a-a 2] L+^-[15+4a-l8a2+12a3-a4 ]} -? R e—n—p(L i j-ip[4-3a 2+2a 3]L 3-i7 T[31a+10a 2-6a 3+10a 4+d 5]L 2+ 32^[l+a] q IT 1 4 4 0 ^ 8^[280-ll85a-987a 2+2065a 3-90a i |-525a 5+85a 6-3a 7]L -^Q1|-[225-120a-524a2-2l6a3+390ai|+440a5+l80a6+24a7+a8]}J (48) where L = -| (1 - a) - y - In 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 of equation 47 and 48 Is equivalent to equations 45 and 46 r e s p e c t i v e l y . Tomotika and Aoi (73) have i n d i c a t e d that the s o l u t i o n s obtained by Sidrak (75) f o r the drag f o r c e on an e l l i p t i c a l c y l i n d e r , and that by Davies (74) f o r a f l a t p l a t e , are i n e r r o r . P i e r c y and Winny (72) obtained an expression f o r the drag f o r c e on a f l a t p l a t e up to terms 2 i n Re which i s i n agreement w i t h the corresponding ex-pr e s s i o n of Tomotika and A o i . The drag force as given by equation 47 and 48 can be assumed to be v a l i d up to Re ^ 2. Imai (76) considered the uniform flow past an e l l i p t i c a l c y l i n d e r at an a r b i t r a r y angle of i n c i d e n c e , h i s r e s u l t s f o r the s p e c i a l cases of flow p a r a l l e l to the major and flow p a r a l l e l to the minor a x i s being i n agree-ment wi t h equation 47 and 48. For high Reynolds numbers, Oseen Ts asymptotic theory f o r Re -* °° can reproduce the patterns of r e a l flow to a r a t h e r s u r p r i s i n g extent, according to M u l l e r (77 ) . Oseen (26) a p p l i e d the asymptotic theory to the 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 the theory f o r flow past a sphere, while Hocking (78, 79) studied the flow past a c i r c u l a r d i s c set per-pendicular to the flow. Tamada and Miyagi (80) studied the flow past a f l a t p l a t e set perpendicular to the stream and obtained an approximate formula f o r the drag f o r c e per u n i t length of the p l a t e at f i n i t e Re: 23 P = \ p U 2 L [TT + 8 r( \ ) R e " 3 7 4 ] (49) where L i s the p l a t e width, Re = u L/v and r i s the Gamma f u n c t i o n . The f i r s t term of equation 49 i s the asymptotic value of P f o r Re -*• °° . An improvement on equation 49 was a l s o made by Miyagi (8la, 8lb) f o r Re of 4, 8, 12, 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 the drag f o r c e experienced by a f l a t p l a t e set along the streaming flow at intermediate Reynolds numbers, using boundary l a y e r theory. B. Swarms I t i s n a t u r a l to suppose that the disturbance due to a body immersed i n a f l o w i n g f l u i d spreads i n the flow f i e l d and that the presence of more than one body i n an otherwise uniform flow gives r i s e to a mutual i n t e r -a c t i o n between the immersed bodies. Many attempts have been made to develop a r e l i a b l e t h e o r e t i c a l r e l a t i o n s h i p between the concent r a t i o n of - such a "swarm" and the fo r c e exerted on a given body w i t h i n the swarm, i n order to p r e d i c t , f o r example, the s e t t l i n g r a t e i n the case of sedimentation, or the pressure drop across a packed bed of p a r t i c l e s . The basic equation f o r creeping flow through a granular bed i s given by Darcy's (94) e m p i r i c a l equation of 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 L' (50) where U i s the s u p e r f i c i a l v e l o c i t y (empty tube v e l o c i t y ) , K i s the p e r m e a b i l i t y of the bed and AP'is the pressure drop across the bed depth L'. Emersleben (95) and S l i c h t e r (96) t r i e d , w ith not much success, to j u s t i f y Darcy's equation on t h e o r e t i c a l grounds. Blake (98) accepted D u p u i t 1s (97) assumption that the i n t e r s t i t i a l v e l o c i t y through a granular bed equals U/e,where e i s the p o r o s i t y of the bed (volume of pore space per u n i t volume of bed) and by dimensional a n a l y s i s extended equation 50 to k vi S • L f where k has become known as Kozeny's constant and S i s the p a r t i c l e surface area per u n i t volume of the bed. Kozeny (99) subsequently derived the above equation by assuming that a packed bed c o n s i s t s of a group of 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 that the t o t a l i n t e r n a l surface and the t o t a l i n t e r n a l volume are equal to the p a r t i c l e surface and the pore volume, r e s p e c t i v e l y (100). Introducing the •hydraulic r a d i u s , m, defined as the r a t i o of the volume occupied by the f l o w i n g f l u i d to the wetted s u r f a c e , gives S = e / m (52) combining equations 50, 51 and 52 y i e l d s K = m2 e / k (53) L e t t i n g S y be the s p e c i f i c surface of the p a r t i c l e s then S = S y ( 1 - e ) (54) and s u b s t i t u t i n g equation 54 i n t o equation 51, U = C K ] ^ 5-5 (55) L y k ( 1 - e r v Equation 55 i s known, as the Carman-Kozeny equation. Kozeny r e a l i z e d t h a t , owing to the tortuous character of the flow through packed beds, the length of the equivalent channel should be Lg , where L e > L T . Hence k = k Q ( L e / L' ) 2 , where k 0 i s the c a p i l l a r y constant; k Q = 2 f o r a c i r c u l a r channel. Davies (101), P i e r c y et a l . (102), P a i r and Hatch (103) and Ratkowsky and E p s t e i n (104) showed that k D f o r various n o n - c i r c u l a r channel shapes considered l i e s i n g e n e r a l , between 1.7 and 3.0. The r a t i o L ' / L* i s termed e the t o r t u o u s i t y . The Kozeny constant k i s a f u n c t i o n of 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 equation are that i t overcorrects 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 to changes i n degree of packing (105) and that i t i s i n v a l i d f o r high Reynolds numbers and f o r high p o r o s i t i e s (105). A review on the Carman-Kozeny equation and some twenty other e m p i r i c a l c o r r e l a t i o n s i s given by E p s t e i n (105). 26 For low Reynolds numbers flow, 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 . This method, i n simple terms, i n v o l v e s a piece-wise matching of boundary c o n d i t i o n s on the boundary su r f a c e s , i n c l u d i n g the container w a l l s , using a set of p a r t i a l s o l u t i o n s . A general d i s c u s s i o n on t h i s t e ch-nique i s given by Happel and Brenner ( 8 ) . Smoluchowski (84, 85, 86) considered an assemblage of n f a l l i n g spheres having a cubic arrangement both i n an i n f i n i t e stagnant medium and i n a container. For the l a t t e r case, the drag force experienced by a given sphere i n the assemblage i s given by where h i s the distance between the centres of two spheres i n the assemblage. Burgers (87), McNown and L i n (88) and L i n (89) obtained approximately the same expression as above f o r a d i l u t e cubic assemblage ( a/h >>1) . The concentration of the s p h e r i c a l bodies i n a cubic assemblage i s given by P = 6n y a U ( 1 + 2.6 a/h ) (56) c = 4TT 3 ( a / h )3 (57) and equation 56 t h e r e f o r e becomes P = 6i y U a ( 1 + 1.61 c 1/3 (58) 27 Pamularo (92) extended the technique of r e f l e c t i o n s and obtained the drag force experienced by a sphere i n a d i l u t e assemblage of c u b i c , rhombohedral and random sus-pensions as F = 6ir p U a ( 1 + 1.91 c 1 / 3 ) (59) F = 6 i n U a ( 1 + 1.79 c 1 / 3 ) (60) and P = 6TT u U a C 1 + 1.3 c 1 / 3 ) (61) r e s p e c t i v e l y . The r e f l e c t i o n technique becomes extremely tedious f o r concentrated suspensions and i s t h e r e f o r e not commonly used f o r such suspensions, i t s a p p l i c a b i l i t y being confined to d i l u t e systems. Another important technique used to evaluate the drag force on a p a r t i c l e i n a swarm of other p a r t i c l e s i s the " c e l l method". This method i s based on the concept that an assemblage of p a r t i c l e s can be d i v i d e d i n t o a number of 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 . This method i s more a p p l i c a b l e than the method of r e -f l e c t i o n s to concentrated suspensions, e s p e c i a l l y where the w a l l e f f e c t due to the v e s s e l c o n t a i n i n g the suspension i s n e g l i g i b l e . Cunningham (118) used the c e l l model to p r e d i c t the drag f o r c e F f o r a swarm of spheres. He considered that the swarm could be represented by a r i g i d sphere surrounded by a r i g i d outer envelope. This model presents the d i f f i c u l t y that the p o s i t i o n of the outer envelope i s e m p i r i c a l l y determined. 