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A new variational principle in fluid mechanics Purcell, Anthony 1981

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A NEW VARIATIONAL PRINCIPLE IN FLUID MECHANICS by ANTHONY PURCELL B.Sc, Dalhousie University, 1975 M.Sc, Dalhousie University, 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA MAY, 1981 (T) Anthony P u r c e l l , 1981 i n In present ing 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 of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I fur ther agree that permission for extensive copying of t h i s t h e s i s for s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representa t ives . It i s understood that copying or p u b l i c a t i o n of t h i s thes is for f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion. Department of The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date jLjg ^ j (98/ - i i -ABSTRACT A new v a r i a t i o n a l p r i n c i p l e i n f l u i d mechanics i s presented, based on a generalized version of the conservation of p a r t i c l e l a b e l constraint. The v a r i a t i o n a l p r i n c i p l e represents an extension of the work of Clebsch (1859) and C.C. L i n (1959) and for the one-component case i t describes a perfect f l u i d with a f i n i t e density of vortices ; for the two-component f l u i d i t yields the Khalatnikov equations for rapidly rotating superfluid ^He. In the l a t t e r case two p a r t i c l e l a b e l constraints are needed, which express the p o s s i b i l i t y of l a b e l l i n g both an element of normal f l u i d and a superfluid vortex, averaged over many vortices. In addition a negative result for a v a r i a t i o n a l formulation of viscous f l u i d s based on a generalized p a r t i c l e l a b e l constraint i s given. - i i i -CONTENTS Page Abstract i i L i s t of Figures v Acknowledgements v i Introduction 1 Chapters 1. A Review of the V a r i a t i o n a l P r i n c i p l e f o r a Perfect F l u i d i n Adiabatic Flow 1-1 An H i s t o r i c a l Outline 5 1-2 The Equations of Motion 7 1-3 The Lagrangian 9 1-4 Symmetry Transformations and Conservation Laws H 1-5 The I n f i n i t e s i m a l "Gauge" Transformations . . . . 12 1-6 The G a l i l e a n Transformations 14 1- 7 An A l t e r n a t i v e Lagrangian 16 2. Consequences of Relaxing the Conservation of P a r t i c l e Label Constraint 2- 1 The "Gauge" Invariance Problem 18 2-2 A Theory of Hydrodynamics with d S / d t 4 0, ds/dt 0 and 3p/3t + V-(pV) = 0 . . . 20 2-3 Interpretation of the Equations of Motion . . . . 25 2- 4 A Theory of Hydrodynamics with dz/dt f 0, ds/dt = 0 and 3p/3t + V«(pV) = 0 . . . 27 3. The V a r i a t i o n a l P r i n c i p l e f o r the Landau Two-Fluid Equations 3- 1 Introduction 31 3-2 The Landau Two-Fluid Equations 33 - i v -Page 3-3 Z i l s e l ' s V a r i a t i o n a l P r i n c i p l e 35 3- 4 Symmetries and Conservation Laws 39 4. A V a r i a t i o n a l P r i n c i p l e f o r Superfluid Helium with V o r t i c i t y 4- 1 Introduction 43 4-2 The Hydrodynamic Equations 44 4-3 The Lagrangian ^6 4-4 Interpretation of z ( x , t ) , V^ and the R e s t r i c t i o n B' = 2 5 2 4- 5 Symmetries and Conservation Laws 54 5. Lin's Constraint Generalized to Include Higher Order Derivatives 5- 1 An Extension of the Results of Chapter 2 . . . . 58 5-2 A Negative Result f o r Viscous F l u i d s 60 5-3 Conclusion 6 3 Bibliography 65 Appendices A. Proof of Clebsch's Lemma 67 B. Interp r e t a t i o n of the Equations of Motion 68 C. Turbulent Solutions of the Equations of Motion . . . ' J D. Measurement of B' from Second Sound in Rotating 4He 79 -v-LIST OF FIGURES Page Figure 1 Rotating Superfluid ^He 68 Figure 2 B' versus T 79 - v i -ACKNOWLEDGEMENTS I wish to thank my supervisor, Prof. F.A. Kaempffer for h i s assistance and kindness during my stay at U.B.C. F i n a n c i a l assistance i n the form of a K i l l a m Predoctoral Fellowship and a Postgraduate Fellowship from the Natural Sciences and Engineering Research Council of Canada i s g r a t e f u l l y acknowledged. -1-INTRODUCTION The modern form of v a r i a t i o n a l calculus i s attributed to Euler (1701 - 1783) and Lagrange (1736-1783) although the very f i r s t results date back to Hero of Alexandria c i r c a 140 A.D. One of the problems of v a r i a t i o n a l calculus i s to fi n d a functional L [ ^ ( x , t ) , x , t ] , c a l l e d the Lagrangian density, such that the Euler-Lagrange equations SL/Sip = 0 are equivalent to the equations of motion. I t i s straightforward to prove that the vanishing of the v a r i a t i o n a l derivative of L i s equivalent to the requirement that the action A = JJ L dxdt be an extremum for a l l variations of U)(x,t) with fixed boundary values. The basic mathematical framework was completed by Noether (1918) who showed e x p l i c i t l y how symmetry transformations of the Euler-Lagrange equations are connected with conservation laws. The chief d i f f i c u l t y i n v a r i a t i o n a l calculus i s that a Lagrangian may not exist when the equations of motion are expressed i n terms of a given set of variables; i n such a case the equations of motion must be rewritten i n a transformed set of variables. For the electromagnetic f i e l d the decomposition of E and B i n terms of the potentials A and q> i s w e l l known; however i n general there i s no way of knowing i n advance which transformation, i f any, w i l l bring the equations of motion into the form of the Euler-Lagrange equations. The advantages i n constructing a Lagrangian formalism are three-fold: (1) Noether's theorem provides a convenient connection between symmetries and conservation laws; (2) symmetry arguments applied to the Lagrangian give a systematic way of extending the equations of motion; and (3) the v a r i a t i o n a l equations may be easier -2-to solve d i r e c t l y than the usual form of the equations of motion. In f l u i d mechanics (3) i s e s p e c i a l l y true. For instance the v a r i a t i o n a l equation associated with the mass density p i s the most general form of the B e r n o u l l i equation, which can be used to discuss properties of f l u i d flow without f i n d i n g exact s o l u t i o n s . In summary, a v a r i a t i o n a l formulation of hydrodynamic systems i s extremely u s e f u l . More than one hundred years elapsed between