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A search for vortex rings in liquid helium Chapman, David Spencer 1966

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i A.SEARCH. FOR VORTEX RINGS IN LIQUID HELIUM . by DAVID SPENCER.CHAPMAN B.Sc. , The U n i v e r s i t y of B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL.FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1966 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t 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 , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d s t u d y . I f u r t h e r a g r e e t h a t 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 p u r p o s e s may b e g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a 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 n o t 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 D e p a r t m e n t o f P h y s i cs 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 V a n c o u v e r 8 , C a n a d a D a t e May 25, 1966. i i Abstract While endeavouring to produce macroscopic vortex rings i n l i q u i d helium by a drop method, a general study of the formation of vortex rings when a l i q u i d drop f a l l s into a stationary bath of the same l i q u i d was made. Preliminary investigations were.made using room temperature l i q u i d s with a wide range of surface tensions, d e n s i t i e s and v i s c o s i t i e s . A cryostat was designed to study vortex r i n g formation i n l i q u i d nitrogen, l i q u i d helium I, and l i q u i d helium II. A numerical method, invo l v i n g v o r t i c i t y and Stokes stream function as parameters, for the so l u t i o n of non-steady, r o t a t i o n a l , viscous flows i s outlined. , Experimental r e s u l t s confirm the reported existence of optimum dropping heights from which the drop develops into a superior vortex r i n g . These optimum heights are analysed, by a photographic study, i n terms of the l i q u i d drop o s c i l l a t i o n . It i s found that optimum vortex rings are formed, i f the drop i s spherical at the moment of contact with the bath, and i s changing from an oblate spheroid to a prolate spheroid. Vortex rings were detected i n l i q u i d nitrogen but not i n l i q u i d helium. i i i Table of Contents Chapter Page I Introduction History 1 L i q u i d Helium: A - T r a n s i t i o n 1 Two-Fluid Model 2 Thermal E f f e c t s 3 Film Flow 3 I r r o t a t i o n a l Flow 4 Quantized C i r c u l a t i o n 4 Detection of a Quantum of C i r c u l a t i o n 4 Quantized Vortex Rings 5 Bubble Chamber 6 II Theory Vortex Motion 7 Dynamics 7 Vortex Ring 9 Formation of Vortex Ring 9 Surface E f f e c t s : Model 12 Stokes Stream Function 13 Numerical Methods 13 I I I Experiment Design 15 Room Temperature Apparatus 15 Cryostat 17 Procedure 21 Measurements 21 Typical T r i a l 22 Photography 22 IV Results and Discussion L i q u i d Properties 23 E f f e c t of Tip Geometry on Drop Volume 25 Optimum Dropping Height 30 Drop O s c i l l a t i o n *C 30 Correlation Between Optimum Heights and T 33 Cryogenic Liquids 38 Surface E f f e c t s 39 Anomalous Results 40 V Conclusions 42 VI Bibliography 44 L i s t of Tables Table Page I Properties of Liquids 24 IT Surface Tension and Density of L i q u i d Helium 39 L i s t of Figures Figure Page 1 C l a s s i c a l Vortex Line 8 2 Gross Section of a Vortex Ring 8 3 Vortex Formation by Cylinder Discharge 11 4 L i q u i d Drop Entering a Bath 11 5 Drop Tips 16 6 Apparatus for L i q u i d Nitrogen and Helium I 19 7 Apparatus for Liq u i d Helium II 20 8 Drop Volume for a F l a t Tip 26 9 Drop Volume for a Conical Tip 27 10 Drop Volume for a Spherical Tip 29 11 h-D Plot of F i r s t Optimum Heights 31 12 F i r s t Optimum Height versus T//° g 32 13 O s c i l l a t i o n of a F a l l i n g Drop 34 14 Drop Shape and Motion 34 15 Drop Shape and Motion for Vortex Ring Formation 35 16 h-D Plot Showing Two Optimum Heights 37 V Acknowledgments I; wish to thank Dr. P. R. C r i t c h l o w f o r the suggestion and s u p e r v i s i o n of the pr o j e c t described; i n t h i s , t h e s i s . I would a l s o l i k e to thank: Dr. R. Stewart f o r h i s h e l p f u l , d i s c u s s i o n s and suggestions; Mr. R. Weissbach, Mr. G. Brooks, and Mr. J . Lees f o r a i d i n c o n s t r u c t i o n of the apparatus; Mr. D. Reid f o r h i s help w i t h the photography; A l l i e d Chemical and DuPont of Canada L t d . f o r the generous donation of fluorocarbons. 1. INTRODUCTION History " ' . Vortex rings were f i r s t reported by Rogers (1858), i n a paper t i t l e d "On the formation of r o t a t i n g rings of a i r and l i q u i d s under c e r t a i n conditions of discharge". A few years l a t e r , i n t e r e s t i n vortex rings grew from a set of lecture demonstrations by Professor T a i t . T a i t con-structed a large rectangular box with a c i r c u l a r aperture i n one end and a f l e x i b l e membrane covering, the other. A suitable tap on the membrane caused a vortex ri n g to shoot out from the aperture and traverse the width of the lecture room. A number of these experiments are described by Thomson (1867) and B a l l (1868). Considerable i n t e r e s t i n the motion and properties of simple vortex r i n g systems continued u n t i l the turn of the century. Smoke rings were projected i n paths i n c l i n e d to one another at any angle or to pass at various distances. Many of these experiments were repeated i n l i q u i d s by Nbrthrup (1912) who discharged a cylinder or cylinders; of dyed l i q u i d into a clear tank. A hundred years a f t e r T a i t ' s o r i g i n a l demonstrations, i n t e r e s t i n vortex rings has again been renewed by the discovery of macroscopic quantum e f f e c t s i n l i q u i d helium. Liq u i d Helium; - T r a n s i t i o n L i q u i d helium,boils at 4.2°K. The temperature of the l i q u i d can be lowered by reducing the vapour pressure above i t s surface. At 2.18°K l i q u i d helium undergoes a "lambda" t r a n s i t i o n at which b o i l i n g appears to cease. The name lambda t r a n s i t i o n a r i s e s from the c h a r a c t e r i s t i c p r o f i l e of the s p e c i f i c heat curve which resembles the Greek l e t t e r 7^. The 2 t r a n s i t i o n temperature i s r e f e r r e d to as the " A - p o i n t . Above the 7\-point l i q u i d helium acts as a. normal f l u i d and i s c a l l e d "helium I " . In c o n t r a s t , below the A - p o i n t l i q u i d helium e x h i b i t s many remarkable p r o p e r t i e s very d i f f e r e n t from usual l i q u i d s and i s denoted as "helium I I " . Keesom and van den Ende (1930) f i r s t observed that helium I I would flow e a s i l y through very f i n e cracks. An experiment by K a p i t z a (1938) measuring the flow of l i q u i d helium I I through narrow channels y i e l d e d a v i s c o s i t y of l e s s than 1 0 " 1 1 poise; A l l e n and Misener (1938). however, measured the damping of an o s c i l l a t i n g d i s k and obtained the very d i f f e r e n t v i s c o s i t y value of 10"^ poise. Two-Fluid Model Two years l a t e r , T i s z a (1940) and c o n c u r r e n t l y Landau (1941) proposed a t w o - f l u i d model f o r l i q u i d helium below the T^-point, which explained the v i s c o s i t y dilemma. This t w o - f l u i d model suggests that l i q u i d helium I I c o n s i s t s of two, mutually i n t e r p e n e t r a t i n g f l u i d s , the "normal f l u i d " of d e n s i t y /°n, and " s u p e r f l u i d " of d e n s i t y ^ s . At any point i n space the l i q u i d d e n s i t y /° may be w r i t t e n as The r a t i o f I Ps depends s t r o n g l y on temperature. The s u p e r f l u i d may be considered as l i q u i d at 0°K; a c t u a l l y i n a s i n g l e quantum s t a t e which corresponds -to the condensed phase of the B o s e - E i n s t e i n l i q u i d . As such, i t has zero v i s c o s i t y and zero entropy. The normal f l u i d i s a s s o c i a t e d w i t h thermal e x c i t a t i o n s , thereby c a r r y i n g the e n t i r e l i q u i d entropy. I t has f i n i t e non-zero v i s c o s i t y a r i s i n g from c o l l i s i o n s between the e x c i t a t i o n s . 