28 Uchida (90) i n v e s t i g a t e d a cubic assemblage using a cubic model, and obtained f o r a d i l u t e swarm of spheres P = 6TT U U. a ( 1 + 2.1 c 1 / 3 ) (62) Kawaguchi (91) used another c e l l model f o r spheres by consi 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 ) - 1 (63) where M = 0.9095. f o r a cubic assemblage and M = 1.1458 f o r a body-centred l a t t i c e . For high d i l u t i o n , i . e . c 0 , equation 63 becomes F = 6n y U a ( 1 + c l / 3 ) ( 6 4 ) Happel and Ast (125) obtained a s i m i l a r expression. Hasimoto (93) constructed a p e r i o d i c array by repeating a bas i c c e l l . He replaced each sphere by a point force r e t a r d i n g the flow and obtained expressions f o r P using F o u r i e r s e r i e s f o r simple, body-centred and fa c e -centred cubic arrangements. His r e s p e c t i v e expressions f o r the creeping flow regime are: F = 6-rr y U a ( 1 F = 6TT y U a ( 1 F = 6 T T V 1 U a ( 1 - 1.76 c 1 / 3 + c - 1.5593 c 2 r 1 ( 6 5 ) 1.791c 1 / 3 + c - 0.329 c2 T 1 ( 6 6 ) - 1.791c 1 / 3 + c - 0.302 c 2 ) 1 (67) 29 When c 0 . , equation 65 becomes P = 6 T T y U a ( l + 1.76 c 1 / 3 ) (68a) and equation 66 and 67 becomes F = 6 u y U a ( l + 1.79 c 1 / 3 ) (68b) I t seems c l e a r that the general expression f o r the drag f o r c e i n d i l u t e suspensions has the form P = 6ir y U a ( 1 + q c 1 / 3 ) (69) The value of q does not vary much between authors and geometric arrangements, i n d i c a t i n g that the p a r t i c l e arrange-ment i s not very important i n uniform d i l u t e suspensions. Hasimoto (93), using the same technique as f o r spheres, obtained an expression f o r the forc e per u n i t length on a square array of c i r c u l a r c y l i n d e r s : E = i ^ y U p — (70) - ± l n ( c A ) - 1.3105 + c A When c ->• 0,the above expression becomes F 1 ln(c/Tr) - 1.31 U 1 J "2 Brinkman (106) obtained 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 the 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 concentrated systems. The drag f o r c e a c t i n g on a sphere i n the swarm is. given 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 . This c e l l model considered the flow past a sphere embedded i n a packed bed. The pressure drop i s summed from two sources, namely, that given by Darcy's equation and that given by the creeping flow equation, so that VP' = - + yV v' (73) Brinkman reasoned that when the p o r o s i t y e-^  1.0 the se-cond term of equation 73 becomes the dominant one, and when e i s 01 5- | the f i r s t term i s the dominant one. In terms of the dimensional stream f u n c t i o n , equation 73 could be reduced to the form E % ' = (constant) E 2 i | / (74). Since Darcy's equation i s e m p i r i c a l , I t f o l l o w s that 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 extension of Brinkman's model was made by Spielman and Goren (107) f o r c y l i n d e r s i n four d i f f e r e n t arrangements: a) A l l c y l i n d e r s normal to flow, b) A l l c y l i n d e r s p a r a l l e l to flow, c) , two-dimensional random d i s -t r i b u t i o n i n planes p a r a l l e l to flow and d) three-dimen-s i o n a l random d i s t r i b u t i o n . The range of e given was 0.7 < e < 0.997-Richardson and Zaki (108) employed a c e l l model f o r sedimenting spheres. P a r t i c l e s were assumed to be s e t t l i n g i n such a way that they are a l i g n e d one above the other i n hexagonal-type tubes. They considered two p a t t e r n s : c o n f i g u r a t i o n I assumed that the spheres i n adjacent h o r i z o n t a l l a y e r s are the same distance apart v e r t i c a l l y 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 that the spheres i n adjacent h o r i z o n t a l l a y e r s touch each other. The r e -s u l t s of c o n f i g u r a t i o n I I are i n good agreement with Brinkman's model f o r 0.6<e<0.95. For the case of c y l i n d e r s the method of r e -f l e c t i o n s cannot be used d i r e c t l y due to Stokes' paradox. The c e l l models of Hasimoto (93) and of 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 e a r l i e r . Emersleben (95) obtained an approximate s o l u t i o n f o r laminar flow p a r a l l e l to c i r c u l a r c y l i n d e r s i n square array using a second order 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) subsequently obtained 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 along 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 of p a r a l l e l c i r c u l a r cy-l i n d e r s Miyagi (121), using Stokes' approximation, and Tamada and Fujikawa (122), using Oseen's approximation, , obtained f o r flow perpendicular to the c y l i n d e r s at low Reynolds number - ln(a/h) - 1.33 + •(. i r / 3 ) (a/h) where h i s the distance between the centres of neighbouring c y l i n d e r s . For a/h<<l , equation 75 becomes very c l o s e to that given by Hasimoto, equation 71» where f o r a square array c = Tra /h • This shows that f o r d i l u t e arrays the pressure drop across s e v e r a l rows of c y l i n d e r s i s equal to the sum of pressure drops across i s o l a t e d rows. For high values of Re, Tamada and Fujikawa employed numerical a n a l y s i s to obt a i n F. For 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 (123) gives p = _?JLJLU_ r ! _ 2 a / h ]"5/2 ( 7 6 ) 2 /2 Kuwabara (124), using Oseen's approximations, t r e a t e d the case of flow- past an i s o l a t e d row of p a r a l l e l e l l i p -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 to flow F = 4 T r y U [ S + i ( l + a 2 ) - ! J — + 2 T Q r 1 (77) d Re b) minor a x i s p a r a l l e l to flow F = 4TT y U [ S + | ( 1 - a 2 ) - + 2 T C ] _ 1 (78) c) c i r c u l a r c y l i n d e r s 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 - 2 T ] _ 1 M d Re . . ° J = 4IT y U [ l n ( d/7T ) - 2 s(3) (Re d / l 6 f r ) 2 + . . . ] - 1 (80) e) 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 ] " 1 = 4TT y U [ 1 - ln(ir/d) - 2 5(3) (d Re/l67r) 2+ . . . ] _ 1 ( 8 l ) where. S = l n [ 8(l+a 2)/Re ] -y , Re =. 2(l+a2) Up/y p Y i s Euler's constant,(1+a ) i s the major semi-axis, 2 (1+a ) i s the minor semi-axis, c(x) i s Riemann's zeta 2 f u n c t i o n , d = h/(l+a )and T G = 27r/(dRe)+ | l n ( d Re/l67T) + £ - 5(3) (d Re/l67r) 2+ + | ? ( 5 ) Cd Re/l67r) 4 - (82) C(3) and 5(5) are both of 0|1| . 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 given by Kuwabara f o r d = 10. Another c e l l model was developed by Happel (126) to p r e d i c t the drag force exerted on a sphere f o r a s e d i -menting swarm. A random assemblage i s considered to c o n s i s t of a number of c e l l s , each of which contains a s p h e r i c a l envelope, such that the r e l a t i v e volume of f l u i d to 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 to be f r i c t i o n l e s s . The s o l u t i o n of the creeping flow equation (equation 6) i n s p h e r i c a l coordinates, as given by equation 7, y i e l d s i j / = | U s i n 2 6 ( A 2 r 2 + B 2 r + C 2 / ' + D 2 r - 1 ) (83) The constants A 2, B 2, C 2 and D 2 are determined from the n o - s l i p c o n d i t i o n s at the surface of the inner sphere, the zero shear s t r e s s c o n d i t i o n at the outer 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 given by 5 P = 4TT y U a — 2 y , + 3 (84) - 2 y + 3Y - 3Y + 2 where y i s the r a t i o of the inner sphere radius to that of the outer sphere, and the volumetric concentration c, i s therefore given by y 3 • Equation 84 reduces to 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 values of y between 0 and 1. For small values of y , i . e . d i l u t e sy-stems, i t reduces to 35 P = 6ir y U a ( 1 + 1.5 c 1/3 (85) I t i s c l e a r that the above expression has the same form • as that given by equation 69. Happel's f r e e surface model gave e x c e l l e n t agreement i n the d i l u t e range with the ex-perimental data of McNown and L i n (88) and the t h e o r e t i c a l expressions of Smoluchowski (84, 85, 86) and Kawaguchi (91) For the intermediate range of c o n c e n t r a t i o n , l e s s favorable agreement was observed w i t h the experimental data or e m p i r i c a l equations of Hanratty and Bundukwale ( 1 1 1 ) , . Hawksley (modified) ( 1 0 9 ) , Richardson and Zaki ( 1 0 8 ) , Adler ( 1 2 7 ) , Steinour (114), Mertes and Rhodes ( 1 2 8 ) , Wilson ( 1 1 7 ) , Verchoor (129) and Happel and E p s t e i n (130),and the t h e o r e t i c a l d e r i v a t i o n of Brinkman ( 1 0 6 ) . But i t should be mentioned that the experimental data of the various workers i n t h i s range are not i n agreement e i t h e r , as i t appears that a unique r e l a t i o n s h i p between r e l a t i v e v e l o c i t y and concentration does not e x i s t i n the intermediate range of c o n c e n t r a t i o n , due p o s s i b l y to agglomeration and c i r -c u l a t i o n . For high concentrations good agreement was again obtained with the e m p i r i c a l equations of Richardson and Zaki ( 1 0 8 ) , Hawksley (modified) ( 1 0 9 ) , Carman-Kozeny (100) and F a i r and Hatch ( 1 0 3 ) . Subsequently Smith (131, 132) 36 used Happel's f r e e surface model f o r the a n a l y s i s of the d i f f e r e n t i a l s e t t l i n g of spheres of d i f f e r e n t s i z e s . Happel (133) a l s o used h i s f r e e surface model f o r a n a l y z i n g flow perpendicular and p a r a l l e l to arrays of c i r c u l a r c y l i n d e r s . His s o l u t i o n of the f u l l y developed laminar flow equation i n c y l i n d r i c a l coordinates f o r these two cases are, 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 _ (87) 4y2 - Y^ - 3 -4 l n Y -1 where y i s the r a t i o of the rad i u s of the outer f r i c t i o n -l e s s c y l i n d e r to the inner s o l i d c y l i n d e r . The con-2 c e n t r a t i o n c i s then given by Y . I t i s i n t e r e s t i n g to note that both equations 86 and 87 show zero drag f o r c e per u n i t length as Y' -> 0 , thus g i v i n g r i s e to Stokes' paradox f o r the 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 of t h e Kozeny con-stant as obtained from equations 86 and 87 with that obtained by Emersleben (95) f o r square arrays and by Sparrow and Lo 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 obtained f o r the d i l u t e range, i n d i c a t i n g that geometrical arrangement does not e f f e c t the 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 the d i l u t e range according to the various 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 are a l s o i n general agreement with the experimental data of Galloway (143) and E p s t e i n (142) and of Gunn and D a r l i n g (144). The numerical values f o r k by Sparrow and L o e f f l e r (119) at lower f r a c t i o n a l 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 free surface model. At the same time as Happel developed the fr e e surface model, Kuwabara (134) suggested that i n s t e a d of im-posing zero shear s t r e s s at the outer envelope, one can im-pose the c o n d i t i o n of zero v o r t i c i t y at t h i s outer en-velope. For the case of creeping flow past spheres,he obtained f o r the drag f o r c e .  