the simultaneous development of v a r i a t i o n a l calculus and f l u i d mechanics and the discovery by Clebsch (1859) of a Lagrangian f o r i s e n t r o p i c , incompress-i b l e f l u i d s . Clebsch's representation of the v e l o c i t y f i e l d i n terms of the Monge p o t e n t i a l s , introduced by Monge (1787), succeeded i n overcoming the two d i f f i c u l t i e s i n formulating a v a r i a t i o n a l p r i n c i p l e for f l u i d s , namely the occurrence i n the equations of motion of n o n - l i n e a r i t i e s and f i r s t order d e r i v a t i v e s of the v e l o c i t y f i e l d . Because of the symmetry properties of the v a r i a t i o n a l d e r i v a t i v e , i t i s impossible to obtain odd order d e r i v a t i v e s of the v e l o c i t y f i e l d V(x,t) as a r e s u l t of v a r i a t i o n s with respect to V(x, t ) . Clebsch's incorporation of both non-linear and f i r s t order d e r i v a t i v e terms i n a v a r i a t i o n a l p r i n c i p l e i s absolutely unique i n c l a s s i c a l f i e l d theory. Bateman (1929) and Lamb (1932) extended the Lagrangian to include compressible, i s e n t r o p i c flows. The adiabatic case was solved by C.C. L i n (1959) who recognized that the conservation of p a r t i c l e l a b e l constraint (Lin's c o n s t r a i n t ) , an expression of the p o s s i b i l i t y of l a b e l l i n g an element of f l u i d , must be e x p l i c i t l y incorporated i n the v a r i a t i o n a l p r i n c i p l e . The p h y s i c a l consequence of in c l u d i n g Lin's constraint i s the appearance of non-zero v o r t i c i t y i n the absence of entropy gradients. - 3 -Because of the d i f f i c u l t i e s with f i r s t order derivatives and non-l i n e a r i t i e s mentioned previously, extensions of Clebsch's v a r i a t i o n a l p r i n c i p l e are extremely d i f f i c u l t to f i n d . The purpose of th i s thesis i s to present such an extension based on a generalized version of Lin's constraint. The physical interpretation of the resu l t i n g theory i s that of a f l u i d with a large number of vortices present, where a l l the hydrodynamic variables have been averaged over regions containing many vortices. In i t s two-fluid version the v a r i a t i o n a l p r i n c i p l e yields the Khalatnikov equations for rapidly rotating superfluid ^He. After reviewing the adiabatic Lagrangian i n Chapter 1 , the consequences of relaxing Lin's constraint for a c l a s s i c a l one-component f l u i d without changing the conservation of mass equation are examined i n Chapter 2 . I t i s found that such theories represent a macroscopic (compared to the mean vortex separation) description of a f l u i d with a large number of vortices present. As necessary background material Chapter 3 reviews Herivel's v a r i a t i o n a l p r i n c i p l e f o r the Landau two-fluid equations. Chapter 4 presents a new v a r i a t i o n a l p r i n c i p l e for the Khalatnikov equations of rapidly rotating superfluid ^He. I t i s found necessary to use two constraint equations, the usual Lin's constraint associated with the normal ve l o c i t y f i e l d and the other constraint expressing the p o s s i b i l i t y of l a b e l l i n g a superfluid vortex, averaged over many vortices. Chapter 5 concludes with an extension of the v a r i a t i o n a l p r i n c i p l e to higher order derivatives and with a negative res u l t for viscous f l u i d s , namely that a generalized Lin's constraint by i t s e l f i s not s u f f i c i e n t to generate the additional viscous terms which occur i n the Navier-Stokes momentum equations. -4-To summarize, the new v a r i a t i o n a l p r i n c i p l e s presented i n t h i s thesis are given by Eqs. (2-6), (2-40) and ;(5-l) which describe one-component f l u i d s with a density of v o r t i c e s , and by Eq. (4-7) which yields the Khalatnikov equations for rotating superfluid ^He. -5-CHAPTER 1 A REVIEW OF THE VARIATIONAL PRINCIPLE FOR A PERFECT FLUID IN ADIABATIC FLOW 1-1 An Historical Outline The equations of motion of a perfect f l u i d were f i r s t developed 1 2 by Euler (1751) and Lagrange (1781) . More than one hundred years 3 later Clebsch (1859), using a representation for the velocity f i e l d k introduced by Monge (1787), succeeded in finding a Lagrangian for the incompressible, isentropic (constant entropy) flow of a perfect f l u i d . Clebsch proved that the isentropic, incompressible equations ijf = - I VP , U = 0 (1-1) dt p where V i s the f l u i d velocity, d/dt = 3/3t + V-V, P(x,t) is the pressure and p is the mass density, can be solved in terms of three scalar functions tJ)(x,t), m(x,t) and i|)(x,t) (the Monge potentials) which satisfy * - f c + ^ , £ + | i + . { i + * V * - 0 . £ - f -0 (1-2) Furthermore Clebsch showed that the equations v"«V = 0 and dm/dt = di|;/dt = 0 are the variational equations of the Lagrangian density L = | | + m|^ + % (V^-rmV^)2 (1-3) which arise from variations in if) and m respectively. This result 5 6 was extended by Bateman (1929) and Lamb (1932) to compressible, -6-7 8 i s e n t r o p i c flows. Taub (1949) and H e r i v e l (1955) attempted with only p a r t i a l success to generalize the v a r i a t i o n a l p r i n c i p l e to the a d i a -batic case ds/dt = 0, where s i s the entropy density. 9 I t was C.C. L i n (1959) who pointed out that Herivel's v a r i a t i o n a l p r i n c i p l e yielded only a subset of the solutions of the Euler equations, those for which VxV = 0 when s = constant, and who supplied the necess-ary a d d i t i o n a l c o n s t r a i n t . L i n observed that even i f the Lagrangian coordinates z:(x,t) do not appear i n the Euler equations, only those v e l o c i t y f i e l d s for which the Lagrangian coordinates could be found should be used i n the v a r i a t i o n a l p r i n c i p l e . L i n incorporated t h i s constraint into the v a r i a t i o n a l p r i n c i p l e i n the form of the conserva-t i o n of i d e n t i t y of p a r t i c l e s equation dz(x,t)/dt = 0, where z(x,t) i s the i n i t i a l p o s i t i o n of a f l u i d p a r t i c l e located at x at time t . 