{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"Aggregated Source Repository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Chung, David Yih","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"Date Available","value":"2011-11-28T18:46:52Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"Date Issued","value":"1962","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree (Theses)","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"Degree Grantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"This thesis describes the investigation of hydrodynamic properties of pure superfluid flow in liquid helium II by observing the motion of suspended particles in a special experimental arrangement called 'Superfluid Wind Tunnel'.\r\nThe flow properties of pure superfluid in different velocity regions have been investigated by using particles made with a suitable mixture of hydrogen and deuterium gases as indicators.\r\nTwo critical velocities, Vp,c and Vs,c, corresponding to 0c and 0t of the oscillating sphere experiments (Benson and Hallett (1956)) have been found. Below Vp,c, the superfluid flow is a perfect potential flow of zero viscosity. Above Vp,c quantized vortex lines are created, therefore the pure superfluid flow breaks down. On the other hand in the vicinity of Ys,c, the starting point of fully developed turbulence, the magnitude of turbulent fluctuations has a maximum which confirms Feynman's prediction (1955) about critical velocity.\r\nA rough calculation shows that the velocities of a particle, which obtains energy from a segment of quantized vortex line, are of the same order as that of experimental values. This suggests that by this way, other than Vinen's (1961) vibrating wire experiment, the quantization of superfluid circulation in units of h\/m might be verified by visual observations.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"Digital Resource Original Record","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/39296?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"Full Text","value":"SUSPENSION OF PARTICLES IN THE SUPERFLUID WIND TUNNEL by DAVID YIH CHUNG B.Sc., National Taiwan University, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics j, We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1962 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia, Vancouver 8, Canada. Department Date \/ , ABSTRACT This thesis describes the investigation of hydrodynamic properties of pure superfluid flow i n l i q u i d helium II by observing the motion of suspended p a r t i c l e s in a s p e c i a l experimental arrangement c a l l e d 'Superfluid Wind Tunnel'. The flow properties of pure super f l u i d i n d i f f e r e n t v e l o c i t y regions have been investigated by using p a r t i c l e s made with a su i t a b l e mixture of hydrogen and deuterium gases as indicators. Two c r i t i c a l v e l o c i t i e s , V p ^ c and V s c , corresponding to 0 and 0 t of the o s c i l l a t i n g sphere experiments (Benson and H a l l e t t (1956)) have been found. Below V p^ c, the superfluid flow i s a perfect p o t e n t i a l flow of zero v i s c o s i t y . Above V p^ c quantized vortex l i n e s are created, therefore the pure superfluid flow breaks down. On the other hand i n the v i c i n i t y of Ys>c> t a e s t a r t i n g point of f u l l y developed turbulence, the magnitude of turbulent fluctuations has a maximum which confirms Feynman's prediction (1955) about c r i t i c a l v e l o c i t y . A rough c a l c u l a t i o n shows that the v e l o c i t i e s of a p a r t i c l e , which obtains energy from a segment of quantized vortex l i n e , are of the same order as that of experimental values. This suggests that by t h i s way, other than Vinen's - i i -(1961) v i b r a t i n g wire experiment, the quantization of superf l u i d c i r c u l a t i o n i n units of h\/m might be v e r i f i e d by v i s u a l observations. - i i i -ACKNOWLEDGEMENTS It i s a great pleasure to express my gratitude to Dr. P. R. Critchlow who suggested the problems attempted i n t h i s thesis, for his active help and many informative comments. I am indebted to Dr. D. V. Osborne for his valuable assistance and discussions toward the con-clusion of the thesis. I also wish to acknowledge the valuable help and encouragement I received from Dr. J. B. Brown, p a r t i c u l a r l y in the la s t stage of the project. I am also g r a t e f u l to Mr. R. Weissbach for h is help and the production of l i q u i d helium; to Mr. J. Lees for performing the glass blowing of the apparatus and to Mr. K. P. Lee for his assistance throughout most of the experiments. F i n a l l y , the support given by National Research Council of Canada i n t h i s project i s g r a t e f u l l y acknowledged. - i v ~ TABLE OF CONTENTS Page Abstract i i Acknowledgements iv L i s t of I l l u s t r a t i o n s and Tables v i i Chapter I. Introduction 1 II. Theory of Liquid Helium II 5 2.1. Two F l u i d Model 5 2.2. Thermomechanical and Mechanocaloric E f f e c t s 9 2.2.1. London's Formula 10 2.2.2. Experimental Fact 11 2.3. Isothermal Flow Through Narrow Channels and C r i t i c a l V e l o c i t y 13 2.4. Feynman's Theory of Quantized Vortex Line and Turbulent Superflow 17 III. Experimental Arrangements 23 3.1. Apparatus 23 3.1.1. Vacuum System and A u x i l i a r y Equipment 23 3.1.2. The Superfluid Wind Tunnel and Superfluid V e l o c i t y Measurements 25 3.1.3. Formation of P a r t i c l e s 27 3.2. Experimental Procedure 29 3.2.1. Cryogenics 29 3.2.2. The Veloc i t y Measurements 30 (i) Superfluid V e l o c i t y Measurements 30 ( i i ) P a r t i c l e V e l o c i t y Measurements 31 -v-3.2.3. Temperature Measurements 33 IV. Experimental Results 34 4.1. Results of Flow Velocity (V g) Measurements 34 4.2. Results of P a r t i c l e V e l o c i t y (V p) Measurements 36 V. Discussion 38 5.L Theoretical Background 38 5.2. Discussion of Results 40 5.2.1. Flow V e l o c i t y (V_) Measurements 40 5.2.2. P a r t i c l e V e l o c i t y (V p) Measurements and Flow Properties of Pure Superflow 41 (i) In the Region V_<1 V_ \u201e 48 P P \u00bb C ( i i ) In the Region V < V p < V 49 ( i i i ) In the V i c i n i t y of V s 50 (iv) In the Region V p > V g c 52 VI. Conclusions 54 Bibliography 57 - v i -LIST OF ILLUSTRATIONS Figure To follow page 1. The energy-momentum spectrum of phonons and rotons 8 2. Diagram of apparatus f o r thermomechanical e f f e c t 10 3. The graph for volume flow against heat supplied 12 3a. The graph for flow v e l o c i t y v s versus power input 12 4. The vacuum system 23 5. The apparatus for superfluid wind tunnel 25 6. The superfluid v e l o c i t i e s (v s) and the mean p a r t i c l e v e l o c i t i e s (v ) versus power input (1st run) 35 7. v s and v^ versus power input (2nd run) 35 8. v s and Vp versus power input (3rd run) 35 9. The in t e n s i t y of turbulence versus power input 51 LIST OF TABLES Table 1. Experimental values of v s and v^ 35a 2. Data on standard deviations and in t e n s i t y of turbulence as a function of mean p a r t i c l e v e l o c i t i e s v p 51a 3. Theoretical values of p a r t i c l e v e l o c i t i e s v p for p a r t i c u l a r values of p and q 48 - v i i --1-4 CHAPTER I INTRODUCTION Since the discovery of peculiar hydrodynaniic properties of l i q u i d helium II i n 1939 by Kapitza, and Allen and Misener, many theories have been proposed to explain the phenomena, of which the two f l u i d model seems to be most successful. By t h i s model, the flow properties of l i q u i d helium can be described i n some d e t a i l . The two f l u i d theory assumes the l i q u i d to be an intimate mixture of two components each capable of independent movement. One of them, the normal f l u i d , i s a 'gas' composed of the thermal excitations (phonons and rotons) moving randomly through the whole l i q u i d , and i t may be accelerated by ordinary viscous forces of the gas-kinetic type. The supe r f l u i d contains the r e s i d u a l momentum and k i n e t i c energy of the l i q u i d and behaves as a perfect f l u i d at absolute zero without entropy or v i s c o s i t y . Using t h i s form i t i s possible to describe quantitatively the flow of l i q u i d helium II at not too high v e l o c i t i e s , the e s s e n t i a l properties being the i n v i s c i d behavior of the superfluid, the free interpenetration of the two components and the thermodynamic mutual force which drives normal f l u i d down temperature gradients. If the superfluid has zero v i s c o s i t y i t should be -2-lmpossible to set i t i n r o t a t i o n from a condition of r e s t , but t h i s can be done and the explanation of t h i s anomaly i s found i n the t h e o r e t i c a l r e s u l t of Feynman (1955) that c i r c u l a t i o n i s quantized i n units of h\/m (h i s Planck's constant, m i s the mass of a helium atoms) and i n the inference that v o r t i c i t y i s l o c a l i z e d in vortex-lines of diameter comparable with the interatomic spacing. Once v o r t i c i t y i s introduced into the superfluid i t can be increased by elongation of the l i n e s i n a kind of turbulent flow and, although most of the superfluid w i l l be in i r r o t a t i o n a l motion, the motion induced by the vortex-l i n e s w i l l appear r o t a t i o n a l on the macroscopic scale. H a l l and Vinen have made an important extension by pointing out that the excitations of the normal f l u i d w i l l be scattered i n the v e l o c i t y f i e l d of the vortex l i n e s and that t h i s w i l l cause a mutual f r i c t i o n i f the vortex-lines are moving r e l a -t i v e to the normal f l u i d . In a ser i e s of papers they have presented very convincing evidence i n favor of the notion that mutual f r i c t i o n between the tv\/o f l u i d s i s e n t i r e l y caused by t h i s e f f e c t . Once the quantized vortex l i n e s are created, i . e . above c r i t i c a l v e l o c i t y , they w i l l interact with the excitations of the normal f l u i d . Therefore mutual f r i c t i o n appears. The existence of a mutual f r i c t i o n provides the simplest explanation of the existence of a c r i t i c a l flow v e l o c i t y above which the simple two f l u i d theory i s no longer v a l i d . -3-Pellam et a l . (1957, 1962) have developed a 'Pure Superfluid Wind Tunnel' within which only superfluid component can move, while the background normal f l u i d remains stationary. They have proven experimentally (for not too high superfluid v e l o c i t y ) that the pure po t e n t i a l superflow exerts zero-force but a prescribed torque on an object placed i n th e i r superfluid wind tunnel. However, in measuring the l i f t on an a e r o f o i l placed i n a uniform stream of superfluid, the l i f t should be proportional to the c i r c u l a t i o n round the a e r o f o i l . But the measurements showed no evidence for quantization. In t h i s thesis, a pure super f l u i d wind tunnel s i m i l a r to that of Pellam et a l * i s used. By observing the motion of suspended p a r t i c l e