F = 6TT y U a [ 1 + | i y + | Q Y 2 - J ^ Y 3 + |Y 4 + |§Y 5 + Y 7+ §=- Y 8 ] [ 1-Y]- 3 [ 1 + | Y + |Y 2+ |Y 3 T 2 (88) and f o r creeping flow perpendicular to c i r c u l a r c y l i n d e r s , he obtained f o r the drag f o r c e per u n i t length - In y - 0.75 + Y - Y /4 Kuwabara (134) obtained the drag f o r c e f o r the case of spheres using the 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 are obtained when one uses the pressure 38 and shear s t r e s s d i s t r i b u t i o n at the sphere surface to c a l c u l a t e P. The drag f o r c e using the l a t t e r method gives F = 30ttuU a ( 9 0 ) * 5 - 9Y + 5y d - Y Equation 90 gives a s l i g h t l y higher value f o r F than equation 88. The experimental work on flow perpendicular to 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 that Kuwabara's model represents t h e i r e x p e r i -mental data very w e l l , while Happel's f r e e surface model f o r flow perpendicular to an assemblage of c y l i n d e r s i s at variance with t h e i r data. On the other hand, Happel's model f o r spheres gives b e t t e r agreement with the a v a i l a b l e e x p e r i -mental data on flow past c l u s t e r s of spheres (126) than Kuwabara's model. 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 considerations alone to decide which of the two models i s the more r e a l i s t i c , although i t has been pointed out that Kuwabara's model, u n l i k e Happel'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 to higher Reynolds numbers. Galerkin's method was used by Snyder and Stewart ( l 4 l ) to solve the creeping flow equation f o r both a simple and a denser (orthorhombic) arrangement of spheres, and ob-tained f r i c t i o n f a c t o r s w i t h i n 5% of those obtained ex-* The reason f o r t h i s discrepancy i n F i s that 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 to the drag-producing stresses at the surface of the sphere alone,as assumed by Ku-wabara,but a l s o to the stresses on the outer envelope. 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 stacked spheres, and 9-1 % , lower than those p r e d i c t e d by Happel (126). There are s e v e r a l other formulae, many e m p i r i c a l , f o r p r e d i c t i n g the drag f o r c e , or more simply the de-v i a t i o n from Stokes flow 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) gives a review of the various formulae. 2. Review of Numerical Work I t i s evident that the s o l u t i o n of the Navier-Stokes equation v i a Stokes, Oseen and Proudman and Pearson expansions f a i l s to describe the flow f o r higher Reynolds numbers. Approximate s o l u t i o n s using the G a l e r k i n method have been t r i e d . This 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 obtained by deter-mining the t r i a l f u n c t i o n parameters such as to s a t i s f y the Navier-Stokes equation 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 having a large number of 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) and by F l u m e r f e l t and S l a t t e r y (34). The optimum t r i a l f u n c t i o n s may be determined by v a r i a t i o n a l c a l c u l u s or 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 to evaluate unknown parameters i n h i s assumed stream f u n c t i o n s f o r flow past a sphere i n an unbounded f l u i d . His recommendations are that one should 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 t n e r e s u l t s from another stream f u n c t i o n at 10< Re < 70 • Without p r i o r knowledge of the answer, however, t h i s s l i c i n g could not be accomplished. Hamielec and 4o Johnson (37), Hamielec et a l , (38) and. Nakano and Tien (hO) used t h i s technique to evaluate the viscous flow around f l u i d spheres at intermediate Reynolds numbers. A f l u i d sphere w i t h a . v i s c o s i t y approaching i n f i n i t y i s , of course, a r i g i d sphere. F l u m e r f e l t and S l a t t e r y (3*0 presented an improvement on the G a l e r k i n method using an a u x i l i a r y v a r i a t i o n a l technique given by S l a t t e r y (39). Such methods are not completely r e l i a b l e , however, as they depend l a r g e l y on the choice of the t r i a l f u n c t i o n . Bourot (137, 138) employed the method of l e a s t squares f i t t i n g of the biharmonic f u n c t i o n to known boundary co n d i t i o n s f o r creeping flow of a sphere i n a c y l i n d r i c a l envelope. Bourot (136) developed the method f o r a spheroid, but d i d not perform the numerical work. This technique r e q u i r e s the general 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 that the outer 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 flow i n ducts and f o r flow 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 used by Thorn (42, 43, hh) to .solve the Navier-Stokes equation f o r flow past c y l i n d e r s at Re = 10, and h i s method has been used by Kawaguti (35) f o r flow past spheres at Re = 20 and c y l i n d e r s (46) at Re = 40. This method i s very tedious and i t was converted i n t o a r e l a x a t i o n method by Fox (47, 48, 49), by A l l e n and Dennis (53) and by Southwell et a l . (50). The flow past c y l i n d e r s was solved by A l l e n and Southwell (54) 41 f o r Re up to 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 of t h e i r method to spheres f o r Re up to 20. Kawaguti (52) i n d i c a t e d that the work of 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 the r e l a x a t i o n method i s the approximation of the d i f f e r e n t i a l equations by t h e i r equivalent d i f f e r e n c e equations, which r e l a t e the values of the f u n c t i o n and the d e r i v a t i v e s i n v o l v e d at a -point to those of neighbouring l a t t i c e p o i n t s . Jenson (55), i n h i s work on spheres, solved the Navier-Stokes equation by s p l i t t i n g i t i n t o two s i m u l -taneous second order equations, using the stream f u n c t i o n and the v o r t i c i t y . Jenson approximated the v o r t i c i t y near the surface by a t h i r d order polynomial which could a l s o be derived by using Taylor's expansion c o r r e c t to Re t h i r d order, and he used an exponential transformation f o r r a d i a l distance from the surface. He obtained the stream f u n c t i o n , v o r t i c i t y , pressure v a r i a t i o n at the. sur-. face and drag c o e f f i c i e n t f o r Re up to 80 (56) Rhodes (57), L e C l a i r and Hamielec (58) and Hamielec et a l (59, 60) extended Jenson's technique with the a i d of d i g i t a l com-puters to inc l u d e higher 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 con 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 high to a c c u r a t e l y represent an i n f i n i t e medium. Hamielec and Raal (6l) adopted Jenson's technique 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 to 500, assuming that the steady s t a t e form of the Navier-Stokes equation s t i l l holds at t h i s Reynolds number. They were c a r e f u l to consider 42 the parameters that a f f e c t the accuracy of the numerical s o l u t i o n and. they have obtained the most accurate numerical values of t h e drag f o r c e to date. Several other workers (62, 63, 64) have obtained' s o l u t i o n s of the unsteady s t a t e form of the Navier-Stokes equation 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 flow along a f l a t plate'has been obtained by Dennis and Dunwoody ( 4 l ) . * * .** A recent paper by Rimon and Lugt (164), published a f t e r the present manuscript was completed, deals with the p a r t i c u l a r case of an oblate spheroid, performing s i m i l a r c a l c u l a t i o n s to the present work using a time-dependent numerical technique. Their r e s u l t s , which were l i m i t e d to Re = 10 and Re = 100, agreed w e l l with the corresponding r e s u l t s i n t h i s t h e s i s . 