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Critical velocities and generation of vorticity in liquid helium II in large pipes of rectangular cross-section… Slater, Williams James 1971

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CRITICAL VELOCITIES AND GENERATION OF VORTICITY IN LIQUID HELIUM II IN LARGE PIPES OF RECTANGULAR CROSS=SECTION AT 1.40 K 1 by WILLIAM JAMES SLATER B. Sc. a University of Alberta, 1958 M. Sc., McMaster University, 1960 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1971 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h . - C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f m y D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ^V^kyS 1 T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a D a t e 11 May 7 / I l l ABSTRACT C r i t i c a l velocities have been measured in liquid helium at 1.40 K for eounterflow 3 superflow arid eounterflow superimposed on superflow. The measurements were made in five pipes having rectangular cross-sec-tions and ranging i n size nominally from 0.2 x 1.2 x 10 cm to 1.2 x .1.2 x 10 cm. The turbulence was detected by means of negative, ion currents perpendicular to the flow. The results are interpreted in terms of creation of quantized vortex lines. For superflow, the c r i t i c a l velocities were slightly higher but within the experimental range of those measured for eounterflow. For the case of superflow-plus-counter f low a second c r i t i c a l point was observed, i n addition to the one corresponding to superflow only. Considerable data was also collected regarding the nature of the ionic current attenuation effects for supercritical flow. The delay times between when the flow rate was raised to or above the c r i t i c a l velocity and when the current attenua-tions were f i r s t observed ranged from more than five minutes for the smallest pipe to three-quarters of a minute or less for the largest one. When the pipe width i s greater than the ion source width the attenua-tions may be explained by Vinen's theory of vortex generation and decay but for smaller pipe widths the attenuation buildup and decay times are greatly increased. A method for increasing the sensitivity of the detection method for large pipes is also outlined, CONTENTS Page List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . v i i i L i s t of Figures . . . . . . . . . . . . . . . . . . . . . . . . . i - x Acknowledgements . . . . . . . . . . . . . . . . , , , xi . CHAPTER 1 INTRODUCTION 1 CHAPTER 2 THEORY 2.1 Excitations i n Liquid Helium II . . . . . . . . . . . . . 3 a. Elementary excitations: phonons and rotons . . . . . 3 b. Vortex lines . . . . . . . . . . . . . . . . . . . . . 3 c. Vortex rings . . . . . . . . . . . . . . . . 5 2.2 Thermodynamic Effects; The Two=Fluid Model . 5 a. Generation of counterflow . . . . . . . . . . . . . . . 5 b. Generation of superflow . . . . . . . . . . . . . . . 7 2.3 Velocity Profiles, Pressure and Thermal Gradients i n Pipes of Rectangular Cross-Section . . . . . . . . . . . . . . . 7 a. Velocity profile of the superfluid component . . . . . 7 b. Velocity profile of the normal component . . . . . . . . 8 c. Pressure and thermal gradients for counterflow . . . . 9 2.4 C r i t i c a l Velocities and Dissipation Forces . . . . . . . . 12 a. Superfluid component c r i t i c a l velocity v S s C and fr i c t i o n force term "Fs . . . . . . . . . . . . . . . . 12 b. Relative c r i t i c a l velocity V r -c and mutual f r i c t i o n force term "F s n . . . . . . . . . . . . . . . . . . . . . 14 c. Normal component c r i t i c a l velocity v"hsc and f r i c t i o n force term T^; C r i t i c a l Reynolds numbers . . . . . . . 17 2.5 Ions i n He II . . . . . . 2 . . . . . . . . . . • . . . . 18 a. Production of ions; Po and Am . . . . . . . . . 18 b. Nature of ions in He II 19 c. Interaction between ions and the elementary excitations; ionic mobilities . . . . . . . . . . . . . . . . . . . 19 d. Interactions between ions and vortices . . . . . . . . 21 e. Space charge effects - Theoretical . . . . . . . . . . 23 f„ Space charge effects - Experimental . . . . . . . . . 32 g. Motion of ions i n pipes during flow . . . . . . . . . . 33 CHAPTER 3 EXPERIMENTAL DETAILS 3.1 Outline of Apparatus . . . . . . . . . . . . . . . . . . . . 37 a. Counterflow . . . . . . . . . . . . . . . . . . . . . . . 37 b. Superflow . . . . . . . . . . . . . . . . . . . . . . . . 37 c. Combination of counterflow and superflow . . . . . . . . 39 3.2 Details of Wind Tunnel Aparatus . . . . . . . . . . . . . . . 39 a. Main superleak section . . . . . . . . . . . . . . . . . 39 b. Measurement of superflow . . . . . . . . . . . . . . . . 42 3.3 Details of Pipes . . . . . . . . . . . . . . . . . . . . . . . 44 a. Construction . . . . . . . . . . . . . . . . . 44 b. Preparation of the ion sources . . . . . . . . . . . . . 46 3.4 Ionic Current Measurement . . . . . . . . . . . . . . . . . . 48 3.5 The Gryostat . . . . . . . . . . . . . . 49 a. Vacuum system . . . . . . . . . . . . . . . . . . . . . . 49 b. Temperature measurement and control . . . . . . . . . . . 49 c. Leads inside cryostat . . . . . . . . . , , 51 d. Shielding the cryostat . . . . . . . . . . . . . . . . . 52 e. Apparatus immersion controller . . . . . . . . . . . . . 53 3.6 Experimental Procedure . . . . . . . . . . . . . . . . . . . 54 CHAPTER 4 DATA AND RESULTS 4.1 Outline of the Measurements . . . . . . . . . . . . . . . . . 55 4.2 C r i t i c a l Velocity Determination by Observation of Ionic Beam Deflection During Counterflow . . . . . . . . . . . . . . . . 55 4.3 C r i t i c a l Velocities and Attenuation Effects for Counterflow . 56 a. Pipe A . . . . . . . . . . . . . . . . . . . . . . . . . 56 b. Pipe B 61 c. Pipe C d. Pipe D e. Pipe E 61 61 62 f. Summary of c r i t i c a l velocity data for counterflow using detection of attenuation method . . . . . . . 4.4 C r i t i c a l Velocities and Attenuation Effects for Superflow . 66 4.5 Comparison of C r i t i c a l Velocities with Those of Previous Workers . . . . . . . . . . . . . . . . 69 4.6 Data for Gounterflow Superimposed on Pure Superflow . . . -73 4.7 Details of the Attenuation Effects . . . . . . . . . . . . 74 a. Increase of attenuation during supercritical flow =~ General characteristics . . . . . . . . . . . . . . . 74 b. Attenuation as a function of flow velocity . . . . . . 75 (1) Rate of attenuation increase for Pipe A . . . . . . 75 (2) Attenuation for Pipe C . . . . . . . . . . . . . . 80 c. Effect of turning off applied voltage; Hysteresis . . 83 4.8 Comparison of the Attenuation Data with Vinen!s • Relations . . . . . . . . . . . . . . . . 88 CHAPTER. 5 CONCLUSIONS 5.1 C r i t i c a l Velocities . . . . . . . . . . . . . . 92 5.2 Attenuation Effects . . . . . . . . . . . . . . . . . . . 93 CHAPTER 6 PROPOSALS FOR FURTHER WORK 96 APPENDICES 1. Calculation of kinetic energy factor, fT,„ _ 98 O J ' KEsmaxs D 2. Calculation of f^, a for pipes shorter than the transition length; . • 104 3. Pipe dimensions at room temperature . . . . . . . . . . . . 107a 4. Estimation of total currents from oC -sources i n He II . . 108 5. Calculation of trapping lifetime for negative ions in vortex lines at 1.40 K and E = 75 V/cm by Pratt and Zimmermann's method . . . . . . . . . . . . . . . . . . . . 113 6. Development of shields for Pipe. D to enhance current attenuation 116 7. C r i t i c a l velocity data for counterflow at 1.40 K . . . . . . . . 117 8. C r i t i c a l velocity data for superflow at 1.40 K . . . . . . . . . 123 9. Current attenuation at Electrode 4 for Pipe C . . . . . . . . . . 125 10. Superleak=tight "grease" for low temperature joints . . . . . . 128 11. Build-up and decay of vortex lines . . . . . . . . . . . . . . . 132 REFERENCES 134 LIST OF TABLES .Table Page 3.3.1 Pipe dimensions and cross-sectional areas at liquid He temperatures 45 4.3.1 Summary of c r i t i c a l velocity data for counterflow 63 4.3.2 Calculation of temperature differences between the ends of the pipes for laminar flow at the measured c r i t i c a l velo- 64 cities for counterflow. 4.4.1 Summary of c r i t i c a l velocity data for superflow 67 4.5.1 Values of v g . c .d^ for Fig. 4.5.1 70 4.7.1 Rate of increase of current attenuation for Pipe A 78 4.7.2 Current attenuation for Pipe C 81 A l . l Evaluation of f T _ for representative values of a/b 103 KE,max,D A2.1 Calculation of f ^ Q and comparison with those for circu-lar and equilateral triangular cross-sections 107 A3.1 Pipe dimensions at room temperature 107a A7.1 C r i t i c a l velocity data for counterflow 118 A8.1 C r i t i c a l velocity data for superflow 124 LIST OF FIGURES Fig. Page 2.5.1 Model used for calculation for local potential due to ions trapped in vortices 26 2.5.2 Schematic diagram of pipe showing deflection of ionic beam 34 3.1.1 Superfluid wind tunnel apparatus 38 3.1.2 Pipes used i n the wind tunnel apparatus 40 3.3.1 Detail of source electrode and source holder 47 3.5.1 Gryostat vacuum system 50 4.2.1 Ionic deflection data for Pipe D 57 4.3.1 Typical records for Pipe A 58 4.3.2 Observed normal component c r i t i c a l velocity vs hydraulic diameter for counterflow 65 4.4.1 Observed superfluid component c r i t i c a l velocity vs hydraulic diameter for superflow 68 4.5.1 Gomparison of results with those of previous workers 71 4.7.1 Rate of increase of current attenuation vs |v n " v s | f° r Pipe A 79 4.7.2 Current attenuation vs heat power density for Pipe C 82 4.7.3 Current at Electrode 4 vs applied voltage for Pipe A after space charge established 84 4.7.4 Current at Electrode 4 vs applied voltage for Pipe E after space charge established 85 4.7.5 Current at Electrode 4 vs applied voltage in absence of attenuation effects 87 LIST OF FIGURES (continued) A4.1 . Gurrent vs average applied electric f i e l d for radial geometry (Hereford and Moss) 111 A4 = 2 Total current vs applied electric f i e l d for Pipe D 112 A9.1 Schematic diagram showing deflection of ionic beam for high normal component velocities 126 A10,l Apparatus to test superleak sealing properties of glycerine-white lead mixture 130 ACKNOWLEDGEMENTS I would like to thank Dr. P. R. Critchlow for suggesting this project and supervising the i n i t i a l experiments. Dr. M. J. Crooks supervised the fina l experiments and the preparation of this thesis. Dr. P. Matthews, Dr. J. B. Brown and Dr. B. Howard also helped to cl a r i f y many points through their questions and comments. Mr. R. Weisbach and Mr. G. Brooks provided considerable technical assistance in the construction of the cryostat, Mr. J. Lees produced the glassware and suggested a number of recipes and procedures. Particular thanks are due my wife, Judy, for her patience and under-standing. Miss Donna Brooks typed the thesis in i t s fi n a l form, I also wish to acknowledge the financial assistance given me by the National Research Council through a Studentship (1962=1965) and research grants (67-1134 and 67-2210). The Physics Department of the University of British Columbia also provided considerable financial support through Assistantships. CHAPTER I  INTRODUCTION Since superfluidity i n liquid helium at temperatures below 2.2K was discovered in 1938, a great many experiments have been carried out i n order to determine the properties of this unusual f l u i d , i n particular, to measure the " c r i t i c a l velocities" of flow at which changes in the flow patterns occur. Most of the f l u i d properties have been described in terms of a model in which the helium liquid i s considered to consist of two interpenetrating components, each with i t s own density and inde= pendent velocity f i e l d . Whereas the "normal" component behaves similarly to a classical f l u i d , the "superfluid" component has zero viscosity and carries zero entropy provided the velocity of flow i s not too high. At sufficiently high velocities both components interact with one another and with the walls of the container; a major problem has been to deter= mine how and to what extent each of these interactions influence the flow characteristics of the liquid. In terms of the two=fluid model, the flow in the pipes may be any one of several modes including: a. both components moving together; b. the superfluid and normal components moving in opposite directions in such a way that the net mass flow i s zero. (This type of flow i s called "heat flush" or "counterflow".); c. the superfluid moving with the normal component being held stationary with respect to the walls of the pipe.(This type of flow i s referred to as "superflow" in this work.); d. a combination of the second and third cases above, i.e., the components moving in opposite directions with the net mass flow not equal to zero. Because of the experimental conditions used in this work, this type of flow i s hereinafter referred to as "counterflow superimposed on superflow". One of the most popular methods for determining c r i t i c a l velocities is simply to set up various flow conditions i n a pipe and to observe at what velocity or velocities abrupt changes in the pressure and temperature gradients occur. These abrupt changes are usually associated with the on-set of turbulence. But this turbulence appears to be a special type: quantized vortices are generated continuously in the liquid helium at supercritical velocities. Two interesting properties of these vortices from the viewpoint of the present work are their relatively long lifetime in the liquid after the flow has been stopped and their a b i l i t y to interact with both positive and negative ions under suitable conditons. Thus an ionic current may be used as a probe to detect the onset of the turbulence. The aim of the present experiment was to conduct measurements of c r i t i c a l velocities for counterflow, superflow and counterflow superimposed on superflow i n pipes of rectangular cross-section ranging in size nominally from 0.2 x 1.2 x 10 cm to 1.2 x 1.2 x 10 cm. For the largest size, electrical insulators (shields) were placed adjacent to ion collecting electrodes for a number of runs to improve the sensitivity of the measurements, In addition to measuring c r i t i c a l velocities, two effects observed during supercritical flow were studied^ a. attenuation of the total negative ion current; b. hysteresis effects due to build-up of the charge trapped in the vortices. Most of the measurements were conducted at 1.40 K; a few runs at 1.60 K yielded data which were similar in nature to those at the lower temperature. CHAPTER 2 THEORY 2.1 Excitations in Liquid Helium II a. Elementary excitations: phonons and rotons 2.1 .-1-.) Landau . . "first showed that the unique properties of liquid helium II could be explained by describing the atomic motions of the liquid in terms of (thermal) elementary excitations. He stated that the energy spectrum of these excitations had two branches corresponding to phonons and rotons. In the two-fluid model, the normal component velocity, v n, is identified with the dr i f t (or mean group) velocity of these excitations. At 1.4 K, the rotons (which exist only in the region of the dispersion curve minimum) are the dominant excitation, accounting for about 75 percent of the total entropy of the liquid. The re-maining entropy i s due to phonons having energies in the linear region of the dispersion curve, i.e., for small wave numbers. b. Vortex lines When a vessel containing He II is set rotating, both the normal and superfluid components rotate with the vessel except at very low angular velocities. The superfluid component, however, is irrotational, i.e., except at singularities' in the liquid • ' . " . These singulari-ties consist of an array of lines parallel to the axis of rotation, each having a circulation of curl V g - 0 (2.1,1) K = n h co0 m oo0 where h = Planck's constant; m = He atomic mass; n = positive integer, usually equal to one; u>0= angular velocity of the rotating vessel. These are called quantized vortex.;lines because their velocity fields correspond to those of classical vortices and their circulations are quantized in units of h/m. Each line ends either at the surface of the liquid, on some part of the apparatus or at the intersection with another line. The energy required to create a vortex line i s so high that the probability of thermal excitation i s negligible; the lines therefore make no significant contribution to the thermodynamic properties of He II. Their presence also does not affect the excitation spectrum of the phonons and rotons since they are so widely spaced (as viewed on the atomic scale). Vortex lines also are formed during flow at supercritical velocities in tubes and channels, although the mechanism of creation s t i l l i s not understood (See Section 2.4,). Here the lines are not aligned in any sort of array, but form a tangled mass. Nevertheless, the quantum condition i n Equation (2.1.2) s t i l l holds. Recently, a number of workers have reported finding quantized . _ 2.1.3) levels of circulation at K = (l/2)(h/m) and (3/2)(h/m) ( , a l -though the probability of observing them seems to be considerably less 2.1.4) than that for integer values,of h/m. Also Sanders and Weinreich have suggested why the circulations observed in vibrating wire exper-ments sometimes appear to vary continuously between the integer quantized values. 2 15") Wave function models " .' ' of vortices.are not yet sufficiently developed to the point where they can be used to predict c r i t i c a l velocities or the nature of ion-vortex interactions. c. Vortex rings The creation of quantized vortex rings in He II was originally proposed as a mechanism to explain some of the results obtained during recent heat flush experiments '. 1- These workers noted that as the velocity,of both positive and negative ions was increased (by increasing the electric f i e l d ) , the ionic mobilities decreased in steps. But the existence of these discontinuities has been confirmed in the other cryogenic liquids, He I, N„, Ar, as 2 18) well as CCl^ at room temperature so they cannot be attribu-ted to creation of the rings. Nevertheless, conclusive data on the existence of the rings 2 1.9) were obtained by Rayfield and Rief ' . i . N- using a time of flight method. They found that the ionic velocities and energies both increased as the f i e l d was increased u n t i l a c r i t i c a l value was reached, above which the velocity varied inversely as the f i e l d strength. This relationship corresponds closely to one between velocity and energy for classical vortex rings. This c r i t i c a l velocity was interpreted by the authors as being one in which the ions gain sufficient energy between collisions with the elementary excitations to create vortex rings sufficiently large that the ions can keep up with them. , 2.1.10, 2.1.11). v . ^ , Recent work has given further support to the vortex ring model but the mechanism of their creation is not known completely. 2.2 Thermodynamic Effects; Generation of Flow a. Generation of counterflow For counterflow there is no transfer of mass and so the mean superfluid component velocity v s m is related to the mean normal n • - 2.2.i*J component velocity v by n ,m - — J E T 1 - vn>^ (2.2.1) Is where ^ and ^ are the densities of the normal and superfluid com-ponents respectively. The mean velocities must be used in this expres-sion because the velocity profiles for the components are not the same, For low velocities the only dissipative force present i s that due to viscosity of the normal component. Assuming that the pressure gradient is constant along the pipe and ignoring the inertia effects for the present, the mean normal component velocity along the pipe i s P ' V n , m |3 U . 