s i n t h i s pure superflow, the hydro-dynamic properties of turbulent superflow have been investigated. A general review of the current theory for l i q u i d helium II i s presented i n Chapter I I 4 In p a r t i c u l a r , a detailed description of the idea of quantized vortex l i n e s i n turbulent supe r f l u i d flow i s given i n section 2.4. Details of the apparatus and of the experimental pro-cedure w i l l be described in Chapter I I I , and the f u l l experimental r e s u l t s w i l l be presented in Chapter IV. A detailed discussion of the behavior of pure superflow i n d i f f e r e n t v e l o c i t y regions w i l l be described i n terms of quantized vortex l i n e s which have been shov\/n i n section 2.4. F i n a l l y , a conelusional remark and the suggestion f o r future work w i l l be given i n Chapter V. -5-CHAPTER II THEORY OF LIQUID HELIUM II 2.1. Two F l u i d Model Liquid helium exhibits quantum mechanical properties on a large scale; no other substance remains l i q u i d to such a low temperature to exhibit the e f f e c t s . Helium d i f f e r s from a l l other substances in that i t i s l i q u i d even down to 0\u00b0K under i t s vapor pressure. C l a s s i c a l l y , at 0\u00b0K a l l motion stops, but quantum-mechanically t h i s i s not so. Helium stays l i q u i d as London (1938) pointed out, because the interatomic forces are very weak and the quantum zero-point motion i s l a r ^ e , and the atomic mass i s small to keep i t f l u i d down to absolute zero. When l i q u i d helium i s cooled down below 2.1S\u00b0K, the so-called A-point, i t behaves in a completely d i f f e r e n t fashion than i t does above the A-point. The t r a n s i t i o n from l i q u i d helium I -l i q u i d above A-point, to helium II involves no change in latent heat, no discontinuous change in volume, but a s p e c i f i c heat anomaly. The word A arises from the peculiar shape of s p e c i f i c heat versus temperature curve. One of the most remarkable properties of helium II i s i t s s u p e r f l u i d i t y , which was discovered independently by - 6 -K a p i t z a (1933) and by A l l e n and Misener (1938). L i q u i d helium II can flow through very narrow channels of the order 10\"^  cm. w i t h a v e l o c i t y of many cm.\/sec, correspond-i n g t o a v i s c o s i t y of the order 10\"*'*'* p o i s e . The flow a l s o behaves almost independent of pressure head. I t i s o f t e n convenient to c o n s i d e r l i q u i d helium II as i f i t were a mixture of two f l u i d s . But we must note \"that t h i s p i c t u r e i s nothing more than a method of d e s c r i p -t i o n . A c t u a l l y i n quantum l i q u i d , such as helium I I , i t i s p o s s i b l e t o have s i m u l t a n e o u s l y two motions each of which i s a s s o c i a t e d w i t h i t s own e f f e c t i v e mass, siich t h a t the sum of these masses i s equal t o the t o t a l mass of l i q u i d , namely = J 3 ^ + f*s where ^ ( d e n s i t y of normal p a r t of the l i q u i d ) e x h i b i t s the c h a r a c t e r i s t i c s of an o r d i n a r y l i q u i d , w h i l e J\"^ ( d e n s i t y of s u p e r f l u i d p a r t of the l i q u i d ) e x h i b i t s the s u p e r f l u i d p r o p e r t i e s . The motions of normal and s u p e r f l u i d take p l a c e without any t r a n s f e r of momentum between them (as long as the r e l a t i v e v e l o c i t y i s s m a l l ) . With t h i s p i c t u r e , the s o - c a l l e d two f l u i d model, the p e c u l i a r b e h a v i o r s of helium II may be e x p l a i n e d . One of the most important parameters i s the r a t i o _Pn\/jP . At 0\u00b0K t h i s r a t i o i s z e r o and i n c r e a s e s as tem-p e r a t u r e i n c r e a s e s , f i n a l l y r e a c h i n g a value of u n i t y at the >k-point. Thus the t r a n s i t i o n from helium II to helium I i s c h a r a c t e r i z e d by the disappearance of the s u p e r f l u i d component. A n d r o n i k a s h v i l i (1946) u s i n g a p i l e of o s c i l l a t i n g d i s c s performing t o r s i o n a l o s c i l l a t i o n s i n l i q u i d helium II measured f*\/f a t d i f f e r e n t temperatures. The r e s u l t s agreed w i t h the t h e o r e t i c a l p r e d i c t i o n s . Because there i s no completely s a t i s f a c t o r y theory of the l i q u i d s t a t e , i t i s tempting to t r e a t a l i q u i d e i t h e r as a very imperfect gas or as a broken down s o l i d . The t h e o r i e s of l i q u i d helium t o be d i s c u s s e d i n the present s e c t i o n are London's theory based on the phenomenon of B o s e - E i n s t e i n condensation i n an i d e a l gas and Landau's theory from the theory of the s o l i d s t a t e . Using B o s e - E i n s t e i n condensation London showed t h a t at a c r i t i c a l temperature Tc the atoms t>egin t o condense i n t o the lowest energy s t a t e . The f r a c t i o n so condensed v a r i e s from 0 at T c to 1 at 0\u00b0K. The atoms i n the ground s t a t e are a s s o c i a t e d w i t h the s u p e r f l u i d component, w h i l e the atoms i n the e x c i t e d s t a t e s are a s s o c i a t e d with the normal component. The zero entropy of the s u p e r f l u i d component i s thereby immediately e x p l a i n e d . The condensa-t i o n temperature f o r an i d e a l gas with atomic mass and d e n s i t y of l i q u i d helium would be at 3.14\u00b0K, which i s not very f a r from the observed A-point of 2.1S\u00b0K. Landau, on the other hand, used a completely d i f f e r e n t s t a r t i n g p o i n t . He emphasized the f a c t t h at helium behaved as a background f l u i d i n v\/hich e x c i t a t i o n s move. At a b s o l u t e zero he p i c t t i r e d helium as a p e r f e c t f l u i d -8-which may flow f r l c t i o n l e s s l y as a pure p o t e n t i a l flow. If heated, the energy produced excitations i n the l i q u i d which c o l l i d e with the walls and with each other character-ize the normal f l u i d component. Further, Landau supposed there were two kinds of excitations, phonons and rotons, which can move f r e e l y within the l i q u i d helium. His analysis leads to an energy spectrum of the type shown i n figu r e 1. The lowest energy of excitations are phonons, whose energy \u00a3 equals pc, where p i s the momentum and c the speed of sound. Above these, separated by an energy gap A . , are those c a l l e d rotons. He supposed l a t e r that E r o t o n - - + <P - P o > 2 A i where p Q i s zero-point momentum, p. i s e f f e c t i v e mass. By applying s t a t i s t i c s one can derive expressions for the thermodynamics quantities and . A l l these quantities are sums of two contributing terms, one for rotons and one fo r phonons. Landau predicted a pronounced change i n the properties of the l i q u i d at about 1\u00b0K below which rotons v\/ould r a p i d l y vanish. The confirmation of t h i s change by the steep increase of the v e l o c i t y of second sound (thermal wave) was a triumph f o r his interpretation. At low enough temperature where the phonons are dominant, the d i r e c t description by means of phonons of long mean free paths seems appropriate. -9-But i n the neighborhood of A-temperature both views lead to d i f f i c u l t i e s . For an i d e a l gas the Bose-Einstein type of condensation predicts a jump i n the deri v a t i v e of s p e c i f i c heat rather than s p e c i f i c heat i t s e l f and more or less a r t i f i c i a l assumptions are required to adjust t h i s discrepancy. On the other hand i t i s d i f f i c u l t to see how, in Landau*s picture, an assembly of those excitations can gradually approach the whole normal l i q u i d at the A-point. However, the two f l u i d model, i n one or other of i t s several forms, has met with a good deal of success i n explaining the peculiar and unique properties of l i q u i d helium II. 2.2. Thermomechanical and Mechano-Calorie E f f e c t s The f i r s t experimental observations of the thermo-mechanical e f f e c t were made by A l l e n , P e i e r l s and Uddin (1937) who i n attempts at measuring the heat conduction of l i q u i d helium II found a peculiar flow of l i q u i d occurring i n the opposite d i r e c t i o n to the flow of heat. The iaechano-caloric e f f e c t i n helium II r e f e r s to the fa c t that when helium II flows from the helium bath to a container through a narrow channel or a tightly-packed plug cooling of the container i s observed. However, the f i r s t theory connected v\/ith these f a c t s was formulated by H. London (1939). The quantitative r e s u l t s from experiments 2 0 Figure I. The energy-momentum spectrum of phonons and rotons to fo l low page 8 10-were f i r s t obtained by Kapitza (1941). 2.2.1. London's Formula H. London (1939) used purely thermodynamical approach by treating the phenomenon of the thermo\u2014mechanical e f f e c t as a r e v e r s i b l e cycle s i m i l a r to that presented by a thermal-electric c i r c u i t . Consider the system (see figure 2) as a r e v e r s i b l e heat engine where i n the heated reservoir the r i s e of l i q u i d helium II produces a pressure difference __\\ p between the two volumes which d i f f e r in temperature by T. Assuming S to be the difference in entropy between the l i q u i d passing through the c a p i l l a r y and that doing work i n the return path from the higher to the lower l e v e l , London obtained the general r e l a t i o n (2.1) where j i s helium density at t h i s temperature. The heat of transport Q which i s supplied to one reservoir and liberated i n the other i s then given by Q - T A S. (2.2) Assuming the helium flow through the c a p i l l a r y c a r r i e s no entropy, which corresponds to zero Thomson heat i n thermo-e l e c t r i c analogy, the two equations (2.1) and (2.2) become Liquid \u2014 W v Figure 2. Apparatus for thermomechanicaI effect to follow page I 0 11-f s (2.3) Z\\ T and Q - TS (2.4) respectively, where S r e f e r s to the t o t a l entropy of the l i q u i d helium. The quantitative confirmation of these equations was obtained by Kapitza (1941) two years l a t e r . 