10 By using Weber's transformation i t follows that Herivel's v a r i a t i o n -a l p r i n c i p l e supplemented with Lin's constraint for the i d e n t i t y of p a r t i c l e s includes a l l solutions of the Euler equations. Following Lin's c r u c i a l step a number of papers appeared extend-10 ing the v a r i a t i o n a l p r i n c i p l e . These include S e r r i n (1959) and 11 Eckart (1960) on adiabatic flow, a s p e c i a l r e l a t i v i s t i c formulation 12 13 of adiabatic flow by Penfield (1966), Seliger and Whitham (1967) on v a r i a t i o n a l p r i n c i p l e s i n continuum mechanics, general r e l a t i v i s t i c Ik 15 treatments of adiabatic flow by Schutz (1970) and Schutz and Sorkin 16 (1977) and a v a r i e t y of rigorous mathematical r e s u l t s by Rund (1976) . Other generalizations include v a r i a t i o n a l p r i n c i p l e s f o r magneto-17 18 19 hydrodynamics by Calkin (1961), Katz (1961) and P e n f i e l d and Haus (1966) and f o r a number of two-fluid systems (see Chapter 3 for d e t a i l s and references). Worth noting are several negative r e s u l t s -7-for variational principles yielding the Navier-Stokes equations : 20 21 22 23 Millikan (1929), Bateman (1931), Gerber (1950) and Bailyn (1980) 1-2 The Equations of Motion The equations of motion of a perfect f l u i d in adiabatic flow are 10 + well known and are given in terms of the Eulerian variables (x,t) (the are just the spatial coordinates, t i s the time) by |£ + V-(pV) - 0 (1-4) 3F " ° d - » which represent the conservation of mass, entropy and momentum of the fl u i d respectively. The variables V, p and s and the material deriva-tive d/dt have been defined previously while the pressure P (p,s) and the temperature T (p,s) are defined in terms of the internal energy density e (p,s) by the Gibbs relation de = Tds + ( P / p 2 ) d p (1-7) In the Eulerian variables (x,t) the velocity V(x,t) i s simply regarded as a vector f i e l d which obeys Eqs. (l-4)-(l-6). In the Lagrangian variables (z,t) the f l u i d flow i s described in terms of particle paths x = x(z,t). If z is fixed while t varies then x(z,t) maps out the path of a f l u i d particle i n i t i a l l y at z. For fixed t, x(z,t) gives a mapping of the region i n i t i a l l y occupied by the fl u i d -8-into i t s p o s i t i o n at time t . Assuming that i n i t i a l l y d i s t i n c t points remain d i s t i n c t implies that x(z,t) possesses an inverse z = z(x,t) which i s the i n i t i a l p o s i -- » - - » • t i o n of a f l u i d p a r t i c l e with p o s i t i o n x at time t (x and z denote the values of the functions x(z,t) and z(x,t) r e s p e c t i v e l y ) . This implies that x = x ( z ( x , t ) , t ) and z = z ( x ( z , t ) , t ) and hence use of the chain r u l e y i e l d s the i d e n t i t i e s Sx 1 8z^ 9x^ 9 z k r i k Q N : r — : r = 6 v i - o ; 9z J 9x 9 Z 1 9x J where x(z,t) and jj(x,t) are assumed to possess continuous d e r i v a t i v e s up to t h i r d order i n a l l d e r i v a t i v e s ; i , j , ... = 1,2,3; repeated i n -dices are summed and unnecessary indices are omitted. In the Lagrangian p i c t u r e of f l u i d flow the v e l o c i t y of a f l u i d p a r t i c l e V(z,t) i s defined as i = 9x X(z,t) _ dx^ n _ _ . 9t ~ dt K ' M u l t i p l y i n g Eq. (1-9) by 9z J/9x X and summing i y i e l d s the equivalent form d z ^ ( x ' t } = 0 (1-10) dt which j u s t states that the i d e n t i t y of the f l u i d p a r t i c l e s i s conser-ved during the motion. Note that Eq. (1-9) and use of the chain r u l e imply df(x,t) = 9 f ( x ( z , t ) , t ) (1-11) dt 9t - 9 -and hence d/dt and 3/3z commute. Eqs. (1-4)-(1-6) and the Lin's con-st r a i n t Eq. (1-10) w i l l henceforth be referred to as the hydrodynamic equations which w i l l be shown to be equivalent to a v a r i a t i o n a l p r i n -c i p l e i n the following section. 1-3 The Lagrangian 10 The following discussion i s due to Serrin i n the Handbuch a r t i -c l e . The Lagrangian density for the hydrodynamic equations i s given by L = % pV2 - pe(p,s) - a{|£ + v"-(pV)} + PB ^ + p Y j (1-12) where the dependent variables are p.s.V^z and the Monge potentials a,6,Y while the independent variables are ( x , t ) . The v a r i a t i o n a l equations are obtained by setting the v a r i a t i o n a l derivatives 6X/6i^a = 3L/3*" - V-(9L/8(v"4;a))- 3(3L/3(3*°7 3t))/3t = 0 where the i | / a are the dependent variables; for a review of v a r i a t i o n a l p r i n c i p l e s i n mathematical physics see H i l l (1951). Variations of the Monge poten-t i a l s <*,B,Y j u s t give Eqs. (1-4) , (1-5) and (1-10) respectively while the other variations give 6V: V = - Va - RVs - Y J ^ Z ^ (1-13) 6p: || + % V 2 - e - j = 0 (1-14) 6s: |f = - T (1-15) 6z: |j = 0 (1-16) -10-Eqs. (1-4),(1-5),(1-10),(1-13)-(1-16) w i l l be col lec t ively referred to as the variational equations of L . When use is made of Clebsch's lemma which follows as an identity from the definit ions d/dt = 9/9t + V«V and V = - Vex - £Vs - y^Vz"' (see Appendix A for a proof) then substi-tution of Eqs. (1-14)-(1-16),(1-5),(1-10) into Eq. (1-17) gives = _ v"(e + -) + TVs = - - V P (1-18) dt p p which is just Eq. (1-6). Hence a l l solutions of the variational equations are also solutions of the hydrodynamic equations. The converse statement can be proven using Weber's transforma-t i o n . Eq. (1-9) or equivalently Eq. (1-10) implies the identity dT<v jJ4) • # 4 + v 3 ^ 1 7 - # 4 + i i ^ v 2 ) ( 1 " 1 9 ) dt l dt l dt l dt l . l 9z 9z 9z 9z 9z Substituting Eq. (1-6) into Eq. (1-19) gives A ( VJ *d) = - L - <* V2 - e - % + T (1-20) d t 9Z 1 9Z 1 P 9Z 1 By defining a = / [e + — - % V 2 ]dt and 6 = / Tdt (the integration is O P 0 carrried out by constant z) then i t immediately follows that Eq.(l-20) can be written as 4z (^4 tV j + V j a + gV js]) = 0 (1-21) d t 9Z 1 -11-Since a(z,o) = 0, 3(z,o) = 0 and x(z,o) = z Eq. (1-21) can be inter-grated as V = - Va - &Vs - y^Vz^ (1-22) where Y J = - V J(z,o). Eq. (1-22) is just Eq. (1-13) and i t i s easy to verify that a,3 and y as defined above satisfy Eqs. (1-14)-(1-16) respectively. Hence the variational equations of L are equivalent to the hydrodynamic equations. 1-4 Symmetry Transformations and Conservation Laws Before considering the transformation properties of the specific Lagrangian given by Eq. (1-12) a more general treatment is needed. 