2 . 2 . ) where P' = pressure between the ends of the pipe; 1 = length of the pipe; /y|^ = coefficient of viscosity for the normal component; |3 = constant-depending on cross-sectional size and shape of the pipe. Now the normal component velocity is determined by the rate of heat flow Q into the He II: Q = T V n ^ A^o S (2.2.3) where T = temperature; •• A = cross-sectional area; ^° = liquid helium density; (= ^  + ); S = entropy per unit mass of the liquid helium; and the pressure difference is given by (thermomechanical effect) A T where A T = temperature difference between the ends of the pipe. By combining the last three equations, we obtain * A T « ^ / > (2.2. S) The exact form of the factor 3^ a n c* t n e i n e r t i a l effects w i l l be considered i n Section 2.3. b. Generation of superflow If two baths of He II are joined by a superleak and the tempera-ture of one of them is raised, the superfluid component w i l l flow through the superleak so as to reduce the temperature difference. Experimentally this relationship cannot be used directly since the temperature difference required to cause quite large flow rates are minute and the losses i n the superleak are not easy to calculate. Instead, a collection vessel is included so that the volume rate of flow -r— is measure A t fl u i d component is d directly. Then the mean velocity of the super-- p A A t ( 2 ' 2 ' 6 - ) 2.3 Velocity Profiles, Pressure and Thermal Gradients i n Pipes of Rectangular Cross-Section a. Velocity profile of the superfluid component The condition that the superfluid i s irrotational (Equation (2.1.1)) infers that the velocity profile i s a plane perpendicular to the flow direction. But there remains the question regarding 2.3.1) the boundary conditions at a wall. In a paper by Nozieres i t i s pointed out that the superfluid component probably does not interact with the walls un t i l the "Landau c r i t i c a l velocity" for the creation of rotons is reached (60 m/sec for temperatures not too close to the -point) and so the profile does, in fact, remain flat up to this point. b. Velocity profile of the normal component For steady laminar flow i n a pipe of rectangular cross-section and dimensions 2a x 2b the normal component velocity i s given by (ignoring , _ . 2.3.2) end effects) where -• a ^ x ^  a; -b <1 y b . The rate at which the series in this expression converges depends on the ratio a/bjit i s more convenient to choose the dimensions so that a/b > 1. 2.3.2) The mean velocity i s r j ^ ( 2 . 3 . 2 . ) c. Pressure and thermal gradients for counterflow In the case of superflow, calculation of pressure gradients are of no concern since the rate of flow can be measured directly. However, for counterflow the velocity must be calculated by Equation (2.2.3), so i t i s important to have an idea of the magnitude of the losses which have not yet been included. The pressure difference, P', i n Equations (2.3.1) and (2.3.2) is that due to viscous forces for the normal component only. To this must be added the pressure difference, P1', required to accelerate both components to their f i n a l velocities i n the pipe (the entrance losses), as well as that, P'1', associated with the exit losses. Thus the total pressure difference between the ends of an actual pipe is For the normal component, the value of P'' is obtained from consideration of the rate of flow of kinetic energy in the pipe p = p 1 4 - P " - h p i l l T 2.3.3) J - <x ) -b This can be written in a more convenient form since con-stant and the denominator is equal to the volume rate of flow: The subscriptQ indicates rectangular cross-section. where f „ depends on the ratio b/a only. KE,max,n The calculation of this factor i s discussed i n Appendix 1. When a viscous fl u i d enters a pipe, the velocity profile is i n i t i a l l y f l a t except immediately adjacent to the walls. At sdme distance (the transition length, 1^) from the entrance (depending on the hydraulic diameter and flow velocity), the profile attains that specified i n Equation (2.3.1). The only data that seems to be available i s that for pipes of 2.3.3) circular cross-section ' " , where the length of transition has been deter-mined to be 1 ^ = 0.13 d Re (2.3.6) T,0 v where d = pipe diameter; Re^ = Reynolds number (See Section 2,4.c). If the pipe is so short that the final velocity profile i s not reached, the factor f does not equal the maximum value indicated in Equation (2.3.5) KE but has a intermediate value between 1.0 (at the entrance) and fT.„ . But KE,max i t can be closely estimated through calculation of the ratio ^E,^^,Q/C^rv^x. / v n , ™ ) | (2.3,-7) where v = value of v at (x,y) = (0,0); n,max n \ >J/ \ > /> subscript 1^, denotes value for pipe length 1 > 1^,. This ratio varies between 1.017 and 1.029 (only 1.2 percent) for a l l values of a/b. For pipes of circular cross-section the corresponding value of (2.3.7) is always unity; even when the cross-sectional shape is an equilateral triangle, the value is only 1.052. Details of the calculations are given in Appendix 2. There is yet one factor which must be included i n the above equations. The viscous losses i n the transition length are actually slightly greater per unit length of pipe than i n the region where the f i n a l velocity profile exists. A theoretical calculation by Langhaar * ° has yielded the value 1,14 for circular cross-section; this value i s very close to that obtained experimentally. No similar study appears to have been done for pipes of rectangular cross-section so the kinetic energy factors have been taken as (using Equation (A2.4)): fKE, max,Q 1 ° 1 4 fKE,max,a (2.3.8) 1 - - - - i - K ^ n When the cross-section of a pipe suddenly enlarges, the excess kinetic energy due to reduction of the fluid velocity i s not con-verted back to pressure as would be expected from Bernoulli's Equation. Instead this energy is lost i n fr i c t i o n a l effects and so 2 ' 3 , 5 ) * , M = o U . * a . J o ) Exit losses can be reduced by shaping the exit so that the cross-section increases gradually. However, since most of the entrance loss discussed above is not due to fr i c t i o n a l forces, the main effect of shaping the entrance is to permit higher c r i t i c a l Reynolds numbers to be attained. It should be noted that for superfluid component at subcritical velocities, the kinetic energy does, i n fact, contribute to the - - . A 2.3.5) pressure at the exit and so p i i i _ p i i s s (2.3.11) The entrance and exit losses for the superfluid component may there-fore be ignored. From the foregoing we now see that at low velocities in pipes of rectangular cross-section P = p V p " b L 3 -n-s- a. f r ~ x A ? or using the geometrical factor introduced i n Equation (2.2.2) P r - ^ . V n > m +- <~ (2.3. .3) and instead of (2.2.4) and (2.2.5), A T - ~~^S (2.3. 2.4 C r i t i c a l Velocities and Dissipation Forces In the equations for motion for the two-fluid model, three f r i c t i o n force terms are usually included. This section outlines the nature of these forces and the c r i t i c a l velocities corresponding to 2.2,1) each of them. It i s a summary from Keller (pp. 275-345) ' * and several recent papers as indicated. a. Superfluid component c r i t i c a l velocity v g ^ and fr i c t i o n  force term F g The c r i t i c a l velocity for motion of the superfluid component seems to be associated with the formation of quantized vortices. Some progress has been made on a quantum mechanical approach to this problem. The energy per unit length of vortex line, as calculated from an analysis of the wave function of the superfluid for cylindrical geometry, is correct; but the theory has not yet been developed sufficiently to permit the prediction of c r i t i c a l velocity as a function of pipe size Of the semi-classical vortex line models that have been suggested, the one which would appear to be most promising i s one 2 4 2) by Craig . H e states that when the velocity at some point near the wall of a pipe or channel reaches a c r i t i c a l value, part of the kinetic energy i s used to create a vortex line. This:line causes the flow velocity to be reduced at that point and increased everywhere else i n the plane perpendicular to the flow. If the volume flow i s great enough the length of the line w i l l therefore increase un t i l i t forms a ring around the perimeter of the pipe. The corresponding c r i t i c a l velocity i s given by where d = hydraulic diameter; H B and C are constants. By suitably choosing B and C Craig was able to f i t this expression out that the constants cannot be calculated independently at the present time and are d i f f i c u l t to estimate; hence the physical arguments leading to Equation (2.4.1) are not easy to justify. An empirical relationship has been determined by de Bruyn to the experimental data very well. However, i t turns Ouboter et a l . 2.4.3), a) -•J m 5/4- "5>£C C 2.4-. 2) a) In cgs units the constant ~ 1 cm 5/4 sec -1 Both Equations (2.4.1) and (2.4.2) agree very well with the -9 -2 experimental data over the range 10 d 10 m. The f r i c t i o n force term F , which i s the f r i c t i o n a l force per unit s volume on the superfluid component because of i t s interaction with the boundaries for velocities ~> v^ ^, i s d i f f i c u l t to determine experimen-ta l l y because i t s magnitude i s small. Basically the procedure involves measuring the difference in chemical potential (usually hydrostatic pressure) between the ends of a capillary as a function of v^ with the normal component held stationary. As w i l l be explained in Section 2.4.b, i t i s likely that the mutual f r i c t i o n force term F masks F i f * _ sn s d H > 1 0 = m . b. Relative c r i t i c a l velocity c and mutual f r i c t i o n force term F — • S n The mutual f r i c t i o n force which i s the dissipation force per unit volume between the superfluid and normal components, is zero when there i s no v o r t i c i t y present.. In addition, i f the vorLicity i s u o l homo-geneously distributed throughout the flowing superfluid the existing mutual f r i c t i o n theory is not expected to be v a l i d . This latter con-dition i s usually expressed by requiring that the average spacing between the vortex lines 1 i s less than the pipe diameter d. If this v 2.2.1) condition i s satisfied then where = a constant dependent.on temperature; v 1 mean value of v i (v - v )s= V R , r,m r ? s n From this we can determine for what values of d the mutual f r i c t i o n force can be expected to be operative. 2.4.4) At 1.40 K Vinen calculates the value 1 I v I = 9.2 x 10'7 m 2/sec v I r,m I (2.4.4) Gonsider the case in which v = 0 so that v = v „ Now no vorticity _ n r s w i l l be created unless v ^ v as was discussed earlier. There-s s,c fore by substituting these conditions and the empirical relation (2.4.2) into (2.4.4) we obtain: d^ > 1.9 x 10°5m (2.4.5) If the normal component i s moving i n the opposite direction to the superfluid motion (as in counterflow) the minimum value of w i l l be even less. Experimentally F g n is much larger than F^ for supercritical flow i f Equation (2.4.5) i s satisfied. 2.4.4) Vinen also states that the mutual f r i c t i o n force, is directly proportional to the total length of vortex line L per unit volume of He liquid, i.e., F = B B ' ° " K ~ - • L where B = the constant in the mutual f r i c t i o n force equation for rotating He II. He then goes on to show that for sufficiently high vorticity densities the rate of change of line length per unit volume is ^ S / -where "X^  , * X ^ a n < l "*X^ = parameters of order unity and s l i g h t l y dependent on temperature; V = an empirical c o e f f i c i e n t strongly dependent on temperature (roughly as T^) . The f i r s t term gives the rate of line generation due to the mutual fri c t i o n ; the second takes into account the fact that the turbulence must be initiated in some (unknown) way; the third states the rate of decay due to the interaction of the lines with one another; and the last term gives the rate of decay due to interaction of the lines with the boundaries (walls of the pipe). For high vortex densities, the second and last terms are negligible d U and so the equilibrium line density (i.e., for •—— = 0),is d t At 1.40 K the approximate values of the parameters and constants are 2.1.2, 2.4.4, 2.4.3), X = 0.29; v d L d L A = 1.0 for positive —r-r- and 0.25 for negative (Vinen was ' i v d t d t unable to explain this difference.); B = 1.4; 7.54 x 10~2; -27 m = 6.646 x 10 kg; "fc = 1.055 x 1 0 = 3 4 J sec; and so (with "X = 1.0): 11 0 9 / L =9.3 x 10 .v sec /m (2.4.9) o r From this the mean distance between vortex lines can be calculated 1 = L " 1 / 2 = 1.04 x 10"6' v" 1 m2/sec (2.4.10) v o r which agrees quite well with the result i n Equation (2.4.4) There is qualitative agreement between Vinen 1s theory and experi-ment. A major d i f f i c u l t y i s that i t inadequately accounts for the on-set of the mutual f r i c t i o n at the c r i t i c a l velocity. c. Normal component c r i t i c a l velocity v and f r i c t i o n force _ c —i- n , c = — term F^; C r i t i c a l Reynolds numbers In Section 2.3, the pressure difference required to maintain a steady, sub-critical (laminar) flow of the normal component was determined (Equation (2.3.12)). Now we consider the c r i t e r i a for calculating the c r i t i c a l velocity v n c and the extra f r i c t i o n a l effects observed when the normal component velocity exceeds this 2.2.1) The definition of Reynolds number comes from the principle of dynamic similarity; for liquid IfeIXj."t is written in terms of either the normal flu i d velocity or the pressure gradient (ignoring the kinetic energy effects) Note that the total f l u i d density appears here instead of the normal component density; theoretically this should be true only when the relative velocity v i s small compared to both v and v r r n s but experimentally this condition would seem to be unnecessary 2 . 4 . 5 , 2 .4 .6) The velocity, of flow or pressure gradient at which the flow becomes turbulent corresponds to the c r i t i c a l Reynolds number which usually takes on values of Although this condition was originally obtained for pipes of circular cross-section, i t also holds quite well for rectangular, triangular or other cross-sections. The lower limit of 1200 seems to be characteristic for a l l fluids and entrance geometries; but an upper limit in excess of 10,000 may be observed i f the entrance to the pipe is properly flared and free from irregularities. In isolated cases, upper limits • '>• 2 3 3) of even 25,000 have been obtained for Re . 2.4.7) 2.4.8) v , c Chase * ' and Tough * ' both have proposed alternative re-lations for describing the c r i t i c a l velocity condition for counterflow in large pipes but neither have received wide acceptance. Above the c r i t i c a l velocity the additional f r i c t i o n a l effects per unit volume in the normal component are accounted for by the term F . 2.5 Ions in He II ^ j • * . r, 210 . A 241 a. Production of ions; Po and Am Ions can be conveniently produced i n He II by two methods: 2 5 1) injection of electrons through photoelectric emission ' ' and 2.5.2) ionization by radiation from radioisotopes . Large currents — 6 ( ~" 10 ampere) may be obtained using an incandescent tungsten 2.5.3) filament " , but the heat transfer to the helium in this method ( 0 . 1 watt) makes i t unsuitable for study of flow phenomena. The main advantages of the radioisotope method are that either positive or negative ions are equally available from the same source, and no add-iti o n a l apparatus i s required either inside or outside the cryostat once the source is installed. Hereinafter, only the radioisotope method w i l l be considered. Alpha emitters, particularly P o 2 ^ and Am 2 4\ are the most popular ion sources i n He II. The path length of the alpha-particles from both of these is approximately 0.26 mm and so a l l the ionization occurs immediately adjacent to the source. The ionic current available depends on several factors: source strength, source thickness, thickness of the protective coating over the source, and recombination rate of ions in the ionization region. The total ionic current which can be expected is estimated in Appendix 4. b. Nature of Ions i n He II The positive ion i s believed to consist of a cluster of atoms having an effective mass of about 40 helium masses. This result, which has been obtained using a classical electrostriction model , indicates that this mass i s only slightly temperature-and pressure-dependent and that the "solid radius" of the ion i s (5.0 ± 0.1) R 2'5*5). The negative ion, on the other hands i s thought to consist of an electron trapped i n a cavity or "bubble" having a radius of 15.4 R at zero pressure ^°^°^) o This radius i s a function of 2.5.7) pressure ' * but i s practically temperature independent. Its value at: 1.4 K at the saturated vapour pressure i s only slightly 2 5 8) lower than this value. A more recent determination •' ' ' has o " v provided a*yalue of 19.5 A,but here.-there-is some uncertainty about the model chosen. Using a different approach, Soda ^""'"^ calculated the negative ion radius to be about 12 to 14 R which supports the experiment value of 15.4 R. co Interactions between ions and the elementary excitations; ionic mobilities Using a hard-sphere model for the positive ion to describe the scattering, the average momentum per meter transferred from each 2.5.5) ion to the excitation gas i s given by ^+ ~~ ^=>+, pU onov\ P-|- ; r o + o n ^ = io (x+T 4.74- MO <nj_e ( 2 . S . 0 _ 4° —18 2 where momentum-transfer cross-section ((2.38 = 0.05) x 10 m ) due to roton scattering at the minimum of the roton energy spectrum; ^/k= roton energy, gap (8,67 0.04) K; + -10 a + = positive ion radius (5.0 = 0.1) x 10 m. At 1„40 Ks this i s p + = C3,37 x i o " , < 0 f 2.^ 1 XlcT'*) k<3 sec - 1 = 2 . 6 S X I o ' J k g sec"* 2 5 5) Now the ionic mobility i s related to p + by ' ° f ( 2 . 5 . 3 ) where e + = ionic charge; and so /A + = fc,08xio"s rv» aV ' s e t ' (2 .5 ,4-) Experimentally s the positive ion mobility at 1.40 K is found to be 2.5.10) jl^^ =. (^.i ± o.l) x/o"S"P n' 1- V*"1 5ec _ l (2.S.S) which i s i n good agreement with Equation (2.5.4); this i s parti-cularly notable since the value changes by a factor of about 10^ between 0.5 K and the /V-point. 2 5 6 2 5 11) The negative ion may be treated i n a similar manner ' * , The result agrees quite well with the measurements of Reif and Meyer 2.5.10) . , J; at 1.40 Kj 2.1.7) Bruschi et a l l , * * found that at certain velocities,,the mobilities of both positive and negative ions decreased i n steps, (gee Section 2.1.c0 At 1.40 K this effect f i r s t occurs for a positive ion velocity of 5.2 m/sec and a negative ion, velocity of 1.5 m/sec. 4 4 The corresponding fields are 8.5 x 10 and 4.3 x 10 V/m respectively, Thus i f larger fields are used, the experimental results may be ad= versely affected. d. Interactions between ions and vortices I f a vessel i s f i l l e d with He II and set rotating, an ionic current travelling perpendicular to the axis of rotation i s attenu= *. A • • A* - 2.5.12) ated according to • ' where i = current for no rotation ( <jJ0 = 0\; o n^ = density of vortex lines; <J~ = capture eross-seetion for the ions by the vortices; 1^  = distance travelled by the current. This equation assumes that there i s no buildup of charge i n the vortices, i.e., the ions are drawn out along the lines as fast as they are captured. At 1.40 K, saturated vapour pressure and an electric f i e l d of 7.5 x 10 3 v/m' 2' 5 , 1 2* CT ((Theoretical) = 5.7 x 10~8 m (2.5.8) '•fr* 8 CT (Experimental >s = (5.7 - 0.3) x 10° m (2.5.9) If a significant fraction of the ions escape from the vortices before they can be drawn along the vortices out of the range of the collector, the observed cross-seetion i s lower than that given above. The effect may be determined from 2»5.8, 2.5.12X = e = e (z . 