43 CHAPTER I I I Formulation of F i n i t e D i f f e r e n c e Equations and Boundary Conditions. The choice of s p h e r o i d a l and e l l i p t i c a l co-ordinates i n t h i s study f a c i l i t a t e s the fo r m u l a t i o n of the boundary c o n d i t i o n s . In a d d i t i o n , the exponential p r o p e r t i e s of these coordinates give a f i n e l a t t i c e near the surface of the p a r t i c l e i n general and near the t i p s i n p a r t i c u l a r , and a coarse l a t t i c e f a r away from the surface. As the i n f l u e n c e of the p a r t i c l e on the. flows i s mainly manifested near the su r f a c e , e s p e c i a l l y around the tips', and as t h i s i n f l u e n c e decreases w i t h distance from the su r f a c e , the l a t t i c e thus produced served the purpose of 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. Spheroids A..1 Formulation of the F i n i t e D i f f e r e n c e Equations The Navier-Stokes equation f o r steady s t a t e flow as expressed i n orthogonal c u r v i l i n e a r c o o r d i n a t e s , equation 1, reduces to VF'V- ( coshg s i n n r 1 . 5(V ,E'V ) c (smh C+cos n ) 2 E* ifrtcosh g s i n n )" ^ 3 ,C cosh g s i n n ) c 4 (sinh 2S+cos 2n) 9(£>r,) and (91) 44 12 2 2 E = —5 5 =— { — T T - - tanh £ l - = - + — = r - c o t n -|— } c ( s i n h ^ + c o s n) H 2 5 3 n 2 9 n (92) f o r oblate s p h e r o i d a l coordinates, the metric co-e f f i c i e n t s f o r these coordinates being given by c { s i n h £+cos n} and h, = 1. c . (94) 3 c cosh E, s i n n where c i s a c h a r a c t e r i s t i c length of the coordinate system. The minor semi-axis, b, and the major semi-a x i s , a, of the spheroid are given by a = c cosh E, (95) b = c sinh £ (96) £ Q being the value of K at the surface of the spheroid. The f o l l o w i n g dimensionless q u a n t i t i e s are now defined: ib = IJJ'/U a 2 , Re = 2a U/v and E 2 = c 2 E* 2 , where Re i s the Reynolds number and U i s the v e l o c i t y of the undisturbed stream. The Navier-Stokes equation then becomes sech E E 2 ( E 2 ^ ) = ge t cosh g s i n n ] C | i |_ { £ ± } a d sinh 2 C+cos n 5 n c o s h ^ s i n ^ cosh C s i n n 45 with „2 1 r d 2 , . r 3 • 3 3 . 3 T. ( 9 8 ) E = - = — { — - t a n h £ ~-=- + — ~ - cot n ~— } s i n h 2 ? + c o S 2 n 3 ? 2 3 ? 3 n 2 3 n I t i s u s e f u l to s p l i t equation 97 i n t o two equations, one p i n terms of if ^  and n and the other i n terms of E \p, E, and n to give . _ „ 2 / , r . >. Re r coshg s i n n n sech £ E ( t, cosh £ s i n n ) = p— L p2 p - J ' a s i n h € +cos n r 3ifr 3 r g 1 3ip 3 : r C i f q q N L 3g 3n c o s h g s i n n 3n 3? ^  cosh £ sinn and E2\\J = r, cosh £ s i n n s e c h 3 g a ( 1 0 0 ) where £ i s the dimensionless' v o r t i c i t y defined as t c, = K a/U,the dimensional v o r t i c i t y being £f = h 3 E'V ( 1 0 1 ) For a spheroid which i s almost s p h e r i c a l , cosh 5 becomes very large 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 f u n c t i o n s are introduced: G = ? cosh g s i n n /cosh £ ( 1 0 2 ) P = 5 cosh£ / cosh £sinri ( 1 0 3 ) 3, 4 6 Equations 99 and 100 then become Re sech £ a.r cosh g s i n n 3^ 9F 3ifr 3F 2 s i n h 5+cos n 2 J L 3? 3n 3n 3C (104) and (105) r e s p e c t i v e l y . The s o l u t i o n of the above two s i m u l -taneous d i f f e r e n t i a l equations i n the stream f u n c t i o n and i n the modified v o r t i c i t y G could be achieved by using Jenson's r e l a x a t i o n technique (55). This tech-nique i n v o l v e s approximating 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 equations which r e l a t e the values of neighbouring p o i n t s on a flow g r i d . The s o l u t i o n of these equations i s followed 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 . 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 Taylor's s e r i e s expansion c o r r e c t to the second order, one obtains f o r a f i e l d f u n c t i o n Q : Considering l a t t i c e spacing A i n the ^ - d i r e c t i o n 32Q = Q(I,J+1) + Q(I,J-1) 2Q(I,J) (106) 47 Figure 1. Oblate sp h e r o i d a l mesh system. 48 92Q = Q(I+1,J) + Q(I-1,J) 2Q(I,J) ( 1 0 ? ) 9n 2 B 2 B 2 9Q _ Q(I,J+1) - Q(I,J-I) ( R. 9£ 2 A U U O j _9Q = Q(I+1,J) - Q ( I - l . J ) (109) 9n 2 B Using equations 106 - 109, the d i f f e r e n t i a l equation 104 and 105 take the 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 + 2/B 2) = G(I,J+1) [ (2 - A tanh C ( J ) ) / 2 A 2 ] + G ( I } J - 1 ) { 2 + A ? t a n h S ( J ) > + G ( I + 1 , J ) { 2 - B cot n ( I ) } 2 A 2 B + G(I-1,J){ 2 + B C g t n ( I ) } - 5s- sech 5 cosh 5(J) s l n n ( I ) 2 2 ^ r [t|»(I,J+l) - i K l , J - l ) ] [ F ( I + l , J ) - F ( I - 1 , J ) ]  1 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+2/B 2) = iJ;(I,J+l) 2 - A tanh g(I) 2 2 A + i|>(I,J-l) 2 + A tanh g ( J ) + i p ( I + l , J ) 2 - B cot n ( D 2 2 A + Tp(I-l, J ) 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 equations f o r an oblate spheroid, and t h e i r s o l u t i o n f o r given boundary c o n d i t i o n s would f u r n i s h values of the stream f u n c t i o n and the. v o r t i c i t y at each poi n t of the g r i d i n the flow f i e l d . Equations 110 and 111 reduce to those obtained by Jenson (55) f o r a per-f e c t sphere when £ ->- <» equations could be obtained from those of an oblate spheroid by r e p l a c i n g each sin 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 the computational point of view t h i s i s e x a c t l y equivalent to changing each sinhg 2 2 to coshg , coshg to sinhg and cos n to - cos n • By t h i s procedure the term a w i l l then represent the length of the minor semi-axis and the Reynolds number w i l l be based on the minor a x i s . For both p r o l a t e and oblate spheroids For a p r o l a t e spheroid, the f i n i t e d i f f e r e n c e 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 Conditions The numerical a n a l y s i s i s confined between an inner surface represented by the spheroid i t s e l f and by an outer envelope which, l i k e the s p h e r o i d a l s u r f a c e , coincides with one of the s p h e r o i d a l coordinate g r i d l i n e s as shown i n f i g u r e 2. As the two flow equations are of second order, four boundary co n d i t i o n s are to be s a t i s f i e d . For an oblate spheroid the boundary c o n d i t i o n s f o r are: f o r n = 0 along BA , $ = 0 n = ir along NO , ^ = 0 a x i s of symmetry £ = £ along AMN, ^ =.0 | surface of spheroid cl At the outer boundary, £ = £^ the flow i s assumed to be T 2 2 2 a streaming p a r a l l e l f l ow, g i v i n g ^ = ± s i n n cosh r sech E, , 2 D a The boundary co n d i t i o n s f o r r, are : f o r n = 0 along BA , r, = 0 n = TT along NO , £ = 0 a x i s of symmetry 2 2 E \p cosh £ £ = S a along AMN , ? = a a ° ' s i n n 5 = 5 b along BRO , ? = 0 case I £ = ? b along BRO , c = C _ 0 case I I Figure 2. Oblate spheroid with i t s outer envelope. The boundary c o n d i t i o n f o r C at E, - E, , 3. which o r i g i n a t e s from equation 1 0 0 , can be as f o l l o w s : Near the surface of the spheroid i t i s assumed that the stream f u n c t i o n can be f i t t e d by a cubic equation having the form i> = a (? - Ka )2 + $ U - ? a ) 3 (112) The above equation s a t i s f i e s the n o - s l i p boundary con-d i t i o n s assuming that the constant value of \b at the sur-face i s zero f o r any value of the constants a and 3 . Equations 98 and 105 combined give 2 c o s h £ a 9 2 . 3 2 , 9 . G = — _ _ { — % . _ tanhCTr + — £ - cot n * (113) sinh^ + c o s n 8^ 5 3n 2 3 n At the s u r f a c e , the n o - s l i p conditions noted above r e q u i r e that Sn 3 r ) 2 35 and equation 113 t h e r e f o r e reduces to G(I , 1 ) = cosh 5 a 2 ' 2 s i n h £ +•cos n H 2 (114) a Where J = 1 i s the value of J at the surface of the spheroid, D i f f e r e n t i a t i n g equation 112 twice with respect to E, y i e l d s . 2 •2-4 = 2a + 6g ( 5 - ? ) (115) 35 a Hence at £ = £ , 9, as 2 2a (116) Introducing equation 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 at J = 2 and J = 3 i n equation 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 according to Taylor's s e r i e s to the t h i r d order, gives a = 8 * ( I > 2 > I » ^ > 3 ) • (118) 4 A and equation 117 becomes cosh2£ [ 8 i f ( I , 2 ) - \|»(I,3) 1 G(I,1) = a — , ; (119) 2 A 2 (sinh2£ +cos 2n ) : . a . which i s equivalent to cosh2£ [ 8 i K l , 2 ) - <Kl,3) 1 ? ( I , 1 ) = a ? = : ( 1 2 0 ) 2 A ( s i n h L+cos n) s i n n a This equation gives the value of £ at the surface of the spheroid (£ = £ ) and could a l s o be a r r i v e d at by ex-a panding d i r e c t l y as a T a y l o r . s e r i e s expansion" c o r r e c t tc the t h i r d order, as subsequently shown f o r e l l i p t i c a l c y l i n d e r s . The boundary c o n d i t i o n for . £'at £ = ? b f o r Case I I which corresponds to Happel's boundary c o n d i t i o n s , can be derived as f o l l o w s : The dimensional v o r t i c i t y , obtained by expanding equation 101, i s given by r ' _ i_ y, r3 r h l h 3 3^* 3 r h 2 h 3 3^' . . c - h , h 2 { ^ C—u^- af 3 + t—hr 3TT ] } ( 1 2 1 For the oblate spheroids h1 = h 2 and equation 121 becomes h?{lr[h> If ] + (122) or 2dhi 3^ + h i f ' (123) The dimensional shear s t r e s s i s given (10) by T • " h7 Ii < If'" - * 177 In -1 h '" ' h ' If' ( 1 2 M For the oblate spheroids, h t= h g and equation 124 becomes 55 5n - 2 h l h 3 ^ It- - h 2 9 h 3 ^ / 95 95 1 95 95 hi h 3 2 » H 2 ;- 9 h r ~ , • 0 9 h 3 ~ . » 0 . 2 . i j.o u v, 1 9^ , , 2 3 9^ , , 2 , 9 ib 1 d 9 n 9n 1 9n 9n 1 3 2 on (125 ) S e t t i n g = 0 and adding equation 125 to equation 123 y i e l d s , 2 , r „ ah, „,« 9h , _ , f TC = 0 5n o *, 2 . d .ib , „ , 2 3 9i|> . 0 . . = 2 h, h 3 — f + 2 h x 9 — 9 ^ + 2 h xh 3n 3 3n 3n 3 h , . . 