2.2.2. Experimental Fact Kapitza's paper investigated the flow properties of helium II through a narrow s l i t under the influence of a temperature difference. The apparatus (Figure 2) was an insulated container which communicated with helium bath v i a s l i t and which was f i l l e d with helium II. By putting the inner l e v e l lower than the bath l e v e l i t became cooler as the inner l e v e l was going up (mechano-caloric e f f e c t ) . A temperature difference was produced which hindered the further flow of the l i q u i d and an equilibrium state could be established quickly. However, i f heat was generated inside the container i t would be f i l l e d with helium II quickly,(thermo-mechanical e f f e c t ) . By generating a known amount of heat i n the container such that no temperature difference arose, and also by knowing the volume of the l i q u i d which flows through the s l i t i t was possible to determine the difference between the s p e c i f i c heat content -12-of the l i q u i d flowing through the c a p i l l a r y and the heat' content i n the container. The smallest gap used was about 1.3 x 10 cm. which was presumably narrow enough to pass only the superfluid component. In his experiment the bulk flow rate in the container as a function of the applied power q was measured. At the same time the temperature differences between the container and the helium bath were observed. At the beginning the l i q u i d l e v e l s in both containers were made the same so that the difference i n hydrostatic pressure can be neglected. The dependence of the flow v e l o c i t y v and the temperature d i f -ference A . T, on the rate of supply of heat, q, are shown i n Figure 3. For small q, T was immesurably small, the bulk flow rate was l i n e a r l y proportional to q. Under these con-d i t i o n s i t was necessary to supply an amount of heat q to each gram of l i q u i d emerging from the s l i t i n order to warm i t up to the temperature of the bulk l i q u i d at the entrance of the s l i t . The flow was under zero pressure head and zero temperature difference. Therefore a temperature difference seems to play a si m i l a r r o l e to a pressure difference i n superfluid flow. Between 1.3\u00b0K and 7v-point (in recent years i t had been extended to 0.1\u00b0K by Bots and Gorter (1S56)) Q was equal to TS within the accuracy of the experiments, where S was the t o t a l entropy per gram of the bulk l i q u i d and the conclusion was that the superf l u i d component flowing through the c a p i l l a r y 0 2 4 6 4 8 10 heat supplied(q x 10 cal.\/sec) Figure 3. Volume flow versus heat supplied power input (mW) Figure 3a. Flow velosity V s versus power input to follow page 12. -13-c a r r i e d no entropy, as London predicted. At a c e r t a i n heat input corresponding to a c r i t i c a l v e l o c i t y the bulk flow rate started to f a l l off and a tem-perature difference was observed. The flow rate was no longer proportional to the heat input. There was l i t t l e doubt that at t h i s point the l i n e a r v e l o c i t y of flow through the c a p i l l a r y reached the c r i t i c a l value beyond which motion of the helium was accompanied by losses. Presumably at t h i s c r i t i c a l flow rate the s u p e r f l u i d character i n the c a p i l l a r y broke down and excess heat was required to provide for t h i s viscous i r r e v e r s i b l e process. The subject of c r i t i c a l v e l o c i t y w i l l be discussed i n the next section 2.3. 2.3. Flow Through Narrow Channels and C r i t i c a l V e l o c i t y 2.3.1, Isothermal Flow through Narrow Channels It i s not easy to obtain s a t i s f a c t o r y quantitative data on isothermal flow through narrow channels. It i s also t e c h n i c a l l y d i f f i c u l t to produce very narrow channels (down to 10 cm. wide) with a uniform cross-section and t h e i r cross-sections cannot be measured accurately. Moreover, only the superfluid component flows strongly through narrow channels and the heat defect of t h i s component gives r i s e to temperature difference, which i n turn reacts back on the flow, so that i t i s very d i f f i c u l t to ensure that the flow i s r e a l l y isothermal. The most complete of the early -14-raeasurements are those of Allen and Misener (1339). Their method of producing the narrowest channels i s of interest and has been extensively used by other experi-menters (Brown and Mendelssohn (1947)). They placed a bundle of f i n e s t a i n l e s s s t e e l wires (about 1000 wires of \u20142 about 6 x 10\" cm diameter) i n a n i c k e l - s i l v e r tube and drew the tube through a succession of s t e e l dies. In t h i s way a channel width of the order of magnitude 10\"\u00b0^ cm. can be obtained. Glass c a p i l l a r y were also used by them to study flow through wider channels. The method of experi-ment was the straightforward one of attaching the glass c a p i l l a r y or w i r e - f i l l e d tube to the lower end of a glass reservoir and observing the rate of emptying or f i l l i n g of the reservoir when i t was p a r t i a l l y immersed i n a bath of l i q u i d helium II. For the wider c a p i l l a r i e s , at temperatures very near the A-point, P o i s e u i l l e ' s law was approximately obeyed, presumably because the flow of the normal component predominated. Otherwide the dependence of the rate of flow on the pressure head, the width of the channel and the length of the channel was very complicated. Quantitatively, the r e s u l t s can be explained by superimposing the viscous flay of the normal component on the more complicated flow of the supe r f l u i d component. The nature of superfluid flow can be seen from the observations at lower temperatures, say 1.2\u00b0K where j*n\/f i s about 3% and the flow of the normal component can be -15 ignored. For the channel of width 10\"\u00b0 cm., the v e l o c i t y i s almost independent of the presure head and i t i s t i s u a l l y assumed t h a t the same v e l o c i t y c o u l d e x i s t under zero p r e s s u r e head. As the channel width i n c r e a s e s from 10 1 t o 10 J cm. the r e i s a g r a d u a l t r a n s i t i o n from i d e a l s u p e r f l u i d i t y t o the e n t i r e l y d i f f e r e n t s i t u a t i o n encountered i n wide c a p i l l a r i e s , where the mean v e l o c i t y i s a marked f u n c t i o n of the p r e s s u r e head. The p r e s s u r e head needed t o produce a g i v e n v e l o c i t y i s a d i r e c t measure of the f r i c t i o n a l f o r c e s opposing the flow. A c r i t i c a l v e l o c i t y V g Q i s not e a s i l y d e f i n e d f o r the wider channels. Selow V s c the flow would then be s u b j e c t to no f r i c t i o n a l r e t a r d a t i o n and would be t r u l y s u p e r f l u i d . However above V there i s a s, c p r o g r e s s i v e b r e a k i n g down of the s u p e r f l u i d i t y as the f r i c t i o n a l f o r c e s b e g i n to b u i l d up i n a markedly n o n - l i n e a r f a s h i o n as the v e l o c i t y i n c r e a s e s . A l l e n and Misener (1939) a l s o made experiments on the flow of helium II through powder packed tubes, u s i n ^ powder of p a r t i c l e s i z e about 10 co. diameter. They found a marked dependence of the volume flow on p r e s s u r e head at a l l temperatures and consequently i t co u l d be concluded that the f e a t u r e s of flow through powders were not n e c e s s a r i l y the same as f o r the flow through c a p i l l a r y tubes. Such c o n c l u s i o n s have been drawn by Dowers, Chandrasekhar, and Mendelssohn (1950, 1953) from experiments w i t h powders, and -16-by Bowers and White (1951), and by White (1951) from experiments on flow through porous membranes. They have confirmed the fact that pressure gradient i s present throughout the length of the column of the packed powder. 2.3.2. C r i t i c a l V e l o c i t i e s The hypothesis of a c r i t i c a l v e l o c i t y (in Kapitza*s experiment, see figure 2) implies that below a certain value the f r i c t i o n a l force i s n e g l i g i b l y small whereas for large values f r i c t i o n a l force can be rather large. There-fore the v e l o c i t y (V g) versus power input (I) (see Figure 3a) curve must have a disc o n t i n u i t y i n i t s f i r s t d erivative at V_ = V 4 + . These considerations r e f e r to the s c r i t i c a l v e l o c i t y V s d i r e c t l y after the heat supply i s started. For, during a constant heat supply I, the V s values change as a function of time which makes the s i t u a t i o n more complicated. But t h i s V s ( t ) function can also be calculated and agreement with experiment i s found (Winkel (1955)). As i s obvious from experimental reasons, i t w i l l always be impossible to decide whether or not the f i r s t derivative of the I-V s curve r e a l l y shows a disc o n t i n u i t y at V g = V g^ c. One can only say that the derivative of I changes rather f a s t at V s = V s^ c. Furthermore i t must be noted that the c r i t i c a l e f f e c t i s related to the supe r f l u i d v e l o c i t y . For small v e l o c i t i e s (V g < V g^ c) the general flow mechanism of helium II i s described by a two f l u i d model. -17. But at high v e l o c i t i e s (V g > V g c ) the introduction of a mutual f r i c t i o n a l force F as a reasonable approximation i s necessary. H a l l and Vinen (for d e t a i l s r e f e r to H a l l (I960)) have further investigated t h i s f a c t . They interpret the mutual f i r c t i o n as the interaction between the e x c i t a -tions of normal f l u i d and the quantized vortex l i n e s (which w i l l be described i n the next section). Once the flow v e l o c i t y i s over c r i t i c a l v e l o c i t y , quantized vortex l i n e s are created. These vortex l i n e s w i l l interact with the excitations of the normal f l u i d , therefore the mutual f r i c t i o n a l force F appears. The existence of a mutual f r i c t i o n F provides the simplest explanation of the existence of a c r i t i c a l v e l o c i t y above which the simple two-fluid theory i s no longer v a l i d . Atkins (1959) has given a l i s t of c r i t i c a l v e l o c i t i e s from d i f f e r e n t experiments. However, the largest channel width mentioned i s 0.4 cm. which gives V g > c = 0.033 cm.\/sec. In the present setup, for a tube of about 1 cm. i n diameter, the c r i t i c a l v e l o c i t y might be of the order 0.01 cm.\/sec. or even l e s s . 