5.io) where T"' = CT = observed eross=section; cross=section i n the absence of escape; P = probability of escape per unit time; X =. trapping lifetime; "t = time from instant of capture; here i t i s the time taken for the ions to be drawn out of the range of the collector. In Equation (2.5.12), P acts effectively as a cutoff for observa-tion of ionic capture; at saturated vapour pressure s this cutoff occuss at about 1.0 K for positive ions and 1.7 K for negative ions. Thus only negative ion currents are attenuated by the presence of vortices above 1.0 R and no attenuation i s observed above 1.7 K for either positive or negative ion currents. 2 5 8) Pratt and Zimmermann " * ' have developed this approach further 9 obtaining a relation for negative ion trapping time as a function of temperature, pressure and ionic radius ^ « ^ e k B l" J (2.5.1.) where £•_= electronic charge; m. = effective ionic mass; /-A_= ionic mobility; CL_= effective hydrodynamic radius of the ion; = characteristic frequency for the vibration of an ion i n the potential well of the vortex line i n presence of electric f i e l d , , V rv>-Lcx_/ \ 2, TV rr* / %,a = potential well depth of vortex line i n absence of electric , f i e l d , V = "saddleppoint" between potential well and asymptotic region away from well i n presence of electric f i e l d j "-'"2 , > / n e . E 0-q= radius of the vortex line core. The difference V = V i s just the effective depth of the well i n the presence of the electric f i e l d . L_ i s strongly dependent on temperature and pressure, and moderately on electric f i e l d . By extrapolating the results of Pratt and Zimmermann the well depth and trapping lifetime at 1.40 K, saturated vapour pressure and 3 E = 7.5 x .10 V/m are found to be (Appendix 5) •(V - V^/k^ = 67.9 K (2.5.12) ^_(1.40°K, 7.5 x 10 3 V/m) «• 4.5 x 106 sec 52 days 1 (2.5.13)-When these workers calculated the ionic radius from the experi= mentally determined value of the exponential term i n Equation t/2.5.11) they obtained = ( i q . s t o . O A (2.<s. I 4 - ) which i s about 20 percent higher than that measured by Zipfel 2 5 6^  and Sanders . However, i f they then substituted this value into the f i r s t part of (2.5.11) the resulting value of was 4 higher than that observed by a factor of 10 . Thus i t would appear that their model i s actually only qualitatively correct. e. Space charge effects ° Theoretical In liquid He II space charge limitation can occur through two mechanisms. F i r s t l y , (in analogy with the vacuum diode) the retarding f i e l d due to the current beam may limit the number of ions leaving the source. Secondly, a significant amount of charge may be= come trapped i n the vortex lines and thus reduce the f i e l d at the source. The effect of charge i n the current beam may be evaluated from 2 5.12) (for plane-parallel geometry) where K' = dielectric coefficient; £ 0 = permittivity of vacuum; •J = current density; = ionic mobility; V--'^ = applied voltage; •s = distance between source and collecting electrodes; G = constant ( O ^ C ^ 00 ) . For large voltages or small currents, C i s large and E i s nearly constant ( i . e . , no current reduction occurs)'. Complete space-charge limitation occurs for C = 0 (E = 0 at the source electrode). The actual lowest permissible f i e l d for negligible charge limitation depends on the ion source strength. Ions trapped i n vortices can also limit the current by reducing the f i e l d at the source. As the charge density i n the vortex lines increases, however, the effective well depth (Equation 2.5.12) Kfe;, w i l l be reduced; therefore the trapping time w i l l get shorter. Even-tually a dynamic equilibrium state w i l l be reached i n which the rate of ion escape i s equal to the rate of capture. To determine when this w i l l occur, we set up a model which combines the ideas contained i n 2 4 4) 2 5 8) the theories of Vinen * * ' and of Pratt and Zimmermann * " . The potential energy of an ion i n a vortex line may be con-sidered i n three parts as that due to: (1) well-depth for an isolated charge; (2) f i e l d of adjacent ions i n the same line; and (3) f i e l d of a l l other ions. It wi5£L. •b-ei'..-iassu.medvthac::^ : (1) the vortex lines do not bend appreciably over a length equal to the average distance between them; (2) the density of vortex lines i s constant throughout the liquid, i.e., the hydrodynamic forces are much greater than the coulomb forces; (3) the density of charge i n the lines i s constant throughout the liquid; (4) ""the mean distance between the ions i n each line i s much less than the average distance between the lines. The f i r s t two assumptions are the same as those that Vinen makes i n the development of his theory. ' The second and fourth ones w i l l be checked at the end of the calculation. A sketch of the model i s shown i n Fig. 2.5.1 (a). Here the lines are drawn parallel although i n the actual case the orienta-tion w i l l be random. Nevertheless, i t can be seen that the ions i n adjacent lines (as well as those at distances > 1 i n the same v line) w i l l tend to affect the potential of the overall region rather than the well depth. The potential V g due to a number i of charges q^ at distances r^ i n liquid helium i s where = permittivity of liquid helium. This may be approximated by a line having a charge density ,7\ = e/1 O O O O O O O - Q - O O O P o o o a. Discrete charges in vortex lines Fig. 2.5.1 Model Used for Calculation of Local Potential Due to Ions Trapped in Vortices (for singly charged ions) and so V - = - T ^ - ^ - T (2-5-. 1-7) e 4- i r e r< v y where r^ = inner limit of charge; r^ = outer limit of charge; Since i n this approximation each point charge has been replaced by a line charge of length l g , r^ may be taken as lg/2 and as 1 ; also since the line charge extends i n both directions, a factor 2 must be included (See Fig. 2jjS^ lT"(t>)}. Because of assumptions (2) and (3) above, the f i e l d due to the trapped charge w i l l be one-dimensional, i.e., i t s effect w i l l be the same as that for a sheet of charge density oriented parallel to the electrode surfaces. For a line density L q and the line charge density e/1^ the equivalent sheet charge density i s where s i s the distance between the source and collecting electrodes. V 2 2.4.4X Since L = 1 , o v ' The electric f i e l d at the electrode surfaces due to this charge distribution w i l l be Now-consider -the case i n which -the source electrode i s at potential "V ^ and the collecting electrode i s grounded. This means that the trapped charge consists of electrons. For the con-dition that the f i e l d at the source electrode be zero (i.e., so •that the current across .the pipe i s zero-=See Section 4.7.e) we must have an applied f i e l d E -c c appl of pi d * S e and so (using Equation (2.5.20)) V a p p l ~ 2 6 2 i i y i e . or Substituting this into (2.5.1:8) yields Ve. ~ Therefore, In this equation, Ve'and-V ^ may be determined experimentally (Section 4.7.e) and so.1 may be calculated by t r i a l and error. An estimate of 1 for this calculation can be obtained through Equations (2..4.<4)-- or -.(2.4..10). In-this model, the Coulomb -forces have been ignored (Assumption (2)^ to see.the .effect -'of this,' we now consider the forces-on the vortices due (1) the hydrodynamis forces; (2) the trapped charge i n the electric f i e l d . If a vortex line i s subject to a force f^,"ithwiliamovetinua direction perpendicular to f with a velocity u (i.e., relative to the superfluid V s2.1 2> component) given by the Magnus effect where K = circulation per unit length of line (= h/m). In this case, the force i s that due to the net electric f i e l d on the charge trapped i n the li n e . I f there i s no relative motion between the superfluid and normal components, the mean f r i c t i o n force on the vortex line i s given by Equation (2.4.6) i n the form But this f r i c t i o n force i t s e l f w i l l produce a Magnus effect velocity u^ parallel to f , i.e., 1 v (s> — i = - C )(- ^ r ) ( - A ) v (2 ,5 .27) I f the normal component i s moving with respect to the superfluid the average effect w i l l be the same since the resulting increase i n f^ for a particular line w i l l he offset by a decrease of the same magnitude for a vortex line of opposite circulation. To obtain f , we must f i r s t calculate the net f i e l d between the electrodes. The applied f i e l d i s given by Equation (2.5.22); the f i e l d due to the trapped charge may be determined from Poisson's Equation (for negative charge) where ^ = charge density per unit volume (= positive quantity). Integration yields where x = 0 at the source electrode and s at the collecting electrode; C = integration constant. The boundary condition i s that the total f i e l d due to the trapped charge and applied voltage ~V j_ i s zero at the source electrode, i.e., ( E ) = ( E * + E A = O Since the charge density per unit volume i s (Equations (2.5.17) and (2.5.23)) the total f i e l d i s found to be - r- ^ r - / ^ ^ P P ' " ^ ^ P P L V - ^ 7>\l \ K (2.532.") This leads to the force per unit length of vortex line (Equation(2.5.23)) f - f - 2 & V « P » ' ^ 4-6iyV,^ * (2.S-.33) which when substituted into Equation (2.5.23), U,= ^ - f x (2.5.34-) This last equation permits one to estimate how long i t takes a given fraction of the vortices to be drawn toward and annihilated at the collecting electrodes: T. d e c o y ^ s , e. ca, y In this last calculation, the approximation has been made that the f i e l d at a l l points remains constant during the annihilation whereas the f i e l d actually increases as .the-vortices .reach the -electrode-. However, the error w i l l not be large for x > s/2. (See Section 4.8)--f. Space charge effects - Experimenta.1 2.5 ,2V Gareri, Scaramuzzi and Thomson * * found i n their counterflow experiments that when the c r i t i c a l velocity (corresponding to the c r i t i c a l Reynolds number) was exceeded, the total negative ion current flowing across the channel decreased according to: hi where —^ = fractional change i n the ionic current; Q . = heater power; Q^. = heater power corresponding to the c r i t i c a l velocity; A = cross-sectional area of channel; E = electric f i e l d ; C 1 = a positive constant. The positive ion current was unaffected. They interpreted this as being due to an effective decrease in mobility of the negative ions because of their interaction with the superfluid v o r t i c i t y . 2 5 13) Vicentini-Mi ssbntvrt:. and Cunsolo ° ° ' also found a lower c r i t i c a l velocity i n their counterflow experiments for which there was no attenuation of either positive or negative ion currents. They sug-gested that this velocity may correspond to that at which the production of vortex lines i s i n i t i a t e d . (See Section 2.4.a) g. Motion of long i n pipes during flow From the foregoing, we can visulize how the average motion of ions w i l l be affected by the flow of He II i n a pipe of rectangular cross-section. Consider the two-dimensional ease i n Fig. 2.5.2. The le;ft wall consists of a single ("source" or "emitting") electrode with a line -source of uniform intensity. The right wall consists of several ("collecting") electrodes insulated from one another. The collecting .electrode directly opposite the <x-source i s shorter i n length than the source i t s e l f . When the normal component i s moving towards the bath end as shown and a uniform electric f i e l d i s applied between the electrodes, the mean normal component velocity may be calculated from •-'•2) 14-^ l4 ^ where d. = width of Electrode 4; 4 yU. = ionic mobility; E = electric f i e l d ; i ^ - current collected by Electrode 4 when the He II i s stationary; s = emitting electrode - collecting electrodes distance; A i ^ A i , . = change i n current collected at Electrode 3 and 5 re-spectively as the He II flow rate increases from zero;, To 3ath To Electrometer (or Grounded). Source Electrode To Voltage Supply To Heater A Figc 2.5,2 Schematic Diagram of Pipe Showing Deflection of Ionic Beam The mean normal component velocity may also be determined from the heat power causing the counterflow motion (Equation (2.2.3)]. This provides a check on the operation of the apparatus. In an actual (three-dimensional) pipe, the -source width may be less than the pipe width. The portion of the f l u i d traversed by the ionic beam would therefore have a mean velocity greater than that over the whole cross-section of the pipe {Section 2.3.b)„ Nevertheless, the mean velocity i n Equation (2.5.37) would differ from the actual value by a constant factor dependent only on the ratio; source width/pipe width. The theoretical value of this •factorAmay be calculated from Equations (2.3.1) and (2.3i2) for pipe lengths greater than the transition length ((Equation (2.3.6|, IT a. G = (2.5.38) where y • = half-width of the radioactive source; s and checked ,by combining data from Equation (2.5.37) and (2.2.3), i.e., Equation (2.5.37) should therefore be replaced by (^.5.4-o) when the source width i s less than the pipe width provided the pipe i s sufficiently long. Above the c r i t i c a l velocity at which the liquid becomes turbulent (and, consequently, large numbers of vortex lines begin to be created), the situation changes., I f the ions i n the beam are positive then at 1.4 K .they w i l l continue to yield data regarding the mean normal com-ponent velocity. However, i f the beam i s composed of negative ions, a significant number w i l l become trapped producing a space charge and causing the total current to be attenuated. The effective mobility of those ions no.t trapped w i l l be reduced so :they w i l l he carried along further by the elementary excitations than i n the non-turbulent ease. . Also, i f the vortices have a net motion along the pipe, .a small fraction of the current w i l l be transported with them. Let us now consider the case i n which only the superfluid component i s moving. Below the c r i t i c a l velocity, no deflection of either the positive or negative current beams w i l l be observed at 1.4 K. In fact, this provides a sensitive check that the normal component i s stationary. Above the c r i t i c a l velocity, the positive ion current w i l l continue to be unaffected but the negative ions w i l l become trapped i n the vortices as i n the counterflow case. The total negative ion current w i l l be attenuated but i f the vortex lines have a net motion along the pipe^ some increase may be observed at the electrodes i n the direction of the vortex motion. ..CHAPTER 3 EXPERIMENTAL DETAILS ' 3cl Outline of Apparatus The wind tunnel apparatus was similar to that of Chung and 3 l i l ) Gritehlow " ; a diagram i s shown i n Fig. 3.1.1. The test section., one end of which opened directly into the bath, was connected to two superleaks i n series. These led into the graduated vessel used for measuring the rate of flow when the superfluid component was being made to move independently of the normal component. This-vessel emptied v i a a superleak and a.length of narrow-bore tubing into the bath. The capillary at the top of the graduated vessel permitted the He gas to escape as the vessel was f i l l i n g but had a sufficiently small cross-sectional perimeter that the film flow was negligible,. Heaters were located at both ends of the main superleak section, and near the exit of the graduated vessel as shown. The apparatus could be operated to give any of the following flow configurations: a. Counterflow This was generated by activating'Heater A adjacent to the test section. The He II flow rate was determined by measuring the power input into the heater and checked by noting the deflection of the ionic beam i n the pipes as explained below. There was no movement of helium i n the apparatus except i n the test section i t s e l f . b. Superflow Pure superflow was generated by activating Heater Bi The flow rate was determined by measuring the rate of f i l l i n g of the C a p i l l a r y Tubing (2 m m ) B a t h Level G r a d u a t e d Vessel Ground Glass J o i n t s C o p p e r - G l a s s Sea l Source Elo .f rode <x- Source -Col lect ing E lec t rodes T h i n - W a l l Sta in less -Steel Tubing Kovar -G lass S e a l Heater A b r a s s £j Compressed Powder Cm Fig. 3.1.1 Superfluid Wind Tunnel Apparatus graduated vessel. When the vessel was f u l l , Heater G was turned on to empty It i n preparation for another flow measurement. c o Combination of eounterflow and superflow This was accomplished by -activating-both Heaters A and B with Heater G being turned on as required. The usual procedure followed was to set the superfluid i n motion at a steady flow rate and then turn .on Heater A to various predetermined powers. 'The .pipe inside the test section were similar i n design-to the 2.5.2-) "channels" employed by Gareri et a l . " except that both ends ' were open as shown schematically' i n Fig. 3.1.2. The cross-sections • were rectangular i n shape with ;the electrodes forming two. of the walls • and' Plexiglas slabs forming- the outer two;. The emitting' electrode -con-tained the•radioactive source and was connected to a regulated DC power supply which could be varied from 0 to - 300 volts with respect to ground. The collecting electrodes consisted of several sections separated by insulators; one or more ;of these sections were connected 10 ' \ . through a 10 ohm resistor to ground, the current being determined by measuring the voltage drop across the resistor with a vibrating reed electrometer. The other collecting electrodes were connected directly to ground. The maximum voltage drop across the resistor was about 50 mill i v o l t s for a potential difference of 90 volts between the emitting and collecting electrodes, so the electric f i e l d was not significantly modified by the current measurements. 3.2 Details of Wind Tunnel Apparatus a. Main superleak section Two superleaks were used i n series to avoid any heat leakage from Heater A into the test section. Koehler and Pellam ^ " ^ ' l ) found that i f only one superleak was used, sufficient heat was conducted past i t Sec. A-A Wall and Support for electrodes Pipe 2 <1 O-*~* — S o u r c e Slectrode r A Collecting Electrodes Shields (Pipe E only) Source and Source Holder (See Fig. 3.3.1) -Cellulose Tape Insulator c>.ec. 0 L -B J L Pipe C Pipe B Cm Pipe A Fig. 3.1.2 Pipes Used in the Virind Tunnel Apparatus'. PIexislas Brass with Gold Plate >ssibly through the metal walls of the easing) to cause some counterflow i n the test section. Any possible heat leaks are thus "grounded" by the metal tubing between the superleaks. Each superleak consisted of a short length of brass pipe i n a) which aluminum oxide powder had been packed using a hydraulic 8 2 2 press (packing pressure ~ 4 x 10 newtons/m «. 