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 9 4 , » 3 2 2 T5n= ® c cosh 5 s i n n (sin h 5+cos n) 3 n 2 _3ij/ 9n ' ... . "\'2L' " 2 9n s i n h 5 +cos n _ C o t n 1 ^ - + c o s n s l n n s i n h 5 cosh 5 ^ 1 ' • v 2 r , 2 95 s i n h 5^ +cos n (126 ) At 5 = 5^  » the stream f u n c t i o n i s f 1 2 2 2 if)' = 75- U c cosh £ b s i n n (127) D i f f e r e n t i a t i n g w i t h respect to n y i e l d s . _Oi -:3n 2 2 U c cosh £ b s i n n cosn (128) and 2 » 2 2 2 2 U c cosh £ b ( cos n- s i n n ) (129) S u b s t i t u t i n g equations 128 and 129 i n t o equation 126 and rendering a l l q u a n t i t i e s dimensionless y i e l d s w 0 2 cosh — 2 2 s i n n ( s i n h g^+cos n) - cosh £ b s i n n 2 2 cosh £ b cos n s i n n 2 2 s i n h £ b + cos n sinh £ b cosh g b, ^ sinh £ b + cos n (130) and by equation 102, G n T C n = 0 2 cosh £ b 2 2 ^ (sinh £b+cos n) \ 2 - cosh s i n n 2 2 cosh £, cos n s i n n + ;—: P_ 2 2 s i n h £b+cos n 2 s i n h cosh £ 2 2 s i n h g^+cos n 3jjj_ 9? (131) value of | | r 5=5 b 57 Equation 131 gives the modified, v o r t i c i t y G at the outer boundary when the shear s t r e s s at t h i s boundary i s equal to zero, i . e . the envelope i s f r i c t i o n l e s s . The can be found by using Newton's d i f f e r e n t i a t i o n formula. Again the boundary c o n d i t i o n s f o r a p r o l a t e spheroid could be obtained by r e p l a c i n g sinh 5 with i cosh 5 and cosh 5 with i sinh 5 • The boundary c o n d i t i o n f o r 5 = 0 at 5 = 5 , b as given by case I , was used f o r the study of flow past i s o l a t e d spheroids. For the c e l l models used to represent swarms of spheroids, case I r e f e r s to Kuwabara's zero v o r t i c i t y model, while case I I r e f e r s to Happel's f r e e sur-face model. Both were used f o r the study of flow through swarms of spheroids. B. E l l i p t i c a l C y l i n d e r s B . l Formulation of the F i n i t e D i f f e r e n c e Equations The Navier-Stokes equation f o r steady s t a t e flow as expressed i n orthogonal c u r v i l i n e a r c o o rdinates, equa-t i o n 1 ., reduces to v. E'V = — i ^ i ^ ) ^ > ( 1 3 2 ) ' c ( sinh 2 5 + s i n 2 n ) ' d {^" T ] ) and E ' 2 = - 2 \ — { — 2 + — 2 > ( 1 3 3 ) c ( s i n h 5 + s i n n) 95 9n 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 rdinates, with the un-di s t u r b e d flow being p a r a l l e l . t o . the x- or. major a x i s . The metric c o e f f i c i e n t s f o r t h i s coordinate system are given by Happel and Brenner (8) as , hj = h 2 = p — ^ p YPT (134) c [ sinh^S + cos n ] ' h 3 = 1 where c i s the c h a r a c t e r i s t i c length of the coordinate system. The minor semi-axis, b, and the major semi-a x i s , a, of the e l l i p t i c a l c y l i n d e r are given by a = c cosh (135) a. b = c sinh E (136) Ct E 0 being the value of £ at the surface of the c y l i n d e r . The f o l l o w i n g dimensionless q u a n t i t i e s are now define d : \|> = /Ub , E 2 = c 2 E' 2 and Re = 2bU / v , where Re i s the Reynolds number and U i s the v e l o c i t y of the undisturbed stream along the major a x i s . The two-dimensional flow: Is; then governed by : 2_ E \ = 1 , : { 3 i 9 E f i _ 91 9 E 2 ! } ( 1 3 ? ) Re fi w . . 2 r x . 2 l3£ 3n 3n 9? ' K ± 5 { ) s i n h E+sm n . and 2 • '2 E 2 = p-{ ^ -+^-5-} ; (138) sinh E+sin n 3? •. 3n • 59 Equation 137 can be s p l i t i n t o two equations, one i n terms of y 5 and n and the other i n terms of £ , £ and n to give ' 2 s i n h 2 ? + s i n 2 n 3« ^ ^ 3* U 3 9 ) and E2TJ, = x, / s i n n 2 5 a (140) where z, i s the dlmensionless v o r t i c i t y defined as ? = ? b/U, the dimensional v o r t i c i t y being h E'V = E'V (141) The s o l u t i o n of the two simultaneous equations, 139 and 140, i n the stream f u n c t i o n and i n the v o r t i c i t y z, could be achieved by using Jenson's r e l a x a t i o n technique (55) as f o r the case of spheroids. Considering l a t t i c e spacing A i n the n - d i r e c t i o n and B i n 5 - d i r e c t i o n , as shown i n f i g u r e 3, and using Taylor's s e r i e s expansion c o r r e c t to the se-cond order as given by equation 106 - 109, the d i f f e r e n t i a l equations 139 and 140 take the f o l l o w i n g d i f f e r e n c e forms: G(l,J)(2/A 2+2/B 2) = G(*»J+1)+G(I.J-1) , 0(1+1,J)+0(1-1,J) A 2 B 2 - Re \ r ^ { I , J + Y ) - t y { I ^ - x ) i r G ( I + l , J ) - G ( I - l , J ) n 2 [ L 2 A J L 2 B J _ r ^ ( I + l , J ) - i J ; ( I - l > J ) G(I,J+1)-G(I,J-1)-,1 2 B J L 2 A J J (142) and ^ ( I , J ) ( 2 / A 2 + 2 / B 2 ) = <HI,J+1WI,J-1) A , ^(1+1, J ( 1 - 1 , J) G(I,J) (sinh2£(J)+sin2n (I) 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 c y l i n d e r s G = r; . For flow p a r a l l e l to the minor a x i s , the f i n i t e d i f f e r e n c e equations could be obtained from equations 142 and 143 by r e p l a c i n g each sin h £ by i cosh E and each cosh E w i t h i s i n h E . By t h i s procedure the term b w i l l now represent the major semi-axis and a the minor semi-a x i s , but the Reynolds number i s again based on the pro-j e c t e d length perpendicular to the flow d i r e c t i o n . The s o l u t i o n of the above f i n i t e d i f f e r e n c e equations f o r given boundary co n d i t i o n s r e s u l t s i n the knowledge of the flow d e t a i l s . When E+°° , equations 142 and 143 become i d e n t i c a l to those obtained by Hamielec and Raal (6l) f o r a c i r c u l a r c y l i n d e r w i t h the exponential transformation £-5 a z r - e a s e where r i s the r a d i a l d i stance f o r the c y l i n d r i c a l coordinates and z the r a d i a l distance i n t h e i r transformed coordinates. B.2 Boundary Conditions The computational f i e l d i s confined between an inner surface (the e l l i p t i c a l c y l i n d e r ) and an outer en-velope which c o i n c i d e with the inner and outer e l l i p t i c a l coordinate 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 four boundary c o n d i t i o n s d e f i n i n g the flow are given by having two boundary c o n d i t i o n s f o r each of ^ and C. For flow along the major a x i s the boundary con-d i t i o n s f o r are: f o r n = 0 along BA , i|) = 0 a x i s of symmetry n = ir along NO , if = 0 E = E & along AMN, if = 0 | surface of c y l i n d e r At the outer boundary, £=5 b , the flow i s assumed to be streaming p a r a l l e l f l o w , g i v i n g ^ = s i n n sin h E b cosech E a The boundary c o n d i t i o n s f o r C are: for n = 0 along BA , z, = 0 n = TT along N , z, = 0 a x i s of symmetry E = E 0 along AMN, z, = E 2if s i n h 2 E „ a a E = E b along BRO, ? = 0 case I E = E b along BRO, z, = z, Q case I I T En The boundary c o n d i t i o n f o r z, at E = £ can be a expanded as f o l l o w s : The v o r t i c i t y z, i s given by equation 140, which when com-bined w i t h equation 138 leads to sinh 2E„ ~2. .2 a 2 2 2 2 s i n h E+sin n 9E 3n { 3_1 + 9_4 } (144) At the surface of the c y l i n d e r , if i s a constant f o r a l l .2 values of n , l e a d i n g to = 0, so that equation 144 becomes 3n 64 s i n h E a 2 2 s i n h 5 + s i n n a o 2 ^ 3 5 2 (145) 5 = 5. The s t r e a m f u n c t i o n n e a r t h e s u r f a c e c a n be e x p r e s s e d as T a y l o r ' s s e r i e s e x p a n s i o n s t o t h e t h i r d o r d e r : 5 = 5. 2 2 A d j | j 2 d 5 2 5 = 5. , A 3 d 3 ^ (146) 5 = 5, i ( ; ( 5 a+2A ) = ^ ( 5 a ) + 2A | | 5 = 5 + 2 A 2 d5 a 5=5, + | A 3 m F d 5 3 (147) 5=5, A t t h e s u r f a c e o f t h e c y l i n d e r , ^ = f|" = 0 J w h i c h s a t i s f i e s t h e n o - s l i p c o n d i t i o n a t t h i s s u r f a c e . C o m b i n i n g e q u a t i o n s 13,,.. y i e l d s 5 = 5„ 146 and 147 t o e l i m i n a t e d — | d 5 3 M5 +A) - M5 + 2A) = 2 A 2 ^-4 a a d5 (148) 5 = 5. E l i m i n a t i n g —^!=j-d5 f r o m e q u a t i o n 145 by u s i n g t h e above 5 = 5 a e q u a t i o n y i e l d s ? ( I 5 D = s i n h 2 5 a 2 A 2 [ s i n h 2 5 + s i n 2 n ( D ] 3, [ 8 ^ ( l , 2 ) - ^ ( I , 3 ) ] (149) The same r e s u l t c o u l d be o b t a i n e d by u s i n g t h e method o u t l i n e d p r e v i o u s l y f o r t h e s p h e r o i d . The b o u n d a r y c o n d i t i o n f o r £ a t 5 = 5 ^ f o r c a s e I I 65 which corresponds to. Happel's fr e e surface model, can be derived as f o l l o w s : Equating the shear s t r e s s given by equation 124 to zero y i e l d s where h 3 = 1 and h1 = h 2 . Introducing the values of dh1 9hj hi , - j r — and -^r— obtained from equation 134 i n t o the on 95 above equation gives 2 2 9 y _ 9 y 2 s i n h 5 cosh 5 ,9i[> 2 cos n s i n n 9^  s ~ r2 „ 2 " . . . 2 r x . 2 95 . u 2 r . . 2 9n U 5 i ; 95 ori sinh 5 + s m n sinh 5 + s i n n At the envelope s u r f a c e , 5=5 , the dimensionless stream f u n c t i o n i s given by \p = sinrisinh5L 0 cosech ^ (152) Using equation 152 to ob t a i n |1 and i _ | then 9 n 9n2 2.,. s i n n 'sinh5^ 2 s i n h 5 b cosh 5 b 9 ijj 3S2 2 2 95 r _ r s i n h 5 s i n h 5 K + s i n n s — a D 5=^ 2 2 cos n. s i n n .sinh 5K + 2 2 " ( 1 5 3 ) s i n h 5^+sin n The v o r t i c i t y at 5 = 5 b as obtained from equation 144 i s 2 s i n h 5 *>2. .2, r _ a r 9_JJ. . 9_1 -, ' ^£ = £ 2 2 2 2 ^ ^b si n h 5 b+sin n 9n 95 (154) ^ b Combination of equations 153 and 154 leads to 66 'x r = 0 2 sinh E sinh E, . a . 2 2 sinh E^+sin r| - 2 sin.n cos T) s i n n 2 2 sinh E^+sin n cosh E, sinh E,. ^ , ^b _b 3jp_ . .2- . . 