3 Many of the unusual phenomena of l i q u i d helium I I can be r e a d i l y e xplained by the t w o - f l u i d model. 1 Thermal E f f e c t s (mechanocaloric, fountain) Daunt and Mendelssohn (1939) ; observed that when l i q u i d - h e l i u m I I flowed out of a v e s s e l through a f i l t e r of packed emery powder, the tempera-ture of the remaining l i q u i d rose. This i s e a s i l y e x p l a i n e d , because only the s u p e r f l u i d component, which c a r r i e s no entropy, i s able to flow, through the f i l t e r . Therefore, the remaining l i q u i d has greater entropy per u n i t volume, and warms. On the other hand, i f heat i s supplied to a v e s s e l which i s connected by a narrow c a p i l l a r y to a surrounding bath, then s u p e r f l u i d w i l l move from the colder bath to the warmer v e s s e l . A temperature d i f f e r e n c e A T between the v e s s e l and the bath w i l l produce a thermomechanical pressure head A P = /» S A T where ./° i s the l i q u i d d e n s i t y and S i s the entropy. This phenomena, . discovered by A l l e n and Jones (1938),. i s commonly known as the f o u n t a i n e f f e c t . F i l m Flow I f l i q u i d helium I I contained i n two beakers i s i n i t i a l l y at d i f f e r e n t levels', then i t i s observed that the l i q u i d l e v e l s w i l l become equ a l i z e d without any communicating tube. between the v e s s e l s . Now any s o l i d surface i n contact w i t h l i q u i d helium I I i s covered by a t h i n f i l m about 100 atomic l a y e r s , t h i c k . I t i s understood that t h i s , t h i n f i l m a cts l i k e a narrow channel through which s u p e r f l u i d may flow. The ra t e at which the fi-lm creeps i s independent of the pressure head: or length of f i l m , but i s determined by the temperature and the, minimum periphery above the upper l i q u i d l e v e l over which, the f i l m must pass. The l a c k of dependence on 4 pressure head i n d i c a t e s that the f i l m flows at some c r i t i c a l v e l o c i t y . I r r o t a t i o n a l Flow Landau (1941) postulated that s u p e r f l u i d flow should always be i r r o t a t i o n a l . That i s c u r l v s = 0 where v g i s the s u p e r f l u i d v e l o c i t y . This c o n d i t i o n i s seemingly v i o l a t e d by r o t a t i n g beaker experiments (Osborne, 1950; Reppy, Depatie and Lane, 1960) which show that s u p e r f l u i d can r o t a t e w i t h the v e s s e l . However, i f the flow i s i r r o t a t i o n a l throughout most of the s u p e r f l u i d , i t i s s t i l l p o s s i b l e to have h i g h l y l o c a l i z e d regions of v o r t i c i t y . These re g i o n s , which are e s s e n t i a l l y s i n g u l a r i t i e s i n the v e l o c i t y f i e l d , may be thought to be s i m i l a r to the v o r t e x l i n e s of c l a s s i c a l hydrodynamics (Lamb, 1895). Quantized C i r c u l a t i o n A p p l i c a t i o n of quantum mechanics by Onsager ,(1949) and l a t e r by Feynman (1955), p r e d i c t e d that l i q u i d helium flow should e x h i b i t macroscopic q u a n t i z a t i o n . I f steady flow, of the f l u i d , i s d e s c r i b a b l e by a s i n g l e wave f u n c t i o n " t ~ , then by r e q u i r i n g that "T' be s i n g l e valued, we f i n d that i t s phase should change.by a m u l t i p l e of 21T around a c l o s e d path. This i s equivalent to saying that the c i r c u l a t i o n k of the f l u i d be quantized, i n u n i t s of h/m. That i s = n (h/m) where h i s Planck's constant and m i s the mass of a helium atom. Detection of a Quantum of C i r c u l a t i o n The d e t e c t i o n of such a s i n g l e quantum of c i r c u l a t i o n has been reported by Vinen (1958). He s t u d i e d the. modes of transverse v i b r a t i o n of a t h i n wire s t r e t c h e d along the a x i s of a r o t a t i n g beaker of helium I I , 5 I f there i s zero c i r c u l a t i o n , the wire,may v i b r a t e i n two perpendicular d i r e c t i o n s and thus w i l l possess doubly degenerate modes. Non-zero c i r c u l a -t i o n w i l l remove the degeneracy because of the Magnus e f f e c t , and w i l l produce two c i r c u l a r l y p o l a r i z e d modes s p l i t by a measurable frequency. P r e l i m i n a r y r e s u l t s produced c i r c u l a t i o n s not equal to i n t e g r a l m u l t i p l e s of h/m, but these were i n t e r p r e t e d i n terms of p a r t i a l attachment of a free v o r t e x to the wire. However, when the wire was subjected to a l a r g e amplitude v i b r a t i o n , thereby removing p a r t i a l v o r t e x l i n e s , the remaining c i r c u l a t i o n was u s u a l l y equal to j u s t h/m. Quantized Vortex Rings The hollow core of a v o r t e x l i n e need not be s t r a i g h t . In p a r t i c u l a r the core may be bent around to form a c l o s e d c i r c u l a r v o r t e x r i n g , and t h i s c o n f i g u r a t i o n should a l s o e x h i b i t quantized c i r c u l a t i o n . R a y f i e l d and R e i f (1963), using ions i n l i q u i d helium as micro-scopic probe p a r t i c l e s , r e p o r t evidence f o r such quantized v o r t e x r i n g s i n 210 s u p e r f l u i d helium. Helium atoms were i o n i z e d by c* p a r t i c l e s from a Po source and the d r i f t v e l o c i t y of the r e s u l t i n g ions was measured under d i f f e r e n t e l e c t r i c f i e l d s . The experiments, conducted at temperatures ranging from 0.28°K to 0.60°K, y i e l d e d the f o l l o w i n g strange r e s u l t s : (a) the v e l o c i t y of the c h a r g e - c a r r i e r was about 10"^ times lower than expected f o r a helium i o n , (b) the v e l o c i t y was roughly i n v e r s e l y p r o p o r t i o n a l to energy, (c) i n a transverse e l e c t r i c f i e l d , the angular d e f l e c t i o n of the c h a r g e - c a r r i e r was four times greater than i t would be f o r a s i n g l e free i o n . The unusually low v e l o c i t y suggested that the observed charge-c a r r i e r c o n s i s t e d of not one, but thousands of atoms forming some w e l l 6 defined l a r g e s c a l e e n t i t y , i n f a c t a v o r t e x r i n g . A charge-carrying v o r t e x r i n g would have the hydrodynamic s t a b i l i t y to move through the f l u i d as a s i n g l e u n i t , and would adequately e x p l a i n a l l experimental r e s u l t s . Furthermore, from the dynamics of a c l a s s i c a l v o r t e x r i n g and using the measured values f o r v e l o c i t y and energy, the core r a d i u s and c i r c u l a t i o n can be c a l c u l a t e d . This y i e l d e d a core r a d i u s of atomic-dimen-sions (1.28 A ) , and a c i r c u l a t i o n equal to 1.0 x 10 cm. /sec., j u s t one quantum u n i t h/m. Bubble Chamber Quite r e c e n t l y , Edwards (1965) has reported the probable observa-t i o n of quantized v o r t e x r i n g s i n a helium I I bubble chamber. A f l a t p l a t e w i t h a small c i r c u l a r aperture i n the middle was placed near the bottom of the helium I I chamber. When the c y l i n d e r piston., was withdrawn, some of the helium was drawn through the aperture. Photographs at subsequent time i n t e r v a l s showed some s t a b l e e n t i t y moving i n the chamber at v e l o c i t i e s of a few centimeters per minute. The objects i n the photographs were i d e n t i -f i e d as v o r t e x r i n g s , but i t has not yet been confirmed whether the c i r c u l a t i o n i s quantized. In view of these developments, : i t seemed u s e f u l to t r y to produce macroscopic v o r t e x r i n g s ( i e : of a few m i l l i m e t e r s diameter),by another method i n l i q u i d helium and to measure t h e i r p r o p e r t i e s . 