30 tons/in ). Layers of cotton wool and brass gauze were l a i d .over the compressed powder to give the superleaks better mechanical' s t a b i l i t y . They were joined by a length of stainless steel tubing as shown and con= nected to -.the-rest of the-apparatus by standard metal-glass seals.. To determine the effectiveness of the superleaks, air flow .rates were measured through them at room temperature and-the effective porosity calculated using the•procedure of Forstat and 3 2 2 ^  Reynolds '* '* . For both superleaks the results were approximate!] the same, viz., Mean -channel radius = r = 16 R (3..2..1) No. of channels =N = 1.6 x 10 (3.2.2) Since for this size of channel the superfluid c r i t i c a l velocity i s •(Equation (2.4..2)' v =t 0..43 m/sec (3.2.3) •S,.C i ' ' • .'• the .maximum volume flow rate through the superleak i s 2 -5 3 N t t r v ~ 5,5 x 10 .., in ./ sec (3.2.4) s s c • : • 2 which i s greater by a factor of more than 10 than the flow rates actually used i n the experiments. .AlgOq abrasive powder having mean particle size of 0..3 .micron.; manufactured by Linde Air Products Go. (Type A); Micro Metallurgical Ltd., Thornhill, Ontario, Catalogue No. MM 219 During helium runs, the superleaks could be checked by noting the deflection of ions when only the superfluid component was flowing through the test section,, For laminar flow there should be no deflection. For higher power inputs to Heater A, a long term variation was noted i n the ionic currents, but this i s attributed to spreading of the ionic beam. (See Section 4..7.b(2).) b. • Measurement -of superflow The rate of superflow was determined by the rate of f i l l i n g of the + 20 ml graduated vessel which was accurate to - 0.1 ml. Baring the experiment's, the scale could be read easily through the s l i t s of the silvered dewars.. The. superleak at the bottom of the graduated vessel was composed ©f the same grade of aluminum oxide powder as that used i n the main superleaks. F i r s t l y , a wad of cotton wool was pushed into the ground glass joint t o the neck between the joint and the vessel i t s e l f ; secondly, a charge of the powder was tamped i n , f i l l i n g the volume of the female section of the joint; and f i n a l l y , a second wad of the cotton wool was placed over the powder with a small amount of nai l polish to hold i t i n position. The fountain pump which consisted of this superleak and Heater G wa.s used to empty the graduated vessel between measurements of the superfluid flow rate. During periods when the measurements were not being made, the power input into Heater G was held at an intermediate value so that the vessel remained about half f u l l ; this procedure helped stabilize the f l u i d pressure at Heater B (by preventing the vessel from overflowing and reducing fluctuations i n the relative heights between the apparatus and the bath level), and permitted better temperature control of the helium bath. The efficiency of the windtunnel may be expressed i n terms of the superfluid component volume rate of flow as a function of power input to Heater B, This relationship was quite sensitive to the depth that the apparatus was immersed i n the bath (Section 3.5.e) as well as being slightly dependent on the size of pipe i n the t e s t s e c t i o n ( f o r supercritical f l o w ) . Typically, the flow would commence at a power input of about 4 mW and increase at the rate of approximately 1.8 -5 3 • • x 10 m / see-W. c. Heaters The heaters were of "Advance" a l l o y , B&S gauge #38, nylon-a) covered wire .Heater's A and:fi-each had resistances of 565 ohms and Heater C, 190 ohms. The leads of Heater A were of #30 copper wire and led out through the test section where they were connected to #30 Advance nylon-eovered wire. The leads to Heater B were en-ti r e l y of the #30 Advance wire and led via the 2 mm tubing out the top of the graduated vessel. The leads of Heater C were similar to those of Heater B and led out through the 2 mm tubing used to empty the graduated vessel. Lead-acid (6 V) batteries were used as the power supplies for the heaters. The measurement of the total power to Heater A was the only one that had to'ite made accurately since this determined the component velocities i n counterflow. For this heater, an extra lead was included so that the voltage drop across the heater i t s e l f could be measured directly. This voltage was measured with a VTVM ^  c) a l l the other voltages and currents were monitored with Avometers Cu-Ni alloy wire, manufactured by Driver-Harris Go., Harrison, N. J. Radio Corp. of America, Camden, N. J,; Advanced Voltohmyst, Type WV-75A Automatic CoiljjWinder and El e c t r i c a l Equipment Co. Ltd., London, England; Model 8 Universal Avometer 3.3 Details of Pipes a„ Construction The electrodes were cut from brass sheet and electroplated by a a) '' commercial firm,,. The gold, surfaces were not mirror smooth but had a fine texture—this roughness probably was not significant compared to the discontinuities between the collecting electrodes and effects of the sharp edges at the entry and exit of each pipe. A scale draw-ing of the finished electrodes is given i n Fig. 3.1.2. The radioactive source (in f o i l form) was trimmed to the required size and soldered-onto a small piece which fitted into the emitting electrode; when in place the surface was flush with that of the electrode. The collecting electrodes were separated by a single layer of "Scotch" tape, which had excellent insulating, properties for voltages up to at least 100 volts at both room and cryogenic temperatures„ A small copper lug was soldered on the back of.each electrode for the purpose of connecting the electrical leads. Four pairs of Plexiglas wall pieces were prepared as shown in Fig. 3.1.2a The surfaces forming the interior of the pipes were polished with "Brasso", which produced a mirror smooth finish. Small blocks of Plexiglas were shaped to f i t at the top of each pipe so as to minimize the gap between the exterior of each pipe and the Pyrex wall of the windtunnel section. The leads to the electrodes also passed through these blocks. The f i f t h pipe size was made by placing Plexiglas shields adjacent to the collecting electrodes as shown in Fig. 3.1.2 to enhance the de= tection of turbulence i n Pipe D. (See Appendix 6). The measured (room temperature) dimensions are given in Appendix 3 and the calculated dimensions at liquid helium temperatures are tabu-lated i n Table 3 .3.1. a \ 'Hudson Plating Go. Ltd.., 275 West 5th Avenue, Vancouver, B. C. TABLE 3.3,1, PIPE DIMENSIONS AND CROSS-SECTIONAL AREAS AT LIQUID HE TEMPERATURES a Mpe Electrodes-Electrode Distance (cm) PlexiglagrPlexiglas Distance (cm) -c) Hydraulic Diameter (cm) Area for Normal Component; Flow (cm ) Area for Super-fluid. Component Flow } (cm ) A 1.21 - 0.01 0.219 - 0.004 0.371 - 3.6% 0.265 - 2.7% 0.360 - 2.0% B 1.20 - 0.01 0.436 - 0.004 0.640 - 2.6% 0.523 - 1.8% 0.618 - 1.5% C 1.21 ± o.Ol 0.812 t o .005 0.972 - 2.2% 0.983 - 1.4% 1.08 . * l . 3 % D 1.21 - 0.02 1.20 -0.01 1.20 3.7% 1.45 - 2.5% •1.55 - 2.3% E 1.21 - 0.02 1.21 - 0.01 1.08 - 5.3% e^ 1.12 - 5.3% f ) 1.30 - 3.6% 1.40 3,3% E b) Distance between source electrode and shield faces (cm) 1.05 -t 0.02 Thickness of shields (cm) ^ ) 0.155 - 0.002 Width of gap between shields (cm)*^ 0.21 - o.OI Length of a l l pipes (cm) 9.9 - 0.05 Measured (room temperature) dimensions are tabulated in Appendix 3. Uncertainties i n dimensions reflect measured variations between ends of the pipes. Hydraulicjliameter, cL = (4) (Cross-sectional area) / (Cross-sectional perimeter). -2 2 Cross-sectional area of gap between pyrex and Plexiglas calculated to be 9.5 x 10 cm ; uncertainty in calculation of size of gap assumed to be negligible. Includes gap between shields. Excludes gap between shields. o •£> b. Preparation of the ion sources Most workers who have conducted experiments with ions i n He II have 210 used Po as the ion source. However, this radionuclide has the disad-vantage that i t tends to contaminate i t s surroundings unless great care has been taken i n depositing i t on the substrate, and a protective layer of some other material has been laid over i t (e.g., Ag or Au). Newly 210 deposited Po , i f placed in a vacuum chamber, w i l l contaminate the entire interior of the chamber within a few minutes. Commercially prepared sources therefore must be treated with unusual care and be discarded i f the protective layer shows any evidence of deterioration. For this reason and the fact that i t has a more convenient h a l f - l i f e , 241 Am has become popular recently with experimenters, even though i t also emits a considerable amount of ^ -radiation. Americium-241 was already available in f o i l form which was suitable a) for the experiment . The active area was only 3 mm wide so the strip could be trimmed to a convenient width for mounting. The f o i l was soldered onto the gold plated piece of the emitting electrode as shown in Fig. 3 .3.1. Supplied by the Radiochemical Centre, Amersham, Buckinghamshire, England, Catalogue No. AMM.l which is a f o i l strip of total width 25 mm, having an active width of 3 mm and a linear activity cjf 30 microcuries/cm; the cover-ing i s 3 micron thick Au (5.8 milligrams/cm ). Fig..3.3.1 Detail of Source Electrode and Source Holder (Al l dimensions in cm) Brass with Gold Plate 3.4 Ionic Current Measurement The emitting electrode was connected through a voltage divider he b) a) and reversing switch to a regulated DC power supply . The voltage at this electrode was monitored continuously, with a VTVM One or more of the collecting electrodes (in parallel) were connected through a 10 ohm resistor to ground, with the voltage developed across the resistor being measured with a vibrating reed c) electrometer . The electrodes not used at a particular time were grounded. This procedure had negligible effect on the current dis-tribution across the pipes since the effective resistance between 13 the emitting and collecting electrodes was •typically about 2 x 10 i ohms. The output of the electrometer was fed via a rheostat to a d) 3 chart recorder , A 4 x 10 F electrolytic capacitor across the input of the recorder effectively shunted out any noise having frequencies greater than about one Hz. The power to the DC supply3 electrometer, VTVM and chart recorder was regulated e^. a) Lambda Electronics Corp., Corona, New York,. Model 28 b^ Hewlett-Parkard Co., Palo Alto, Calif. Model 4lOB c) Applied Physics Corp., Monrovia, Calif, "Carey" Vibrating Reed Electrometer, Model 31CV. The 10^ ohm resistorwas contained within the preamplifier of tbe, electrometer. ^ Minneapolis-Honeywell Regulator Co., Toronto, Ont. Model 153X18= VH-II-III-118 J Sola Electronic Co., Chicago, 111, Cat. No. 5006 a. Vacuum system A schematic diagram of the vacuum system is shown in Fig, 3,5.1. The cryostat was connected, via a two-inch vacuum line (and bellows to a) reduce vibration) to a large mechanical pump A , The two manometers provided the primary means of temperature measurement. The backing b vacuum for the manometers was obtained by the diffusion pump and c) mechanical pump B which produced pressures of less than 1 micron d) Hg. This backing pressure was measured by the McLeod gauge A with the Pirani gauge B e'^ . being used mainly for leak testing when necessary b. Temperature measurement and control The primary means of determining and setting the He bath tempera-ture was by measurement of the vapour pressures and use of the "1958 4 3.5.1) He Scale of Temperatures" . Two manometers connected in paralle were employed, one f i l l e d with mercury and the other with butyl phthalate. Since the density of Hg at various temperatures is well 3.5.1) known • * •, the second manometer was calibrated against the one containing the mercury during each run. The liquid level differences f) + in the manometers were measured with a cathetometer to within = 0.05 ; mm. The correction for the local value of gravitational acceleration was included 3.5.2)^ a) b) Kinney Manufacturing Co., Boston, Mass. Model CVM 8-6=10; later denoted Model KC-46. Provided by J. Lees, Glassblower, UBC Physics Dept. ^Central Scientific Co., Chicago, 111. "Hyvac-2" Cat. No. 91305 d ^  Edwards High Vacuum, Sussex, England. "Vacustat" Model 2-G e) ' Consolidated Electrodynamics Corp.., Rochester, N. Y. Type GP-110 f) N Gaertner Scientific Co., Milwaukee, Wis., Chicago, 111 Once the desired bath temperature had -been established i n each a) t r i a l , a .carbon composition resistor having a nominal resistance of 33 ohms was used to monitor -the temperature. At 1.40 K the resistance excluding leads was about 5900 ohms and the sensitivity b ^  about 17 ohms per mil l i k e l v i n . A Wheatstone bridge together with c) + a "nu l l " detector enabled the resistance to be read to - 1 ohm. The temperature was controlled, manually by means of one of the •large valves in the pumping line. In some of the earlier t r i a l s , a 3.5.3) monostat ° ' was included i n the main pumping line but was found to be ineffective when varying amounts of power was dissipated in-side the cryostat by the heaters. The manostat was therefore later removed. With these techniques, the bath temperature could be regulated •+ easily to within - 0.5 millikelvin at 1.40 K. c, Leads inside cryostat The superfluid apparatus could be moved vertically about 25 cm inside the cryostat i n order to keep the bath level on a fixed point on the apparatus during measurements using pure superflow. (See Section 3.5..e). The leads had to be designed to permit' this move-ment and yet be relatively noise-free; irregularities i n movement of the leads would cause jumps in signal because of capacitance •effects, The leads were of #30 Advance wire (see Section 3 .2.c) except immediately adjacent to Heater A. About 250 cm of wire was allowed Allen-Bradley Co., Milwaukee, Wis. Carbon composition, 33 ohm, 0.1 watt Leeds and Northrup Co., Philadelphia, Pa. Guarded Wheatstone Bridge, Cat, No. 4735; 150 kilo-ohm resistor included in^battery circuit to keep current through resistor in He II bath below 10 ampere. Minneapolis-Honeywell Regulator Co,, Philadelphia, Pa. Portable Brown Electronic Null Indicator, Model 104W1 for each lead i n the variable length section. F i r s t l y three or four of these lengths were wound together counterclockwise i n a "rope", then two or more of these ropes were wound clockwise into a single larger rope. The leads to the collecting electrodes were kept separate from those to the heaters and thermometers. Finally each of the larger ropes were formed into spirals by winding around a 3/8 inch diameter rod. It was found that such a spiral would extend elastically from about 10 to 35 em at both room and cryogenic temperatures. The leads terminated at the top of the cryostat at 7-pin miniature tube sockets which were connected,, i n turn, to 7-pin kovar-glass vacuum seals mounted i n the cap of the cryostat. At the apparatus, the con-nections to the heaters and thermometers were soldered joints; those to the electrodes of each of the pipes were through miniature connectors to allow for ease of assembly. d. Shielding the cryostat The magnitude of the currents being measured by the vibrating reed -12 electrometer were usually of the order of 10 ampere or less. Before shielding was placed around the cryostat, the detecting circuits were sensitive to any ele c t r i c a l or mechanical disturbance i n the labratory. Considerable electrostatic noise was also generated by the boiling liquid nitrogen between the dewars. These problems were effectively overcome by wrapping the helium a) dewar with brass gauze , being careful to insulate the portion of the gauze which overlapped i t s e l f (to reduce possible eddy currents) and grounding this at one point to the cryostat supporting frame. Shielded cables were employed i n a l l the external circuits with the shielding in each being connected to ground at one point only. Gauze shielding placed around outside the nitrogen dewar was not nearly as effective. e. Apparatus immersion controller As the liquid He evaporated during a run, i t was necessary to keep the apparatus immersed i n the bath at a constant depth, parti-cularly during periods when the superfluid component was being driven by Heater B. I f the bath level "dropped relative to the apparatus, the pressure head over which the helium had to be pumped increased and so the volume flow rate would decrease. For low flow rates, variations i n the relative heights as small as one mm caused significant changes i n flow rate. The problem was complicated by. i . The evaporation rate of the bath decreased as the bath level f e l l in the dewar. i i . The evaporation rate depended on the power inputs to , the heaters. i i i . During measurements of superfluid component flow, as the level inside the graduated vessel rose, the out-side bath level lowered. Then as the vessel was emptied, the bath level would rise again. •J -.Tq overcome this problem, an immersion controller was constructed. This consisted of a motor geared to a 1/4 inch stainless-steel tube which moved up and down through a sliding vacuum seal in the cap of the cryostat. The windtunnel was attached to the lower part of this 3.5 .4) tube. A capaeitor-commutated parallel inverter * ' operated the motor satisfactorily over rthe range from 10 to 100 Hz. This mech-anism provided good control to, well within 1 mm but had to be moni-tored almost continuously. To further reduce the error associated with the change in bath level while the graduated vessel was.filling, the power to Heater C was normally set so that the vessel remained approximately h a l f - f u l l . Then just before a measurement of the superfluid flow rate was to be made, the power to Heater C was increased so that the vessel was emptied. Consequently, the lower half of the vessel would f i l l more rapidly than the upper half but the average f i l l i n g rate would yield an accurate determination of the flow rate. Because the depth of im-mersion of the apparatus :could not be monitored during these measure-ments^ the motor speed for the height-control mechanism had to be ad-justed very carefully before each measurement was made, .6 Experimental Procedure The special procedure employed, during precooling because, of the presence of the glycerine-white lead mixture i n the apparatus i s described i n Appendix 10. Measurement of the rate of f i l l i n g of the graduated vessel (during determinations of the superfluid flow rate) was done by eye through the dewar s l i t s . A cathetometer could not be used here because the apparatus was normally moving downward during these measurements. For both the eounterflow and superflow measurements, the heater powers were usually increased in steps u n t i l the c r i t i c a l velocities were reached; occasionally the heaters were turned off between each increase i n power but this procedure greatly increased the time required to make a c r i t i c a l velocity determination, and did not influence the results significantly. For measurements in which the counterflow was superimposed on pure superflow, the superflow was initiated at a preselected velocity and the counterflow turned on at a later time and increased u n t i l the desired effects were observed. After each measurement of a c r i t i c a l velocity, the temperature of the helium bath was raised above the *X=point before another deter-mination was attempted. CHAPTER 4  DATA AND RESULTS 4.1 Outline of the Measurements Several types of measurements were made; a. Detection of the c r i t i c a l velocities for counterflow by observing changes in the ionic currents at Electrodes 3 and 5 as functions of the power input to Heater A; b. Detection of the c r i t i c a l velocities for counterflow, superflow and superflow-plus-counterflow by observing the flow rates at which the total currents began to be attenuated; c. Observation of the rates of attenuation increase and total attenuations as functions of flow rates above the c r i t i c a l velocities; d„ Observation of hysteresis in the attenuation effects. The data from these measurements are considered i n turn i n the following sections. 