2 3E sinh E^+sin n ^ £=£> ( 1 5 5 ) The value of | 1 5 = 5v i s found by using Newton's formula f o r d i f f e r e n t i a t i o n . The above expression, equation 155> gives ' e x p l i c i t l y the v o r t i c i t y at the outer boundary when the shear s t r e s s at t h i s boundary i s zero. Once again the boundary co n d i t i o n s f o r flow along the minor a x i s could be obtained by r e p l a c i n g each s i n h E with i coshg and cosh E wi t h i sinh E • Since the two simultaneous d i f f e r e n t i a l equations to be solved are of second order, four independent boundary conditions were s u f f i c i e n t to provide a unique s o l u t i o n i n each case. CHAPTER IV 67 Formulation of Drag C o e f f i c i e n t Equations and the Pressure D i s t r i b u t i o n The d e r i v a t i o n of the drag c o e f f i c i e n t equations f o r an oblate spheroid w i l l be given i n d e t a i l as i t w i l l serve as an example of the general method. A. Spheroids G o l d s t e i n (10) gives the dimensional shear s t r e s s i n general c u r v i l i n e a r coordinates as T E n = T n £ = y h hi 2 L_ (h, V J + l l l ' L - ( h i Yl ) 9E 71 h 1 3n ^ (156) where v^ and v^ are the dimensional v e l o c i t i e s i n the n - a n d E - d i r e c t i o n s r e s p e c t i v e l y . At the surface of the spheroid, the n o - s l i p c o n d i t i o n s give 9vl . . = v = Vj. '= 0 9n - n and equation 156 becomes En 9v = F = v n 1 K =E a 3E (157) The v o r t i c i t y i n general c u r v i l i n e a r coordinates i s de-f i n e d as 1 = Vxv = hjhjh. h. 3E v, h. "n h. 9 9n h. -<J> h. 9_ 9<f> v h. (158) 68 For axisymmetrie and two-dimensional flows a l l de-t r i v a t i v e s i n 0 are zero and v^ =0. The only non-vanishing v o r t i c i t y i s that i n the 0 - d i r e c t i o n and i s _» t given by z, = i z, , where ? ni L 8?. n 2 8Tf 5 ] (159) and hj = 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 co-ordi n a t e s . Using the n o - s l i p condition at the surface (5 = 5 . ) , equation 159 becomes a , 8v c' =h — ^ ^5=5 n i 85 5=5 (160) a S u b s t i t u t i n g equation 157 i n t o equation 160 y i e l d s 5n = y S£=5 5=5 K ^a ^a (161) Making a l l terms dlmensionless by p u t t i n g t r = T ' /2pU 2-5n 5n and z, = . s'a/U > the dlmensionless l o c a l shear s t r e s s at the surface i s then given by 5n = 4 c ? _ r / Re 5='5a (162) The surface of the oblate spheroid f o r z = 0 i s given by y x with 2 2 c sin h 5 a 2 2 c cosh 5 and y = c sin h 5 cos n cL x = c cosh 5 s i n n a (163) (164) (165) 69 The surface area of the s t r i p on which the shear s t r e s s i s considered to act i s 2ir x ( d x 2 • + d y 2 ) 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 area of the oblate spheroid. The drag force i n the d i r e c t i o n of the streaming flow a c t i n g on the whole surface due to shear s t r e s s alone i s then given by P = 2TT x ( d x 2 + dy 2 ) 1 / 2 cos e (166) 5 = 5 a The drag on a body i s u s u a l l y expressed in.terms of a dimensionless drag c o e f f i c i e n t drag force i n the flow d i r e c t i o n 'D (Projected area normal to the) ( d i r e c t i o n of flow ) x k i n e t i c pressure From equation 166 the s k i n drag c o e f f i c i e n t then becomes (167) 1U1 2TT x (dx 2 + d y 2 ) 1 / 2 T^|g = g a C Q S DS (TT C 2 c o s h 2 £ ) ( |p U 2 ) 70 Now cos 6 = — 2 1 / 2 ( d x 2 + d y 2 ) 1 7 2 dx = c cosh 5 cos n dn , a dy = - c sinh £ s i n n dn , .a. and making use of equation 162 and equation 167 y i e l d s CDS = I? t a n h ? a / s i n 2 n C 5 = C d n ( l 6 8 ) The above equation gives the s k i n ( f r i c t i o n ) drag co-e f f i c i e n t f o r an oblate spheroid. In the l i m i t i n g case of sphere, tanh '£ • i s u n i t y and equation 168 becomes a i d e n t i c a l to that given by Jenson ( 5 5 ) f o r a p e r f e c t sphere. For 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 equation 168 shows a zero s k i n drag c o e f f i c i e n t , i n conformity w i t h simple geometrical con-s i d e r a t i o n s . For a p r o l a t e spheroid, tanh? a i s t r a n s -formed as usual to coth £ , but i t then becomes d i f f i c u l t a to p r e d i c t C^g i n the l i m i t of a very long needle, f o r which Cr_ r 0 and coth B, -* CD . ^ a a The pressure d i s t r i b u t i o n around an oblate spheroid i s obtained by considering, the i ^ and i ^ d i r e c t i o n s of the Navier-Stokes equation. The Navier-Stokes equation as given by equation 3 can be transformed (8) to 71 - ' 1 - - l - - ' 2 = v x K = i V P + ^ V v + v V x £ P . 2 ( 1 6 9 ) by u s i n g s i m p l e v e c t o r 1 i d e n t i t i e s and t h e c o n t i n u i t y t e q u a t i o n , V.v = 0 . L e t us now c o n s i d e r t h e i component o f e q u a t i o n 1 6 9 t e r m by t e r m . By d e f i n i t i o n , we h a v e v x z = h h h 1 2 3 V h l i n / n 2 ± / h 3 V, V V ! t t r, r/h ? /h £,/h ( 1 7 0 ) As m e n t i o n e d p r e v i o u s l y f o r a x i s y m m e t r i c and t w o -d i m e n s i o n a l f l o w s , zl ~ g' = 0 P u t t i n g = c , t h e t e r m i n t h e i - d i r e c t i o n on t h e l e f t - h a n d s i d e o f e q u a t i o n 1 6 9 i s - h ^ The f i r s t and s e c o n d t e r m s on t h e r i g h t - h a n d s i d e o f e q u a t i o n 1 6 9 y i e l d h* 3P' + I h ^ p 3n 2 " 2 3n where V i s d e f i n e d by e q u a t i o n 4. By d e f i n i t i o n , we have V x ? = h ! h 2 h ; 9/9E i /h„ n 2 9/9n 1 / h 3 9/34) 4 / h i C n / h 2 ( 1 7 1 ) and t h e t h i r d t e r m i n t h e i - d i r e c t i o n on t h e r i g h t - h a n d n s i d e o f e q u a t i o n 1 6 9 becomes -v h 1 h 3 | I ( ? V h 3 ) 72 The i^-component of equation 169 thus becomes h • 1 -' 2 v n i n 3 3£ (172) _ i 2 1 At the su r f a c e , v =0 , = 0 and f o r an oblate spheroid, hj = h 2 , so that equation 172 becomes 9P 9n 5=5 = y { 95 a 5=5, 3 35 ( l / h 3 ) } (173) Introducing the dimensionless pressure P = 1 t P - P . st__ (173a) s u b s t i t u t i n g f o r h 3 and rendering a l l terms dimensionless, then equation 173 becomes Re 9P T~ 9n _ 3? 5=5. 95 + C r_ p tanh 5 5=5 4 a a (174) I n t e g r a t i o n of equation 174 along AS of f i g u r e 2 gives n 4 I <• H 0 P " P ° " Re I [ 95 + ? r _ r tanh 5 Q ] dn (175) 5=5a ? _ 5 a a where P Q i s the dimensionless f r o n t a l stagnation pressure (at n = 0) and P i s the dimensionless pressure at point S on the surface of the spheroid. The dimensionless stagnation pressure P c i s ob-tai n e d by c o n s i d e r a t i o n of the i^-component of equation 169. Using '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 side of equation 169 Is 'h-2v^  c' , since once again ^ = ^' and ^ =' 5^ = 0 f o r axlsymmetric and two-dimensional flows. The f i r s t and second terms on the right-hand side of equation 169 y i e l d f o r the i ^ - d l r e c t i o n h ' -' 2 ~ dK 2 1 H The l a s t term of equation 169 gives h 2 h 3 7 ^ - ( ? / h 3) . The i ^ .-component of equation 169 thus becomes h*V,B r l r + I h i H ' + v h* h 3 In ^'/h, ) (176) Considering flow along the a x i s BA-shown i n f i g u r e 2 gives r v =0 (no-cross flow) n t v. =0 ( a x i a l symmetry) _«2_.2 _,2 _.2 _ i 2 Hence v = vc + v + v, = v r , 5 n <p 5 and equation 176 along n = 0 becomes 3 v £ , 3P (177) where a l l q u a n t i t i e s have been rendered dlmensionless and the values of h , , h 2 and h 3 f o r an oblate spheroid have been s u b s t i t u t e d . I n t e g r a t i n g along BA y i e l d s B dC (178) n=0 74 As point B l i e s on the outer boundary and point A on the surface of the spheroid (stagnation p o i n t ) , i t f o l l o w s r e s p e c t i v e l y that P B =0, v B = 1 and P A = P D, v A = 0. Equation 178 therefore becomes E, J U cot n+ f^ - ) d£ (179) p ° = 1 + Si" / U c o t n + In" ) ^a Along t he a x i s n = 0, cotn = °° and t, = 0, using L* H o s p i t a l ' s r u l e , equation 179 becomes EVj d£ (180) P = ! + §_ fn F ° 1 Re J ~r\ n=o Equation 180 gives the dimensionless f r o n t a l s tagnation pressure and equation 175 gives the dimensionless pressure d i s t r i b u t i o n around the surface of t h e oblate spheroid . I t i s i n t e r e s t i n g to note that P becomes anti-symmetrical about n =TT/2 as Re + o and that the unityterm i n equation 180, which a r i s e s from the i n e r t i a l term of the. Navier-Stokes equation, does not appear when the i d e n t i c a l a n a l y s i s i s made on the creeping flow equation. Using diagram 1, the form (pressure) drag co-e f f i c i e n t i s given by 2TT x ( dx + dy ) ' s i n 6 P c D p = - L -2 - j - . (181) TT c cosh E a 75 2 2 1 / 2 where s i n 0 = dx / ( dx + dy ) and at the Surface, x = c cosh £ s i n n £1 dx = c cosh £ cos n dn , 3. rendering equation l 8 l TT C D p = J P s i n 2 n dn ( 1 8 2 ) 0 The t o t a l drag c o e f f i c i e n t i s obtained by adding the s k i n and form drag c o e f f i c i e n t s . The c o r r e s -ponding equations f o r a p r o l a t e spheroid could be derived by the usual transformation. B. E l l i p t i c a l C y l i n d e r s F o l l o w i n g the a n a l y s i s given f o r an oblate spheroid, equation 1 6 2 i s a l s o a p p l i c a b l e to an e l l i p t i c a l c y l i n d e r , f o r which the shear s t r e s s at the surface i s t h e r e f o r e T £n = 4 ? /Re ( 1 6 2 ) The surface of the e l l i p t i c a l c y l i n d e r at z = 0 i s given by 2 2 2 y 2 + = 1 d 8 3 ) c s i n h £ c cosh E a a 76 with y = c sinh £ s i n n cl (184) and x = c cosh £ cos n a (185) The surface area of the s t r i p on which the shear s t r e s s i s a c t i n g i s £ ( d x 2 + dy 2 ) 1 / 2 , dy U U Diagram 2. 