7 THEORY Vortex Motion Consider a v e l o c i t y f i e l d described by a potential function $ , § = c tf where c i s a constant and Cp i s the usual azimuthal angle. In the xy plane equipotential l i n e s form a pencil of rays through the o r i g i n , while stream-l i n e s form c i r c l e s with centers at the o r i g i n . The r a d i a l v e l o c i t y v r and the tangential v e l o c i t y Vw> are given Thus the v e l o c i t y at every point i s perpendicular to the radius vector and i t s magnitude i s inversely proportional to the distance from the o r i g i n . This form of motion i s c a l l e d a vortex. Dynamics The dynamics of vortex motion were developed by Helmholtz (1868) and can be summarized as follows: 1. No f l u i d p a r t i c l e can have r o t a t i o n i f i t did not o r i g i n a l l y rotate. 2. F l u i d p a r t i c l e s which at any time are part of a vortex l i n e , always belong to the same vortex l i n e . 3. The strength of a.vortex i s always constant. Vortex filaments therefore, must eit h e r be closed tubes, or end on the bound-ar i e s of the f l u i d . The strength or c i r c u l a t i o n k of a vortex i s the value of the l i n e integral, of the v e l o c i t y taken around a closed path p surrounding the vortex core' r —^ k = ^ p v . d l 8 Schematic i l l u s t r a t i o n showing a v o r t e x r i n g i n c r o s s - s e c t i o n . The whole v o r t e x ring, moves with v e l o c i t y v i n the a x i a l d i r e c t i o n . 9 Vortex Ring I f a v o r t e x l i n e i s bent i n t o a c i r c l e then i t i s known as a vortex r i n g . Suppose the v o r t i c i t y i s concentrated near a c i r c l e of radius R and i s d i s t r i b u t e d over a c i r c u l a r c r o s s - s e c t i o n of ra d i u s a where a « R. The energy E, impulse P and t r a n s l a t i o n a l v e l o c i t y v are given (Lamb, 1895) i n terms of R, a, and the c i r c u l a t i o n k by: v _ k2R/°, In 8R 7 \ 41TR <• A 4 > P = 7T /> kR 2. Vortex Ring Formation Vortex r i n g s can be produced by at l e a s t three, methods: (a) d i s k impulse (b) c y l i n d e r discharge (c) l i q u i d drop I f a c i r c u l a r d i s k submerged i n a l i q u i d bath i s given an impulse, then a v o r t e x r i n g w i l l form behind the d i s k . I f the d i s k i s then d i s s o l v e d away, t h e o r e t i c a l c o n s i d e r a t i o n s by Taylor (1953) show that a knowledge of the r a d i u s and i n i t i a l impulse of the d i s k i s s u f f i c i e n t to determine the v e l o c i t y , r i n g diameter and core diameter of the r e s u l t i n g v o r t e x r i n g . The most common experimental method of producing v o r t e x r i n g s i n f l u i d s " i s by c y l i n d e r discharge. A sharp tap on the p i s t o n causes a p o r t i o n of the f l u i d to be e x p e l l e d from the c y l i n d e r through a c i r c u l a r aperture on the c y l i n d e r face. V e l o c i t y d i s c o n t i n u i t i e s which would e x i s t at the perimeter of the aperture i n a. non-viscous f l u i d become a l a y e r of v o r t i c e s when the v i s c o s i t y i s not zero. (see F i g . 3) > I f the time of the impulse 10 on the piston i s short, then vortex elements, having equal and opposite c i r c u l a t i o n , form on oipposite ends of the diameters. In two dimensions the motion of each vortex element on the end of a diameter i s governed by the v e l o c i t y f i e l d of the other, and so the pair move with constant v e l o c i t y at r i g h t angles to the l i n e joining, them. Since each vortex element around the aperture i s influenced by the remaining elements, a v e l o c i t y i s given •to. the whole r i n g . Experiments by Reynolds (1876)' indicated that such vortex rings t r a v e l with constant impulse for considerable distances. Vortex r i n g formation by l i q u i d drops f a l l i n g into a bulk l i q u i d at rest was f i r s t reported by Rogers (1858), but was studied i n some d e t a i l by Thomson and Newall (1885/). A summary of the l a t t e r paper i s presented here. They considered a l i q u i d drop f a l l i n g i n t o a bath, not n e c e s s a r i l y the same l i q u i d , without splashing. The drop retains i t s shape and, when just inside the bath, experiences at i t s surface a f i n i t e a l t e r a t i o n i n tangential v e l o c i t y over a very small distance. This gives r i s e to a vortex f i l m . The drop tends to become disk-shaped,, and at the same time the v o r t i c i t y d i f f u s e s inwards and outwards. I f the v i s c o s i t y i s such, that when the drop i s i n a disk shape i t i s f u l l of vortex motion, then the unstable vortex disk w i l l break up into a stable vortex r i n g . . . Furthermore, Thomson's experiment showed that the q u a l i t y of a vortex r i n g changes considerably with changes i n dropping height. The upper l i m i t for vortex r i n g formation corresponds to splash conditions, . about 5 cm. for most l i q u i d s . Below, t h i s maximum dropping height^ the formation of a vortex r i n g depends, on the shape of the drop as i t contacts the bath surface. 11 aperture (a) (b) FIG.. 3 Vortex formation by cyl i n d e r discharge. The surface of d i s c o n t i n u i t y of a non-viscous f l u i d (a) changes into a layer of v o r t i c i t y when the v i s c o s i t y i s not zero (b). (a) (b) (c) FIG. 4 L i q u i d drop entering a bath of the same l i q u i d . Pressure gradients r e s u l t i n g from surface curvature are shown i n (c). 12 Surface E f f e c t s Since v o r t e x r i n g s are formed when a drop i s placed on the bath s u r f a c e , i t seemed that surface t e n s i o n e f f e c t s , although ignored by Thomson and Newall, could be quite important i n v o r t e x r i n g formation. Furthermore surface t e n s i o n e f f e c t s lead to a very simple p h y s i c a l model f o r t h e o r e t i c a l c o n s i d e r a t i o n . The surface energy E S which a s p h e r i c a l drop< of r a d i u s r gives up upon e n t e r i n g a bath i s simply E S = T 4fT r 2 . The k i n e t i c energy E of the same drop a f t e r free f a l l through a distance h i s A 3 E = ~1T r /«>gh For drops and dropping, heights i n t h i s experiment the r a t i o E G/E i s of the order of u n i t y . In p a r t i c u l a r when the drop i s formed so that i t detaches from the t i p and touches the bath simultaneously, surface energy i s about twice the k i n e t i c energy*for a l l l i q u i d s . Model Figure 4 shows a s p h e r i c a l drop (a) before contact, (b) during contact, and (c) a f t e r contact w i t h a l i q u i d surface. I f surface t e n s i o n i s indeed important then we can consider the model as seen i n Figure 4 (c) at time zero. Two processes w i l l help to e s t a b l i s h the d e s i r e d c i r c u l a t i o n f o r v o r t e x r i n g formation. The curvature of the f r e e surface i s such that the center of the drop,is a region of r e l a t i v e high pressure whereas the periphery of the drop i n the plane of the bath surface i s a r e g i o n of low pressure. The f l u i d elements i n the center of the drop are thus a c c e l e r a t e d along the pressure gradients as shown ( F i g . 