4.2 C r i t i c a l Velocity Determination by. Observation of Ionic Beam Deflection  during Counterflow In the early stages of this project, a series of runs were carried out for counterflow in which the mean normal component velocity was measured as a function of power density input. This 2.5.2) technique was the same as that used by Gareri et a l . " ; i t consisted essentially of measuring the change in current at Electrode 3 or 5 as the power input to Heater A was increased, and calculating the velocity using Equation (2,5.40). Only the results for Pipe D are given here since very few data-were collected for the other pipes. Because the transition length, 1^, was much greater than the pipe length over the whole range of velocities (see, for example. Table 4.3.2), G was taken to be unity. The results, which are plotted in Fig. 4.2.1, are apprdximately i n accordance with the 2.5.2) observations of Gareri et a l . although the difference between the positive and negative ion data (i.e., detection of the c r i t i c a l velocity) i s not as noticeable here. Because of this, the attenuation method described i n the next section was used almost exclusively i n later t r i a l s . 4.3 C r i t i c a l Velocities and Attenuation Effects for Counterflow As was explained in the last section, a considerable number of runs were done for Pipe D to measure the c r i t i c a l velocity and observe any other phenomena, including attenuation, which may have been present. However, this pipe size was found to be quite insensitive to these effects. Later i t was discovered that large ionic current attenuations could be observed i n Pipes A and B (and to a lesser extent in Pipe C) for flow rates above the c r i t i c a l velocity and that Pipe D could be made sensitive by placing insulating shields adjacent to the collecting electrodes (Pipe E). Once i t was established what type of effect should be expected i t was possible to obtain data for Pipe D as well. a. Pipe A Tracings of typical records for Pipe A (Fig. 4.3.1) illustrate the nature of the data collected. For Electrode 3 (the upper tracing), the signal would rise to a new constant level whenever the power input to Heater 3 was subcritical, then drop back to the i n i t i a l level when the heater was turned off. But when the heater power was raised to a supercritical value (commencing at 13 minutes from the beginning of the tracing i n this case), the current would rise to the level expected, then shortly thereafter begin decreasing steadily. This was interpreted as the point at which the flow became turbulent. When the heater was later turned off (at 29 minutes), the signal would drop to a level somewhat below the i n i t i a l value and then begin to increase at a very slow rate. Thus the 10 8 6 4 2 0 1 1 a . E l e c t r o d e 3 U - 3.0 x \0"*A (T r ia l Z 4 C ) t J HA off HA on (5.6) HA orv (8.3) V. p F i off _L HA | 0 ^ 10 15 2 0 25 3 0 3 5 4 0 4 5 5 0 c a> o CD > 7 _ 1 1 1 1 1 1 1 1 6 — 5 H A on (2.4) J — H A increased (3.7) -4 H A increased (4.7) \ 3 b. Electrode 4 — 2 U = 3.1 x ICf^A \ H A off — 1 (Trial 4 0 A ) v — 0 1 1 1 1 1 " 1 1 10 15 2 0 25 3 0 3 5 4 0 4 5 5 0 5 4 3 2 H A on( l l ) H A off I I c. Electrode 5 i 4 - 3 . 0 x I0""A (Tr ial 24C) HAon (13) HAoff ^ _L H A on (\8) _L 0 5 10 Fig. 4.3.1 15 3 0 3 5 4 0 20 2 5 Time (min) Typical Records fo r P i p e A ( H A = Heater A ; Values in brackets a re m e a n normal component velocities in I0"*m/sec.) 4 5 5 0 mechanism causing the attenuation had a relatively long lifetime (of the order of an hour or longer) provided the applied voltage was left on. By turning off the applied voltage for only a short time as shown, the attenuation mechanism could be made to dissipate quickly allowing the current to increase more rapidly to the i n i t i a l value. This record therefore indicates that the (normal component) =3 =3 critic a l - v e l o c i t y i s between 5.6 x 10 and 8.3 x :10 m/sec, and that the vortex structure developed during the supercritical flow is relatively stable for this size of pipe. The lower tracing in the Figure shows that the overall effects observed using Electrode 5 were similar to those for Electrode 3 except, of course, that the signal level would drop instead of increase to a new level whenever Heater A was turned on. This record also illustrates another way in which the onset of attenu-ation due to supercritical flow could be detected. When the heater was activated at 21 minutes the current decreased as ex-pected (for subcritical flow) and then remained steady u n t i l the heater was shut off. But the signal did not rise a l l the way to i t s i n i t i a l level. Furthermore, when the heater was turned on again at 33 minutes, the signal level dropped, then began decreasing steadily immediately, i.e., there was not a delay of a few minutes as had been observed i n the record for Electrode 3 above. Thus, i n = 2 this case, the c r i t i c a l velocity appeared to be between 1.1 x 10 = 2 and 1.3 x 10 m/sec; even though the attenuation between 21 and 28 minutes was very small, the flow was s t i l l supercritical. The record for Electrode 4 (the middle tracing) is much simpler since no deflections are normally -observed for subcritical flow when the power input to Heater A i s changed. (The slight jump i n signal when the heater was turned on at the ten minute mark i s not considered significant). Here the attenuation effect did not manifest i t s e l f u n t i l Heater A had been on at a power ^ the c r i t i c a l value for 4 1/2 minutes. The signal f e l l rapidly thereafter u n t i l the heater was turned off at 39 minutes. In this =3 record the c r i t i c a l velocity i s seen to be between 3,7 x 10 and =3 4.7 x 10 m/sec. If a l l the collecting electrodes were connected together the observed signal was similar to that for Electrode 4 except that the attenuation effect was not as large. For this reason, much of the c r i t i c a l velocity data were, collected using this electrode only, particularly for the large pipes since in these the point at which the attenuation began was more d i f f i c u l t to detect. Once the c r i t i c a l velocity was exceeded, further attempts to measure the c r i t i c a l velocity consistently yielded values at least 20 percent lower than the i n i t i a l one. Also the rate of attenuation increase for the same heater power input was less. This hysteresis effect persisted for long periods; i f the heater power and applied voltage were both turned off for 20 minutes after the c r i t i c a l velocity had been exceeded, a subsequent measurement of the c r i t i c a l velocity showed that this hysteresis was s t i l l strong. For this reason, during c r i t i c a l velocity-measurements the helium bath was cycled through the -point after each determination. In the record for Electrode 4 (Fig. '4.3.1), the i n i t i a l signal level was decaying slowly even before the heater was turned on. This data sample was obtained about eight months later than those examples for Electrodes 3 and 5. This decay was, in fact, observed .•- i n a l l measurements i n the later t r i a l s and i s attributed to a slight deteriora= tion i n the plated surfaces of the electrodes. Polarization charges would collect on the electrodes when the applied voltage was on, reducing the total electric f i e l d across the pipe. It was observed that for Pipe A particularly, the attenuation did not begin immediately when the heater power was raised to the c r i t i c a l value. Usually the delay was about three minutes but on one occasion a value of more then five minutes was observed at a relatively high power level. The values appeared to be independent of the heater power. Generally, the delay was less i f the power input was increased gradually to the c r i t i c a l value than i f the heater was turned off for a short period before increasing the power level. However, once the c r i t i c a l velocity was exceeded, no delay times were observed in subsequent measurements unless the bath temperature was cycled f i r s t l y through the ^\~point. b. Pipe B The data were similar to that for Pipe A except that the attenuation effects for supercritical flow were not as pronounced. c. Pipe G For supercritical flow i n Pipe G, the current collected at Electrode 4 did not decrease continuously as i t did for Pipes A and B, but would drop within a minute or two to a lower level depending on the flow rate where i t would remain relatively constant. This effect i s similar to that observed by Careri 2 5 2) et a l . in.their apparatus . " , except that they measured the attenuation of the total current across the pipe. d. Pipe D Detection of the attenuation was quite d i f f i c u l t for Pipe D. I f the power input to Heater A was increased i n small steps, there was no sudden drop i n signal at Electrode 4 corresponding to a c r i t i c a l velocity. Rather, as the power input reached relatively high values, the noise level would gradually increase. When a c r i t i c a l flow velocity was detected, the delay between the instant that the heater power was raised above the c r i t i c a l value and that at which the attenuation was f i r s t noted was usually less than one minute and seldom more than about one and one-half minutes. Data was also collected for this pipe at 1.60 K, and an applied voltage of 25 V instead of 90 V. The c r i t i c a l velocity appeared to be about 20 percent lower at the higher temperature but was not i . -62 affected by the change in f i e l d . The sensitivity of the measurements appeared to be significantly improved at the lower voltage, nevertheless. At 25 V s the current i s close to being space charge limited (See Section 2.5.e.), so any increase in the charge.density across- the pipe would reduce further the total current observed, e. Pipe. E - • Although the c r i t i c a l velocity for Pipe E was approximately the same as for Pipe D, the sensitivity for detecting the point at which the. attenuation began was approximately the same as for Pipe B, Also the attenuation increased steadily during supercritical flow as for the two smallest pipes. Furthermore, the measured delay time was usually about two or three minutes—considerably longer than that for Pipe D. f. Summary of c r i t i c a l velocity data for counterflow using the  detection of attenuation method •xn Table 4.3.1 and Fig. 4.3.2, the average c r i t i c a l velocities for (1.400 =0.001) K and V , "= (90 - 1) V are presented. These are the appl upper limits of the values required to i n i t i a t e turbulence since i t i s like l y that slightly lower values could be obtained by allowing for the possibility of longer delay timesf^ No attempt was made to measure the velocities at which the. turbulent flow reverts to laminar flow. The error bars indicate the ranges i n the observations. (See Appendix 7 for details). Included i n the Table are the ranges of observed delay times and calculated values of Re . Delay times of three to five minutes were V j C common for Pipe A but these became shorter as the pipe size increased. For Pipe D i t was often d i f f i c u l t to detect any delay at a l l , but this may be due to the fact that the attenuation i t s e l f was d i f f i c u l t to discern. For Pipes A, B and E the Re^ ^ 's are close to those expected for classical fluids but for the other two pipes the values are some-what higher. ' This statement refers to the results obtained with the present ap-paratus and under the present conditions. These are not necessarily upper limits in any absolute sense. In Table 4.3.2 the calculated temperature differencesbetween the ends of the pipes for laminar flow at the measured c r i t i c a l velocities are presented along with various quantities discussed i n Chapter 2. The values of AT were of the order of 10 i n a l l cases. TABLE-4.3 .1 SUMMARY OF CRITICAL VELOCITY DATA FOR COUNTERFLOW & Pipe v n,m, c -3 (10 m/sec) Corresponding v s (10 ^ m/sec) Re v sc Maximum range i n v and n,m, c Re v, c (Percent) Delay .Time (min) A 4.33 2.60 1560 t 16 1 to > 5 B 2.27 1.56 1410 t 25 1 1/2 to 4 •' C 4.75 3.53 4460 ~ ± 25 1/2 to 4 D 3.45 2.64 4000 t 8 :<M/4 to •3/4 E 2.45 1.86 2570 ± 1 4 1/2 to 2 See Appendix 7 for Data. CALCULATION OF TEMPERATURE DIFFERENCES BETWEEN THE ENDS OF THE PIPES FOR LAMINAR FLOW AT THE MEASURED CRITICAL VELOCITIES FOR COUNTERFLOW i-i Pipe See Note A .. CC )D : : E a/b 5.53 2.75 1.49 1.01 1.15 a ^KEsmaXjQ ^ " v (10 m/sec) •1.72 4.33 1.92 2.27 2.09 4.75 2.14 3.45 2.13 2.45 b - c Re v sc 1560 1410 4460 4000 2570 c 1 T (m) 0.75 1.17 5.6 6.2 3.6 d 1/1T 0.13 0.085 0.018 0.016 0.028 (v /v ), 1.57 1.44 1.10 1.08 1.20 •e ^Vmax, p. ^Vm5 .q ^ 1^ 1.68 1.88 2.06 2.09 2.09 . f F K E , n 1.41 1.40 1.11 1.09 1.22 g f • K E , • 1.52 1.63 1.13 1.10 1.26 h /D"v2 f 12 (n n,m. •: K E , • (10 = 4 N/m2) 1.55 0.46 1.39 0.72 0.41 "\T\~ Vn,m, 1 @ 2.28 0.35 0.29 0.084 0.089 (10° 4 N/m2) P T (10" 4 N/m2) 3.83 0.81 1.68 0.80 0.50 i A T (10° 8 K ) 2.0 0.42 0.88 0.42 0.26 J a) b) c) d) e) Table 3.3.1 Appendix 1 Table 4.3.1 Equation (2.3.6) and Table 3.311 Ref. 2.3.3 f) g) h) i ) J) Obtained during numerical intergration calculation ^KE,max,D Equation (A2.8) Equation (2.3.9) Equation (2.3.13) Equation (2.3.14) S 4 <? Pipe A o 3 V Pipe C Pipe D Y 9 Pipe B Y P i p e E Fig. 4.3.2 Observed Normal Component Critical Velocity vs Hydraulic Diameter for Counterflow 6 8 d H ( I0" 9 m) 10 12 4.4 C r i t i c a l Velocities and Attenuation Effects for Superflow For superflow the rates of attenuation at the c r i t i c a l velocities were much less than i n the case of counterflow; i n fact, for the larger pipes i t was very d i f f i c u l t to detect any attenuation of the currents at a l l . On the other hand, the c r i t i c a l velocity for Pipe A occured at such a low volume rate of flow that the small varia-tions in the apparatus height relative to the bath level caused large uncertainties in the flow rate measurements. Otherwise, the attenuation effects observed were similar to those for counterflow. At supercritical velocities, the signal level would decrease continuously for Pipes A, B and E whereas for the other two pipes i t would move down to a new, relatively constant level. As the pipe size increased there was a tendency for the delay times to decrease. A single measurement made using V ^ = 25 V for Pipe D indicated that the method might be slightly more sensitive at reduced fields. Unlike the case for counterflow, the sensitivity of the measurements for Pipe E seemed to be only slightly better than that for Pipe D. The observations are summarized in Table 4.4.1 and Fig. 4.4.1. (See Appendix 8 for individual measurements). Because of the scarcity of data and because the c r i t i c a l velocities appeared to be f i e l d independent, the value for V ^ = 5^ v i s included for Pipe D. As for counterflow, these are the upper limiting velocities required to i n i t i a t e turbulence rather than those needed to support i t . The error bars i n Fig. 4.4.1 indicate the observed range of values i n each case. These results indicate that the c r i t i c a l velocity was independent of the hydraulic diameter, within accuracy of the data. SUMMARY OF CRITICAL VELOCITY DATA FOR SUPERFLOW & Pipe v" ' ' ; s s C ( 1G m/sec) Maximum range in V s,c (Percent) Delay Time (min) A 4.0 + 45 4 1/2 B 2.9 + 40 3 C 5.0 t 24 2 1/4 D 5.1 t 47 3/4 to 2 1/2 E 2.8 t 3 4 4 See Appendix 8 for Data. The main limitation i n this method i s that i t is relatively insensitive; at the c r i t i c a l velocities obtained, the attenuation was usually just detectable, particularly for Pipes C and D. The question naturally arises whether or not the normal component was actually stationary during the superflow measurements. For this reason checks were carried out periodically to ensure that the super-leak section was s t i l l effective, and that extraneous temperature gradients were not interfering with the flow. These consisted merely of making measurements of current changes at electrodes.other than No. 4 to determine i f there were sudden changes in current similar to those observed during counterflow when the flow rate was changed. None were seen in any of the runs. However, gradual increases in current were often observed at supercritical velocities, particularly at Electrodes 2 and 3. This i s to be expected, of course, since the attenuation effects measured for a l l electrodes were smaller than those for Electrode 4 only, i.e., the attenuation of the current at Electrode 4 was caused partially by spreading of the ionic beam. £ 4 O Fig. 4.4.1 Observed Superfluid Component Critical Velocity vs Hydraulic Diameter for Superflow Pipe A PipeC 0 0. PipeB PipeD 0 - J PipeE 6 3 8 10 d H (!0- 3m) 4.5 Comparison of C r i t i c a l Velocities with Those of Previous Workers The mean superfluid component c r i t i c a l velocities calculated in Tables 4.3.1 and 4.4.1 are summarized in Table 4.5.1 and plotted in Fig. 4.5.1. This graph has the format used by de Bruyn Ouboter 2 4 5) et a l . to i l l u s t r a t e their contention that the normal com-ponent flow becomes turbulent classically for counterflow at 1.4 K -3 in channels having diameters larger than about 10 m; i . e . s the c r i t i c a l velocity under these conditions corresponds to a Reynold's Number (Equation (2.4.11)) between approximately 1200 and 2300. Curves of constant Re are horizontal straight lines on this v plot. The superfluid c r i t i c a l velocities obtained during the counterflow experiments have been adjusted to show their true relationship with respect to the values.of Re^ shown. This adjustment consisted essentially of removing the correction applied earlier because of the gap between the Pyrex test section and the Plexiglas walls of the pipes. The values published by previous workers have been tabulated 2.1.2) 2.2.1) recently by Wilks " ' and Keller ' , and are plotted merely as solid points i n this figure. For diameters larger than about 10 ~* m these separate into two branches: the upper one corresponds roughly to those experiments in which only the superfluid component was in motion, whereas the lower one corresponds to those i n which there was counterflow or the whole liquid was moving. Some excep-tions to this pattern w i l l be pointed out later. 2.4.2) The solid.line on the graph is Craig's ' ' relation (Equation (2.4.1)) derived from consideration of the energy required to produce a vortex ring around the periphery of the channel, VALUES OF v FOR FIG. 4 .5 .1 Pipe (10=3m) Counterflow v a) s,adj (10 m/sec) 1-^ s,adj (10 ^m^/see) Superflow s ,c (10 m/sec) s ,c -6 2 (10 m /see) A 3.71 + 3.6% B 6.40 + 2.6% C 9.72 + 2.2% D 12.0 + 3.7% E 10.8 + 5.3% 3.5 16% 1.8 - 25% 3.8 - 25% 2.8 8% 2.0 t 14% 1.3~i 20% 1.2 - 28% 3.7 - 27% 3.7 - 12% 2.2 - 19% 4.0 - 45% 2.9 - :40% 5.0 t 24% 5.1 - 47% 2.8 = 34% 1.5 - 49% 1.9 = 43% 4.9 - 26% .6.1 - 51% 3.0't 39% a) Vs,adj ^ fri /(s ^ V n s Note; To plot lines of constant Re r^. AX^ * v 3 6 At 1.40 K, |0/jo= 0.0754; f> = 145 kg/m ; = 1.5 x 10" kg/m-sec This relation is rather insensitive to the selection of the parameters, e.g., B must be varied by a factor of about 28 and C by a factor of 3 at a given value of d i n order to cover the experimental range of v g ^ ( —50 percent) i n Fig. 4 . 