2.4 Feynman's Theory of Quantized Vortex Lines and  Turbulent Superflow The f i r s t theory s a t i s f a c t o r i l y accounting f o r many peculiar hydrodynamic properties of l i q u i d helium II was that due to Landau. It explains s u c c e s s f u l l y the two-fluid -18-raodel and the fact that the superfluid component can undergo f r i c t i o n l e s s flow without i n t e r a c t i o n with either the normal f l u i d or any s o l i d boundaries. However the theory suffered from two serious defects: f i r s t i t predicted that the superfluid flow should always be i r r o t a t i o n a l , whereas t h i s was observed not to be true; and secondly, i t predicted values f o r the c r i t i c a l v e l o c i t y above which superfluid flow ceases to be f r i c t i o n l e s s that were too large. These d i f f i c u l t i e s were removed as soon as i t was r e a l i z e d that, although the i r r o t a t i o n a l condition mast hold throughout most of the superfluid, i t i s p e r f e c t l y possible to have highly l o c a l i z e d regions of v o r t i c i t y i n the superfluid i n the form of what 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 , i . e . in the form of vortex l i n e s or sheets. Furthermore i t was r e a l i z e d that c i r c u l a t i o n in the superfluid should be quantized in units of h\/m (h 3s Planck's constant, m i s the mass of a helium atom), so that the strength of a vortex l i n e should be quantized. The view that superfluid c i r c u l a t i o n i s quantized and the idea of vortex l i n e s were both discussed f i r s t by Onsager (1949) and developed considerably by Feynman (1955). Of these ideas that of the quantized vortex l i n e seems at present to be the most important. It i s possible that the vortex sheet i s sometimes important i n i n i t i a l breakdown -19-of i d e a l s u p e r f l u i d flow at high v e l o c i t i e s . With these ideas Vinen (1961) and Feynman (1955) t r y to explain many problems i n hydrodynamics of helium II i n th e i r review a r t i c l e s * The detailed quantum mechanical problem w i l l not be presented here. We only mention some of the important formulae and r e s u l t s which they developed and are c l o s e l y related to the interpretation of present r e s u l t s which w i l l be discussed i n considerable d e t a i l i n Chapter V. If tjf i s a wave function describing the l i q u i d at r e s t , then the wave function ijf exp |(im\/C) V \u2022 _3 r\u00b1J represents a uniform t r a n s l a t i o n a l v e l o c i t y V of the l i q u i d as a whole. The function where S i s a variable function of positions and S(? i) i s i t s value at the location of the i - t h atom, represents an i r r o t a t i o n a l flow and the v e l o c i t y at any point i s v* => \u2014 grad S (2.2) m The condition that the wave function should s a t i s f y the Bose symmetry means that the phase change should be an in t e g r a l multiple of 2 71, and t h i s leads to the condition on the v e l o c i t y V that dr - p h (2.3) m -20-where p i s any integer. If the v e l o c i t y i s a function of the radius r only, then ' = P l F (2.4) A v e l o c i t y f i e l d of t h i s type i n an i n f i n i t e l i q u i d i s the f a m i l i a r vortex l i n e of c l a s s i c a l hydrodynamics. It i s a c i r c u l a r flow about a l i n e axis, with the v e l o c i t y increasing as the axis i s approached i n such a way that Curl V s \"\u00bb 0 everywhere except at points on the axis. Equation (2.4) describes \"quantized vortex l i n e s \" i n which the angular momentum per helium atom about the axis i s r e s t r i c t e d to an i n t e g r a l multiple of -ii. The energy per unit length of a vortex l i n e with p quanta of c i r c u l a t i o n J-b a 2 - , P , _ _ 1 \u201e (b ) ( 2 - 5 ) J \u00b0 m a the upper l i m i t 'b\u00bb i s related to the s i z e of the container. The lower l i m i t 'a* has the order of magnitude of the interatomic distance. -21 From t h i s formulation, f o r a channel of width d the corresponding c r i t i c a l v e l o c i t y would be v s , c <&> (2.6) This i s a very s a t i s f a c t o r y r e s u l t which agrees with experimental V s ^ c , a t least q u a l i t a t i v e l y . Experimental f a c t shows that when su p e r f l u i d flows s t e a d i l y through a s t r a i g h t channel i t can do so without appreciable f r i c t i o n a l d i s s i p a t i o n provided that i t s v e l o c i t y does not exceed a c e r t a i n c r i t i c a l value V g c . But as soon as the v e l o c i t y does exceed t h i s c r i t i c a l value complicated non-linear f r i c t i o n a l forces begin to appear. It now seems very l i k e l y that t h i s breakdown of i d e a l s u p e r f l u i d i s due to the creation of quantized vortex l i n e s (or vortex sheets), these l i n e s s t r e t c h and spread out into tangled array, so that the s u p e r f l u i d becomes i n e f f e c t turbulent, and the f r i c t i o n a l forces a r i s e from t h i s turbulence. The d i f f e r e n t mechanisms i n creating the vortex l i n e s had been investigated i n more d e t a i l by Vinen (1961). One of the p o s s i b i l i t i e s i s a protuberance can i t s e l f cause the creation of a length of vortex l i n e . Thus a small length of l i n e might be created close to the protuberance of the wall and then be pulled away from i t by the main flow. In our experiment one additional way might be - 2 2 -possible. That i s the vortex l i n e s may be generated from the surface of powder-filled f i l t e r . Similar to the case of an ordinary f l u i d i n a container with a hole at the bottom. When the f l u i d i s sucked hard from the hole, a vortex l i n e with one end attached at the hole i s produced. Feynman had pointed out that more l i k e l y a l i n e gets started somehow and has i t s ends t i e d on the wall. Then the forces of the f l u i d on the rest of the l i n e cause i t to wander about i n such a way that more and more vortex l i n e i s fed out. Similar things could happen inside tubes. If the tubes are very narrow the l i n e w i l l h i t the other surface e a s i l y and be attached by the walls. It can never get very far from a wall. Even i f started somehow i t w i l l f a l l back into the tube walls unless the v e l o c i t y s u f f i c e s to keep i t i n the stream. There-fore the smallest tubes have the highest c r i t i c a l v e l o c i t y . F i n a l l y i n t h i s section, i t i s worth mentioning that Townsend (1961) has attempted to apply modern concepts Of turbulent flow to the macroscopic behavior of l i q u i d helium. Equations of motion are developed f o r a macroscopic v e l o c i t y f i e l d which i s the average of the true v e l o c i t y f i e l d over a region that contains many vortex l i n e s . He concluded that the isothermal flow of l i q u i d helium at large Reynolds numbers i s very s i m i l a r on a macroscopic scale to turbulent flow of a Newtonian f l u i d . He assumes that the flow i s steady with vortex l i n e s moving i n the general d i r e c t i o n of flow i n a quasi-regular array. Unfortunately, thermally induced flow cannot be adequately described by h i s theory. -23-CHAFTER III EXPERIMENTAL ARRANGEMENT The object of the present experiment was to investigate pure super f l u i d flow through a stationary background of normal-fluid. Information about the superflow was obtained by suspending i n a superf l u i d wind tunnel f i n e p a r t i c l e s of a s o l i d i f i e d hydrogen-deuterium mixture. The experimental equipment consisted of a conventional vacuum system, cryogenics and two reserv o i r s for preparing the gas mixture. The su p e r f l u i d and p a r t i c l e v e l o c i t i e s were measured separately, by v i s u a l observation i n each case. The temperature difference between the measuring beaker and the wind tunnel could be made by two carbon thermometers. A l l these arrangements w i l l be discussed i n d e t a i l i n t h i s chapter. 3.1 Apparatus 3.1.1. Vacuum System The cryostat and the a u x i l i a r y equipment used f o r low temperatures were of the usual design. The main components of the vacuum system are shown i n fi g u r e 4. The vacuum system consisted of a si n g l e stage mercury d i f f u s i o n pump T o c r y o s t a t C H g T o c r v o s t a t 4-0 \u00bb 0 0 c.c. D i s c h a r g e T o \u00b0 \u00bb r 5 b o c.c. H g d i f f u s s i o n p u m p 9 V F. 5 L i t e r s 5 L i t e r s T o A i r Pi ra n i p H To A i r L l q uid Nitrogen tra p M. ^ o t a ry P u m p Hg O i l M o n o m e t e r s c i g s j r e 4 . V a c u u m s y s t e m t o f o l l o w p a g e 2 3 preceded by a r o t a r y pump and f o l l o w e d by a l i q u i d n i t r o g e n t r a p T. Mi 'and M 0 were mercury and o i l manometers r e s p e c -t i v e l y . The d e n s i t y of the o i l was 15.3 times s m a l l e r than that of the mercury. F^ and F2 were two 5 l i t e r r e s e r -v o i r s f o r s t o r i n g the mixture of hydrogen and deuterium gases i n the r i g h t p r o p o r t i o n . These gases c o u l d be \"shot\" i n t o the helium dewar through C. The volume of gas admitted at each shot c o u l d be ad j u s t e d by the 500 cc and 100 cc c o n t a i n e r s , and the pre s s u r e by mercury manometer M3. The P i r a n i gauge P and d i s c h a r g e tube D were l o c a t e d i n such a p o s i t i o n that a leak t e s t i n g i n any p a r t ox the system was p o s s i b l e . The o u t l e t II was used f o r pumping the j a c k e t s of the t r a n s f e r siphon and of the helium dewar. A standard double dewar c r y o s t a t was used. The dewars were arranged c o n c e n t r i c a l l y so that the outer formed a l i q u i d n i t r o g e n space around the inner dewar w i t h i n which the l i q u i d helium was s t o r e d . Both dewars were made of pyrex g l a s s and were s i l v e r e d i n such a way th a t two narrow s t r i p s and the bottom were l e f t f o r v i s u a l observa-t i o n s . The helium dewar v\/as c l o s e d at the top by means of a b r a s s cap. A rubber gasket s e a l e d the dewar vacuum t i g h t . The cap had german-silver t u b i n g o u t l e t s f o r the Kinney pump, t r a n s f e r tube, vapour p r e s s u r e l i n e , p a r t i c l e making tube and helium gas r e t u r n l i n e . Both the vacuum system and the cap assembly wex*e b u i l t on a r i g i d Dexion stand. To pump on the l i q u i d helium bath, a l a r g e c a p a c i t y - 2 5 -Kinney mechanical pump had been i n s t a l l e d i n the laboratory and was linked to the cap by a four inch pipe. The rate of pumping could be controlled by means of a small and a big valve i n p a r a l l e l to each other. The pressure over the helium bath could be varied continuously and read by a mercury manometer for higher pressures and an o i l manometer for lower pressures while the other side of the manometers were pumped by a mercury d i f f u s i o n pump. At maximum pumping speed, a temperature of 1.39K was obtained. 3.1.2. The Superfluid Wind Tunnel and Superfluid  V e l o c i t y Measurements The experiment involves a \"superfluid wind tunnel\" (similar to that used by Pellara et a l (1957, 1962> whereby ground state l i q u i d helium can be set into controlled, uniform motion through a background of stationary normal f l u i d . The d i f f i c u l t y of producing a region of pure sup e r f l u i d , r e a l i z a b l e only at absolute zero, i s circumvented by establishing a region of pure super f l u i d flow i n the presence of the motionless normal f l u i d component. (In other words, the average v e l o c i t y of the e x c i t a t i o n i s zero). Within t h i s tunnel investigations of the hydrodynamic properties of pure su p e r f l u i d flow with background of stationary normal f l u i d can be made. The sketch of figure 5 shows the c y l i n d r i c a l region S where the separated v e l o c i t y f i e l d occurs. G i s a glass to follow page 2 5 - 2 6 -tube through which the p a r t i c l e s pass down into the tunnel. (The method of making p a r t i c l e s w i l l be described i n the next subsection 3.1.3). The two superleaks F j and F 2 are made of brass tube f i l l e d with f i n e cerium oxide powder (with an average diameter of the order 10~ 4 cm) pressed by nearly 3 tons of force. Only s u p e r f l u i d component can pass through these. (It was observed that above A-polnt no helium could flow through these superleaks.) The heater H permits the application of the thermo-mechanical e f f e c t to provide a flow of superf l u i d through the arrested normal f l u i d component i n the tube S. F^ and F 2 are sealed together with ordinary solder with an empty space i n between. Two glass tubes T^ and T 2 with Kovar metal sections and K 2 are connected to two ends of these double superleak by woods metal. Part of the glass tube T 2 i s bent smoothly to a horizontal section S where a few marks are made on the glass so that the p a r t i c l e v e l o c i t y (v p) can be measured when p a r t i c l e s move from one mark to the other. This double superleak configuration prevents normal f l u i d flowing back through S, by providing a thermal ground through the Kovar metal sections and through the brass i t s e l f . Scales are made on the tube T j for the purpose of superfl u i d v e l o c i t y (v s) measurements. But l a t e r on a small beaker P of about 2 cc i n volume was connected to the upper end of tube T j with a c a p i l l a r y for v g measurements. A -27-fountain pump B was made to empty the l i q u i d helium in the. beaker after each measurement. V e l o c i t y determinations were obtained d i r e c t l y by timing the f i l l i n g of the beaker or timing the l e v e l r i s i n g over the scale on the tube T j . The r e s u l t s from these two ways were i n quite good agreement. 