4-c). Pressure gradients must 13 c o i n c i d e w i t h the normals at a free surface. Secondly, g r a v i t a t i o n forces p u l l the d r o p : i n t o the bath and the processes o u t l i n e d f o r c y l i n d e r d i s -charge are repeated. S toke s S trearn Function The exact t h e o r e t i c a l s o l u t i o n to the problem of a drop f a l l i n g i n t o a bath i s very d i f f i c u l t s ince i t deals w i t h a non-steady, v i s c o u s , r o t a t i o n a l flow. Because of the a x i a l symmetry, the flow i s best described by a stream f u n c t i o n and t h i s may be used f o r a numerical s o l u t i o n . Stokes 1 stream f u n c t i o n ~i" i s p a r t i c u l a r l y u s e f u l f o r axisymmetric flows. i s a f u n c t i o n of p o s i t i o n P and i s d i m e n s i o n a l l y and n u m e r i c a l l y equal to the r a t e of flow through the surface generated by r o t a t i n g P about the a x i s of symmetry. The r a d i a l v e l o c i t y v r and a x i a l v e l o c i t y v z are .given by The stream f u n c t i o n describes i n a l g e b r a i c form the geometry of the flow p a t t e r n and i t s d e r i v a t i v e s y i e l d the components of the v e l o c i t y v e c t o r . I t i s p a r t i c u l a r l y u s e f u l i n t h i s problem because since "$r i s a s c a l a r , " i t can be added a l g e b r a i c a l l y f o r composite states, of flow. Furthermore i t s existence does not depend on the motion being i r r o t a t i o n a l . Numerical Method Numerical f l u i d dynamics techniques f o r high speed computers have been developed r e c e n t l y which produce t h e o r e t i c a l s o l u t i o n s to very complex problems. Fromm (1963) has o u t l i n e d a method f o r computing non-steady, incompressible, v i s c o u s flows, using a stream f u n c t i o n " ? " and v o r t i c i t y «0 as parameters. The e s s e n t i a l equations i n the c y l i n d r i c a l axisymmetric system are the v o r t i c i t y equation, 14 1*- = -v _ v ll* + \) 7 2 t o and the Poisson equation, where y 2 = ^ - + ^ + ± 4-A l l v a r i a b l e s are r e l a t e d to an E u l e r i a n mesh of c e l l s . The c a l c u l a t i o n may be o u t l i n e d as f o l l o w s : 1. i s chosen to s a t i s f y the i n i t i a l c o n d i t i o n s . 2. v e l o c i t i e s and thus Lit are obtained from . 3. advanced values, of (H are found by a f i n i t e - d i f f e r e n c e form of the v o r t i c i t y equation. 4. the advanced CD i s a source :term i n the Poisson equation. The new values of "$T are found f o r a l l points i n the mesh by successive approximations. 5. steps 2, 3 and 4 are repeated u n t i l the flow has advanced the d e s i r e d time. This method would seem to be a p p l i c a b l e to both the s i m p l i f i e d surface t e n s i o n model, and f o r the more general problem of an o s c i l l a t i n g drop f a l l i n g i n t o a bath at r e s t . 15 EXPERIMENT Design The apparatus b a s i c a l l y c o n s i s t e d of three p a r t s ; a mechanism to form pendent l i q u i d drops, a l i q u i d bath at r e s t i n t o which the drops f e l l , and some detector to i n d i c a t e the formation of v o r t e x r i n g s . A c r y o s t a t was designed i n order to, use cryogenic l i q u i d s such as n i t r o g e n , helium I , and helium I I . Room Temperature Apparatus Various drop, t i p s of brass and glass were attached to a l i f t mechanism, which was used to vary the height of the t i p above the bath. One turn of the l i f t r a i s e d or lowered the t i p 0.16 cm. An overflow method was used to t r i c k l e l i q u i d down the siVles of the t i p s to form pendent drops. The overflow was s u i t a b l y geared so that drops could be formed as slowl y as d e s i r e d . The s p h e r i c a l t i p ( F i g . 5-a) was made from a 6 cm. length of 2 cm. O.D. glass tubing. The bottom of the tube was c l o s e d i n t o a s p h e r i c a l surface w i t h a 1.3 cm. rad i u s of curvature. Two small holes 3, cm. from the bottom allowed f o r the l i q u i d overflow. The c o n i c a l t i p ( F i g . 5-b) was machined from a brass rod 2 cm. i n diameter. The semiangle of the cone was 30°. The surface of the t i p was p e r i o d i c a l l y scored w i t h f i n e emery paper to enhance wetting by the l i q u i d . A brass c o l l a r and interchangeable f l a t t i p i s i l l u s t r a t e d i n F i g . 5-c. S i x such t i p s were made w i t h r a d i i v a r y i n g from 0.19 to 1.01 cm. The bottom of each t i p was a l s o scored w i t h f i n e emery paper. The l i q u i d bath was contained i n a c l e a r g l a s s graduated c y l i n d e r . The v e s s e l was large enough to minimize w a l l e f f e c t s . I n most t r i a l s the drops were dyed w i t h sudan IV, or. methylene blue and therefore the coloured 16 (a) Spherical Tip (glass) (b) Conical Tip (brass) (c) Flat Tip (brass) FIG. 5 Drop Tips 1 7 vortex r i n g was detected d i r e c t l y . In some cases, a thin layer of small p a r t i c l e s was placed on a dish a few centimeters below the bath surface. I f a vortex r i n g approached the dish with reasonable v e l o c i t y , then the p a r t i c l e s were displaced leaving a r i n g imprint. A water vortex r i n g e a s i l y s h i f t e d i r o n f i l i n g s on a dish 8 cm. below the bath surface. Cryostat The cryostat was of conventional design. The helium and the nitrogen glass dewars were s i l v e r e d , but c l e a r s l i t s 1.5 cm. wide provided good v i s u a l observation of the experiment. A l l i n t e r n a l apparatus was mounted on thi n wall s t a i n l e s s s t e e l tubes, which extended above the cryostat cap. By means of s l i d i n g O-ring seals, various parts; of the apparatus could be rotated or l i f t e d . E l e c t r i c a l leads were introduced into the cryostat through kovar seals. Provision was made for the introduction of a v a r i a b l e density hydrogen-deuterium gaseous mixture into the cryostat, which upon s o l i d i f i -c a t i o n would form the detector powder. The storage system consisted of glass vessels and connecting tubing, any part of which could be kept at a given pressure below one atmosphere, (a) Apparatus f o r l i q u i d nitrogen and helium I. This apparatus (Fig. 6) consisted of a drop beaker with a piston, and a detection dish. The piston was lowered by external r o t a t i o n of a st a i n l e s s s t e e l tube, e f f e c t i v e l y geared (60:1 r a t i o ) so as to produce drops slowly. The beaker and piston were both f i l l e d by immersion i n the bath. . Another small glass beaker, containing a #32 gauge Advance wire heater, was occasionally positioned below the drop,beaker for repeated • measurement of drop, volume. 