2k The following discussion can be found in greater detail in H i l l If the variational equations of a Lagrangian ~L[ty] 5 L(ty,Vty, 3ii)/3t,x, t) maintain the same functional form under the infinitesimal transforma-tions L|>] -»• L ' [ i | ) ' ] = L[ty] + SL[ty], if/* •+ ty,a = tya + &tya, x -' x = x + fix, t t' = t + fit and L' [ty' ]d 3x 'dt' = L[^]d 3xdt (the latter con-dition just maintains the numerical invariance of the action) then they are said to be form invariant. This implies S L ' ^ ' J = 6 L . V ] ( 1 2 3 ) ct ct Hence xff is a solution of the equations of motion then so i s ty' and the transformation is said to be a symmetry transformation. A necessary and sufficient condition that Eq. (1-23) hold for arbitrary Ct ct ty (x,t) or equivalently for ty and their derivatives considered as Ct independent variables (not just for those ty which satisfy the -12-equations of motion) i s that the old and new Lagrangians be related by a t o t a l divergence L'[*'] = M*'] + V«6ft + 9 6fi /9t (1-24) o Eq. (1-24) just says that L i s invariant under the transformation, i f 60, = <5fi =0 then L i s said to be form invariant. o I f Eq. (1-24) holds for ^ a and their derivatives considered as independent variables (hence the equations of motion may not be used in v e r i f y i n g Eq. (1-24)) then Noether's theorem gives a conservation law i n the form |^ + V-s = 0 d-25) ot •+ where the i n f i n i t e s i m a l forms of o and s are given by 6° = (L " ^ " +Jr«* a + fino ( 1 " 2 6 ) 3 ( " 9 T } ^ T F 0 9 ( 3 t } 8t = - *£. 6 t + ( L f i - - 6 x\y> a) + — «*° + (1-27) 9V\J; 3Vi(/ 3 Vi^ Hence to test for a possible symmetry transformation which generates a conservation law v i a Noether's theorem either Eqs. (1-23) or (1-24) may be used, with tj; and their derivatives considered as independent variables. 1-5 The In f i n i t e s i m a l "Gauge" Transformations The discussion i n Sec. 1-3 shows that the essential step i n finding a v a r i a t i o n a l p r i n c i p l e for the hydrodynamic equations i s the representation of the v e l o c i t y f i e l d given by Eq. (1-13). However a - I n -d e f i n i t e value of V does not uniquely determine the values of the variables which appear on the R.H.S. of Eq. (1-13). In fact a ,3,Y and ~z may be subjected to "gauge" transformations which do not change the value of V and which keep L invariant and hence the v a r i a t i o n a l equations of L form invariant (transformations of s are not allowed since this destroys the form invariance of Eqs. (1-14) and (1-15)). The form invariance of Eq. (1-13) implies that the i n f i n i t e s i m a l gauge transformations a + a' = a + 6a, g + 3' = 3 + 63, y Y ' = Y + z + z' = z + 6z and V -»• V = V s a t i s f y V' = -Va' - 3*Vs» - y'tfz'i = v = -Va - 3Vs - y^vV (1-28) or equivalently V(6a + yhzh + 63Vs + 6Y JVZ J - 6z jVy j = 0 . (1-29) The form invariance of the other v a r i a t i o n a l equations implies that £ (6a) = £ (63) = £ (6y) = (6z) = 0 (1-30) which have the solutions 6a = 6a(s,y,z), 63 = 63(S,Y,Z), 6Y = 6 Y ( S , Y , Z ) , 5Z = 6z(s,y,z). (1-31) Substitution of these results into Eq. (1-29) yields (e ~ + 63)Vs + (e ^ — + 6y j)Vz : i + (e ^  - 6Z j)VY : ' = o (1-32) d s 8z 3 9y J where for convenience 6a + y^Sz 3 = eG(s,y,z) and e i s an i n f i n i t e s i m a l constant. -14-Since s, y and z are independent variables Eq. (1-32) implies that the i n f i n i t e s i m a l gauge transformations are given by 6a - E G - E Y J ^ , 6B = -e |£ , 6yi = -e , 6z j = e (1-33) 9Y3 9 8 9z J 9 YJ The conservation law associated with the gauge transformations i s (pG(s,Y,z)) + V-(pVG(s,Y,z)) = 0 (1-34) which i s e a s i l y v e r i f i e d from Eq. (1-25). Note that i f 6z = 0 then the gauge transformations take the form G = G(s,z) (1-35) and the conservation law becomes (pG(s.z)) + V-(pVG(s,z)) = 0 (1-36) 1-6 The Galilean Transformations In t h i s thesis a l l equations are invariant under the Galilean trans-formations: s p a t i a l t r a n s l a t i o n , time t r a n s l a t i o n , rotation of coordinates and Galilean boosts. The transformation properties of -»- ->• V,p,s and z are w e l l known but the transformation properties of the Monge potentials a,3 and y are not known a p r i o r i and must be deduced by requiring that the Lagrangian be invariant under the Galilean group (i ) Under the i n f i n i t e s i m a l s p a t i a l displacement x -*• x' = x + 6x where 6x i s an i n f i n i t e s i m a l constant, z -*• z' = z + 6x. The results of Sec. 1-5 show that the Monge potentials a, 3 and y remain fixed apart from a gauge transformation of the type G = G(s,z). From Eq. (1-25) i t i s easy to v e r i f y that the conservation law associated with t h i s -15-symmetry transfomation i s the conservation of momentum equation • ^ § 7 - + V j (pvV + P6 i j) = 0 (1-37) o t ( i i ) Under the time translation t •+ t ' = t + St where 6t i s an i n f i n -i t e s i m a l constant, z -»- z* = z - V(z,o)6t (since z was defined as the i n i t i a l position of a f l u i d p a r t i c l e , a s h i f t i n the time o r i g i n s h i f t s ^ as w e l l ) . The form invariance of Eq. (1-13) gives y -*- y' = y - y J(9V J(z,o)/9z)6t apart from a gauge tranformation of the type G = G(s,z). From Eq. (1-25) follows the conservation of energy equation -£r (pe + h pV 2) + V j([pe + h pV 2 + P]V j) = 0 (1-38) d t ( i i i ) Under the rotation of axes x - > x l = x + 6 0 x x , where 68 i s an i n f i n i t e s i m a l constant vector i n the d i r e c t i o n of the axis of rotation with a magnitude equal to the angle of rotation, V V = V + c6 x V and z - ^ z ' = z + 6 6 x z . The form invariance of Eq. (1-13) gives y y' = y + 5 Q x y } apart from a gauge transformation of the type G = G(s,z). From Eq. (1-25) the conservation of angular momentum equation i s 4~ (PV x x) + V j(pV jV x x) - x x VP = 0 (1-39) d t (iv) Under the Galilean boost x x = x + 6 V t , where 6V i s an o o ->-i n f i n i t e s i m a l constant vector, V ->• V = V + 6V . The form invariance of ' o Eq. (1-13) gives a -*• a' = a - X*6Vq apart from a gauge transformation G = G(s,z). From Eq. (1-25) the conservation of center of mass equation is 4- (pVAt - p x 1 ) + V j ( P v V t - p x V + P t 6 ± j ) = 0 (1-40) a t -16-/ 1-7 An Alternative Lagrangian The v a r i a t i o n a l p r i n c i p l e given by Eq. (1-12) requires that the Lagrangian picture of f l u i d flow be adjoined to the hydrodynamic equations i n the form dz/dt = 0. Seliger and Whitham 1 3 have shown that t h i s i s not necessary. Consider the Lagrangian L = h PV 2-pe(p,s) - a { | | - + v".