3.1.3. Formation of the P a r t i c l e s It i s very i n t e r e s t i n g to suspend v i s i b l e p a r t i c l e s i n l i q u i d helium i n order to observe the peculiar hydrodynaraic properties of helium II. The main d i f f i c u l t y i s that, due to the fact that helium II has exteraely small v i s c o s i t y , a l l the heavier p a r t i c l e s w i l l f a l l to the bottom very ra p i d l y . Chopra and Brown (1957) suggested that with a sui t a b l e mixture of hydrogen and deuterium gases one might get v i s i b l e f i n e p a r t i c l e s suspended i n helium II, provided they had the same density as l i q u i d helium at that temperature. In the present work we discuss the flow properties of these p a r t i c l e s i n a superfluid wind tunnel. The method of making the p a r t i c l e s follows that of Chopra. The detailed description of the p r i n c i p l e may be found i n Chopra's thesis (1957). Since the freezing points of hydrogen and deuterium gases are very close, namely 20.4\u00b0K and 23.6\u00b0K respectively, a c a l c u l a t i o n from t h e i r densities at 4.2\u00b0K showed that the mixing r a t i o H2\/D2 should be 1\/1.11 by volume. -28-In the experiment, the mixing containers (F.^  and F 2 i n figure 4) were evacuated and f i l l e d with 5.5 cm of D 2 gas and 5.0 cm of H 2 at room temperature. The mixture was usually prepared quite a long time before the experiment was performed so that i t was thoroughly mixed at the time of use. The position of heater H' in the glass tube G whose one end was connected to the superfluid wind tunnel i s shown in figure 5. The mixture of gases was admitted through a tube G when a \"shot\" of gases was made. When l i q u i d helium had been pumped to a desired temperature (~ 1.4\u00b0K) a volume of the mixture of gases, either 100 cc or 500 cc, was used depending on the number of p a r t i c l e s needed. The heater, about 1 to 4 watts i n power, was then switched on and off quickly as the gases went into the cryostat from the top of the tube. Without the heater the mixture of gases entered rather slowly (except vhen some pressure was used) and formed a thick coagulated cloud. The heater increased the volume of gases forming the p a r t i c l e s i n cryostat. In t h i s way one can e a s i l y get p a r t i c l e s of various sizes by varying the amount of heat supplied to the heater. When p a r t i c l e s were shot into the tunnel most of them remained suspended i n the l i q u i d helium, but some of them moved toward the wall and stuck on i t . Moreover, some p a r t i c l e s combined together to form a p i l e of p a r t i c l e s . This process took a few minutes, after which no more p a r t i c l e s were l e f t suspended. For the next shot the heater - 2 9 -was switched on and then o f f quickly with less power. The p a r t i c l e s o r i g i n a l l y suspended i n the tube would come down into the tunnel where the p a r t i c l e v e l o c i t y (v p) was being measured. 3.2. Experimental Procedure 3.2.1. Cryogenics After the apparatus for the wind tunnel was set up properly, the helium dewar was put on and pumped hard i n order to get r i d of the foreign gases in the dewar and superleaks. Then the helium dewar was f i l l e d with helium gas at atmosphere pressure. After pumping out the helium ~ dewar jacket to about 1 mm of Hg, the precooling was started by f i l l i n g the outer dewar with l i q u i d nitrogen for about one to two hours. The helium dewar was cooled to l i q u i d nitrogen temperature before the transfer of l i q u i d helium. The 25 l i t e r helium storage vessel was connected with a transfer siphon to the helium dewar and an overpressure of about 3 cm of mercury was applied to the vessel from a helium cylinder. During the transfer the helium return l i n e was l e f t open so that the evaporated helium could return to the gasholder. It took about 15 minutes to transfer s u f f i c i e n t helium, aft e r which the storage vessel was removed and transfer siphon blocked. Then the bypass valve of the pumping pipe was slowly opened and at the same -30-time the return l i n e closed. The l i q u i d helium was then pumped hard by means of the Kinney pump to a temperature of about 1.4\u00b0K at which most observations were made. 3.2.2. Ve l o c i t y Measurements (i) Superfluid v e l o c i t y (v s) measurement The v s measurement has been referred to in section 3.1.2 (see figure 5). When the heater H (550 ohm constantan wire) with a d e f i n i t e power input from a storage battery was switched on, the supe r f l u i d started to move toward the heater through the superleaks and made the l e v e l i n the measuring beaker r i s e . By using a cathetometer the time needed for the l i q u i d l e v e l to cross two marks on the beaker was taken. The current through the heater was measured at the same time. The beaker was then emptied by a fountain pump connected in the bottom of i t so that another v e l o c i t y corresponding to a d i f f e r e n t power input could be measured. From these values one could calculate the supe r f l u i d v e l o c i t y i n the supe r f l u i d wind tunnel. The r e s u l t s w i l l be given i n the nest chapter. The other possible way of measuring the flow v e l o c i t y was by a photographic method. By using a stroboscope (this technique w i l l be discussed l a t e r in t h i s section) with known i n t e r v a l of time between flashes, pictures could be taken with a polaroid camera. The v e l o c i t y could be measured with a t r a v e l l i n g microscope from these multiple -31-exposure films. With t h i s technique one could get more accurate r e s u l t s at higher v e l o c i t i e s . It was also p a r t i c u l a r useful for the following important reason. It was noted i n the section on c r i t i c a l v e l o c i t y (2.3) that once the v e l o c i t y of s u p e r f l u i d exceeded a certain c r i t i c a l value i t was no longer proportional to the power input but became a function of time, making the s i t u a t i o n somewhat more complicated. If one needs to know the precise s i t u a t i o n i n higher v e l o c i t y measurements the photographic method i s highly recommended. In the present work th i s method was not used due to lack of suitable camera in t h i s laboratory. ( i i ) P a r t i c l e V e l o c i t y Measurements As to the p a r t i c l e v e l o c i t y measurements we w i l l discuss three d i f f e r e n t methods here. A. Observation by Cathetometer When the p a r t i c l e s came down from v e r t i c a l tube to the wind tunnel, they would move in the same d i r e c t i o n as the superfluid flow, i f the v e l o c i t y of the l a t t e r was large enough. Using the cathetometer one could measure p a r t i c l e v e l o c i t i e s by noting the time taken by a s i n g l e p a r t i c l e t r a v e l l i n g from one mark to the other along the tunnel. However, the r e s u l t s were not very promising because of the narrow view of the cathetometer and the f a s t motion of the p a r t i c l e s in the tunnel. Therefore the following two ways were t r i e d . -32-B. Photographic Method Manchester and Brown (1955) suggested a photographic method in their measurements on adiabatic o s c i l l a t i o n in f i l m flow. They used a stroboscope which would give the flashes with d e f i n i t e time i n t e r v a l . At the same time a camera focussed on the c a p i l l a r y had i t s films driven by a constant speed motor. But i t was suggested that we could take a multiple exposure for wind tunnel i f the f l a s h could r e f l e c t the l i g h t from p a r t i c l e s toward a polaroid camera. The camera we used was of the DuMont Type 2620 which was o r i g i n a l l y designed for oscilloscope. In order to get better focus the f o c a l length was changed and a close - up lens was also used. By t h i s method depth of focus was very short, making the measurement d i f f i c u l t . Several r o l l s of f i l m (ASA index 3200) were used, but only a few of them came out. It was found that the r e f l e c t i o n of l i g h t from the l i q u i d nitrogen and glass dewar was comparable with that from the p a r t i c l e s . This would have to be reduced i n any future investigation. More-over a s p e c i a l kind of polaroid camera should be used, with distance adjustment, close-up lens, and a single r e f l e c t i o n viewer so that the camera could be focussed conveniently. Due to the lack of t h i s s p e c i a l kind of camera i n t h i s laboratory t h i s powerful method, which would give instantane-ous p a r t i c l e v e l o c i t y in the tunnel, has not been developed any further. -33-C. Direct V i s u a l Observation With a suitable brightness of l i g h t shining on the tunnel in a certain d i r e c t i o n , one was able to measure the time taken for a p a r t i c l e t r a v e l l i n g from one scale to the other. V i s u a l l y one could pick up those p a r t i c l e s which seemed to be of the same s i z e and did not stop on the way. The si z e of the p a r t i c l e s estimated from the picture taken by polaroid camera were about 100 to 500 microns in diameter. Once a p a r t i c l e moved near the wall i t would s t i c k on the wall and stop there. After several t r i a l s t h i s method of observation was found very p r a c t i c a l in our case. Three successive runs were made and re s u l t s recorded which w i l l be discussed in the next chapter. 