5 . 1 . The counterflow data from the present experiment l i e close to the upper limit set by de Bruyn Ouboter et a l . as that corresponding to generation of turbulence due to motion of the normal component. Actually, as was pointed out i n Section 2.4.c, the upper limit for the i n i t i a t i o n of turbulence for a classical f l u i d flowing through a pipe, can have much higher values, depending on the care which i s taken in the fabri-cation of the pipe, particularly at the entrance, and in the execution of the experiment. The data would therefore seem to support their hypothesis. F i g . 4 . 5 . 1 C o m p a r i s o n of Resul ts w i t h Those of P r e v i o u s W o r k e r s O Counter f low (Adjusted values) ® Superflow x Super f low-p lus -counVer f low (Sec 4.6) Constaru RQV • From summaries of Keller tc W i l k s © Ricci 6c Vicentini- iViissoni A Chung 8c Critchlow Y G r i f f i t h s ( u p p e r b o u n d ) • Peshkov 8c S t r y u k o v IO"* IG" 5 10 '* 10" dn (m) The data for pure superflow,, on the other hand, l i e well below that which we would expect from the analysis of de Bruyn Ouboter et a l . ; in fact, the c r i t i c a l velocities are closer to those for counterflow. The reason for this difference in the superflow results is not.clear. Many of the other measurements employed methods in which the superfluid component passed through a superleak before entering the region where the c r i t i c a l velocity was to be detected. This i s analogous to the case of an aeronautic wind tunnel in which baffles are placed ahead of the test section to reduce the gross turbulence due to propeller wash. Perhaps the function of the extra superleak is to prevent vortices i n the bath from entering the test section, but one might expect some vortex gener= ation as the liquid emerged from the superleak i t s e l f because of the large velocity gradients there. Although the plotted points for superflow l i e above those for counterflow i n a l l cases, the ranges of observed values over-lap, This would lead one to conclude that the same mechanism may have been responsible for i n i t i a t i n g the attenuation for both types of flow, i n agreement with the experimental results of 4 5 1) Peshkov and Stryukov ° . However, evidence w i l l be presented in the next section to show that this possibility is unlikely. The situation is actually not nearly as straight forward as 2.4.5) de Bruyn Ouboter et a l . state ' ° . Several workers appear to have detected superfluid component c r i t i c a l velocities which l i e well below the usually observed range of values. Three of these exceptions are: 4.5.2) a. Ricci and Vicentini=Missoni ' " obtained a constant value of v during counterflow experiments with a rectangular s, c cross=section channel between 0.888 and 1.918 K. Re varied v from 6500 to 54 over this temperature range. b. Chung and Critchlow detected two c r i t i c a l velocities in their superfluid wiridtunnel. They thought the f i r s t one corresponded to generation of vortices in the region of each of the particles which were being used as probes, and the second (which is near the upper branch i n the Fig.) announced the breakdown of puas potential flow in the windtunnel i t s e l f . 4.5.3) c. Griffiths' value ° ° is below that of any other experimenter; his apparatus was sufficiently sensitive, however, to determine only the upper bound. More examples could be given. Some of the values shown as solid dots i n the lower branch of Fig. 4.5.1 also appear to be valid deter-minations of the c r i t i c a l velocity for superflow. Thus there is s t i l l considerable controversy regarding the meaning of a l l the data published to date. 4.6 Data for Counterflow Superimposed on Pure Superflow A number of attempts were made to obtain c r i t i c a l velocities for the five pipes with the two types of flow superimposed on one another. 4.5.1) Because of the work done by Peshkov and Stryukov " , i t had been anticipated that the c r i t i c a l value for the total superfluid component velocity would be the same as that for counterflow alone. Since i t was d i f f i c u l t to accurately control very low volume flows using Heater B, practically a l l the early t r i a l s were conducted on Pipes C, D and E. But no attenuation effects were seen unless either velocity was raised above that required when the c r i t i c a l values were being measured independently. Later a new result was obtained using Pipe A. In this run the superfluid component velocity (i.e., Heater B on only) was set at 6.9 x 10 4 m/sec which is well above the c r i t i c a l value. (See Table 4.4.1}). The attenuation commenced almost immediately. Then about 15 minutes later Heater A was turned on and i t s power increased in steps. But no additional rate of attenuation increase was =3 observed u n t i l the normal component velocity reached 2.3 x 10 m/sec, at which time the current suddenly began decreasing at a considerably faster rate. When the heaters were turned off several minutes later the signal level became relatively constant. Because the Re^ corresponding to the second c r i t i c a l point was only 840, i t is believed that the total superfluid component velocity was the controlling factor here as well. This value of v is included in Fig. 4.5.1. It lies above the other values s for Pipe A but, like them, is well below that predicted by Craig's equation. 4.7 Details of the Attenuation Effects In the previous discussion, the attenuation effects have been con= sidered mainly as a means for detecting the c r i t i c a l velocities. But considerable data was also collected regarding the nature of the attenuation i t s e l f ; the findings are presented in this section. a. Increase of attenuation during supercritical flow  General characteristics As has already been outlined i n this chapter, the characteristics of the attenuation were found to vary with the size of pipe used. But for a given size of pipe, the appearance of the effects on the chart records during superflow was similar to that during counterflow except that the magnitudes were con-siderably smaller. The observed characteristics may be summarized as follows: (1) For Pipes A and B, the attenuation increased con= tinuously with time over a.long period (more than one=half hour); then when the flow was stopped the signal began to rise very slowly. Thus the rate of attenuation increase seemed to be the important quantity here, (2) For Pipe C, the signal moved rapidly to lower level when the c r i t i c a l velocity was exceeded. Thereafter i t remained relatively constant u n t i l the flow rate was increased further at which time the signal moved down to a new level. Thus the total attenuation for a given flow velocity seemed to be the important quantity here. When the heater was turned off the signal began to increase slowly. (3) For Pipe D, the attenuation effects were the same as for Pipe C.except that they were smaller in magnitude and the signal i n i t i a l l y rose rapidly when the flow was turned off. (4) For Pipe E, the effects were similar to those for Pipes A and B except that the magnitudes were smaller and the rate of increase in signal was faster here when the flow was turned off. b. Attenuation as a function of flow velocity The attenuation for varying flow rates was studied i n detail for Pipes A and C; the results are as follows. (1) Rate of attenuation increase for Pipe A In Table 4.7.1, data for the rate of attenuation increase are given for values of v and v v . The rate was not ° s I n •> s' measured from the instant that the signal started to drop off. Usually after about one minutes the rate became relatively con-stant for several minutes or longer before beginning.to " t a i l off" particularly i f the flow velocity was quite high, and this i s where the measurement was made in each case. I f a measurement of the rate of signal decrease was taken, then the flow and applied voltage turned off for several minutes so that the signal.returned to i t s i n i t i a l value when the voltage was turned back.on, the rate was usually considerably less :than -78 ( F a l l o w s p . - ~ ? 5 ) the original measurement for the same flow velocity. However, i f a higher flow velocity was used i n the second case the discrepancy (with respect to an i n i t i a l measurement at the higher velocity) appeared to be small. Most of the data,in the Table f a l l into this category. TABLE 4.7.1 RATE OF INCREASE OF CURRENT ATTENUATION FOR PIPE A Counterflow Superflow v - v 1 • n si =3 (10 m/sec) Fractional Rate of Increase of Current Attenuation (10 4 sec •*") Firs t time c r i t i c a l velocity exceeded? V n =3 (10 m/sec) V s (10_4m/se.c) V s =3 (10 m/sec) + + (r m • (- 6%) , (= 7%y ( r 77o) 4.6 3.6 5.0 1.7 i 0.2 Yes 4.6 3.6 5.0 1.3 + 0.2 No 7.7 6.0 8.3 3.3 0.2 Yes 12 9.3 =~ 13 4.7 + 0.2 No 4.8 4.8 1.8 - 0.2 No — 6.1 6.1 2.5 ~ 0.2 No 28 28 18 i 2 No If the rate of attenuation increase i s plotted against v g the data for counterflow separate from those for superflow. However, i f the rate i s plotted, against | v n = v g | as i n Fig. 4.7.1, the data for both types of flow f a l l approximately on 2 0 o CD 1 I o o C 18 16 14 Fig. 4.7.1 Rate of Increase of Current Attenuation vs l\7r,-vs| for Pipe A c CO 0) u> 0 o C O Counterflow ® Superflow — Equation (4.7.1) o <a +-a DC 8 a c o o a i. 8 12 I6 2 0 2 4 Iv^-vJ 0 0 " 5 m/sec) 2 8 3 2 the same curve. Thus, the rate that the attenuation increases i s directly related to the relative velocity between the normal and superfluid components. Because i n counterflow at 1.40 K the normal component moves much faster than the superfluid component, the attenuation would be expected to be much more noticeable than for pure superflow. A linear least squares f i t was calculated using the lower six points i n the plot. The resulting equation was ( | v n = v s l ^ - n m/ sec): -1 Rate of increase of current attenuation (sec ) = •(-1.2 x 10"5 + • 3.8 x 10"2 |.vn-- v |-) (4.7.1) with a root=mean~square deviation of 2.7 x 10 sec . Thus the origin of the plot i s within the extrapolated experimental range f of this line. The deviation from this relation would appear to be quite large for high flow rates. (2) Attenuation for Pipe C For Pipe C, eounterflow data were obtained of the current attenuation for Electrode 4 and for a l l electrodes as shown in Table 4.7.2. For the higher rates the current decreases at Electrode 4 are due mainly to deflections of ions towards the open end of the pipe rather than to attenuation of the current as a whole, i.e., the deflections are sufficiently large that the ions are no longer collected along the whole length of Electrode 4. Fig. 4.7.2 shows that i n both cases, the dependences of the attenuations on the heat power densities are approximately linear. (Lines are least squares f i t s ) . The line for A i ^ g passes with= i n experimental error of the origin. By subtracting the decrease due to attenuation only (as measured by a l l electrodes i n parallel) from the averaged data for Electrode 4 i t might be expected that the resulting line (shown dashed i n the Figure) would give.the decrease i n i ^ due to deflection of the ions up the pipe;only. Howevers when Heater A was turned off the A i ^ value did not return to zero, even though the attenuation of the total current did vanish. TABLE 4.7.2 CURRENT ATTENUATION FOR PIPE C Electrode 4 only Electrode s 1 to 6 ( i 4 = 5.0 x 1 0 = 1 2 A - 2%) ( i l - 6 = 7 ' 2 X = 12 + 10 A I .2%) Power Density Attenuation Power Density Attenuation (102W/m2) (Percent) (lO^/m 2) (Percent) ( i 57.) ( = 5%) 1.29 2.2't 0.1 1.27 0.7 = 0.1 2.18 4.2 t 0.2 2.19 1.3 t o . l 3.40 7.2 1" 0,3 3.40 1,4 - 0.1 4.97 10.8 - 0.3 4.95 2.1 =" 0.2 8.30 18.6 - 0.4 11.7 4.8 t .0,2 0.0 4.2 t .0.4 0.0 0.0 =0.1 (Heater off) (Heater off) This has been checked quantitively using the procedure out= lined i n Appendix 9. The attenuations were, i n fact, greater than expected for a l l supercritical powers. Z8 Because of these observations, i t is believed that, once the c r i t i c a l velocity has been exceeded, the ionic beam has a tendency to spread so that the loss i n current at Electrode 4 is partially balanced by increases in current at Electrodes 2 and 3 as long as the counterflow is maintained. When the flow is stopped, the difference i s made, up by increases at Electrodes 3 and 5. More w i l l be said about this i n Section 4.8. c. Effect of turning off applied voltage; hysteresis For Pipes A, B and C, the attenuation effects would persist for a long time after the flow was turned off. Even for Pipes D and E, the signal would rise only part way to the i n i t i a l value with the remainder of the attenuation decaying relatively slowly. But for a l l the pipes, i f the applied voltage between the source and collecting electrodes was turned off for a period of two or three minutes after the attenuation had been established and then tTirr»p,d on y. the s^ .PTirvl would foA'nd t° be ?t the 3 eve! ?t had before the c r i t i c a l velocity was exceeded. This effect was studied further by making a series of mea-surements of ionic current vs voltage after the attenuation was well established and the flow had 'been turned off. Representative plots.are ishown for Pipes A and E i n Fig. 4.7.3 and 4.7.4. The solid curve i s a f i t through the observed points and the arrows indicate the sequence i n which the measurements were made. The exact shape of the curves for a particular pipe depended on how long the flow had been above the c r i t i c a l velocity before the heater had been turned off, i.e., how much attenuation was present; however, the general nature was always the same. As the voltage was reduced the current rapidly decreased to zero where a) i t remained from then on . Actually, a small negative current (electrons moving towards the source electrode) was observed. This current was probably due to the trapped charge redistributing i t s e l f i n the absence of V ,. Fi 5 . 4,7.3 30 Current at Electrode 4 vs Applied Voltage for After Space Charge Established Pipe A < 2 0 ro T O 10 Observed values (Arrows indicate seqjjen.ee of measurements) . Values pnor to f low. •4. " "appl Current chanqe during super-cr i t ical flow 0 20 4 0 6 0 8 0 V, PP1 (V) Vappl (V) While being decreased, i f the voltage was held at any point more than about 15 seconds the current was observed to begin, rising slowly again. When the voltage was then increased from zero, the current was found to take on values approximately equal to those before the flow was started (Fig. 4.7,3). This means that the mechanism causing the attenuation had dissipated within a short time i n the. absence of the applied voltage. This hysteresis was observed for a l l the pipes-, although the current reduction was usually small for the larger ones. Neverthe= less, even for Pipe D the reverse voltage between the source and collecting electrodes would reach 20 to 30 V (point at which the current vanished as the applied voltage was reduced). To determine the possible role of space charge on the ionic currents prior to the attenuation'., being established, measurements of i ^ as a function of V ^ also were conducted for Pipes B, C and D (Fig. 4.7.5). The experimental values for Pipe D are seen to be approximately linear for V ^ ^> 15 V. The single point for Pipe C f i t s the data for Pipe D. For Pipe B the current i s a l i t t l e less at the highest fields presumably because the width of the ionic source i s close to that of the pipe. Equation (2.5.15) with C set equal to zero (complete space charge limitation) i s also shown. The dielectric coefficient of liquid helium at 1.40 K is 1.0572 4.7.1) ; the effective beam dimensions at Electrode 4 are 3 x 10 mm. (See Section 3.3.b.). The negative ion currents are seen to be space charge limited for V a ^ 15V but for larger voltages, the current i s determined by the f i e l d at the ion source i t s e l f . In Appendix 4 i t is shown that only a small fraction of the available ions are actually drawn across the space/between the electrodes; the remaining ions are neutralized through recombination i n the region of ionization. i — i r / / Fig. 4.7.5 U J / / C3 Pipe B A P i p e C o Pipe D ; — Linear fit to Pipe D dcvta Equation (2.5.15) Current at Electrode A-vs Applied Voltage in Absence of Attenuation Effects 2 0 4 0 6 0 V a P P i (V) 8 0 For a l l the pipes-, measurements of the positive ion current as a function of V ^ yielded values within about 10 percent of the nega-tive ion data except at the lowest voltages. This i s a further indi-cation that the nature of ionic transport across the pipes for large V 1 plays l i t t l e or no role in determining the magnitude of the current; rather i t i s the f i e l d at the source i t s e l f that i s important. The question does arise, however, whether or not the attenuation is due to an effective reduction i n ionic mobility so that the space charge limitation can now be operative at much higher fields. Equation • • ""2 (2.5.15) states that ^°^^ appj_ ^ a ^ c a s e s , and such a curve i s plotted i n Fig. 4.7.3. The disagreement between this curve and the experimental data illustrates that the attenuation effects cannot be explained in this way. 4.8 Comparison of the Attenuation Data with Vinen 1s Relations In Table 4.3.1. we see that the values of v at the observed c r i t i c a l T =3 velocities for counterflow were a l l i n the order of 3 x 10 m/sec. Therefore, the time for the attenuation "to increase to 95 percent of i t s f i n a l value at 1.40 K may be calculated using Equation (All.5) and the values of the constants below Equation (2.4.8); Although for Pipes C and D the attenuation reached equilibrium values quite quickly, for Pipes A, B and E i t was observed to build up over much longer periods of time. It should be noted at this point again that the radioactive source was much narrower than the effective width of the collecting electrodes for Pipes C and D, unlike the situation in the other cases. The significance of this on the mech= anism of thea.attenuati.on is not known, however. There is qualitative agreement, nevertheless, between the i n i t i a l rate of attenuation increase for Pipe A during both counter-flow and superflow (Fig. 4.7.2) and Vinen's equation for the i n i t i a l rate of vortex generation (Equation 2.4.7) (4-. 8.2.) Here the vortex density i s sufficiently high that the i n i t i a t i o n term is negligible but low enough that the decay terms are not yet effective. The length of time that the attenuation effects remain i n the pipes after the flow has ceased i s also a question which i s unsolved. The decay term in Vinen's expression i n this case shows that for 95 percent of the line length to dissipate i s (Appendix A l l with v = -3 r 3 x 10 m/sec): ^ o . o S L . l ~ MO ( 4 . 8 . 