18 The d e t e c t i o n p a r t i c l e s were he l d on a s l i g h t l y concave l u c i t e d i s h 3.0 cm., i n diameter, which could be moved r e l a t i v e to the drop beaker, (b) Apparatus f o r l i q u i d helium I I . The s u p e r f l u i d p r o p e r t i e s of l i q u i d helium I I were u t i l i z e d i n designing the apparatus ( F i g . 7) f o r a p p l i c a t i o n below the > - p o i n t . The drop beaker was f i l l e d using a f o u n t a i n pump, and then emptied by the super-* f l u i d f i l m flow. C a l c u l a t i o n s showed that drops at s u i t a b l e time i n t e r v a l s (10 seconds) could be produced from a 1.0 cm. diameter beaker. A 50 ohm c o i l of #36 gauge Advance wire was imbedded i n a packed cerium oxide powder to produce the d e s i r e d thermomechanical e f f e c t . Power was sup p l i e d to the f o u n t a i n pump from a 3,volt dry c e l l . 19 S l i d i n g Seal for Piston S l i d i n g Seal for Drop Beaker II11 11 1,1.1 11 n n Winch Drive Cryostat Cap Winch Piston L i q u i d Lucite C o l l a r Drop:Beaker Liqu i d FIG. 6 Apparatus for l i q u i d nitrogen and l i q u i d helim I 20 S l i d i n g Seals Fountain Support To Hydrogen-Deuterium Storage Cryostat Gap Fountain Pump Cerium Oxide Heater y. Drop Beaker Bath Level Dish FIG. 7 Apparatus for l i q u i d helium II 21 Procedure P r e l i m i n a r y i n v e s t i g a t i o n s were made to study the formation of vo r t e x r i n g s by the drop method i n l i q u i d s at room temperature. The d i s -tance which a v o r t e x r i n g t r a v e l l e d i n the bath was measured as a f u n c t i o n of dropping height. A photographic study was made of a f a l l i n g water drop i n order to c o r r e l a t e v o r t e x r i n g formation to the l i q u i d drop o s c i l l a t i o n . Measurements Room temperatures were measured w i t h a G.H. Zeal mercury ther-mometer. The l i q u i d n i t r o g e n bath was assumed to be at i t s normal b o i l i n g p o i n t , 77.3°K. Helium vapour pressures were measured with mercury and b u t y l phthalate manometersmercury being 12.94 times, more dense than b u t y l phthalate. Temperatures were then determined from the N.B.S. T 5 3 helium vapour pressure t a b l e s . In determining dropping h e i g h t s , a scal e c a l i b r a t e d i n 1mm. d i v i s i o n s was used f o r the p r e l i m i n a r y i n v e s t i g a t i o n s . A Cenco cathetometer was used f o r more accurate work, i n c l u d i n g a l l measurements i n s i d e the dewars. This reduced p o s s i b l e e r r o r on t y p i c a l dropping heights to l e s s than 1%. Drop volumes were determined by c o l l e c t i n g and measuring the volume of a known number of drops. With the cryogenic l i q u i d s the cathe-tometer was used to measure the l e v e l r i s e i n a 3; mm. I.D. (helium) or a 6 mm. I.D. (nitrogen) glass beaker f o r a known number of drops. Since v a r i o u s dyes were used to co l o u r the l i q u i d drops, the surface t e n s i o n of the dyed and undyed l i q u i d s was checked.by Jaeger's method using a c a p i l l a r y t i p of 0.713, mm. diameter. The surface t e n s i o n was found to change by l e s s than 4% f o r any l i q u i d upon adding the dye. 22 - T y p i c a l T r i a l In a t y p i c a l t r i a l , the bath was f i l l e d and l e f t standing u n t i l the l i q u i d came to r e s t . A dyed p o r t i o n of the same l i q u i d was used to f i l l the drop beaker. For a given height above the bath, a pendent drop was s l o w l y formed on the t i p and l e t f a l l i n t o the bath. I f a v o r t e x r i n g formed, the distance D which the r i n g t r a v e l l e d was recorded. This pro-cedure was repeated four more times at that dropping height. The height was v a r i e d from zero to splashing c o n d i t i o n s i n no more than 0,5 cm. I n t e r v a l s , w i t h c e r t a i n height regions being i n v e s t i g a t e d i n more d e t a i l . For a given dropping height h an average of the f i v e values of D was found and p l o t t e d against h. Drop, volumes were, measured f o r the dyed l i q u i d during each t r i a l . Photography .In-order to c o r r e l a t e v o r t e x r i n g formation to the drop o s c i l l a -t i o n , closeup p i c t u r e s were taken of the water drop during i t s f i r s t f u l l o s c i l l a t i o n a f t e r r elease from a drop t i p . The photographic equipment c o n s i s t e d of a P r a c t i c a 35 mm., s i n g l e lens r e f l e x , camera w i t h a 135. mm, telephoto lens on a 100 mm. lens exten-s i o n tube. The drop was i l l u m i n a t e d from the side by a v a r i a b l e frequency Xenon f l a s h stroboscope. The stroboscope was c a l i b r a t e d using a photo-e l e c t r i c c e l l and o s c i l l o s c o p e , and was shown to have a continuous frequency range from 7 c y c l e s / s e c . to 80 c y c l e s / s e c . and a f l a s h d u r a t i o n not greater than 0,8 m i l l i s e c o n d s . Negatives were projected onto a screen where the images could be copied at a m a g n i f i c a t i o n of about 12, Measurements of the v e r t i c a l and i h o r i z o n t a l dimensions of the drop were taken from the drawing. 23 RESULTS AND DISCUSSION P r o p e r t i e s of the l i q u i d s used are presented i n t a b u l a r form. The e f f e c t of t i p geometry on l i q u i d drops; i s discussed. Results i n d i c a t e the e x i s t e n c e of c e r t a i n optimum dropping heights f o r the formation of vo r t e x r i n g s and these heights are analysed i n terms of the drop o s c i l l a t i o n . L i q u i d P r o p e r t i e s P r o p e r t i e s ; o f the l i q u i d s used are l i s t e d i n ta b l e I. Values f o r T the surface t e n s i o n , /° the d e n s i t y , and the kinematic v i s c o s i t y were obtained from t a b l e s as f o l l o w s : most room temperature l i q u i d s , from the Chemical Rubber P u b l i s h i n g Co. "Handbook of Chemistry and Ph y s i c s " ; f l u r o -carbons, from a Duppnt of Canada t e c h n i c a l brochure "Freon"; l i q u i d helium, from A t k i n s " L i q u i d Helium"; and l i q u i d n i t r o g e n , from the N.B.S. "A Compendium of the M a t e r i a l s at Low Temperatures". In the s i x t h column, V o l . i s the measured drop volume using a s p h e r i c a l g l a s s dropping t i p . The l i q u i d d r o p - o s c i l l a t i o n t i m e . T , was c a l c u l a t e d from t h i s Volume; The dropping height h i s the distance the drop f a l l s and i s measured from the center of the drop when i t j u s t detaches from the t i p , to the center of the drop when i t contacts the bath surface. h-6pt i n column e i g h t of ta b l e I i s the observed f i r s t optimum dropping he i g h t , a term which i s defined and discussed l a t e r i n t h i s chapter. The r a t i o T//° g has been c a l c u l a t e d f o r each l i q u i d and i s seen to change by a - l i t t l e more than two orders of magnitude: 0.65 x; 10" 3 cm. 2 f o r l i q u i d helium at 4.2°K, to 74.3 x 10" 3 cm.2 f o r water at room temper-ature. Without helium, the range i s about one order of-magnitude. 24 TABLE I Properties, of Liquids L i q u i d T V T//° g Vol T h-opt xlO 2 XlO 2 x l O 2 x l O 2 dynes /cm g/cm3 cm2/ sec cm 2 cm3 sec cm water 72 .8 1.00 1. 00 7. 43 9.6 3.94 2.4 fonnamide 58 .2 1.13 3. 32 5. 15 , 6.9 3.98 2.0 benzylamine 39 .5 0.98 1. 63 4. 12 6,7 4.24 2.0 benzene 28 .9 0.88 0. 74 3. 35 4.5 4.02 2.0 acetone 26 .2 0.79 0. 41 3. 39 5.1 4.26 0 butyl alcohol 24 .6 1.04 3, 70 3. 10 4.5 4.18 2.2 ethyl acetate 23 .9 0.90 0. 50 2. 71 3.7 4.06 1,7 freon 113 23 .0 1.57 0. 42 1. 50 1.4 3.36 0.8 methyl alcohol 22 .6 0.81 0. 74 2. 