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 area of the e l l i p t i c a l c y l i n d e r . where I i s the length of the s t r i p i n the 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 given by 2 J T R ft 5n 'DS l ( d x 2 + dy 2 ) 1 / 2 cos 6 (186) where cos 0= ( \ pU 2 ) 2 c sinh £ I dx ( d x 2 + d y 2 ) 1 / 2 77 dx = - c cosh E s i n ndn , and using equation 162, equation 186 becomes TT C DS Re coth 5=5 a s i n n dn (187) 0 The above expression gives the s k i n drag c o e f f i c i e n t 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 the main flow p a r a l l e l to i t s major a x i s . For an e l l i p t i c a l c y l i n d e r with the main flow p a r a l l e l to i t s minor a x i s , coth £ becomes a tanh E and hence i n the l i m i t i n g case of a f l a t p l a t e CL set perpendicular to the flow d i r e c t i o n , C^g -* 0 as tanh £ ->• 0, which i s i n t u i t i v e l y c o r r e c t . deduced from the a n a l y s i s of the oblate spheroid by appro-p r i a t e l y modifying equation 173 and 175, the r e s u l t being since the l a s t term of both equations 173 and 175 becomes zero because h 3 =1 f o r e l l i p t i c a l c y l i n d r i c a l coodinates. The stagnation pressure P G can be s i m i l a r l y obtained from equations 176 and 178, where the l a s t term of equation 176 again reduces to zero f o r h 3 = 1, so that the r e s u l t i n g equation 179 i s modified to a The dlmensionless surface pressure, P, can be n (188) 78 ^b P o = 1 + k/In- I n d ? (189) ,n=o Using diagram 2, the form (pressure) drag c o e f f i c i e n t i s given by ^ P s i n 2 J P s i n 6 H ( dx^ + dy .) • C = 2 (190) u r I 2 c si n h £ a CDF = J P cos n dn (19D o where s i n 9 = ~ — 0 — N >0 (dx 2 + dy 2 ) 1 / 2 and dy = c s i n h E cos n dn cl The expression given by equation 191 i s i d e n t i c a l to that obtained by Hamielec and Raal (.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 , c Drp> i s again ob-t a i n e d by adding C D S and C D p. 79 CHAPTER V Discussion and P r e s e n t a t i o n of Results 1. A p p l i c a b i l i t y of the Navier-Stokes Equations and Choice of the Numerical Technique The steady s t a t e form of the Navier-Stokes equation i s a s p e c i a l case of the general momentum equation governing the flow 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 of the f l u i d are constant. The momentum equation i s derived f o r a continuous f l u i d , even though r e a l f l u i d s are composed of f i n i t e elementary p a r t i c l e s i n continuous motion i n a r e -l a t i v e l y large expanse of space. As the f l u i d under study i s continuous even to the i n f i n i t e s i m a l l i m i t , i t s proper-t i e s such as pressure, v e l o c i t y and temperature can be regarded as continuous and can be mathematically 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 fre e path of the f l u i d molecules, then the momentum equation i s adequate i n d e s c r i b i n g the flow f i e l d . However, f o r small body dimensions, the Brownian Movement of the f l u i d molecules c o n t r i b u t e s to the flow f i e l d . Lapple (148) has shown that the c o n t r i b u t i o n to the drag fo r c e experienced by a 10y diameter sphere from Brownian Movement i s about 0.5%, whereas f o r a l u diameter sphere i t .is over 130$, of the g r a v i t a t i o n a l l y induced drag. This i n d i c a t e s that the a n a l y s i s i n the present work i s only 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 large enough that 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 of the Navier-Stokes equation i s used, which means that there i s complete symmetry between "top" and "bottom" of the flow f i e l d and hence that 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 could not be detected. The main o b j e c t i v e of t h i s work i s to f i n d the flow 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 spheroids 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 using the c e l l models, r a t h e r than to i n v e s t i g a t e and experiment w i t h various numerical techniques. Jenson's r e l a x a t i o n technique has been found (58-61) to be very adequate and s u c c e s s f u l ' i n a c c u r a t e l y d e s c r i b i n g the flow f i e l d f o r both c i r c u l a r c y l i n d e r s and spheres. I t s adaptation to spheroids and e l l i p t i c a l c y l i n d e r s f o l l o w s q u i t e r e a d i l y . 2. R e l a x a t i o n Procedure The r e l a x a t i o n technique used i n v o l v e d c a l c u l a t i n g new values f o r if.(I,J) and G(I,J) from the four adjacent p o i n t s l y i n g on the orthogonal l i n e s c r o s s i n g at ( I , J ) , a s shown i n f i g u r e 1 f o r spheroids and f i g u r e 3 f o r e l l i p t i c a l c y l i n d e r s . The new values of and G are c a l c u l a t e d by means of equations 110 and 111 f o r the spheroids and 142 and 143 f o r the : e l l i p t i c a l c y l i n d e r s . As the r e l a x a t i o n procedure i s the same f o r the spheroids and the c y l i n d e r s , the technique f o r the former only i s discussed i n d e t a i l . In the. i t e r a t i v e pro-cedure f o r s o l v i n g the d i f f e r e n c e equations, r e l a x a t i o n parameters a g and ag were introduced to s t a b i l i z e the com-putations as f o l l o w s : 81 1^(1,J) = ^ n _ 1 ( I , J ) + a s [ i[»*(I,J) - * n _ 1 ( I , J ) ] (192) G n ( I ' J ) = G n - l ( I ' J ) + a g [ G n ( I > J ) ~ G n - l ( I ' J ) ] ( 1 9 3 > where the s u b s c r i p t n denotes the nth i t e r a t i o n and super-s c r i p t * s i g n i f i e s the unmodified r e s u l t of the nth i t e r a t i o n . The proper choice of a and a were very important f o r con-s g t r o l l i n g the 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 of u n i t y , i . e . the new value c a l c u l a t e d f o r point ( I , J ) i s used unal t e r e d i n the next i t e r a t i o n . I t has been shown by Porsythe 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 g r e a t l y improves the ra t e of convergence. Also they suggest that i t i s b e t t e r to be s l i g h t l y on the high side of the optimum a than on the low side f o r higher r a t e of con-vergence. In the present problems equations 111 and 143 are l i n e a r , and i n general a value of 1.8 was assigned to a f o r spheroids and 1.8 - 1.6 f o r e l l i p t i c a l cylinders.. On the other hand, equations 110 and 142 are l i n e a r only when Re i s equal to zero, and t h e i r " n o n l i n e a r i t y " becomes more pronounced as Re in 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 increased f o r the s i n g l e bodies, i n g order to obt a i n convergence. The present work s u b s t a n t i a t e s the comments made by Burggraf (153) that the r e l a x a t i o n parameter a appearsto be f a i r l y i n s e n s i t i v e to 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 of 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 of these c a l c u l a t i o n s , p o i n t s were considered from the surface of the spheroid to the 82 outer boundary, s t a r t i n g w i t h n=0 at the f r o n t to n=tr at the r e a r . Once a c y c l e was completed, the next c y c l e was s t a r t e d from n = TT proceeding down to n=G,points being taken from the outer boundary to the s o l i d s urface. Corrected values of 4* and G were used as soon as they became a v a i l a b l e . However, i f the cor r e c t e d values obtained during a given 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 that the optimum value of a would be l e s s than that of the former method, and con-g sequently the number of i t e r a t i o n s would be increased f o r proper convergence. Rhodes (57) used the l a t t e r method and found that he r e q u i r e d an excessive 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 of flow past a.sphere at Re = 100. The number of i t e r a t i o n s f o r the l i m i t i n g case of a sphere using the, former method was 1000 as obtained i n the present work. During the course of the work i t was found that large. values of a introduced an o s c i l l a t i o n of the v o r t i c i t y at the s u r f a c e , and that such an o s c i l l a t i o n sometimes l e d to divergence. For t h i s reason a separate r e l a x a t i o n parameter, a ,was introduced to the v o r t i c i t y at the surface. This gs made i t p o s s i b l e to assign a higher value to a than was pre-v i o u s l y p o s s i b l e , a being smaller than a.. Values of a and a f o r various values of Re and aspect r a t i o , A.R., are gs given i n t a b l e s 2 and 3 f o r spheroids and t a b l e s 4 and 5 for' e l l i p t i c a l c y l i n d e r s . 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 values to if and G on the g r i d from the creeping flow equation f o r a sphere i n the case of the spheroids and from the 83 Table 2 R e l a x a t i o n f a c t o r s f o r oblate spheroids Re A.R. a g a gs No. i t e r a t i o n s f o r convergence 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 0.999 0.5 5 0.9 0.5 5 0.5 0.5 5 0.2 0.5 0.3 400 0.3 400 0.3 450 0.3 450 10 0.999 0.3 10 0.9 0.3 10 0.5 0.3 10 0.2 0.3 0.15 1750 0.15 2100 0.15 2200 0.15 2300 20 0.999 0.25 20 0.9 0.25 20 0.5 0.25 20 0.2 0.20 0.13 1100 0.13 1050 0.13 1200 0.10 1300 50 0.999 0.2 50 0.9 0.2 50 0.5 0.2 50 0.2 0.15 100 0.999 0.08 100 0.9 0.08 100 0.5 0.08 100 0.2 0.08 0.01 9-0.2 0.8 0.1 900 0.1 1000 0.1 1000 0.08 . 