( PV)}+p3 + PY ff" + pH(Y,z,t) (1-41) where a , $ , Y and z are to be interpreted as Monge potentials and H i s an arbitrary function of y, z and t. Variations of a and (3 give Eqs. (1-4) and (1-5). The other variations give 6V: V = - va - 3^s - yVz (1-42) 6p: ^ + isV 2-e - -+Y47-+H=0 (1-43) dt p dt 6 s : 4f = " T t 1 " 4 4 ) at The Clebsch lemma gives while substituting Eqs. (l - 4 3 ) - ( l - 4 6 ) , (1-5) into Eq. (1-47) gives £ = _y-(e + I _ H + y M) + TVs - |= Vz + Yv" f» « - I VP (1-48) dt p ' 9y 9z ' 9y p which i s just the conservation of momentum equation (1-6). Hence a l l solutions of the v a r i a t i o n a l equations of L are also solutions of -17-Eqs. ( l - 4 ) - ( l - 6 ) . The converse statement can be proven using Pfaff's theorem which -> ->-states that an arb i t r a r y 3-vector V + BVs can be written i n the form V + 3Vs = - Va - yVz . (1-49) Since 3 i s arb i t r a r y i t can be chosen such that d3/dt = -T. Clebsch's lemma and Eq. (1-6) then give which can be rewritten using Eq. (1-5) and d3/dt = -T as * £ + * " - « - f + £ ) + £ y 2 - i | ? Y . „ ( 1 - 5 1 , Self-consistency conditions imply that a, y and z must s a t i s f y Eqs. (1-43), (1-45) and (1-46) respectively. Hence the v a r i a t i o n a l p r i n c i p l e given by Eq. (1-41) i s completely equivalent to Eqs. (1-4)-(1-6) and no reference to the Lagrangian picture of f l u i d flow i s needed. Note that the addition of the term H(Y,z,t) means that the Lagrangian i s invariant under the smaller group of gauge transformations G = G(s). This completes the review of the v a r i a t i o n a l p r i n c i p l e for adiabatic flow. -18-CHAPTER 2 CONSEQUENCES OF RELAXING THE CONSERVATION OF PARTICLE LABEL 2-1 The "Gauge" Invariance Problem As shown i n Chapter 1 (see Eq. (1-12)) the Lagrangian for the perfect f l u i d i n adiabatic flow i s j u s t the k i n e t i c energy minus the in t e r n a l energy plus a sum of Lagrange m u l t i p l i e r s times the constraints 3p/3t + V* (pV)=0, ds/dt=0 and dz/dt=0. Variation of L with respect to V gave the representation for V, then use of Clebsch's lemma and the v a r i a t i o n a l equations for p,sfz,a,& and y yielded the conservation of momentum equation. In fact the precise form of the constraint equations for p,s and z given above i s c r u c i a l to the success of the v a r i a t i o n a l p r i n c i p l e . For instance consider the Navier-Stokes entropy production equation pds/dt = kT _ 1 V 2T + T ^ V ^ T ^ where T±J'= Xv'-vV^-l- p ( V ^ + V ^ V 1 ) , k,X and u=constant. If t h i s constraint i s incorporated i n the v a r i a -t i o n a l p r i n c i p l e simply by using a Lagrange m u l t i p l i e r B then the Lagrangian i s Variations of a , y and z are unchanged while variations of P,s ,B and V give L' = % pV2 - pe(p,s) - a{|£ + V-(pV)} + B{ ds dt - kT_ 1V 2T - T W V T 1 3 } + P Y d F (2-1) (2-3) (2-2) 66: p 4r = kT _ 1 V 2T + T _ 1 V V T 1 J (2-4) -19-( 2 A straightforward c a l c u l a t i o n y i e l d s an equation for V ( 2 which i s not the Navier-Stokes momentum equation unless the RHS of Eq. ( 2 - 6 ) vanishes. This constraint would have to be added to the Lagrangian with no guarantee of a solution to the resulting closure problem. In addition the Monge potential 3 which was introduced as a Lagrange m u l t i p l i e r and has no unique physical interpretation, cannot be eliminated from the R.H.S. of Eq. ( 2 - 6 ) by using the v a r i a t i o n a l equations of L'. The conclusion i s that the form of the constraint equations for p,s and z determines whether the Monge potentials a,3 and y (which have no unique physical interpretation) can be eliminated from the equation for dV/dt. The l a t t e r case w i l l be summarized by saying that the equation for dV/dt i s "gauge" invariant i . e . the Monge potentials a,3 and Y c a n be eliminated i n terms of V,p,s,z and th e i r derivatives. Sec. 2-2 w i l l explore an extension of the v a r i a t i o n a l p r i n c i p l e discovered by the author i n which the equation for dV/dt i s "gauge" invariant while the conservation of p a r t i c l e label and entropy equa-tions are modified and the conservation of mass equation remains unchanged. In Sec. 2-3 the equations of motion are interpreted as describing a f l u i d with a f i n i t e density of vortices where the -20-hydrodynamic v a r i a b l e s V,p,s and z have been averaged over a region containing many v o r t i c e s . A v a r i a t i o n a l p r i n c i p l e f o r which dV/dt i s "gauge" invariant and only the conservation of p a r t i c l e l a b e l equation i s changed i s given i n Sec. 2-4, i t i s found that because of the requirements of G a l i l e a n invariance such a theory must be non-linear i n the gradient of the Monge p o t e n t i a l y. 2-2 A Theory of Hydrodynamics with dz/dt ^ 0, ds/dt ^ 0 and 3p/3t + V'(pV) = 0. Consider the Lagrangian L = % pV 2 - pe(p,s) - pg(u) - a { | ^ + V-( PV) } + pB jj| + p Y j (2 -7) where the dependent v a r i a b l e s are V,p,s,z,a,8 and y> ^ E - VgxVs -Vy^xVz^ and g(w) i s an a r b i t r a r y function of ID = [to| . After the v a r i a t i o n of V* i s c a r r i e d out to = V*V and the - pg(w) term i n L may be interpreted as adding a v o r t i c i t y dependent cont r i b u t i o n to the i n t e r n a l energy e(p,s) of the f l u i d . This i s s i m i l a r to an assumption 25 made by Khalatnikov and Bekarevitch (KB) i n deriving the equations of su p e r f l u i d helium with a f i n i t e density of sup e r f l u i d v o r t i c e s where the i n t e r n a l energy of the f l u i d i s allowed to depend on the supe r f l u i d v o r t i c i t y . The following i d e n t i t i e s w i l l prove h e l p f u l i n f i n d i n g the v a r i a t i o n a l equations of L |r- (- p g ( U ) ) = Vs.vx(p ! £ § ) , ! - (~Pg) = " VB.vx( p (2-8) O p dOl (0 O S dO) 0) with analogous expressions for y and z. By using Eqs. (2-8) the -21-v a r i a t i o n a l equations are e a s i l y shown to be <5V: V = - Va - BVs - y J V z J (2-9) 6a: |£- + V-(pV) = 0 (2-10) 6 p : a t + B o f + y J f r + h v 2 " e ' \ ~ g ( u ) = ° ( 2 ~ 1 3 ) 63: p £ - - * . . * x < p | j £ > (2-12) 6 s : p | i . . ^ . V K ( p | j | ) _ p T (2-13) ^ p d s l . _ ^ x ( p | j 5 ) (2-14)-6 z i : P | l i = . V - V X ( p | j | ) (2-15) Since the representation f o r V remains unchanged Clebsch's lemma Eq. (1-17) i s unchanged. Substi t u t i o n of