85 4.6 4.42 1,7 freon 11 22 .0 1,48 0. 28 1. 49 1.8 3.78 0 ethyl, alcohol 21 .4 0.79 1. 51 2. 67 4.5 4.43 1.9 freon 114-B2 21 .0 2.16 0. 33 . o, 99 0.98 3.45 1.0 isopentane 13 .8 0.62 0. 33 2. 26 4.1 4.68 0 nitrogen 8 .3 0.81 0. 20 1. 05 1,2 3.71 1.3 helium (1,8°K) 0 .32 0.15 0. 01 0. 22 0.12 3.04 -helium (4.2°K) 0 .08 0.13 0. 03 0. 06 - - -25 E f f e c t of Tip Geometry on Drop Volume t I Harkins and Brown (191?) showed that the mass m of a l i q u i d drop which forms sl o w l y from the f l a t t i p of r a d i u s r i s V . ' mg = 2 f f r T f ( r / a ) ':' (1) where a = (2T//° g), i s the c a p i l l a r y constant. The drop weight c o r r e c t i o n f.(r/a), i s a f r a c t i o n a l q u a n t i t y having a cubic dependence on r/a. Their r e s u l t s were i d e n t i c a l f o r g l a s s t i p s and f o r brass, t i p s : of the same geometry. Results obtained w i t h water d r o p l e t s using s i x f l a t brass t i p s : of d i f f e r e n t r a d i i agree w i t h Harkins and Brown w i t h i n experimental e r r o r . Drop volumes were then measured f o r f i v e l i q u i d s , T//*g ranging from 9.7 x 1(T 3 to 7.5 x 10" 2 cm.2, using a, f l a t brass t i p of r a d i u s 2.35 mm. The mass of the drop can be expressed as the product of d e n s i t y times volume i n equation ( 1 ) , and. f ( r / a ) , c a n be obtained from Harkins and Brown's paper. Figure (8) shows good agreement with equation ( l ) . Now l e t the f l a t t i p be replaced*by an i n v e r t e d cone of h a l f angle . T h e o r e t i c a l c o n s i d e r a t i o n s by Brown and McCormick (1948) r e v e a l t h a t , provided the angle of contact between the l i q u i d and surface of the cone i s the same, a l l drops forming on a cone of f i x e d 9- are s i m i l a r i n shape at the unstable stage. The drop,volume V should then be given by V - (T//°g) 3/ 2 $(©-) where ^>($) i s some constant dependent only on 9- . A \ l o g - l o g p l o t of drop,volume versus T/P g f o r a brass c o n i c a l t i p w i t h ©• equal to 30° i s shown i n f i g u r e 9. The experimental points are s c a t t e r e d and depart s l i g h t l y from the p r e d i c t e d slope of 3/2. However, the theory considers the l i q u i d - s o l i d contact angle to be constant, i n p a r t i c u l a r equal to zero, but t h i s c o n d i t i o n was probably not s a t i s f i e d I 26 FIG. 8 Drop volumes for a f l a t brass t i p . Experimental points are ( l e f t to r i g h t ) , freon 114-B2, carbon t e t r a c h l o r i d e , ethyl acetate, acetone and water. 2 7 0.10 0.01 _ Drop Volume Z (cm. 3) slope 3/2 J L 0 . 0 / 0.10 T//°g (cm. ) FIG. 9 Drop volume for a c o n i c a l brass t i p . Experimental points are ( l e f t to r i g h t ) freon 114-B2, freon 113, ethyl alcohol, benzene, formamide and water. Theoretical l i n e of slope 3/2 i s shown. 28 In t h i s experiment. The contact angle i s i n general dependent on the contamination of the s u r f a c e , and may be greater than zero i f the l i q u i d i s evaporating or i f the g l a s s surface has become dry. The drop, volumes f o r Freon 113, Freon 114-B2, and e t h y l a l c o h o l should have been obtained i n a saturated atmosphere of t h e i r own vapours to prevent evaporation. Furthermore, the time i n t e r v a l f o r a drop.to grow to the unstable stage was approximately 10 to 15 seconds and not a few minutes as recommended by Harkins and Brown. Figure 10 shows drop volume against T//° g f o r a s p h e r i c a l g l a s s t i p . These p r e l i m i n a r y i n v e s t i g a t i o n s were only meant to i n d i c a t e the approximate r e l a t i o n s h i p , between drop volume and T//°g under v a r i o u s t i p geometries and t h e r e f o r e the usual precautions f o r accurate surface t e n s i o n work were not taken. 29 0.10 Drop Volume \- (cm.J) 0 . 0 / 0.00/ I J I I I I o.oi 0.05 T//?g (cm. 2) FIG, 10 Drop volume for a spherical glass t i p . 30 Optimum Dropping Height A l i q u i d drop f a l l i n g into a bath of the same l i q u i d w i l l u s ually form a vortex r i n g f o r a l l dropping heights between 0 and 4 cm. However, for c e r t a i n dropping heights the vortex r i n g travels much further i n the bath than for other dropping heights. Let the q u a l i t y of a vortex r i n g be measured by the v e r t i c a l distance D which i t travels i n the bath before stopping or breaking up. On a graph with D as the ordinate and h as the abscissa, those dropping heights for which D i s a l o c a l maximum.may be c a l l e d optimum dropping heights, or h-opt. C h a r a c t e r i s t i c h-D behaviour i s shown i n figure 11 for f i v e d i f f e r e n t l i q u i d s . Most l i q u i d s portray one l o c a l maximum at a dropping height between 1 and 3 cm. Since the peaks are f a i r l y broad, values of h-opt are accurate to about ± 1 mm. or an error of 5% for a t y p i c a l drop-ping height of 2.0 cm. Given a l i q u i d , i t would be useful ;to be able.-to~predict the dropping height which corresponds to a maximum i n D. Figure 12 shows the p o s i t i o n of the f i r s t optimum dropping height as a function of T/Z'g. A l i n e has been drawn to indicate the trend i n the graph. The p o s i t i v e slope r e s u l t s not only from an increase i n o s c i l l a t i o n time but also r e f l e c t s a T/./° dependence i n the nature of the drop detachment. For any one l i q u i d the-magnitude of D at h-opt i s roughly l i n e a r with dropvolume. The extrapolated D intercept i s non-zero. Drop O s c i l l a t i o n Time Raleigh ,(1897) derived an o s c i l l a t i o n time T for the v i b r a t i o n of a l i q u i d mass about a spherical shape to be 31 FIG. 11 h-D p l o t showing f i r s t optimum.height f o r (A) freon 113, (B) freon 114-B2, (C), methyl a l c o h o l , (D) benzylamine, and (E), b u t y l a l c o h o l . 32 33 where V i s the volume of the l i q u i d sphere. This r e l a t i o n s h i p was v e r i f i e d by photographic i n v e s t i g a t i o n of a f a l l i n g water drop ( F i g . 13),using 75 c y c l e / s e c . stroboscopic i l l u m i n a -t i o n . Figure 14 shows the e c c e n t r i c i t y a/b of the drop as a f u n c t i o n of dropping h e i g h t , a and b being the h o r i z o n t a l and v e r t i c a l diameters r e s p e c t i v e l y , a/b> 1 corresponds to an oblate spheroid, a/b< 1 to a pro l a t e spheroid, and a/b=l to a s p h e r i c a l shape, h i n d i c a t e s the p o s i t i o n of the center of the drop. Dropping heights were r e l a t e d to time i n t e r v a l s by assuming free f a l l f o r the drop. For a water drop* of .062 cm.3 volume, the d i f f e r e n c e i n f a l l .times between heights of 0.9 cm. and 2.9 cm. f o r which the drop i s s p h e r i c a l i s .034 sec. The c a l c u l a t e d o s c i l l a t i o n time f o r the same drop i s ,032 sec. This i s considered to be reasonable agreement since the i n i t i a l d i s t o r t i o n has a large amplitude and i s not symmetrical about the e q u i l i b r i u m shape. The process i s p a r t i a l l y a r e d i s t r i b u t i o n of mass and not a true s p h e r i c a l o s c i l l a t i o n . C o r r e l a t i o n Between Optimum Heights and T In f i g u r e 15, the l i q u i d drop o s c i l l a t i o n i s shown together w i t h an h-D p l o t of the f i r s t l o c a l maximum f o r d i s t i l l e d water using a s p h e r i c a l g l a s s t i p . Dropping height h i s used as a common or d i n a t e . The a c t u a l s i z e of the drop i s represented by a sphere of .062 cm.3 volume. In f i g u r e 15-C:it may be seen that v o r t e x r i n g s are best formed when the dropping height is.1.7 em. For t h i s dropping height the drop i s s p h e r i c a l i n shape when i t . t o u c h e s the bath ( F i g . 