3 ) The interesting point to note here i s that this i s the approxi-mate time for the attenuation to dissipate i n a l l cases provided V , was turned off. Thus i t is apparently the presence of the appl r r ions together with the applied electric field, that grossly affects the rates of build-up and decay of the vortex lines i n the pipes. From Equation (2.5.35), the time for approximately half of the vortices to annihilate at the collecting electrodes i s (using the values 1 = 1.40 x 10° v = m/sec (Equation (2.4.10), V v r 3 appl (observed reverse voltage) = 70 V and v = 3 x 10 m/sec) ; decay 4 - J 3 ^ £ i v V a p p , 2 1 c U y * C 4 .8 .4" ) But because the f i e l d increases at each point as the charge i s removed, the velocities w i l l be slightly higher. Even i f we take the velocity at x = s (Equation 2.5.34) the decay time for half the vortices i s s t i l l T d e c a y ~ 1 5 d a ? S ( 4' 8° 5> The very slow rate at which the attenuation decreased when the flow was turned off suggests that this analysis may have some merit but the question of why the dissipation of the vortices through mutual interaction is inhibited remains unanswered. For the larger pipes, particularly, even though the current at Electrode 4 was reduced considerably during supercritical flows the total"current was affected only very l i t t l e . This effect could not have been produced by a two-dimensional space charge since the total current had been shown to be dependent on the applied f i e l d at the source i t s e l f . The exact shape of the space charge would appear to depend quite strongly on the relative widths of the source, pipe and effective area of the collecting electrodes. As was pointed out i n Chapter 2, the attenuation of the negative 2.5 ion current which was observed by Careri, Scaramuzzi and Thomson depended on the heat power density i n excess of the c r i t i c a l value (i.e., (Q - Q c)/ A = = s e e Equation (2.5.36)). However, i n the present experiment the attenuation was found to be proportional to the power density i t s e l f (Fig. 4.7.3), The difference between these results seems to be due to more than variations i n the space charges developed; -in fact, their result indicates that there may be cases i n which the relative velocity between the superfluid and normal components (and hence the mutual f r i c t i o n force) is not the dominant factor i n the generation of high densities of vortices i n large pipes and channels. CHAPTER 5 CONCLUSIONS 5 .1 C r i t i c a l Velocities a. The method of detection of negative ion current attenuation is satisfactory for measuring c r i t i c a l velocities in large pipes (3.7 x 10 m < hydraulic diameter <• 1.2 x 10 m) for counterflow, superflow and counterflow superimposed on counterflow. The sensitivity of the method increases as the hydraulic diameter decreases and so this technique probably can be extended to pipes and channels consider-ably smaller than the range employed. b. C r i t i c a l velocities measured during pure superflow i n a l l the pipes at 1.40 K were consistently higher but within the experimental range of the respective values measured during counterflow. Further, the value yielded by the t r i a l i n which counterflow was superimposed on superflow was found to be lower than expected from the corresponding Reynolds number calculated from the normal component velocity and the total f l u i d density. Therefore, the present experiment did not substantiate the hypothesis that for counterflow in large pipes the c r i t i c a l velocity is controlled by In fact, there i s no theoretical reason why this relation should apply to counterflow since =;v.c:. si--L:s£:jit-i-"..'it the relative velocity |^v - v n | is.not small compared to either v or v . s n c„ During the t r i a l i n which counterflow was superimposed on superflow., two c r i t i c a l velocities were observed, both of which are believed to be associated with the superfluid component. d. The mean superfluid component c r i t i c a l velocities for the five pipes were found to be Counterflow: (2.4 =0.3) x 10 = 4 m/sec Superflow: (4.0 1" 2.8) x 10 = 4 m/sec The uncertainties are the means of the observed variations i n each case. The c r i t i c a l velocity values for the individual pipes were not sufficiently precise to suggest any dependence on pipe size. These values represent upper limits for the c r i t i c a l velocities since i t is believed that slightly lower values could be obtained i f the possibility of longer delay times j/M'as'- allowed for. J 2 Attenuation Effects a. For the pipes i n which the ion source width was greater than the effective width of the collecting electrodes (Pipes A, B and E), the attenuation increases continuously over a period of time of an hour or more for flow rates close to the c r i t i c a l velocity. For the smallest pipe in the series, the i n i t i a l rate of attenuation increase was observed tq be directly proportional to the relative velocity between the superfluid and normal compon= ents. Although this dependence is i n agreement qualitatively with Vinen's theory of vortex line generation, the times for the system to reach equilibrium during the flow, and for the ^ attenuation effects to decay after the flow has been turned off are much longer than those predicted by him. V • •' b. For the pipes i n which the ion source width was narrower than the effective width of the collecting electrodes (Pipes C and D), the attenuation was observed to increase rapidly to a level depending on the flow rate whereupon it , thereafter remained constant u n t i l the flow rate was changed to a new value. Here the total al } See footnote on page 62. attenuation was directly proportional to the flow rate, but since these measurements were made for counterflow only i t is not known which of the three velocities, v^, v g or v , is the controlling one. The times for build-up and decay of the attenuation agree approximately with the Vinen relations for these pipes. c. The delay time between the instant the flow rate was increased to above the c r i t i c a l value and that at which the attenuation was f i r s t observed decreased on the average as the hydraulic diameter of the pipes increased. d. The decay of the attenuation i n a l l the pipes is faster than that predicted by a theory based solely on the migration of vortices across the pipes due to the combined action of the applied electric f i e l d and Magnus forces. e. Placing shields over the collecting electrodes i n a large pipe so that their effective width i s made less than that of the ionic source enhances the attenuation effects and hence improves the sen-s i t i v i t y of the measurements. f. When the applied f i e l d i s turned off the attenuation mechanism in a l l the pipes dissipates within a few minutes. However, some hysteresis effects remain for a long time which cause further deter-minations of c r i t i c a l velocity, rate of attenuation increase, etc. to yield lower values unless the temperature of the liquid He i s f i r s t l y cycled through the \-point. g. The attenuation of. the current measured at a single electrode directly across from the ionic source is greater than that for the total current i n a l l cases, but particularly for the pipes i n which the source is narrower than the effective width of the collecting electrodeso This is attributed to spreading of the ionic beam, i' h. In the absence of the attenuation?effects, the ionic current across the pipes was determined by the f i e l d strength at the source electrode rather than by the nature of the ion transport„ CHAPTER 6  PROPOSALS FOR FURTHER WORK 1„ Measurements of c r i t i c a l velocities employing the ion detection method should be extended to smaller pipe sizes. Various pipe lengths should be included with the ion sources located at various points along the pipes. Exploratory measurements for "larger pipes might also be useful (with shields as in Pipe E to obtain the required sensitivity) but here unless the lengths were corresponding greater there would be consider-able uncertainty as^ to/the meaning of the data. 2. Determinations should be made of the minimum flow rates required to main-tain turbulence in the pipes. I f the turbulence is not classical these c r i t i c a l rates might approach zero. =8 3. A differential temperature measurement system sensitive to 10 degree at 1.40 K should be developed i n order to detect possible lower c r i t i c a l velocities i n the large pipes. The results described herein suggest that the superfluid c r i t i c a l velocity might be multi=valued. The possi= b i l i t y that these may correspond to various levels of vortex line cir= culation (including half=integer values==See Section 2.1.b) has not yet been shown for flows through pipes and channels. One reason for this i s that the minute temperature differences between the ends of larger-pipes and channels are d i f f i c u l t to measure for velocities below that at which gross turbulence occurs (Table 4.3.2). For example, in 4.5.2) Ricci and Vicentini-Missoni's work * , the sensitivity of the ther-mometers was 10 K which is three orders of magnitude greater than the values in the table. Thermometry techniques have improved considerably since then but probably are s t i l l not sufficiently developed to detect possible effects at these lower c r i t i c a l velocities. 4. The present apparatus should be modified to include: a. interchangeable collection vessels of smaller sizes (down to 1 ml or less) to permit precise measurement of very low flow rates; b. a collection basin for the liquid He leaving the graduated vessel via the fountain pump (Heater C), This i s to reduce distur= bances in the bath above the opening to the test section of the apparatus, especially when measurements are made over long periods; c„ an automatic bath level control precise to at least 0,1 mm. The level would be maintained by raising or lowering a displacement volume in the bath during actual measurements of superflow. (The present system for moving the apparatus should also be retained). d. a manostat precise to 10 ^  degrees or better for heat input loads of 0 to 10 mW and 10= degrees for 10 to 200 mW. 5. Further experiments should be carried out to determine conditions which affect the c r i t i c a l velocity data for a particular pipe, e.g.: a. possible dependence on delay time; b„ sensitivity to external disturbances; c. dependence on surface roughness of pipe entrance and walls. APPENDIX 1 CALCULATION OF THE KINETIC ENERGY FACTOR, f. KE.max., • The evaluation of Equation (2.3.5) can be done by either analytic or numerical integration. These methods w i l l be discussed in turn together with values obtained from each for the pipes used in the experiment. a. Analytic integration First the form of the series is simplified by setting = (2.1 + 1) 2b ( A l - l ) so that Equations (2.3.1) and (2.3.2) become P ' 4- ^ ( |) m i . y r^i X (PH.e ) and V P V z Z By letting M denote the series term i n Equation (A1.2), we can write the last factor i n (2.3.5) i n the form: | (f-y-- M ) d x d y 1 = O The numerator is evaluated by expanding 7 and integrating term by term ( [ dy - 4 cxb"7 ( R l - O H y ^ * ^ = - ^ a ^ (A I' "7 ) — ^  i, = o ( A 1.8) ( r> 1.9") -a — b 0 0 jfcuyX b m . ( * |. I<L ) 3 b h dxty = 4 3 t z ^ fz — -a -b (A 1-13) - a - b V. 1--° CJ-o ^ L W J ) (^r^j)-<>^^ia ^ J I K V I J ^ -a - t <=° / \ w | ) I ^ 1*1 , - I I 1 1 \ 4-(ft 1. 1 5 ) Collecting terms and simplifying gives the numerator for f : b W L ( - L - - ! 4 - - J V . — . — . — ! > This expression and Equation (A1.4) have been evaluated for representa= tive values of a/b; see Table A l . l , b. Numerical integration " The quadrant ( 0 x ^ a, 0 y <£-b) was divided into 100 equal areas (area boundaries defined by x/a = 0.0, 0.1, 0.2, = -== = •=•, 1.0; y/b = 0.0,, 0,1, ------, 1.0) and the factor V L=0 at the center of each area((x/a, y/a) = (.05, .05), (.05, .15), (.05, .25)s _ = ====9 (.95, .95)). The convenience of using this factor arises from the fact that i t varies only from 0 at the boundaries to 1.0 at the center of the pipe. Thus L is merely the mean of the cubed values divided c r KE, max,EJ by the cube of the mean value. The results for the same quantities a/b used in the analytical integration are included i n Table A l . l for compari-son, The differences depend on the number of series terms calculated i n each method; the numerical integration results are probably the more accurate even though only the f i r s t 21 terms were evaluated. Values of fT,_ _ for the actual pipes used in the experiments are KE,max, Q r r ^ given in Table 4.3,4; these calculations were done using the numerical integration method, TABLE A l . l EVALUATION OF f T r T n FOR REPRESENTATIVE VALUES OF a/b KE^ max, •,  a/b 1.0 1.5 3.0 6.0 10.0 50.0 100.0 a) oO Num( f TrTn _ ) v KE,max,• 0.19445b8 0.76419b8 3.3520b8 8.8350b8 - - --Denom(fT,„ ) • KE 9max sa Q. 088904b8 0.36012b8 1.7548b8 5.0974b8 KE,max,a Analytical inte-gration method 2.187 2.122 1.910 1.733 -- 1.543 f KE jinax, • Numerical inte-gration method 2.135 2.093 1.888 1.703 1.617 1.537 1.537 1.537 Percentage difference 2.4 1.4 1.2 1.7 -- - - — 0.38 a/b = is the case of two infinite fl a t plates: v = t ,v = —— ; \ APPENDIX 2 CALCULATION OF £,„ „ FOR PIPES SHORTER THAN THE TRANSITION LENGTH Since no data for the transition length (for laminar flow) appear to exist for pipes of rectangular cross-section; published results for 2.3.3) circular pipes have been used . Thus the transition length for a pipe of rectangular --cross-section i s taken to be (See Equation 2.3.6) L T S D - °' 1 3 ^ v <A2'L> where d^ = hydraulic diameter = 4ab/(a + b) for a pipe of dimensions 2a x 2b. Values of 1 ,-. at the c r i t i c a l velocities for the pipes used in the present experiment are included i n Table 4.3.2. To determine the value of f _ , we f i r s t estimate the value of ivh , LI v _ /v by means of max, • m, n / V m a x ; • N I / V/mw* , O \ _ . where the subscript J2. denotes the value for a pipe of length k. ; the subscript Jly denotes the value for it ^  y ; the subscript o denotes the value for a pipe of circular cross-section. Since (v /v ). = 2.0, max,o m,o | ( / V r ^ a * > O N 1 -105 In Table A2-I, the calculated values of (— — - J, are given for V v ^ n /XT-several values of a/b. By reconsidering the quantity f T r r, „ obtained in Appendix 1 (fT.„ i s the value of f„_ _ for X JL-r )we see r i KEjinaXja KE,0 1 that the ratio ^ K E ^ r n ^ a - ( A 2 . 4 - ) varies only over a very small range (about 1.2 percent) for a l l values of a/b. Also the corresponding value for pipes of circular corss-section is unity both for JtL = 0 and for -r • Thus by making the assumption that there is a direct proportionality KEj a I oL ( ) - 1 ( A 2 . S ) / V ^ a x ..'a \ X v ^ , a we can write K e ; Q K E ) m a x ) P j V v. ( A 2 , 4 ) or by means of Equation (A2.3), -P. V 4-K E , m<xx ; O / v r r i a x , , O \ ( A 2 . 7 ) uev.can -be obtained Once the transition length, i s known/ the ( '•—— jval 2 3 3) from the compilation by Prandtl and Tietjens . In Table A2.1, calcu= lated values of V^e, m<aX,Q_ | are given along with corresponding data ( V > a \ for pipes having circular and equilateral triangular cross-sections 2.3.2) TABLE A2..-1 CALCULATION OF f n AND COMPARISON WITH THOSE FOR CIRCULAR• AND EQUILATERAL TRIANGULAR CROSS-SECTIONS • a/b • O A 1.0 1.5 3.0 6.0 10.0 50.0 100.0 K^E,max 2.135 2.093 1.888 1.703 1.617 1.537 1.537 1.543 2.000 2.338 (V /v 1 ^ max m->JLr 2.090 2.053 1.857 1.666 1.588 1.501 1.500 1.500 2.000 2.222 ^KE9max 1.022 1.019 1.017 1.022 1.018 1.028 1.029 1.029 1.000 1.052 K^E:,max v v -1 / max / \ V / v Ao m ^ ' 0.022 0.019 0.017 0.022 0.018 0.028 0.029 0.029 0.000 0.052 = 107c% APPENDIX 3 PIPE DIMENSIONS AT ROOM TEMPERATURE TABLE A3 J. PIPE DIMENSIONS AT ROOM TEMPERATURE Pipe Electrode=El,ectrode Distance -1^.* (cm) Plexiglas-Plexigas Distance 1^* (cm) A 1.22 t 0.01 0 .210 ' t 0.004 B 1.21 - 0.01 0.429 - 0.004 C 1.22 t 0.01 0.810 t 0.005 D 1.23 - 0.02 1.20 0.01 E 1.23 - 0.02 1.22 t 0.01 Sot Thi Wic tree electrode = shield distance 1.06 t 0.02 .ckness of shields 0.157 - 0.002 th of gap between shields 0.20 t 0.01 Length of a l l pipes (cm) 10.0 _ - 0.05 * Uncertainties i n dimensions reflect measured variations between ends of the pipes. Note: Thermal contraction factors for calculation of dimensions at liquid helium temperatures are given i n Reference A 3 . 1 . • APPENDIX 4 ESTIMATION OF TOTAL CURRENTS FROM OC°SOURCES IN HE II The mass of a radioisotope required to produce a desired disinte= A4.1) gration rate is t , A W d A (A4-.0 where t, = half=life of the radioisotope; W = atomic weight; N q = Avrogadro's Number*, dA/dt= disintegration rate. If the radioisotope is deposited on a f l a t surface i t s thickness i s ro r wJL (A4.2) where =^> •= density of the radioisotope; w = source width; 1 = source length; and so •t./x W„, d A 241 A4.2) The properties of Am are well, known'-.a; :'.::; for. the source characteristics see Section 3.3.b. From these, — 6 Since the Au protective covering was 3 x 10 m thick, the self-absorption of the o i . -radiation was negligible. But only a small fraction of the crt. -particles pass through the protective layer. Ignoring edge effects, this fraction may be calculated by integrating over the solid angle of the emerging radiation. F = ~~7~ (A4-.S) e - e where d =. layer thickness;,. R = maximum range of oC- radiation. m 241 Substituting in the values for the Am f o i l , F = 0.29 (A4.6) The average energy of the emerging radiation is ^ t » , a v t = E « ^ r z 2 (A4"V> where = i n i t i a l energy; 241 and so for the Am f o i l (E ^  - 5.5 Mev), , ave 2.7 Mev (A4.8) Now, the activity of the c<-source was 45 ywCi. Assuming that the helium atoms are singly ionized by the passage of the radiation and that A4 4") the energy required to create an ion pair i s 30 eV " , we obtain from Equations (A4.6) and (A4.8): Ionization rate = 4.8 x 10^ atoms/sec. (A4.9) • =19 Since the electronic charge i s 1.60 x 10 G, Maximum current across the pipe = 7.7 x 10 = 9 A (A4.10) The actual current observed using this source in Pipe D was -12 3 7.8 x 10 A for a f i e l d of 7.4 x 10 V/m. The difference is due to recombination of the ions i n the region of ionization. Hereford and Moss have measured the current collected from a week P o 2 ^ source 3^ in liquid helium as a function of temperature and electric f i e l d for radial geometry. Some of their results are reproduced i n Fig. A4..1. Since they found that the temperature dependence was slight between 1.2 and 1.5 K, their curve for 1.31 K compared with data collected during the present experiment for Pipe D (Fig. A4.2). Extrapolating the straight line portion of their curve to zero f i e l d as shown in Fig. A4..1, -12 an intercept of about 1.2 x 10 A is obtained; the fractional change - 8 =1 of the current per unit electric f i e l d was 3,8 x 10 (V/m) . For Pipe D the curve for negative ions was found to l i e slightly below that for the positive ions. The dependence is greater for both cases because of space charge effects. For the straight line portions the values are (See Equation (A4.10) for the value of the ionization current); Current Sign Positive Negative The agreement between these results and those of Hereford and Moss is not bad considering the large differences i n source strength, electric f i e l d intensities and geometry. Obviously, this relationship deserves further study. Zero Field Fractional Change per Intercept Unit Electric Field 1.6 x 10° 1 2 A 11 x 10 = 8 (V/m) = 1 0.9 x 10' 1 2 A 12 x 10"8 (V/m) = 1 a) The source strength is estimated to be about 0.08 /ACi from their stated value of the ionization current , -11 ... (1.41 x 10 A) I l l = 1.41 x 10"" A ' 0 8 16 2 4 3 0 E a p p , ( l 0 5 V / m ) Eappl 0 0 3 V/m) APPENDIX 5 CALCULATION OF TRAPPING LIFETIME FOR NEGATIVE IONS  IN VORTEX LINES AT 1.40 K and E = 75 V/CM BY PRATT AND ZIMMERMANN'S METHOD In order to f i t Equation (2.5.11) to the experimental data, Pratt 2 5 8) and Zimmermann ' ' f i r s t separate out the temperature dependence on , V and V (a is assumed to be independent of T): O J f t 5= — ~ — w f l o (A<S. l ) ( s o V. I S o I So since V„^ >> V,„. The subscript "o" denotes that the quantities so CO AO labelled are evaluated at some reference temperature and pressure. Next, they use the empirical formula where ^ _ > 0 o is the negative ion mobility for ( S O A _ A V / (A£.s) where / 2 TT €. / _ >v 4 3 /U_;< r w us. AO The primes on TTE and AV0 indicate that these are not the same as "Ca and A V 0 ( __ = ^ACP ^ e c a u s e °f t n e approximation i n Equation (A5.3). To f i t this to their data, the following relations were used A = 8.10 K for P <Z. 5 atm (A5.7) P = (146.2 - 1.6/T - 8.75 x 103 exp (-9470/T)kg/m3 (A5.8) 3 For E - 3.3 x 10 V/m and a reference temperature of 1.650 K, they obtained T 0 = (1.5 - 0.2) x 1 0 ° 1 2 sec (A5.9) o I = (59.4 - 0.2) K (A5.10) These parameters gave a good f i t for trapping lifetime over the total 3 range of their measurements (10 to 10 sec). To estimate the trapping time at 1.40 K.we f i r s t calculate p and |0 from (A5.8) p (!.