3.2.3. Temperature measurements Two carbon resistances of about 80 ohms at room tem-perature were put in the measuring beaker P (see figure 5) and the wind tunnel T 2. By using a Wheatstone bridge and a high s e n s i t i v i t y galvanometer i t was possible to measure the temperature difference of a few millidegrees at 1.4\u00b0K. The resistances increased up to 8000 ohms at helium temperatures, and the current through them was about 0.1 milliamperes. -34-CHAPTER IV EXPERIMENTAL RESULTS A l l measurements were conducted at a nearly f i x e d tem-perature of 1.4\u00b0K for two reasons. Since 1.4\u00b0K i s the temperature normally reached by the pumping system, this procedure enabled repeated observations to be carried out at the same superfluid density and therefore the same Jg\/J 3 which was 0.93 at 1.4\u00b0K. Also the decreased temperature s e n s i t i v i t y of fs\/f with decreasing temperature made the 93% concentration of superfluid the most r e a d i l y controllable. 4.1. Results of Flow Velocity (v g) Measurements Using the method described i n section 3.2.2 the super-f l u i d v e l o c i t i e s (v s) i n the wind tunnel car; be calculated by the formula f Vo =\u00bb \u2014 V where v i s the ve l o c i t y measured i n the beaker. Taking f s -~\u2014 = 0.93 at 1.4\u00b0K values of v are obtained f o r various f power inputs (see Table 1). In figures 6, 7 and 8 the flow v e l o c i t i e s v g are plotted against the power input W. Some remarks are made here on v s measurements. Due to -35-s l i g h t changes in temperature during the helium run, there i s a small f l u c t u a t i o n in Jg\/j 3 , as mentioned in the begin-ning of t h i s chapter. At 1.4\u00b0K t h i s i s very small. A change i n 0.05\u00b0K gives only 0.5% v a r i a t i o n in f s \/ \/ > \u2022 Therefore the correction due to t h i s factor i s small. The evaporation losses due to the extremely small temperature gradient along the tube are also small f o r the low v e l o c i t y region (vs-<.0.3 cm\/sec). But in the higher v e l o c i t y region ( v g > 0 . 3 cm\/sec) th i s cannot be neglected. A rough c a l c u l a -t i o n for a temperature difference of 10 millidegrees shows that not more than 10% i s l o s t due to evaporation. From the measurement at 1.4\u00b0K and a power input of 20 mW, i t was found that the temperature difference was about 10 millidegrees. The creeping f i l m along the edge of the beaker i s also small. A l l these corrections tend to make the values of v g s l i g h t l y higher than calculated. However, the correction for the difference in cross sectional area between the wind tunnel and the beaker i s about 7%. (The former i s 7% bigger than the l a t t e r . ) This w i l l r e s u l t in a smaller v g- As a r e s u l t of these, the p o s i t i v e and negative corrections almost cancelled each other except for low v e l o c i t y region where evaporation i s not important and 7% correction should be made. Pellam et a l (1957-1962) found that in the superfluid wind tunnel v_ could change suddenly by about 10% for some unknown reason even in the low v e l o c i t y region between 0.01 and 0.1 cm\/sec. In our case, where highly unstable turbulent TABLE 1 Power input Theoretical Experimental 1st run 2nd run _3rd run W(mW) v e(cm\/sec) v s(cm\/sec) v_(cm\/sec) v (cm\/sec) v_(cm\/sec) s P p P 4.95 0.11 0.0815 p a r t i c l e s remained stationary 8.8 0.196 0.145 0.143 - 0.143 11.2 - 0.172 - - -13.8 0.308 0.192 0.232 - 0.2 16.6 - 0.239 - 0.258 19.8 0.44 0.268 0.329 - 0.291 23.2 - 0.276 - 0.345 27.0 0.60 0.292 0.456 - 0.533 30.8 - 0.313 - 0.379 35.2 0.79 0.344 0.453 - 0.561 39.6 - 0.324 - 0.445 44.5 1.0 0.326 0.695 - 0.623 49.7 - - 0.493 55.0 1.22 0.35 0.797 \/ Theoretical 10 20 30 40 5 0 6 0 Power input (mW) Figure 6. V and V v e r s u s power input to f o l l cwpaqe 3 5 0.8 to follow page 3 5 0.9 flow e x i s t s , g r e a t e r f l u c t u a t i o n i n v s would be expected, e s p e c i a l l y i n the higher v e l o c i t y r e g i o n . In a d d i t i o n , i n t h i s region the error i n t i m i n g the f i l l i n g of the beaker with a stop watch i s r e l a t i v e l y i n c r e a s e d due to the curved surface of the helium l e v e l . Moreover as mentioned b e f o r e , v s i s not constant (when v s > v s c ) , but a f u n c t i o n of time. I t i s d i f f i c u l t t o measure t h i s e f f e c t i n the present set up because the measur-i n g beaker i s only 2 cc i n volume. The r e s u l t s are measured o n l y d u r i n g the f i r s t few seconds ( i . e . b e f o r e an e q u i l i b r i u m s t a t e i s reached). A l l these f a c t o r s make the high v s measurements u n c e r t a i n , so that o n l y q u a l i t a t i v e c o n c l u s i o n s can be drawn. 4.2. R e s u l t s of P a r t i c l e V e l o c i t y (v p) Measurements By the method,described i n 3.2.2. the p a r t i c l e v e l o c i t i e s can be obtained d i r e c t l y once the time taken for a p a r t i c l e to cross a d e f i n i t e distance i s known. The data are given in Table 1. It must be noted here that the p a r t i c l e v e l o c i t y Vp measurement was taken when there was more power than 8.8 mW in the heater. Below th i s value the p a r t i c l e s were observed to be stationary. Presumably below a certain v e l o c i t y Vp^ c the flow i s a perfect p o t e n t i a l flow with zero v i s c o s i t y . In t h i s region the energy i s not enough to produce one quantized vortex l i n e . The lowest value of Vp measured are 0.075 cm\/sec. -37-Due to the t u r b u l e n c e of the s u p e r f l u i d flow, e s p e c i a l l y i n the high v e l o c i t y r e g i o n , the p a r t i c l e v e l o c i t i e s are wide l y s c a t t e r e d . The reasons w i l l be d i s c u s s e d i n the next chapter. The mean values are taken by averaging over s e v e r a l p a r t i c l e v e l o c i t i e s w i t h the same power i n p u t , i . e . with the same flow v e l o c i t i e s v s . The r e s u l t s are a l s o p l o t t e d i n the same graph ( f i g u r e s 6,7,3) f o r purpose of comparison. As mentioned i n s e c t i o n 3 . 2 . 2 , the main d i f f i c u l t y was the f a c t t h a t the way of making p a r t i c l e s c ould a f f e c t the i n i t i a l motion of the p a r t i c l e s b e f o r e they entered the wind tunnel, and could a l s o i n f l u e n c e the p r o p e r t i e s of the f l u i d i n the tunnel because of the heat generated i n the p a r t i c l e making tube. I t should be noted t h a t most of the v p measurements were made by v i s u a l o b s e r v a t i o n s . Only p a r t i c l e s of approximately the same s i z e were taken. Those which were bi g g e r i n s i z e , and t h e r e f o r e moving slower, were not recorded. A p e c u l i a r t h i n g must be mentioned here. On a few o c c a s i o n s one or two p a r t i c l e s were seen moving backward as w e l l as forward i n the tu n n e l . But very few of t h i s s o r t were found. An attempt has a l s o been made to f i n d the lowest vs which w i l l make the p a r t i c l e move. But the i n f l u e n c e of the shot d i s t u r b e d the i n i t i a l motion ox the p a r t i c l e s , and t h i s l i m i t c o u l d not be d e f i n e d s h a r p l y . -38-CHAPTER V DISCUSSION In t h i s chapter the theory of vortex l i n e i n s u p e r f l u i d t u r b u l e n c e w i l l be simply d e s c r i b e d ( s e c t i o n 5.1). I t i s on the b a s i s of t h i s theory that the experimental r e s u l t s are i n t e r p r e t e d ( s e c t i o n 5.2). 5.1. T h e o r e t i c a l Background In c l a s s i c a l hydrodynamics, t u r b u l e n t flow at f a i r l y l a r g e Reynolds numbers i s c h a r a c t e r i z e d by the presence of an extremely i r r e g u l a r v a r i a t i o n of v e l o c i t y w i t h time at each p o i n t . The v e l o c i t y c o n t i n u o u s l y f l u c t u a t e s about some mean value , and i t should be noted that the amplitude of t h i s v a r i a t i o n i s i n general not s m a l l i n comparison with the magnitude of the v e l o c i t y i t s e l f . A s i m i l a r i r r e g u l a r v a r i a t i o n of the v e l o c i t y e x i s t s between p o i n t s i n the flow at a given i n s t a n t . The paths of the f l u i d p a r t i c l e s i n t u r b u l e n t flow are extremely complicated, r e s u l t i n g i n an e x t e n s i v e mixing of the f l u i d . Owing to the extreme complexity and i r r e g u l a r i t y of the motion of the molecules, the theory of t u r b u l e n c e has to be a s t a t i s t i c a l theory (Landau and L i f s h i t z (1959), B a t c h e l o r (1953)). No complete q u a n t i t a t i v e theory of tu r b u l e n c e has yet been evolved. -39-Let the mean v e l o c i t y be denoted by u =? V . The d i f f e r e n c e v' v - u between the t r u e v e l o c i t y and the mean v e l o c i t y v a r i e s i r r e g u l a r l y i n the manuer c h a r a c t e r i s t i c of tur b u l e n c e . v' i s c a l l e d the f l u c t u a t i n g p a r t of the v e l o c i t y . T h i s i r r e g u l a r motion superposed on the mean flow may be q u a l i t a t i v e l y regarded as the s a p e r p o s i t i o n of t u r b u l e n t eddies of d i f f e r e n t s i z e s . In other words, the t u r b u l e n t flow behaves to some extent l i k e a laminar flow, but with an e x t r a e f f e c t i v e v i s c o s i t y due to eddies present i n the flow. As the Reynolds number i n c r e a s e s , l a r g e eddies appear f i r s t ; the s m a l l e r the eddies, the l a t e r they appear. For very l a r g e Reynolds numbers, eddies of every s i z e from the l a r g e s t to the s m a l l e s t are present. In the case of helium I I , the development of any q u a n t i t a t i v e theory of s u p e r f l u i d t u r b u l e n c e p r e s e n t s extremely d i f f i c u l t problems, probably as d i f f i c u l t as those encountered i n c l a s s i c a l t u r b u l e n c e theory. However, we may imagine that s i m i l a r s i t u a t i o n s occur i n t u r b u l e n t helium. Throughout most of the s u p e r f l u i d any flow must be i r r o t a t i o n a l , but l o c a l i z e d r e g i o n s of v o r t i c i t y are p o s s i b l e i n the form of e i t h e r quantized v o r t e x l i n e s or vo r t e x sheets. In f a c t the s u p e r f l u i d eddy v i s c o s i t y can be estimated i f vortex l i n e s combine together to make eddies. The r e s u l t s agree with experiment. -40-5.2. Discussion of Results The interpretation of the experimental r e s u l t s w i l l be summarized under two headings based on the idea of quantized vortex l i n e s i n supe r f l u i d flow as developed by Feynman (1955) and Vinen (1961). 