15-A). The drop i s a l s o s p h e r i c a l when the dropping height i s . 0.9 cm. and 2,9 cm. but the corresponding v o r t e x r i n g formation i s i n f e r i o r . 34 1.0 CM. FIG. 13 Drawing of a f a l l i n g water drop a f t e r detaching from a s p h e r i c a l t i p . Negatives were projected onto a screen and copied at a m a g n i f i c a t i o n of about 12. 0.7 i.o A3 a/b a/b> 1 i/b< 1 h (cm.) FIG. 14 The e c c e n t r i c i t y of a water drop i n free f a l l showing drop o s c i l l a t i o n . Drop volume i s 0.062 cm. 3 6 There are therefore two c o n d i t i o n s f o r optimum formation of v o r t e x r i n g s : the drop must be s p h e r i c a l or near s p h e r i c a l upon contact w i t h the bath, and the drop must be changing from an oblate spheroid to a p r o l a t e spheroid. The i n t e r n a l v e l o c i t y of the drop as shown i n f i g u r e 1 5 - B may be described by a Stokes stream f u n c t i o n i f = A i n t . 2 where A i s a constant. Thus f o r optimum vo r t e x r i n g formation, the com-posit e flow at the moment of impact i s of the form T r 2 = (Az - B) — comp. 2 where B i s a constant connected w i t h the t r a n s l a t i o n a l v e l o c i t y of the drop. A second maximum i n D should then appear at a dropping height which d i f f e r s from the f i r s t optimum height by a distance equivalent to one period of o s c i l l a t i o n . This behaviour was observed and i s shown i n f i g u r e 16 f o r water and for e t h y l a l c o h o l using a. c o n i c a l t i p . Values of t , the time i n t e r v a l between optimum h e i g h t s , and T are subject to p o s s i b l e e r r o r s of 1 0 7 c and 5 % r e s p e c t i v e l y . The d i f f e r e n c e s between t and T f o r water ( 8 7 , ) , benzene ( 3 7 0 ) , e t h y l a l c o h o l ( 4 7 0 ) and l i q u i d n i t r o g e n ( 2 7 0 ) are w e l l w i t h i n the experimental e r r o r . These c o n d i t i o n s q u a l i t a t i v e l y agree with Thomson's e x p l a n a t i o n of t h i s phenomena. Since the d r o p . i s changing from an oblate to a pro l a t e spheroid, the i n t e r n a l v e l o c i t y f i e l d i s such as to preserve the s p h e r i c a l shape upon impact. A v o r t e x f i l m w i l l form and then r o l l up i n t o a v o r t e x r i n g when the drop,has become disk-shaped. On the other hand, i f the drop i s changing from a p r o l a t e to an oblate spheroid j u s t before contact w i t h the bath then both the impact and the o s c i l l a t i o n act together to f l a t t e n 37 water: D (cm.) h t H Vol = 0.07 cm.3 = .034 sec. 0 AO .^0 30 H.Q f.O h (cm. ) i e t h y l a l c o h o l : 3 h (cm. ) FIG. 16 h-D p l o t showing two optimum he i g h t s . Values of t and T are shown f o r comparison. 38 the drop. The drop w i l l pass through the d i s k shape before vortex.motion spreads s u f f i c i e n t l y and the v o r t e x r i n g i f formed at a l l , w i l l be of i n f e r i o r q u a l i t y . Cryogenic L i q u i d s Vortex r i n g s were produced i n l i q u i d n i t r o g e n i n accordance w i t h the method o u t l i n e d above. Granules of p a r a f f i n w i t h a d e n s i t y of 0.85 . 3 g,/cm. were..found to be a s u i t a b l e d e t e c t o r . Two optimum.heights were observed at 1.3 cm. and 3.9 cm. This: i s a s e p a r a t i o n of .038 seconds f a l l time, i n good agreement w i t h the c a l c u l a t e d drop o s c i l l a t i o n of .037 seconds. . With h equal to the f i r s t optimum dropping h e i g h t , the maximum depth i n the bath f o r which v o r t e x r i n g s were observed was 4 cm. Vortex r i n g s were d e f i n i t e l y detected, since f o r h between the f i r s t and second optimum heights (2.5 cm.) the drop caused no movement of the p a r t i c l e s on the d i s h 5 mm. below the bath surface. From the d e t e c t i o n method employed, i t i s not c e r t a i n whether the n i t r o g e n v o r t e x r i n g s stopped or went turbu-l e n t at a depth of about 4 cm. Vortex r i n g s were not: observed, i n l i q u i d helium. D i f f i c u l t y was encountered i n f i n d i n g a s u i t a b l e d e t e c t i o n powder. The hydrogen-deuterium mixture, produced by the same method as Chopra and Brown (1957), s o l i d i f i e d i n c l u s t e r s of quite large p a r t i c l e s . Fine, ground cork p a r t i c l e s of d e n s i t y , 3 0.2 g./cm. were t r i e d and found to be too dense. The most s u i t a b l e detector 3 was a powder of b a l s a wood having a d e n s i t y of 0.16 g./cm. . P a r t i c l e motion was observed only when the d i s h was very c l o s e to the bath s u r f a c e ; 2 mm. depth f o r a dropping: height of 5 cm. The surface t e n s i o n of l i q u i d helium, and thus T / g changes co n s i d e r a b l y w i t h temperature as i s shown i n t a b l e I I (Atkins,; 1959). 39 TABLE I I Surface Tension and Density of L i q u i d Helium Temp T /° T//°g-x:ld3 °K dynes/cm. g/cm? cm 2 4.2 0.08 .125 0.65 3.0 0.21 .141 1.52 2.2 0.29 .146 2.03 1.2 0.34 .145 2.39 I t has been noted i n t h i s chapter that D at h-opt i s roughly l i n e a r w i t h drop, volume f o r a given l i q u i d . Since drop .volume i s a f u n c t i o n of T//°g, the maximum drop s i z e and therefore an increased l i k e l i h o o d of observing v o r t e x r i n g s would be found at 2.2°K i n helium I and 1.2°K in; helium I I . At 1.85°K the measured drop,volume was 1.2 x 10 cm. . Eigh t lengthy t r i a l s were conducted w i t h l i q u i d helium,, of which two are described here. In l i q u i d helium I , f o r temperatures between 2.2°K and 3.0°K, the d i s h was set j u s t below ( l e s s than 1 cm.) the bath surface and the dropping height was changed from 0 to 2 cm. i n 2 mm. i n t e r v a l s . F i v e drops were formed at each dropping height but no v o r t e x r i n g s were detected. In l i q u i d helium I I , f o r 1.6°K and the d i s h j u s t below the bath, no v o r t e x r i n g s were observed f o r dropping heights between 0 and 2.5 cm. Surface E f f e c t s I t may be seen from f i g u r e s 11 and 16 that v o r t e x r i n g s are formed even at dropping heights which approach zero. Since surface energy i s approximately twice the k i n e t i c energy f o r a drop placed on a l i q u i d s u r face, t h i s energy must c o n t r i b u t e c o n s i d e r a b l y to the vor t e x r i n g energy. 40 Secondly i f the v o r t e x r i n g energy was obtained completely.