Eq. (2-11) into Eq. (1-17) y i e l d s P |V± + v i p = _ p V l g _ p ( M + T ) v i s + p ds_ v i 3 _ p d y i v i z j + pd£V YJ C 2 ' 1 6 ) dt Y By using Eqs. (2-12)-(2-15) , Eq. (2-16) can be rewritten as P i X l + v 1 ? = - pvS + V 3 - v x ( P |&S)yis _ ^ s . ^ ( p ! * H ) v i B dt 9u) w dui OJ + V'Vx(p |^5 ) V V - Vz j.Vx(p | & H ) 7 V (2-17) ' 9co OJ 9a) 0) The vector i d e n t i t y (1«C)B - (A«B)£ = Ax(BxC") and the representation -22-for V mean that the Monge potentials 8 and y c a n b e eliminated from Eq. (2-17) i n the form p ^ | + VP - - pVg + V-*(p |& S)x Z (2-18) a t o(ji) a) or equivalently dvJ k + v j(P6 i : i + [a) 26 i : i - wV]) = 0 (2-19) dt a) da) Hence the effect of the - pg(ai) term i n the Lagrangian i s to add a symmetric contribution to the stress tensor = —- T^- [u)26^-o)*'ur'1. ID 3D As i n Chapter 1, Eqs. (2-10) ,(2-12), (2-l4) and (2-18) w i l l be c o l l e c t i v e l y referred to as the hydrodynamic equations. Note that the representation for V implies that oi = VxV and hence the hydro-dynamic equations are "gauge" invariant. By using a suitable generalization of Weber's transformation i t can be shown that a l l solutions of the hydrodynamic equations are also solutions of the v a r i a t i o n a l equations (2-9)-(2-15) . If z(x,t) i s assumed to be i n v e r t i b l e and d i f f e r e n t i a b l e then Eqs. (1-8) are unchanged. M u l t i p l i c a t i o n of Eq. (2-14) by 3x^/3z"' gives the equivalent form v i = | x i z i t i _ i [ ^ ( p ^ | ] ^ d t P dO) (0 Hence i t i s V' = V + — Vx(p —) = 3x(z>t) which i s the tangent to p 9u> u> 9t the p a r t i c l e paths x = x ( z , t ) , not V as i n Eq. (1-9). I f d'/dt = 9/9t + V'«V then use of the chain r u l e implies that the analogue of Eq. (1-11) i s -23-^ f ( x , t ) = |^ f ( x ( z , t ) , t ) (2-21) ->-and hence d'/dt and 3/3z commute, not d/dt and 3/3z as in the Lagran-gian picture of f l u i d flow. From Eq. (2-21) follows the identity ± 1 fvJ Sxf. A = d'V3 3xJ j 3x^ _ = dyJ_ 3x^ d t a 1 d t a 1 . 1 d t a *• 3z 3z 3z 3z + [ v V v , £+(V , £ - vS7V] (2-22) 3 Z 1 Substitution of Eq. (2-18) into Eq. (2-2 2) gives ( V J = .L_ (_e- - g + % V 2 + V-(V'-V)) + TV js (2-23) d t Sz 1 3Z 1 P 3Z 1 By defining B = - /. T d't and a = - / [-e- - - g + % V2+V«(V'-V)]d't 0 O P (the integration i s carried out at constant z) and since d's/dt = 0 from Eqs. (2-12), (2-2l) then Eq. (2-23) can be rewritten as ^- 0^4 [V j + V ja + BV js]) = 0 (2-24) d t Sz 1 The l a t t e r equation can be integrated by defining y^ = -V J(z,o) as V j = - V ja - BV js - Y £ V j z £ (2-25) which i s just Eq. (2-9). From the d e f i n i t i o n of a,B and y given above -24-d ta ~ e - - - g + h V 2 + V(V'-V) = 0 (2-26) d'B dt = - T (2-27) = 0 (2-28) By using the d e f i n i t i o n V' = V + — Vx(p |& —) and Eq. (2-25) i t p 9(0 (0 immediately follows that Eqs. (2-26)-(2-28) are i d e n t i c a l with Eqs. (2-11), (2-13) and (2-14) r e s p e c t i v e l y . Hence the hydrodynamic equations are completely equivalent to the v a r i a t i o n a l equations. Note that since the representation f o r V i s unchanged, the "gauge" -y -y transformations of a,B,Y and z have the same form as given by Eq.(l-35). -y -y -y -y i-y n Furthermore since (o = Vx[- Va - BVs - y Vz ] i s c l e a r l y invariant under these transformations the Lagrangian given by Eq. (2-7) i s also i n v a r i a n t . As the reader may e a s i l y v e r i f y from Eq. (1-25) the conservation law which a r i s e s from the gauge invariance of L i s -|z- (pG(s,y,z)) + V.(pV'G(x,Y,z)) = 0 (2-30) which follows from Eqs. (2-10) ,(2-12), (2-14) and (2-15). Since (o i s also a Ga l i l e a n invariant the Lagrangian given by Eq. (2-7) i s invariant under the G a l i l e a n group. The conservation laws which a r i s e from G a l i l e a n invariance can be derived from Eq. (1-25) as |r- (pvS+V^pvV+Pe^+^- l^- [<o 2 <S i j -u )V]) = 0 (2-31) dt (0 0(0 -25-4 - (% pV 2+pe+pg)+V-([% pV2+pe+pg+P]V+ £- |& ux[TVs+V'xu]) = 0 (2-32) d t 0) dO) f f E ^ V v ^ + V ^ e ^ V v V + e ^ V C p f i ^ ^ l & t ^ f i ^ V ] } ) ^ (2-33) d t (i) od) | - ( p v V p x ^ + V ^ p v V t - p x V + P t S 1 ^ -2- | & [ w ^ - w V l t ) = 0 (2-34) d t 0) oli) which represent the conservation of momentum, energy, angular momentum ii k and center-of-mass r e s p e c t i v e l y (e i s the permutation symbol). 2-3 I n t e r p r e t a t i o n of the Equations of Motion The Lagrangian given by Eq. (2-7) d i f f e r s from that of Eq. (1-12) by the ad d i t i o n of a v o r t i c i t y dependent cont r i b u t i o n to the i n t e r n a l energy. This i s analogous to the theory of (KB) which describes the motion of s u p e r f l u i d helium with a f i n i t e density of su p e r f l u i d v o r t i c e s i n which the hydrodynamic v a r i a b l e s are averaged over a macroscopic region containing a large number of v o r t i c e s . In f a c t , i n Chapter 4 a modification of the Lagrangian given by Eq. (2-7) w i l l be used to derive the equations of (KB). This suggests that a s i m i l a r i n t e r -p r e t a t i o n can be made for the hydrodynamic equations given i n Sec. 2-2, that they describe a f l u i d with a f i n i t e density of v o r t i c e s -v -»-where the hydrodynamic v a r i a b l e s V ,p,s and z have been averaged over many v o r t i c e s . Once the form of the t o t a l energy of the f l u i d i s s p e c i f i e d , then the equations of motion follow from the standard technique i n hydrodynamics. Conservation laws for mass,momenturn, entropy and energy are assumed which give s i x equations i n f i v e variables.The r e s u l t i n g self-consistency conditions f i x the hydrodynamic equations;see Appendix B f o r d e t a i l s . -26-Since the turbulent flow of a f l u i d i s characterized by a d i s t r i -bution of v o r t i c i t y i t i s worthwhile to examine the equations of mocion for turbulent solutions. By adopting Kolmogoroff's assump-2 6 t i o n that only the energy d i s s i p a t i o n e 0 (a constant with dimen-sions £ 2 t - 3 ) be used i n the i n e r t i a l subrange of turbulent flow then by dimensional analysis g(u>) = K'e 0ai K' = dimensionless constant. Provided a closure r e l a t i o n i s assumed for the two-point v o r t i c i t y c o r r e l a t i o n function of the form <ui(x+r) «u(x) (u>(x))~3> = K"<w(x+r) •w(x)>~^ (2-35) (K" = dimensionless constant) then the hydrodynamic equations of Sec. 