__•<_ K) =. 1,1-7 x lo"5" k g / m 3 ( A S . I l ) I So then substitute the values into (A5.5) f l.4o * , _ * v / m " 7 . - 9 x l O ^ *e<^ ( A S . I _ ) 3 3 Now when E is increased from 3.3 x 10 to 7.5 x 10 V/m, the trapping 2 5 8) lifetime i s reduced by 62 percent ' " and so « _ STt &cKy% ( A S . 1 4 ) The well depth i s calculated from and V c i n Equation (2.5,11.) using = 19.5 S:and a =1.46 X: o W l . 4 0 L 7 . 5 x l 0 3 V / m " - 6 9 ' 3 K ( A 5 a 5 > W l . 4 0 K , 7.5 x l 0 3 V / m = " 1 ' 4 3 K < A 5' 1 6> and so / V R - V A \ ^ B l ^ ^ . s x / o V/ M 67.9 K (A5.17) APPENDIX 6 DEVELOPMENT OF SHIELDS FOR PIPE D TO ENHANCE CURRENT ATTENUATION A series of t r i a l s were conducted to determine i f the sensitivity of the c r i t i c a l velocity determinations for Pipe D could be improved by placing shields over the collecting electrodes. The idea behind these shields was to produce similar ion collection conditions to those in Pipe A; therefore, the shields were made so that only a 2 mm wide gap down the centreline of the electrodes was le f t exposed. During these t r i a l s , the following materials were tried: i . cellulose tape a^; i i . polythene sheet of thickness about 0.05 mm; i i i . Plexiglas sheet of thickness about 0.16 mm (1/16 inch). The cellulose tape had practically no effect. It appeared that this material probably was sufficiently porous that the ions pass through i t unimpeded. The polythene sheeting seemed to improve the sensitivity somewhat but was d i f f i c u l t to f i x to :the electrodes. -'Available, cements permitted the sheeting to break away from the electrodes at low tempera-tures and grossly interfere with the flow pattern of the He II through the pipe. The Plexiglas sheeting appeared to work well. It was fixed to the electrodes at both ends of the pipe using cellulose tape. The exact configuration of the shields is described i n Section 3.3.a. Canadian Technical Tape Ltd.., Montreal. "Tuck-Tape" --E60 APPENDIX 7 CRITICAL VELOCITY DATA FOR COUNTERFLOW AT 1.40 K The data are in Table A7.1; the following should be noted: 1. Cross-sectional areas for normal component and superfluid component flows are given i n Table 3.3.1. 2. The normal component velocity is given by Equation (2.2.3). 3. The superfluid component velocity i s given by P \ Cro&s-se.c-'t-ior^l area. Car n o r w o l -Clow C . V o S l - s e t K o K \ a l area -Pov •superflow 4. The Reynold's No. is given by Equation (2.4.11) 2 12") 5. At 1.40 K ^ J--^, jo =. / . 4 S x /O kg A 5 = /. 3 2 . x / o * j / k ^ - K ft/ = 1 . 5 4 - X / o " ^ m • sec Symbols i n Table A7.1: Q /A = heat power density; c v = normal component mean c r i t i c a l velocity; n,m,c v g = superfluid component velocity; R ev c = Reynolds number, corresponding to c r i t i c a l velocity. TABLE A7.1  CRITICAL VELOCITY DATA FOR COUNTERFLOW 1. Pipe A T r i a l Delay Time (min) Q <io 2 C/A 2 W/m ) V =n,m,c (10 m/sec) Corresponding V -4 s (10 m/sec) Re ( - 7%) 7%) (- 97o) <i 117c) 24B 4 > 0.906 1.18 3.38 4.49 2.03 2.64 > 1210 1580 24 C 3 1.37 2.02 5.13 7.55 3.08 4.53 1840 2710 25A 1 1.15 1.23 4.28 4.58 2.57 2.75 1540 1640 25B >.5 — 1.86 — 6.94 4.17 - - 2490 40A 4 1/2 .966 1.21 3.61 4.52 . 2.16 2.71 1290 1620 40A 1 - - 1.22 — 4.56 - - 2.74 1640 40B 2 1/2 .475 ,721 1.77 2.69 1.07 1.61 640 960 Mean 0.973 1.35 3.63 5.03 2.18 3.02 1300 1810 Mid-point 1 .16 4.33 2 .60 1560 Variation (Perc< ant) 16 16 16 2 . P i p e B T r i a l Delay Time • (min) Q ao2 C 2 W/m ) (10 _n,m,c m/sec) Corresponding V -4 s (10 m/sec) Re (t 6%) (- 67.) 77=) 37o) 26 4 > 1.07 ^ > 4.00 > 2.76 2480 39A 3 1/2 0.415 .644 • 1.55 2.40 1.07 1.66 960 1490 39B 1 1/2 .424 .658 1.58 2,45 1.09 1.69 980 1520 39C 2 1/2 .635 .933 2.37 3.48 1.63 2.40 1470 2150 39D 2 .333 .511 1.24 1.91 .857 1.31 770 1180 Mean 0.452 0.763 1.69 2.85 1.16 •1.96 1050 1760 Mid-point 0. 608 2,27 1.56 1410 Variation (Percent) 25 25 25 3. Pipe C T r i a l Delay Time (min) •Q CM 2 2 (10 W/m ) V - 9 ' (10 m,c m/sec) Corresponding V -4 s (10 m-sec) Re (- 6%) 6%) (- 8%) 27 2 1/4 1.28 > 4.78 > ^ 3.55 > 4490 28A 4 .834 • — • 3.11 2.31 2930 29 1/2 1.29 1.64 4.82 6.11 3,.58 4.54 4530 5740 Mean 1.29 1.25 4.82 4.67 3.58 3.47 4530 4390 Mid -point 1.27 4.75 3.53 4460 Variation (Percent) — 25 ^-25 -25 TABLE A7. 1 -- continued 4. Pipe D Tr i a l Delay Time (min) (10 Q Jk 2 2 W/m ) (10 m,c m/sec) Corresponding V -4 s (10 m/sec) Re (- 7%) 7%) 9%) (- 10%) 32 < 1/4 > 0.574 0.834 > 2.14 3.11 > 1.63 2.38 > 2480 3610 33 < 1/4 : 1.41 1.57 5.25 5.84 4.01 4.46 6090 6780 33 < 1/4 .560 .807 2.09 3.01 1.59 2,30 2420 3490 33 3/4 - - .800 — 2,99 — 2,28 — 3460 Mean 0.848 1.00 3.16 3.74 2.41 2.86 3660 4340 Mid-point 0.924 3.45 2. 64 4000 Variation (Percent 8 8 8 5. Pipe E T r i a l Delay Time (min) Q /A 2 2 (10 W/m ) (10 V m,c m/ sec) Corresponding V -4 s (10 m/sec) Re (- 8%) 87.) (- H%) e-137=) > e > > 35 2 0.537 0.760 2.00 2.84 •1.52 2,15 2090 2960 37A 2 .623 .726 2.33 2.71 1.76 2.05 2430 2830 37C 1 1/2 .564 .739 2.10 2.76 1.59 2.09 2200 2880 38B 1/2 .542 .767 2.02 2,86 1.53 2.17 2110 2990 Mean Mid-point Variation (Percent 0.567 0.748 0.658 ) 2.11 2.79 2.45 14 1.60 2.12 1.86 14 2210 2920 2570 14 ::. A P P E N D I X 8 C R I T I C A L V E L O C I T Y D A T A FOR S U P E R F L O W A T 1 . 4 0 K The data are i n Table A 8 . 1 ; the following should be noted: 1 . The superfluid velocity is given by Equation ( 2 . 2 . 6 ) . 2 . The cross-sectional area for the superfluid component flow is given in Table 3 . 3 . 1 . . 3 . For the fin a l measurement for Pipe D (Trial 3 3 ) , the v a p p l = 2 5 v « 4 . For the measurement for Pipe E (Trial 3 8 B ) , the bath temperature was 1 . 4 1 K . 5 . Symbols i n Table A 8 . . 1 : A v A t v s ,m = volume of liquid collected in graduated vessel; = time to collect volume AV of the liquid; = mean superfluid component velocity. TABLE A8.1  CRITICAL VELOCITY DATA FOR SUPERFLOW Pipe T r i a l Delay Time (min) Av (ml) A t (sec) V (10 m/sec) Mid-point -4 (10 m/sec) Variation (Percent) (-0.2 ml) (- 0.5 sec) A 40B 4 1/2 1.9 5.1 255 265 > 2.2 J 137=. ^ 5.8 - 6% .4,0 45 B 39B 3 1.9 10.1 199 443 > 1.7 J 127= ^4.0 - 47= 2.9 40 C 31 2 1/4 10.1 10.1 360 141 > 2.8 J 37= ^ 7.2 - 37= 5.0 24 •D 33 33 3/4 2 1/2 10.1 .10.1^ 10.1 262 56 310 ~> 2.7 'J 47= ^•12.. 6 - 47= < 2.3 - 47= 5.1 47 E 38B 4 5.1 (Estimated fr power input) 232 om Heater B >. 1.7 \ 71 ^1 3.8 - 107= 2,8 34 N3 4> APPENDIX 9 CURRENT ATTENUATION AT ELECTRODE 4 FOR PIPE C For large deflections of the ionic beam, current may not be collected along the whole length of Electrode 4. The following i s an estimate of the magnitude of this effect. If Ax, and Z\xr are defined as i n Fig. A9.1, then the deflection 4 5 angle is and since L 4- V ( ° S T A / A V 0 because the normal component velocity becomes less than that given by Equation (2.2.3 ) for supercritical velocities and, as is shown in Fig. 4.7.2, the change in f i e l d at the source is only about five percent even at the highest velocities. (The current emitted from the source is: approximately--proportional to the f i e l d strength at the source i t s e l f -see Section 4.7'.c) Fig. A9.1 Schematic Diagram Showing Deflection of Ionic Beam for High Normal Component Velocities (Also see Fig. 2.5.2. ) Now at 1.40 K, yU._ = (3.5 - 0.1) x 10~ 5 m 2/volt sec (Equation (2.5.6) S = 1.32 x 10 joule/ kg K (Reference 2. .1 .2) 2 3 (° =1.45 x 10 kg/m (Reference 2.1.-2) A = 9.83 x 10 = 5 m2 (Table 3-3.1) d 4 = (1.00 - 0.01) x 10~2 m (Fig. 3.1.2) s = (1.21 t o.Ol) x 10~2m (Fig. 3.1.2) Ax, = (2.5 t 0.5) x 10~3 m (Fig. 3.1.2) 5 ' appl V . = 86 V - 1% (Trial 27) Therefore ±- ^ - (i.as- ± a % ) Q 4 - ( o - 2 S ± ^ i % ) (A9 .4 - ) This expression is valid only for ^ ^ / . / ^ A ^ ^ which means that Q ^"0,14 watts - 297, before any change in the current at Electrode 4 would normally be expected, i.e., taking the possible error into account, no decrease of ionic current would be expected at Electrode 4 for Heater A powers less 3 than about 95 mW (or power densities i n Pipe C less than about 1.0 x 10 W/m2) a ) Larger changes in current could be expected i f A >f_ was for some reason smaller than stated above. For example, i f A x, = 0 in Equations ('A9 ..!:•) and (A9.3) then for Q = 0.14 watts, _ i ^ / i ^ = 267» which is more than sufficient to account, for^the. observations. However, mea= surements of the currents at Electrodes 3, 4, and 5 made during the t r i a l s before the deflection and attenuation data were collected showed that the electrodes were indeed properly aligned (i.e. and nothing was found wrong during inspection of the pipe when the cryostat was dissembled. APPENDIX 10 SUPERLEAK°TIGHT "GREASE" FOR LOW TEMPERATURE JOINTS The ground-glass joint just above the main superleak section f a c i l i t a t e d installation and maintenance of Heater B s and permitted the apparatus to be readily dismantled when necessary for leak testing. The smaller joint at the base of the graduated vessel allowed Heater C and i t s associated superleak to be installed. Below their freezing points, ordinary vacuum greases have coeffieients of thermal expansion which are considerably greater than that of Pyrex. However, i t was found that a mixture of glycerine and white lead (2PbC03'Pb(0H>2) formed a satisfactory seal Al0-l\ A series of tests were carried out using this mixture: Even when the glass parts forming the joint were not mated, the leakage of He II was s t i l l quite slow. These tests also determined the effectiveness of the mixture for joints i n which one of the mating surfaces was not Pyrex. a. Pyrex-Pyrex joint The apparatuses for the Pyrex-Pyrex joint tests are shown i n Fig. AlO.l.a and AlO.l.b. For the ground glass joint, the mixture was used i n the same way as vacuum grease at room temperatures. During the t r i a l s the cryostat could not be evacuated, before precooling because the air pressure inside the apparatus otherwise would have forced the joint apart. No liquid He was observed inside the apparatus even after i t had been immersed in the bath for an hour at temperatures well below the )\ -point. The apparatus i n Fig, AlO.l.b provides information on how greatly the bulk contraction of the mixture differs from that of Pyrex as the temperature is lowered to that of l i q u i d He, When the apparatus was lowered into the bath at about 1.2 K so that the inner section projected only about one cm above the bath level, approximately 14 minutes were required for i t to f i l l . Although this leakage rate is low, i t is s t i l l much greater than that which can be accounted for by film flow through the 0.15 cm diameter opening at the top. Probably, by experimenting further with the composition of the mixture, a good seal could be obtained, b. Pyrex-Plexiglas and Pyrex-Teflon joints This mixture was also used at the top of the windtunnel section to help f i l l the gap between the Pyrex tube and the Plexiglas parts of the inserted pipe sections. Because of the different contraction rates as the tempera= ture was lowered, the leakage through the gap would, cause a significant systematic error i n the calculated superfluid flow rates; for the smallest pipe this could be as high as 26 percent. (See Table -.3_;3. 1) . The apparatus for these tests is shown i n Fig. AlO.l.e. Here the glycerine-white lead mixture was found to be completely ineffective. This i s apparently because the coefficients of expansion of the Plexiglas and A3 1) Teflon are much greater than that of Pyrex , and the adhesive-elastic properties of the mixture are not suitable to compensate for tbis, c. Procedures for using the glycerine-white lead mixture The usual techniques employed in precooling the cryostat had to be modified because water absorbed in the glycerine (from the atmosphere) is expelled at reduced pressures. This effect gives the liquid an appearance of boiling even at pressures of a few cm Hg and would cause the ground glass joints to separate. The procedure followed, therefore, was f i r s t l y to close a l l the valves to the cryostat and pour a small amount of liquid nitrogen into the outer dewar. This condensed the water vapour in the air -inside the cryostat at the bottom of the He dewar where i t would not interfere with observations through the s l i t s . After an hour the outer dewar was f i l l e d with liquid nitrogen. When a second hour had passed, the dewar was evacuated, flushed and f i l l e d with He gas. The normal procedures for precooling and transfering liquid He were followed thereafter. a. Pyrex-Pyrex Ground b. Pyrex-Pyrex Undated C, Fyrex-Plcxiglas and ulass joint J o i n t Pyrex-Teflon Hated Joint iS. AL0.1 Apparatus to Test Superleak Sealing Properties of Glycerine-White Lead Mixture Between runs, the glycerine tended to drain out of the ground glass joints and so the mixture had to be replaced every few weeks. The mixture around the top of the wihdtuanel section (between the Pyrex wall and the Plexiglas parts of the pipes) was renewed im-mediately prior to each run. APPENDIX 11 BUI_D°UP AND DECAY OF VORTEX LINES -2.1.2) Following Vinen's analysis " we set £ = V L 0 . ( A U . O Considering that the second and last terms i n Equation (2,4.11) are negligible, i _ _ = _________ ^ =C,Vi ( i - ^ ) This may be used to calculate the length of time for oLf to attain any value from zero to unity. Selecting c~p = 0.95, X d i (A 11.3) The integral on the right i s simplified by letting 3L^  = 3f, , and using integral tables: 0.97*7 , If", <'-*/> = S.24- ( A M . 4 ) Therefore T = ( £ S*" ( A II.s) For the ease of decay (after the flow has been turned off ) , the length of time required for 95 percent of the line length to dissipate is (See Equation (2.4.7).)' - X „ - k L „ ( A l l - 6 " > or (from Equation (2.4.8)) T OS" l l v. (AM.-7) REFERENCES 2.1.1 L. D. Landau, J . Phys. Moscow JL1, 91 (1947) 2.1.2 J. Wilks, "The properties of liquid and solid helium" (Clarendon Press, Oxford, 1967), Chap. 12 and 13 2.1.3 C- Di Castro, Phys. Letters 24A, 191 (1967/ • 2.1.4 T. M. Sanders, Jr. and G. Weinreich, Phys, Letters 27A, 172 (1968) 2.1.5 G. V. Chester, R. Metz and L, Reatto, Phys. Rev. 175,, 275 (1968) 2.1.6 G Careri, S. Cunsolo and Mr. Vicentini-Missoni, Phys. Rev. 136, A311 (1964) 2.1.7 L. Bruschi, P. Mazzoldi and M. Santini, Phys„ Rev. 167* 203 (1968) 2.1.8 L. Bruschi, G. Mazzi and M. Santini, Phys. Rev. Letters \2J5, 330 (1979) 2.1.9 G. W. Rayfield and F. Rief, Phys. Rev. 136, A1194 (1964) 2.1.10 S. Cunsolo and B. Maraviglia, Phys. Rev. Ij37, 292 (1969) 2.1.11 R. J . Donnelly and P. H. Roberts., Phys. Rev. Letters 23, 1491 (1969) 2.2.1 W. E. Keller, "Helium-3 and Helium-4" (Plenum Press, New York, 1969), Chap, 8 2.3.1 P. Nozi_res, i n "Quantum fluids" (D. F. Brewer, Ed., North Holland Publishing Company, Amsterdam, 1966), page 1 2.3.2 R. Berker, i n "Encyclopedia of Physics, Vol. VIII/2: Fluid dynamics I I " (S. Flugge, Ed., Springer-Verlag, Berlin, 1963), page 1 2.3.3 L. Prandtl and 0. G. Tietjens, "Applied hydro- and aeromechanics" (Dover Publications, New York, 1957), Chap. 3 2.3.4 H. L. Langhaar, J. Appl, Mech. _9, A55 (1942); Reprinted i n Trans. ASME _+, A55 (1942) 2.3.5 D. F, Boucher and G, E. Alves, i n "Chemical Engineers' Handbook" (R. H. Perry, C. H. Chilton and S. D. Kirkpatrick, Ed., McGraw=Hill Book Company, New York, 1963), 4th ed., Section 5 B. K. Jones-, Phys.. Rev. :L77S 292 (1969) •P-. -P. Craig, Phys. Letters -21 , 385 (1966) Reference 2 . 1 . 2 : page 328 and Appendix Tables Al and A5 Wo F, Vinen, Proc. Roy. Soc. (London) A240, 114, 128 (1957), A242, 493 (1957),:A243, 400 (1958) R, de Bruyn Ouboter, K. W. Taeonis and W. M. van Alphen, Prog. Low Temp, Phys. 5, 44 (1967) F. A, Staas, K. W. Taeonis and W. M. van Alphen, Physica 27, 893 (1961) ' C. E. Chase, Phys. Rev. 127, 361 (1962), 131, 1898 (1963) and in "Superfluid helium" (J. F. Allen, Ed., Academic Press, London, 1966), page 215 J. T. Tough, Phys. Rev. 144, 186 (1966) M. A. Woolf and G. W. Rayfield, Phys. Rev. LettersJL5, 235 (1965) G. Careri, F. Scaramuzzi and J. 0. Thomson, Nuovo Cimento 13, 186 (1959) and 13, 957 (1960) G. E, Spangler and F. L. Hereford, Phys. Rev. Letters 20, 1229 (1968) K, R. Atkins, Phys. Rev. 116, .1339 (1959) K. W. Schwartz and R. W. Stark, Phys. Rev. Letters .22, 1278 (1969) C. Zipfel and T. M. Sanders, i n "Proceedings of the 11th Inter-national Conference on Low Temperature Physics" (J. F. Allen, D. M. Finlayson and D. M. McCall, Ed.,.St. Andrews Scotland, 1968), Paper A7.10 B. E. Springett, Reference 2.5.6, Paper A7.9 W. P. Pratt, Jr. and W. Zimmermann, Jr., Phys. Rev. 177, 412 (1969) T. Soda, Progr. Theoret. Phys, (Kyoto) 36, 435 (1966) 2.5.10 F. Reif and L. Meyer, Phys. Rev. .119, 1164 (1960) 2.5.11 G. Baym, R. G, Barrera and G. J. Pethick, Phys. Rev. Letters 22, 20 (1969) 2.5.12 D. J . Tanner, Phys. Rev. 152, 121 (1966) 2.5.13 M„ Vieentini=Missoni and S. Cunsolo, Phys. Rev. 144, 196 (1966) 3.1.1 D. Y. Chung and P. R. Critehlow, Phys. Rev. Letters 14, 892 (1965) 3.2.1 T. R. Koehler and J . R. Pellam s Phys. Rev. 125, 791 (1962) ,3.2.2 H. Forstat and C. A. Reynolds, Phys. Rev. IjOl, 513 (1956) 3.5.1 F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement and J. K, Logan, J. Res. Natl. Bur. Std. (U. S.) 64A, 1 (1960) 3.5.2 From a compilation by the Department of Geophysics, University of British Columbia, February 1964 3.5.3. E. J. Walker, Rev. Sci. Instr. 30, 834 (1959) 3.5.4 B. D. Bedford and R. G. Hoft, "Principles of Inverter Circuits" |John Wiley and Sons,.Inc., New York, 1964) 4.5.1 V. P. Peshkov and V. B. Stryukov, Zh. Eksperim. i Teor. Fiz. 41, 1443 (1961) (English transl.s Soviet Phys,—JETP 14, 1031 (1962)) 4.5.2 M. V. Ricci and M. Vicentini-Missoni, Phys. Rev. 158, 153 (1967) 4.5.3 D, J. Griffiths, Phil. Mag. .17, U09 (1968) 4.7.1 C. E. Chase, E. Maxwell and W. E. Millet, Physica 27, 1129 (1961) A3.1 R. J. Gorruccini and J. J. Gniewek, "Thermal expansion of technical solids at low temperatures = A compilation from the literature" (U. S. Department of Commerce, National Bureau of Standards, Monograph 29, 1961) A4.1 R. D. Evans, "The Atomic Nucleus" (McGraw-Hill Book Company, New York, 1955), Chapter 15 A4.2 R. C. Weast, Ed., "Handbook of Chemistry and Physics" (Chemical Rubber Company, Cleveland, Ohio, 1969), 50th ed, A4.3 D. G. Crowter, U. K. Atomic Energy Authority, Research Group Report RCC - R85 (1960) A4.4 F. L. Hereford and F. E. Moss, Phys, Rev. 141, 204 (1966) A10.1 J. Lees, Department of Physics, University of British Columbia, private communication 

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