-from the k i n e t i c energy of the drop, one would expect an increase i n D f o r successive i n c r e a s i n g h-opt i n any one l i q u i d . This i s not the case. Values of D. f o r the f i r s t three optimum heights i n benzene were found to be 9.0, 10.0 and 8.6 cm. while the drop, k i n e t i c energies corresponding to these optimum heights increased by the r a t i o of 1:3.5:7.2. Thomson and Newall's paper contains an h-D p l o t f o r smaller water drops showing a general decrease i n D w i t h i n c r e a s i n g order of h-opt over f i v e optimum he i g h t s . I t would seem from these observations that surface t e n s i o n e f f e c t are s i g n i f i c a n t and thus the model of v o r t e x r i n g f o r m a t i o n - i n chapter I I has at l e a s t q u a l i t a t i v e j u s t i f i c a t i o n . Anomalous Results I t may be seen from t a b l e I that three room temperature l i q u i d s e x h i b i t e d unusual behaviour. Vortex r i n g s were produced i n isopentane, acetone and freon 11 at near zero dropping h e i g h t s . That i s , i f the droppin height was such that the f u l l y grown drop, touched the bath at the same moment that i t detached i t s e l f from the t i p , then i n a l l three l i q u i d s v o r t e x r i n g s were formed which t r a v e l l e d more than 6 cm. For any greater dropping heights the v o r t e x r i n g d i s i n t e g r a t e d almost immediately a f t e r the drop entered the bath. Reynolds numbers were c a l c u l a t e d f o r a l l drops using the known kinematic v i s c o s i t y , the drop diameter c a l c u l a t e d from volume measurements, and a constant v e l o c i t y . The Reynolds numbers f o r isopentane, acetone and freon 11 are s i g n i f i c a n t l y higher than any other room temperature l i q u i d used, i n f e r r i n g that f o r non-zero dropping heights the flow i s t u r b u l e n t . I f t h i s i s . the sole reason f o r r i n g breakup,, then i t i s s u r p r i s i n g that 41 vortex rings were observed i n l i q u i d nitrogen which has a much higher Reynolds number than a l l of the room temperature^ l i q u i d s . Furthermore, since evaporation effects may be hampering vortex ring formation, experi-ments with acetone, freon 11 and isopentane should be repeated i n a dewar surrounded by an ice bath. F i n a l l y i t would be useful to study more li q u i d s having low v i s c o s i t i e s and therefore high Reynolds numbers since there i s some doubt whether vortex rings can be formed.by a drop method i n such l i q u i d s . 42 CONCLUSIONS The i n i t i a l o bject of t h i s study was to produce v o r t e x r i n g s i n l i q u i d helium by the drop, method. In endeavouring to do so, i t was found that a general i n v e s t i g a t i o n of the formation of vo r t e x r i n g s by a l i q u i d drop f a l l i n g i n t o a bath at r e s t was necessary. In p a r t i c u l a r i t was d e s i r a b l e to determine the optimum dropping heights and the re q u i r e d detector depth f o r any l i q u i d whose p r o p e r t i e s are known. L i q u i d s were chosen so as to cover a uniformly d i s t r i b u t e d and wide range of surface t e n s i o n to d e n s i t y r a t i o . The kinematic v i s c o s i t i e s of the l i q u i d s used, with the exception of l i q u i d helium, ranged from 2 0.002 to 0.037 cm. /sec. Optimum dropping heights were observed.in agreement with Thomson and Newall. For a s p h e r i c a l g l a s s drop t i p of 1.3 cm. radius of curvature, the f i r s t optimum height f o r a l i q u i d was found to l i e near a s t r a i g h t l i n e of slope 0.4 passing through the point (0.2 cm.2, 1.5 cm.) on a l o g - l o g p l o t of h versus l//°g. Successive optimum heights were found t o d i f f e r by time i n t e r v a l s equal to T , the c a l c u l a t e d drop o s c i l l a t i o n time. A photographic study of a . f a l l i n g water drop was made i n order to c o r r e l a t e the optimum heights and drop o s c i l l a t i o n . There are two c o n d i t i o n s f o r optimum v o r t e x r i n g forma-t i o n ; the drop.must be s p h e r i c a l upon contact w i t h the bath, and the drop must be changing from a oblate spheroid to a pro l a t e spheroid. Vortex r i n g s were formed, i n acetone, freon 11 and isopentane, by p l a c i n g the drop.on the bath surface, but at any greater-dropping heights :the vortex r i n g d i s i n t e g r a t e d Immediately. Since Reynolds numbers f o r these three l i q u i d s are s i g n i f i c a n t l y higher than the other room temperature 43 . l i q u i d s , the flow may be turbulent for non-zero dropping heights. Vortex rings were detected, i n l i q u i d nitrogen at. the predicted dropping heights. From the detection method employed,, i t remains uncertain whether the vortex rings stop or the flow.becomes turbulent at a depth of a few centimeters. Because of the low surface tension of l i q u i d helium, drop volumes, and thus magnitude of ring impact on the dish was expected to be very small. A number of t r i a l s were made with l i q u i d helium I and l i q u i d helium II at temperatures for which the drop size was a. maximum, and near the predicted f i r s t optimum height. No vortex rings were detected. Energy considerations, as well as experimental r e s u l t s i n f e r that surface tension cannot be ignored i n the formation of vortex rings. A simple model i s proposed which considers a stationary l i q u i d surface just a f t e r the drop has contacted the bath. Pressure gradients, r e s u l t i n g from surface curvature, and g r a v i t y forces both contribute to the c i r c u l a t i o n required for a vortex r i n g . Numerical methods involving a high speed computer should be employed to provide a t h e o r e t i c a l s o l u t i o n to the problem. Because of i t s s i m p l i c i t y and physical s u i t a b i l i t y , the model based on surface tension should serve as a f i r s t approximation. A second and better approximation involves adding an i n i t i a l v e r t i c a l v e l o c i t y to the f l u i d elements i n the drop, but s t i l l commencing the flow a few moments a f t e r contact. The exact problem of the formation of vortex rings by a drop method may also be solvable since the shape and the. motion of the drop required for optimum vortex r i n g formation, are now known. 44 BIBLIOGRAPHY A l l e n , J . F. and Jones, H., (1938), Nature 141, 243. A l l e n , J . F. and Misener, A. D., (1938), Nature 141, 75. A t k i n s , K. R., (1959), L i q u i d Helium, Cambridge U n i v e r s i t y Press, Cambridge. B a l l , R. S., (1868), P h i l . Mag. 36, 12. Brown, R. C. and McCormick, ,H. , ,(1948) , P h i l . Mag. 39, 420. Chopra, K..L. and Brown, J . B., (1957), Phys. Rev. 108, 157. Daunt, J . G. and Mendelssohn, K. , (1939),' Proc. Roy. Soc. (London). 170, 143. Edwards, M. H., p r i v a t e communication. Feynman, R. P., (1955), Progress i n Low Temperature Physics, Ed. C.J. Gorter, North Holland P u b l i s h i n g Co., Amsterdam, Volume 1. Fromm, J . , (1963), Los Alamos S c i e n t i f i c Lab., Report number LA2910. Harkins, W. D. and Brown, F. E., (1919), J . Am. Chem. Soc. 41, 499. Helmholtz, H., (1868), Monatsber. Akad. Wiss., B e r l i n , 215. K a p i t z a , P. L., (1938), Nature 141, 74. Keesom, W. H., and van den Ende, J . N., (1930), Commun. Leiden, No. 203d, Proc. Roy. Acad. Amsterdam 3_3, 243. Lamb, H. , (1895), Hydrodynamics, Dover P u b l i c a t i o n s , , New York. Landau, L. D., (1941), J . Phys. Moscow 5, 71. Northrup, E. F., (1912), Nature 88, 463. Onsager, L., (1949), Nuovo cimento j5, supplement 2, 249. Osborne, D. V., (1950), Proc. Phys. Soc. (London) 63, 909. Rale i g h , Lord, (1879), Proc. Roy. Soc. (London) 29, 71. R a y f i e l d , G. W. and R e i f , -F. , ..(1963), Phys. Rev.. L e t t e r s 11, 305. Reppy, J . D. , Depatie, D. and Lane, C. T. , (1960), Phys. Rev. L e t t e r s 5_, 541. Reynolds, 0., (1876), Nature 14, 477. 45 Rogers, W. B., (1858), Am. J . S c i . 26, 246. Taylor, G. I . , (1953), J . Appl. Phys. 24, 104 Thomson, J . J . and Newall, H. F., (1885), Proc. Roy. Soc. (London) 39, 417. Thomson, S i r W., (1867), P h i l . Mag. 34, 15. T i s z a , L. , (1940), J . phys. radium 1^ , 165. ,Vinen, W. F. , (.1958), Nature 181, 1524. 

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