5.2.1. v s Measurements The graphs for v s vs. power W (figures 6,7,8) show the same shape as that from the s i m i l a r experiments by Kapitza (1941) and many others. Below a certain power input, the v e l o c i t y i s l i n e a r l y proportional to the power. No tempera-ture gradient appears between tubes Tr> and T^ (see figure 5). In the present case, we have a thermal ground i n the positions of the superleaks; therefore the power used i s higher than that i n Kapitza's experiments, since there i s some heat being conducted into the helium bath and consequently l o s t i n warming up the whole helium. Once the power exceeds a cert a i n c r i t i c a l value corresponding to a c r i t i c a l v e l o c i t y ( v s ^ c ) the v e l o c i t y ceases to increase l i n e a r l y with increas-ing power. At the same time temperature gradient sets i n . The excess heat i s used in the d i s s i p a t i o n process occurring i n the turbulent superfluid and also i n warming up the helium bath. Moreover, the v e l o c i t y becomes a s p e c i a l function of t and also fluctuates about a mean value. In Winkel's (1955) experiment, i t was found at 1.72\u00b0K for d (width of the channel) \u00ab\u00b0 1.5 u. with power input I = 2.15 mW, the time needed to reach an equilibrium state was approximately 4 min. -41-This time would depend on experimental conditions. The same thing might be happening i n our case, but due to the present way of measuring v s , i t was not possible to check i t . If measurements of v s are taken immediately after switching on the power, the values of v s w i l l be smaller than the steady value of v s . v , however, has to be measured some time after the power i s switched on, and i s therefore related to the steady value of v g and not to i t s i n i t i a l value. The calculated values of v s from the thermomechanical e f f e c t are also shown i n figu r e 6 by assuming that there i s no heat leak to the helium bath and no d i s s i p a t i o n process occurs i n the tunnel. From these two curves one can observe how much heat power i s l o s t i n turbulence of superfluid and warming up the helium bath. 5.2.2. Vp Measurmeat and Flow Properties i n the Superfluid Wind Tunnel F i r s t of a l l we want to describe the mechanism which makes the p a r t i c l e s move i n the tunnel. The process of creation of vortex l i n e i n supe r f l u i d turbulence has been discussed i n sections 2.4 and 5.1. The breakdown of i d e a l superflow i s due to the creation of quantized vortex l i n e s (or vortex sheets); these l i n e s s t r e t c h and spread out into a tangled array, so that the superfluid becomes turbulent. A motion involving v o r t i c i t y i s unstable. The vortex l i n e s -42-twist about i n an even more complex fashion, increasing t h e i r length at the expense of the k i n e t i c energy of the main stream. That i s , i f a l i q u i d Is flowing at a uniform v e l o c i t y and a vortex l i n e i s started somewhere upstream, th i s l i n e i s twisted into a long complex tangle further down stream. (The source of vortex l i n e s i s the contact between flowing l i q u i d and the walls which are i r r e g u l a r . Vortex l i n e s are created inside the tube.) To the uniform v e l o c i t y (v s) i s added a complex i r r e g u l a r v e l o c i t y f i e l d . The energy of t h i s i s supplied by the pressure head. These things happen i n helium since, except for distances of a few %. from the core of the vortex, the laws obeyed are those of c l a s s i c a l hydrodynamics. It i s reasonable to expect that the vortex l i n e s (which are quantised) i n the superfluid wind tunnel would drag the p a r t i c l e s made by mixing D 2 and II 2, i f the energy of the vortex l i n e s or sheets acting i s large enough to supply the k i n e t i c energy of the p a r t i c l e s . However Kelvin's theorem, the law of conservation of c i r c u l a t i o n in c l a s s i c a l hydro-dynamics, states that i n a homogeneous f r i c t i o n l e s s f l u i d the c i r c u l a t i o n round a closed f l u i d contour i s constant i n time. Therefore i t must have, i f any, even numbers of quantized vortex l i n e s terminating on a free p a r t i c l e . Half of the l i n e s must have opposite p o l a r i t y but the same strength ( i . e . same quantum of c i r c u l a t i o n ) as the other half. The energy which comes from shoi'tened segments w i l l make the p a r t i c l e move. -43-If the p a r t i c l e s i z e (of the order 0.2 to 1 mm in diameter) i s large compared with the diameter of the vortex l i n e and the distance between the l i n e s , one might expect two or more l i n e s to act upon a single p a r t i c l e . In some cases, a l i n e p a r t i a l l y acting on the p a r t i c l e i s also possible (similar to that of Vinen's experiment (1931) on v i b r a t i n g wires). The p a r t i c l e s w i l l move faster i f the number of vortex l i n e s q and the number of quantum of c i r -c ulation p are larger. A rough c a l c u l a t i o n presented here shows that the v e l o c i t y of a p a r t i c l e of given dimensions i s r e s t r i c t e d around some d e f i n i t e values, i f i t i s dragged by an even number of quantized vortex l i n e s with d i f f e r e n t number of quanta of c i r c u l a t i o n p. Feynman (1955) obtains the formula for energy per unit length of one quantized vortex l i n e as l i n e s a i s a length of the order of the atomic spacing The exact determination of a would require solving the d i f f i c u l t quantum mechanical problem. In almost a l l applica-tions the r a t i o b\/a w i l l be very large, and the logarithm m where j\u00b00 i s the f l u i d density i n atoms per cc m i s the mass fo r one helium atom b i s the distance between the centers of two -44-large enough to be in s e n s i t i v e to the exact value of a. For t h i s reason we w i l l not attempt a detailed evaluation, but simply choose a to be close to the atomic spacing, o say 4.0 A. In a more complicated s i t u a t i o n the value of a w i l l be the same, but the value of b w i l l be some other c h a r a c t e r i s t i c dimension of the apparatus or more usually the spacing between vortex l i n e s which depend on the properties of turbulence present i n the tube. -For a cylinder of l i q u i d rotating at angular v e l o c i t y co \u00bb 1 rad\/sec the vortices are about 0 . 2 cm apart; t h i s makes the value of ln(b\/a) = 14. In the present case, the actual distance between vor t i c e s are not known. But we can s t i l l use ln(b\/a) = 14 as an approximated value. Then tie energy of a quantized vortex l i n e of 1 cm long w i l l be 14 p^ x 10 ergs. If the p a r t i c l e moves a distance dx, at the same time shortens q vortex l i n e s by dx, each vortex l i n e has energy \u00a3 per unit length, then the energy given up by vortex l i n e s i s q Q \u2022 dx = dE. Therefore force on the p a r t i c l e = \u2014 - q \u00a3 . The mechanism of a vortex l i n e giving up i t s energy to the p a r t i c l e may be considered i n thi s way. When a vortex l i n e has one end attached somewhere on the wall, and the other end to the p a r t i c l e , i t mijht contract. The energy of the shortened l i n e segment w i l l be transferred to the K.E. of the p a r t i c l e and increase i t s v e l o c i t y v along the l i n e . -45-Suppose 2, 4, 6, ... q l i n e s attach themselves to the p a r t i c l e and p u l l i t against the viscous resistance of the normal f l u i d . The equation of motion for the p a r t i c l e with mass m and radius r may be written as where q\u00a3is the force on the p a r t i c l e due to q l i n e s . 2 T_ + dx d x .Let \u2014 =\u2022 v . \u2014\u2014. = v dt ' J A 2 d f the equation becomes mv + 6rp2y v = q (5.2) Assume the solutions of the homogeneous equation mv + GTT^y v = 0 (5.3) are -t\/rz t -t\/\"C v H - A e , v H - -A - 1 (5.4) sub s t i t u t i n g Eq. (5.4) into Eq. (5.3) we have - m ^  + A 677-ifr- = 0 m -46-If the general solutions of Eq. (5.2) are of the form - t \/ r v = v H + V p = A e + C then Q \u201e =-r-we have the general solu t i o n v = A e \" t \/ C + SlL (5,6) If v = 0 at t = 0 ( i . e . the v e l o c i t y of the p a r t i c l e i s zero before i t i s attached by the li n e s ) then A . _ SLL we have The value of time constant z. i s (1 - e \" t \/ r ) (5-7) m ^ _~ _ 1 4 3 1 2r* (artjy 7 3 ' ^Tryy 63\u00a3 -47--2 For a p a r t i c l e of radius r = 10 cm moving i n a viscous f l u i d with 1.5 x 10~5 poise (this i s about the v i s c o s i t y of normal f l u i d at 1.4\u00b0S) 2.10\"\"4 x x sec ~ \u2014 sec 1.5 x 63 x 10~5 4 , 7 5 In the Vp measurements, the p a r t i c l e s always moved about one to two cms, i . e . at least two sec, before the data were taken. This i s large compared with 1\/5 sec, so that in most of the cases we have the values of Vp i n the stationary state. But for larger p a r t i c l e s , r i s i n fact -2 a few times 10 cm. The values of XL w i l l be a few sec, therefore v p observed may be only a f r a c t i o n of Vp f i n a l . Consider the case t \u00bb \"C , from Eq. (5.7) and the value of \u00a3 , we get ^ - -2\u2014 x 14 p 2 x 10\"\"8 (5.8) where p i s number of quanta of c i r c u l a t i o n associated with each l i n e . From Eq. (5.8) we can calculate the v e l o c i t i e s v (see the following table) of a p a r t i c l e attached by an even number of vortex l i n e s i n the stationary state with d i f f e r e n t values of p. -48-TABLE 3 q=\u00bb2 q=4 q=S q\u00bb8 p=l 0.146 cm\/sec 0.292 cm\/sec 0.438 cm\/sec 0.584 cm\/sec p=2 0.590 M 1.180 \" 1.770 * 2.360 p-3 1.200 \" 2.400 \" 3.600 \" 4.800 p=4 2.360 \" 4.720 \" 7.080 \" 9.440 Some of these v e l o c i t i e s come out to be very close to our experimental values of Vp. However, i t must be noted here that the calculations stated above are based on Feynman's formulation which, as mentioned by Feynman himself, are not based on a fir m foundation, but merely a reasonable estimate. For t h i s reason, and because of the fact that the exact dimensions of the d i f f e r e n t p a r t i c l e s can not be obtained, and the experimental errors, only q u a l i t a t i v e conclusions can be obtained and w i l l be discussed i n the following subsections for each case. (i) In the Region v p < v p Superfluid helium i s an ideal f l u i d of zero v i s c o s i t y . It does not exhibit turbulence below the c r i t i c a l v e l o c i t y v\u201e _ because of the quantum mechanical e f f e c t . Since -49-v o r t i c i t y i s quantized and cannot begin at as low amount as desired, one must supply enough energy to create the f i r s t one or two vortex l i n e s . Below v p ^ c the f l u i d i s a pure po t e n t i a l flow. Any s o l i d p a r t i c l e suspended i n such a pei'fect flow w i l l not move because there i s no vortex l i n e to drag the p a r t i c l e along. Our r e s u l t a c t u a l l y confirms t h i s f a c t . But due to the present way of making p a r t i c l e s the c r i t i c a l v e l o c i t y (v_ .) i s not accurately defined. ( i i ) In the Region v p c < v p < v s ^ c Above v p ^ c , t h e vortex l i n e s are generated and the number of l i n e s w i l l increase gradually as the power input increases. Most of the l i n e s w i l l move i r r e g u l a r l y towards the heater. The more power input (the higher v s are), the more irr e g u l a r and denser the l i n e s . Once a p a r t i c l e i s attached by two or more l i n e s which w i l l give up energy to the p a r t i c l e s , the p a r t i c l e s w i l l move with a v e l o c i t y v p . As the power input i s gradually increased (therefore v s increased), the p r o b a b i l i t y of one p a r t i c l e being acted upon by more l i n e s (the number i s q) with