1500 0.05 1000 0.05 1150 0.05 1250 0.05 1800 0.6 250 - 400 swarms 84 Table 3 R e l a x a t i o n f a c t o r s f o r p r o l a t e spheroids R e A.R. a a No. i t e r a t i o n s g gs f o r convergence 1 0.9 0.70 0.55 450 1 0.5 0.70 0.55 500 1 0.2 0.60 0.50 400 5 0.9 0.5 0.3 550 5 0.5 0.5 0.3 550 5 0.2 0.45 0.3 600 10 0.9 0.25 0.15 700 10 0.5 0.25 0.15 550 10 0.2 0.25 0.15 900 20 0.9 0.25 0.15 800 20 0.5 0.25 0.15 750 20 0.2 0.40 0.30 600 50 0.9 0.20 0.10 850 50 0.5 0.20 0.10 850 50 0.2 0.15 0.10 650 100 0.9 0.08 0.05 1000 100 0.5 0.08 0.05 1000 100 0.2 0.06 0.04 650 0.01 0.9-0.5 0.6 0.50 400 swarms 0.01 0.2 0.5 0.40 800 swarms 85 Table 4 R e l a x a t i o n 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 with O.R. > 1 Re O.R. a oi No. i t e r a t i o n s g gs f o r convergence 1.0 %1.0 0.6 0.45 1900 5.0 <vi.O 0.2 0.15 2500 1.0 10/9 0.4 0.3 2150 5.0 10/9 0.2 0.15 2950 15.0 10/9 0.08 0.05 2250 40.0 10/9 0.03 0.02 2500 90,0 10/9 0.005 0.003 3500 .5 2 0.65 0.55 1150 1.0 2 0.40 0.3 1750 5.0 2 0.2 0,15 1800 15.0 2 0.06 0.03 ... 2000 50.0 2 0.02 0.01 2150 v2 5 0.6 0.5 1150 1.0 5 0.5 0.4 1500 5-0 5 0.12 0.10 2150 10.0 5 0.05 0.035 1800 20.0 5 0.03 0.015 1950 40.0 5 0.02 0.01 1500 0.01 10/9-5 0.7 0.5 450 - 600 swarms 86 Table 5 Rela x a t i o n f a c t o r f o r e l l i p t i c a l c y l i n d e r s with O.R. < 1 Re O.R. a • a g g No. i t e r a t i o n s f o r convergence 1.0 0.2 0.4 0.3 1900 5.0 0.2 0.3 0.15 2500 15.0 0.2 0.06 0.03 5500 40.0 0.2 0.018 0.012 3500 0.01 0.9-0.2 0.7 0.5 450 - 600 swarms 87 numerical 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 of the e l l i p t i c a l c y l i n d e r s . Using equation m , f o r oblate spheroids a new value of if was c a l c u l a t e d and then modified by equation 192. The new value of if was used i n equation 110 to f i n d an improved value of G, which was subsequently modified by equation 193. This procedure was repeated u n t i l no f u r t h e r change 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 of any f u n c t i o n between successive i t e r a t i o n s . 3. Computer Programs The numerical a n a l y s i s of flow past oblate spheroids, i n c l u d i n g the c e l l models, were c a r r i e d out using 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 spheroids 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 360/67 i n conjunction w i t h Michigan Terminal Systems. Two general programs were w r i t t e n , one f o r the spheroids and the other f o r the e l l i p t i c a l c y l i n d e r s . Each program was w r i t t e n to incorporate a l l p o s s i b l e v a r i a b l e s to be s t u d i e d : oblate or p r o l a t e i n the case of the spheroids, flow past major or minor a x i s i n the case of 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 run. Each program evaluated the s k i n and drag c o e f f i c i e n t s and the dimensionless surface pressure, and i t i n i t i a t e d the contouring of 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 l i s t i n g * The contouring program was 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 of B r i t i s h Columbia. 88 of the two programs i s given i n Appendix IV.. The p o i n t values of the stream f u n c t i o n and v o r t i c i t y f o r a l l the cases are stored i n a master tape i n d u p l i c a t e . The computer times to produce a new value f o r one g r i d p o i n t are 0,000441 and 0.0002 61 second per 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 numerical work In the r e l a x a t i o n procedure, or i n any other technique i n which only point values of the f i e l d f u n c t i o n under con-s i d e r a t i o n are known, misleading r e s u l t s could be obtained from the values of 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 unless great care were taken by c o n s i d e r i n g the various com-p u t a t i o n a l parameters that could i n f l u e n c e the flow f i e l d . These parameters are: (a) the number of Taylor 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 outer boundary f o r s i n g l e bodies and (e) the d i f f e r e n t -i a t i o n technique. Comments on each parameter f o l l o w : (a) The flow equations were transformed to f i n i t e d i f f e r e n c e ' equations by using second order Taylor expansions. Jenson (55), i n h i s work on spheres, used both second and f o u r t h order Taylor expansions f o r 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 values of the drag c o e f f i c i e n t s f o r a l a t t i c e spacing of 12 degrees i n the angular, and 0.2 u n i t s of In r (where r = dimensionless r a d i a l coordinate) i n the r a d i a l , d i r e c t i o n . Although a s i m i l a r i n s e n s i t i v i t y to the order of the Taylor expansion would be l e s s probable f o r a spheroid or 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 order expansion was nevertheless used throughout t h i s a n a l y s i s i n order to keep the com-puter time w i t h i n reasonable bounds. (b) Convergence of the computed values f o r the 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 In the f o u r t h s i g -n i f i c a n t f i g u r e of any f u n c t i o n between successive i t e r a t i o n s . With such a tolerance the drag c o e f f i c i e n t s and surface pressure d i s t r i b u t i o n s would not change by more than 0.2% and 0.5%, r e s -p e c t i v e l y , f o r the case of a spheroid and 0.5% and 1.0$ r e s p e c t i v e l y , f o r the case of 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 . In some cases of low Re and large mean diameter f o r the e l l i p t i c a l c y l i n d e r s , the convergence c r i t e r i o n was taken when no f u r t h e r change occurred i n the f i f t h s i g n i f i c a n t f i g u r e of any f u n c t i o n between successive i t e r a t i o n s . During the course of t h i s work i t was found that while the drag c o e f f i c i e n t s were not very s e n s i t i v e to the degree of convergence of the l a t t i c e p o i n t s , P was very s e n s i t i v e to the degree of con-vergence. (c) The e f f e c t of the mesh s i z e f o r the oblate spheroids at Re: = 50 and Re = 100 was evaluated by changing the 90 g r i d s i z e f o r the angles from ^6° to ^3°, and the r a d i a l step from ^0.06 to ^ 0.03. The r e s u l t i n g changes i n the t o t a l drag c o e f f i c i e n t s f o r the two Reynolds numbers at any aspect r a t i o were w i t h -i n the numerical e r r o r (e.g. approximately 2%). No appreciable change i n the dimensionless f r o n t a l stagnation pressure was detected, but the re a r stagnation pressure was a l t e r e d by about 30$ which, however, represented only a 5% change r e l a t i v e to the f r o n t a l stagnation pressure. For lower Re, the l a r g e r step s i z e was used because the r a t e of change of the flow f u n c t i o n s with step s i z e was not as r a p i d as that f o r higher values of Re. The shape of the wake was found to be almost the same f o r the two mesh s i z e s used except f o r A.R. of 0.2, where the wake length was about 10% smaller f o r the f i n e r g r i d s i z e . For the p r o l a t e spheroids, the step s i z e was found to be more c r i t i c a l , and the r e f o r e a smaller step s i z e was used throughout. A change i n step s i z e of ^6° i n the angular and ^0.04 i n the r a d i a l d i r e c t i o n to ^ 3° and^0.02° r e s p e c t i v e l y , 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 , using the 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. PQ was si m u l -taneously changed by 1% and P n = T r by about 6% r e l a t i v e to P 0. Some se l e c t e d values f o r the drag co-e f f i c i e n t s and P0. are given i n Table 6 . The angle Table 6 Some s e l e c t e d data f o r s i n g l e spheroids Mark Type of A.R. Mean IMxJM Re C D S C D p C D T P D spheroid diameter 3 5 3 5 3 5 dm points points points' points p o i n t s p o i n t s 266 Oblate 0 .2 7. 0 41x65 100 0 .1947 1 .155 1 .145 1 • 350 1 .340 1 .060 1 .061 137 " . 0 .2 17. 11 33x59 100 0 .1755 1 .157 1 .146 1 .333 1 .321 1 .057 1 .058 234 0 .2 17. 11 61x105 100 0 .1666 1 .123 1 .118 1 .289 1 .285 1 .056 1 .057 113 0 .2 17. 11 33x59 50 0 .2928 1 .378 1 • 371 1 .671 1 .664 1 .116 1 .117 165 " 0 .2 17. 11 61x105 50 0 .2855 1 .368. 1 .366 1 .653 1 .651 1 .115 1 .115 97 0 .2 17. 11 33x59 10 0 .9186 2 .985 2 .982 3 .904 3 .900 1 .559 1 .564 96 0 .2 29. 90 33x69 10 0 .9055 2 .940 2 .936 3 .846 3 .842 1 .554 1 .560 128 0 .5 17. 13 33x59 100 3 .747 0 .812 0 .809 1 .187 1 .184 1 .077 1 .078 224 " 0 .5 17. 13 61x105 100 3 .627 0 .769 0 .774 1 .132 1 .137 1 .076 1 .076 111 " 0 .5 17. 13 33x59 50 0 .601 0 .9996 1 .006 1 .600 1 .607 1 .151 1 • 153 164 0 .5 17. 13 61x105 50 0 .586 0 .981 0 .984 1 .567 1 .570 1 .149 1 .150 59 0 .5 17. 13 33x59 10 1 .818 2 .162 3 .971 1 .685 84 0 .5 29- 95 33x73 10 1 .802 2 .169 2 .173 3 .970 3 • 974 1 .682 1 .687 126 0 • 9 17. 10 33x59 100 0 .576 0 .558 0 .557 1 .134 1 .133 1 .103 1 .104 217 " 0 .9 17- 10 61x105 100 0 .554 0 .548 0 .562 1 .102 1 .116 1 .100 1 .100 126 0 .9 17. 10 33x59 100 0 • 576 0