2-2 provide a closed equation for the two-point v e l o c i t y c o r r e l a -t i o n function, which can be solved as -v -s- 2/3 2 /3 2 /3 <(V(x+r) - V(x)) 2> = 9/2(10K'K"/9) 1 z ' r ' (2-36) o Eq. (2-36) agrees with Kolmogoroffs p r e d i c t i o n f o r the i n e r t i a l subrange, provided Kolmogoroff's constant K(K-.5) i s given by K = 9/8(10K'K"/9)2/'3. Hence i t i s possible to model the i n e r t i a l sub-range of turbulent flow with the hydrodynamic equations of Sec. 2-2 provided the form of g(w) i s given by dimensional a n a l y s i s as g = K'£ 0a> - 1 and the closure r e l a t i o n Eq. (2-35) i s assumed. Since t h i s subject i s p e r i p h e r a l to the main topic of t h i s thesis the. d e t a i l s of the foregoing discussion are relegated to Appendix C. The Beltrami d i f f u s i o n equation -27-d (»!) . i s i ^ + i ( ^x f l ) 1 (2-37) at p p p at which follows as an identity from the d e f i n i t i o n of d/dt, when com-bined with Eq. (2-18) i n the barotropic case P = P(p) yields an equation for the v o r t i c i t y c o i V J V,i ( 2._ 3 8 ) dt p p Eq. (2-38) can be integrated using Eq. (2-21) as o>l = «o 3(z,o) 8x1 ( 2 _ 3 9 ) P p ( z , o ) 8 z J i _^  _^  which states that — i s transported with v e l o c i t y V not V as i n the Lagrangian picture of the f l u i d flow. Note that since V«(pV) = v*(pV) then d'p/dt = - pV»V' which can be integrated as p/p(z,o) = J = det *r , just as i n the Lagrangian picture of f l u i d flow. O A 2-4 A Theory of Hydrodynamics with dz/dt 0, ds/dt = 0 and 3p/3t + V*(pV) = 0. To simplify the search for a Lagrangian such that only the con-servation of p a r t i c l e l a b e l constraint i s altered, assume that the representation for V remains unchanged and that the new Lagrangian maintains the invariance under the gauge transformations given by Eq. (1-35). The only expressions which involve the Monge potentials a,3 and y and are invariant under the gauge transformations are 3a/3t + B3s/3t + Y J3z J/9t and - Vex - BVs - Y J V z J (see Ref. 16 for -28-a proof). To keep the conservation of mass and entropy equations unchanged, a d d i t i o n a l terms involving a and 8 cannot appear in the Lagrangian (since v a r i a t i o n of these Monge pot e n t i a l s j u s t y i e l d the conservation of mass and entropy equations). This means that the Monge po t e n t i a l s can appear i n any a d d i t i o n a l terms i n the Lagrangian only in the form u>*Vs = - Vs*(Vy x V z J ) . The representation for V i s unchanged, thus V may not appear i n any add i t i o n to the Lagrangian. Furthermore since 3/3t i s not invar-iant under G a l i l e a n boosts (see Sec. 1-6) no time d e r i v a t i v e s may appear (d/dt cannot be used since t h i s would involve V). Since z z = z - V(z,o)6t under time t r a n s l a t i o n (see Sec. 1-6) the v a r i a -b l e z can appear only i n the term w*Vs. I t i s easy to see that Z'^s i s i nvariant under s p a t i a l t r a n s l a t i o n , time t r a n s l a t i o n , G a l i l e a n boosts, r o t a t i o n s and inversion of coordinates (w -»• -co) however under time inversion co -»• -co and thus co»Vs -*• - co*Vs. Hence to maintain the Ga l i l e a n invariance of the Lagrangian, any a d d i t i o n a l terms in the Lagrangian must have the form h(p,s,V*p,Vs, (co*Vs)2) , where h i s a d i f f e r e n t i a b l e scalar function of i t s arguments, and therefore are non-linear i n . In view of the preceeding disc u s s i o n consider the Lagrangian L" =%pV 2-pe(p,s)-h(p,s , (at-Vs) 2)-a{|f+^.(pV)}+pB ^ + p Y j ^ (2-40) dt dt dt where for s i m p l i c i t y Vp and Vs have been eliminated from h apart from (<o*Vs)2 terms. The v a r i a t i o n a l equations of L" are 6V: V = -Va-gVs-y-'vV (2-41) -29-<5a: •§£• + V.(pV) = 0 (2-42) a t . da ds , j dz J . i TTo P 3h n . o N 6 p : dt P dt Y d t " * V " e " p " 9p" - ° t * " 4 2 0 66: ff = 0 (2-44) 6 s : M . _ T _ l | h _ 1 ^ 3 h j ^ . ^ l ^ j - ( 2_ 4 5 ) d t P 9(^-Vs) 6zj : £*£ . _ 1 ]x^ 8.V (2-47) d t P 3(5-Vs) S u b s t i t u t i o n of Eqs. (2-43)-(2-47) into Eq.(l-17), which i s unchanged, and use of the vector i d e n t i t y ^x(BxC) = (A*C)B - (A«B)C* y i e l d s the conservation of momentum equation 4 - (pV 1) + V ^ p v V + P e ^ - T 4 ) = 0 (2-48) d t where the symmetric stress tensor i s given by = ( h - p ^ ^ J f(w.Vs)6 i j - ( n V r t V s ) ] (2-49) 3 p 8(w-Vs) Mu l t i p l y i n g Eq. (2-46) by 9x1/3z-' y i e l d s the equivalent form + m 9x(z,t) _ 1 ^ 3h ] x ^ s ( 2 _ 5 Q ) 8 t p stf-Vs) which states that v" + p"1 [ dh/8 (OJ*VS) ] xVs = V 1 i s the tangent to the p a r t i c l e paths, not V as i n the Lagrangian p i c t u r e of f l u i d flow. -30-Using the techniques developed i n Sec. 2-2 i t i s straightforward to show that a l l solutions of the hydrodynamic equations (2-42), (2-44), (2-46) and (2-48) are also solutions of the v a r i a t i o n a l equations of L". The conservation law associated with the gauge invariance of L" i s unchanged from Eq. (2-30). The conservation laws for the energy, angular momentum and center-of-mass are given r e s p e c t i v e l y by \ \ - (% pV2+pe+h)+Vj (V j [% pV 2+ Pe+h+P]+v , j(^Vs) ^ — ) = 0 (2-51) 9 t 3(J.7s) | r ( e i j % x j v V A e i J V J v V + e l j V { P 6 M - T k £ } ) = 0 (2-52) | - ( p v H - p x ^ + V ^ p v V t - p x V + P t e ^ - T ^ t ) = 0 (2-53) which follow from Eq. (1-25). Just as i n Sec. 2-3, the hydrodynamic equations of L" may be interpreted as describing a f l u i d with a f i n i t e density of v o r t i c e s (see Appendix C for d e t a i l s ) . Note that both Eqs. (2-48) and (2-18) have stress tensors which depend on the v o r t i c i t y and thus describe non-Stokesian f l u i d s . In conclusion, the re l a x a t i o n of the conserva-t i o n of p a r t i c l e l a b e l c o n s t r a i n t i n a one-component f l u i d i s equiv-alent to v o r t i c i t y dependent contributions to the stress tensor and the energy of the f l u i d . The r o l e of the r e l a x a t i o n of the conserva-t i o n of p a r t i c l e l a b e l constraint for su p e r f l u i d helium w i l l be con-sidered i n Chapter 4, as a prelude to t h i s work Chapter 3 w i l l review 27 Z i l s e l ' s v a r i a t i o n a l p r i n c i p l e for the Landau two-fluid equations. -31-CHAPTER 3 THE VARIATIONAL PRINCIPLE FOR THE LANDAU TWO-FLUID EQUATIONS 3-1 Introduction The su p e r f l u i d i t y of He was f i r