higher number of quanta of c i r c u l a t i o n p i s increased. This makes the average v e l o c i t i e s v p (shown i n figures 6, 7 and 8) increase with increasing power. The values of v p obtained are not equal to that calculated above. Namely 0.146 cm\/sec, 0.590 cm\/sec, 1.20 cm\/sec ... corresponding respectively to p = 1, 2, 3, ... etc. This may be due to one of the following reasons. -50-(a) The p a r t i c l e s are of d i f f e r e n t sizes with d i f f e r e n t diameters. But the calculated values are for r \u00ab 0.1 min. In addition, they are not perfect spheres as has been assumed. (b) The number q may be d i f f e r e n t for d i f f e r e n t p a r t i c l e s because of the uneven d i s t r i b u t i o n of l i n e s in the tube and the d i f f e r e n t contact areas of l i n e s with p a r t i c l e s . (c) I t i s quite possible that a p a r t i c l e may be attached simultaneously by two l i n e s with p \u00ab\u2022 1 and two li n e s with p =\u2022 2 etc. (d) The calculations shown above are only an approximation. Several factors, e.g. radius r, the values of b changed with d i f f e r e n t p a r t i c l e s . (e) The measured v p are act u a l l y average v e l o c i t i e s over a ce r t a i n distance, but not instantaneous v e l o c i t i e s which are probably needed. ( i i i ) In the V i c i n i t y of C r i t i c a l V e l o c i t y v S i C Feynman (1955) makes the following prediction about the region very close to the c r i t i c a l v e l o c i t y v g ^ c . As a r e s u l t of more energy being needed to form more vortex l i n e s i n the stage of the onset of f u l l y developed turbulence, probably the s i t u a t i o n near the c r i t i c a l point w i l l be very complicated and ir r e g u l a r . He predicts that very close to the c r i t i c a l v e l o c i t y when loss just begins, the resistance w i l l be ir r e g u l a r and show flu c t u a t i o n s . -51-These flu c t u a t i o n s are very small however and would be hard to detect. When helium i s driven, just above the c r i t i c a l v e l o c i t y , through an emery powder superleak, some noise should be generated as the various vortex l i n e s suddenly form and pass into the stream. The i r r e g u l a r i t i e s are a r e s u l t of unpredictable quantum t r a n s i t i o n s between states of no vortex l i n e and one with a section of l i n e . In c l a s s i c a l turbulence the quantity i s a measure of the magnitude of the turbulent disturbance. This quantity, known as the \" i n t e n s i t y of turbulence\" may have values i n the neighborhood ox 1 to 10 per cent in ordinary f l u i d flowing i n a tube. (The value < u > i s the mean of the time average of u.) Corresponding to t h i s , the quantity V-^v^p\/Vp may be used as the magnitude of turbulent f l u c t u a t i o n I i n l i q u i d helium. The values of I corresponding to each power input are calculated (see Table 2) for 3 runs we have performed. The graphs are plotted i n figure 9. The important things here are near the c r i t i c a l v e l o c i t y . There are peaks which do show maximum f l u c t u a -tions just above the c r i t i c a l v e l o c i t y . This i s exactly the s i t u a t i o n Feynman (1955) predicted. Furthermore, the graph also shows that there e x i s t s a minimum after the turbulence has f u l l y developed. At t h i s v e l o c i t y the f l u c t u a t i o n i s small (though higher than those from c l a s s i c a l turbulence), and a meta-stable Power Number of input observations W(mW) n 1st run S.8 6 13.8 9 19.8 13 27.0 11 35.2 12 44.5 12 55.0 13 2nd run 16.G 7 23.2 13 30.8 11 39.6 12 40.7 10 3rd run 8.8 6 13.8 8 19.8 23 27.0 5 35.2 7 44.5 20 TABLE 2 Average v e l o c i t i e s v~ n(cm\/sec) Standard deviation(s.d.) (cm\/sec) Intensity of turbulence I to ( vP,n- vp,n)^\/\u00bb s - d - \/ ^ S.d. from the mean (cm\/sec) s.d. \/J n - l ' 0.143 0. 232 0.329 0.45G 0.453 0.695 0.797 0.0243 0.0363 0.0791 0.0855 0.0532 0.1341 0.2077 17 15. 6 24 13.7 11.8 19 2G. 1 SO.0185 \u00b10.0128 \u00b10.0211 \u00b10.0270 \u00b10.0160 \u00b10.0405 \u00b10.0535 0.258 0.345 0.379 0.445 0.493 0.0595 0.10055 0.0468 0.0788 0.0916 23.1 29 12.8 17.3 18.6 \u00b10.0243 \u00b10.0290 \u00b10.0148 \u00b10.0237 \u00b10.0305 0.143 0.20 0.291 0. 533 0. 561 0.623 0.0359 0.0572 0.1012 0.1650 0.2210 0.2093 15.8 28.6 34.6 31 39.4 33.6 \u00b10.0160 \u00b10.0216 \u00b10.0202 \u00b10.0825 \u00b10.0245 \u00b10.0241 to follow page 5 I 4 0 35 3 0 2 5 u o z UJ -12 0 ZD GO or 3 O I 5 UJ r-510 5 h 0 0 3rd run 1 10 2 0 3 0 4 0 POWER INPUT (mW) 5 0 Figure 9. Intensity I v e r s u s power input -52-stato might be attained at t h i s point. When the power input i s higher than t h i s minimum, the f l u c t u a t i o n v \/ i l l increase again. However, the curve for the t h i r d run shows s l i g h t l y d i f f e r e n t behavior. After a minimum i s reached another maximum occurs. This might be due to the averaging from only a few points and are.not enough to represent the s t a t i s t i c a l average of the actual f a c t . (iv) In the Region v p >^ v s c Above v s ^ c the mean p a r t i c l e v e l o c i t i e s Vp are much higher than v g. From figures 6, 7 and 8 i t can be seen that Vp increase with greater power input, but v g does not. This i s because v p are the v e l o c i t i e s of p a r t i c l e s due to the action of vortex l i n e s which represent small l o c a l v e l o c i t y variations superposed on the main flow. Therefore i t i s reasonable to expect that the mean ve l o c i t y of the main flow (vs) w i l l not change with the \\ortex l i n e s present in i t . The excess power are used i n generating more vortex l i n e s probably with high p values without increasing v s . This i s what ac t u a l l y observed i n vp measurements. Nearly a l l the values of Vp are higher than v g. The i n d i v i d u a l values of vp do not equal that obtained from calculations. This may be due to s i m i l a r reasons to those stated i n subsection ( i i i ) . Except in t h i s region there are more vortex l i n e s present in the tunnel with a smaller distance between them. Moreover, the number p may have the values 1, 2 or even 3. At the same time in the higher v e l o c i t y measurements the errors are r e l a t i v e l y high. F i n a l l y , i t i s inter e s t i n g to note that in t h i s region almost a l l the widely scattered v p occur between the t h e o r e t i c a l (curve c on figure 6) and measured (curve A on figure G) v s. It means p and q are r e s t r i c t e d to the following values (see Table 3) for p = 1 , q => 4, 6, 8, 10 p = 2 , q = 2, 4 and p = 3 , q = 2 For greater power input q i s increased, i . e . more pairs of vortex l i n e s are terminating on one p a r t i c l e . -54-CHAPTER VI CONCLUSIONS From the preceding discussion i t i s evident that p a r t i c l e s made with a suitable mixture of deuterium and hydrogen gases can be used to study the macroscopic motion of pure superflow i n l i q u i d helium II with the stationary normal f l u i d as a background. The Gorter-Mellink experiments on heat flow, Hall ' s angular acceleration experiments, and Vinen's vibrating wire experiments can a l l be interpreted by assuming that superflow can break down into turbulent flow, in which the motion of the superfluid consists of a tangled mass of vortex l i n e s . In t h i s experiment i t has been proven that the motion of vortex l i n e s i s actually superimposed on the ide a l super-f l u i d flow, just as the motion of eddies on the main flow in a c l a s s i c a l f l u i d . By v i s u a l observations of the motion of suspended p a r t i c l e s i n the superf l u i d wind tunnel, i t has been found that the vortex l i n e s may p u l l the p a r t i c l e s along the l i n e s i f the energy transferred to the p a r t i c l e s from shortened segments of the l i n e s are large enough. Two c r i t i c a l v e l o c i t i e s , v p c and v s ^ c were found. -55-Below v_ no p a r t i c l e can move, indi c a t i n g that the super-f l u i d flow below a cert a i n c r i t i c a l v e l o c i t y i s i n fact perfect p o t e n t i a l flow of zero v i s c o s i t y . Above VpfC quantized vortex l i n e s are created, therefore the pure superfluid flow breaks down. In the region v p ^ c < v p < v s , c t h e quantized vortex l i n e s with p (number of quanta of c i r c u l a t i o n ) equal to 1 or 2 are possible. The average p a r t i c l e v e l o c i t i e s v p are l i k e l y to be equal to the superfluid v e l o c i t i e s v s . It i s possible that accurate values of the c r i t i c a l v e l o c i t y do not exist i n p r i n c i p l e . For, i f the c r i t i c a l behavior i s of microscopic o r i g i n , due to the creation of quantized vortex l i n e s , i t i s by no means certa i n that i t can always be related to the same macroscopic average v e l o c i t y . In the v i c i n i t y of the c r i t i c a l v e l o c i t y v s ^ c , the stage of onset of f u l l y developed turbulence, Feynman (1955) stated that the s i t u a t i o n must be very complicated and i r r e g u l a r . He predicted that very close to the c r i t i c a l v e l o c i t y when loss just begins, the resistance w i l l be i r r e g u l a r and show large f l u c t u a t i o n s . This has been con-firmed i n the present experiment, namely the fluctuations of turbulence has a maximum near the c r i t i c a l v e l o c i t y v s ^ c (see figure 9). Above v_ \u201e the p a r t i c l e v e l o c i t i e s become widely scattered due to the complicated motions of the quantized -56-vortex l i n e s . The number p for the l i n e s may have the values 1, 2 or even 3. This makes v p higher than v s . A l l the excess heat from the heater i s used i n creating these l i n e s , and a temperature gradient appears. From the calculations for v e l o c i t y of a p a r t i c l e cor-responding to d i f f e r e n t values of p, i t was found that the energy of a quantized vortex l i n e was large enough to make the p a r t i c l e move with an observable v e l o c i t y . One would expect that t h i s method might be a very powerful way of examining experimentally the true picture of the quantiza-t i o n of c i r c u l a t i o n of vortex l i n e s i n superf l u i d flow, either i n a tube flow ( l i k e the present set up) or i n rotat i n g helium. In t h i s way the actual behavior of super-f l u i d flow may be understood. Ilowever, more th e o r e t i c a l works are needed to c l a r i f y the actual microscopic pictures and to provide a better foundation for the possible development of t h i s technique. BIBLIOGRAPHY All e n , J. F., P e i e r l s , R., and Uddin, M. Z. Nature 140, 62 (1937). Allen, J. F. and Misener, A. D. Nature 141, 75 (1939). Allen, J. F. and Misener, A. D. Proc. Roy. Soc. A 172, 467 (1939). Andronikashvili, E. L., J. of Phys. (USSR) JLO, 201 (1946). Atkins, K. R. Liquid Helium (Cambridge University Press, 1959). Batchelor, G. K.. Homogeneous Turbulence (Cambridge University Press, 1953). Benson, C. 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