s t observed by Kapitza i n 1938 who found that l i q u i d helium below T = 2.17°K could flow A through t h i n c a p i l l i a r y tubes with zero resistance. On the other hand experiments with rotating l i q u i d helium showed that the super-f l u i d could not be interpreted as,a c l a s s i c a l one-component f l u i d 2 9 with zero v i s c o s i t y . These two observations led Landau to develop the two-fluid model of su p e r f l u i d i t y as consisting of the flow of two interpenetrating f l u i d s , the entropy carrying normal f l u i d with v e l o c i t y V n and the zero-entropy superfluid with v e l o c i t y V . The Landau two-fluid equations which Landau postulated to describe t h i s model consist of conservation laws for the mass, entropy and t o t a l momentum of the f l u i d and an equation of motion for the superfluid v e l o c i t y for a t o t a l of eight equations. The eight independent variables may be taken as p,s,V n and V^ where p and s are the t o t a l mass and entropy of the f l u i d per unit volume. I t has long been suggested that superfluidity i s a quantum phenomenon which occurs when an appreciable f r a c t i o n of the He1* atoms enter the groundstate i n a Bose condensation giving r i s e to long-range 30 order i n the phase of the wavefunction of the Bose condensate. ->• In f a c t , once the independent variables p,s,Vn and V g and their Galilean transformation properties are specified then the Landau two-fluid equations follow without further recourse to the quantum -32-theory from Galilean invariance arguments and by requiring that the conservation of energy equation be redundant (otherwise when combined with the conservation laws for the mass, entropy and momentum and the equation for the superfluid t h i s would y i e l d nine equations i n eight 31 unknowns, see Putterman for a detailed derivation). A v a r i a t i o n a l p r i n c i p l e for the Landau two-fluid equations was 27 f i r s t given by Z i l s e l i n 1950. Although a l l solutions of Z i l s e l ' s v a r i a t i o n a l equations s a t i s f y the Landau two-fluid equations the converse result does not hold. Z i l s e l ' s representation for the -y -y -y normal v e l o c i t y V implies V x V =0 for ps / p = constant which J n r n ^n i s too r e s t r i c t i v e ( p i s the mass density of the normal f l u i d ) . n 15 Schultz and Sorkin have pointed out that t h i s d i f f i c u l t y may be eliminated by postulating a Lin's constraint for V n i n analogy with the v a r i a t i o n a l p r i n c i p l e for the adiabatic flow of a c l a s s i c a l one-component f l u i d (see Chapter 1). In addition Z i l s e l ' s v a r i a t i o n a l 32,33 p r i n c i p l e has been c r i t i c i z e d on the grounds that x = P n / P ^ s treated as an independent variable i n contradiction with the Landau model. In spite of th i s i t can be shown that Z i l s e l ' s v a r i a t i o n a l p r i n c i p l e supplemented with Lin's constraint for V"n i s completely equivalent to the Landau two-fluid equations. The absence of a Lin's constraint for V g ensures that the superfluid remains i r r o t a -t i o n a l i . e . V x v =0. For a review of these points see the s a r t i c l e by Jackson. Sec. 3-2 reviews the Landau two-fluid equations ; the notation 35 w i l l follow London. The equivalence of Z i l s e l ' s v a r i a t i o n a l p r i n -c i p l e supplemented with Lin's constraint for 1 and the Landau two-f l u i d equations i s proven i n Sec. 3-3 while the symmetries and -33-conservation laws associated with Z i l s e l ' s Lagrangian are discussed i n Sec. 3-4. 3-2 The Landau Two-Fluid Equations 35 From London the Landau two-fluid equations for sup e r f l u i d He 4 are given by | ^ + V - ( p V + p V ) = 0 (3-1) 3t n n s s d ( p s ) + V.( PsV ) = 0 (3-2) o t " d V s s + Vy = 0 , V x V = 0 (3-3) + V j ( p v V + p v V + PS 1 3) = 0 (3-4) dt 3(p V 1 + p V 1) K n n s s 3t ' ' V K s ' s ' s ' K n ' n ' n d where = — + V « V . Eqs. (3-1), (3-2) and (3-4) represent the dt 9t Y conservation of mass, entropy and t o t a l momentum of the f l u i d respec-t i v e l y while Eq. (3-3) gives an equation of motion for the s u p e r f l u i d . The v a r i a b l e s p,s,V and V have been defined previously while n s P n and p g are the d e n s i t i e s of the normal and s u p e r f l u i d components r e s p e c t i v e l y and the t o t a l mass density i s given by p = p n + p g . The i n t e r n a l energy d i f f e r e n t i a l has the form de = Tds + ( P / p 2 ) d p + % (V -V ) 2 d X (3-5) n s where e ( p , s , x ) i s the s p e c i f i c i n t e r n a l energy, T i s the temperature, the pressure P = p(-e + Ts + % ( n^- s^^  2X + and u i s the chemical po t e n t i a l . Note that Eq. (3-5) implies (3e/3x)p,s = % ( V - V J 2 and hence there e x i s t f u n c t i o n a l r e l a t i o n s h i p s of the form p n = P n(p,s, -34-(V -V ) 2 ) and p = p (p,s,(V -V ) 2 ) . The independent v a r i a b l e s of the XI S S S XI s Landau two-fluid equations may therefore be taken as the eight v a r i a -->- ->• bles p,s,V and V . n s Following Z i l s e l the factor T i s defined as 9p 3p ^ + V . ( P T V ) = + v-(p v )] = r (3-1)' dt n n 3t s s Using Eq. (3-1)' i t follows that Eqs. (3-2) and (3-4) can be rewritten i n the convenient equivalent forms d ( s / x ) 77 '--^-Z-r (3-2)' dt P X ^ d V p P - r 2 - 2 + - VP H sVT + V(V -V ) 2 + (V -V ) — =0 (3-4) ' dt p p 2p n s' n s' px Assume that there e x i s t s a Lin's constraint for the normal v e l o c i t y f i e l d of the form d z j ( x , t ) n =0 dt " " (3-6) M u l t i p l i c a t i o n of Eq. (3-6) by 3x 1/3z' ] y i e l d s the equivalent form v i = 3 x X ( z ? t ) ( 3 _ 7 ) n 9t where the function x = x(z,t) i s the inverse of z = z(x,t) and the i d e n t i t i e s given by Eqs. (1-8) s t i l l hold. Eq. (3-7) j u s t states that V n i s the tangent to the p a r t i c l e paths x = x ( z , t ) . For the c l a s s i c a l one-component f l u i d the p a r t i c l e paths x = x(z,t) were associated with the movement of small f l u i d elements. The Landau two-fluid model c o n s i s t s of two interpenetrating f l u i d s and i t i s -35-no longer c l e a r what phys i c a l i n t e r p r e t a t i o n the p a r t i c l e paths have; see Jackson f o r a discussion of t h i s point. In the absence of any further progress on t h i s matter, Eq. (3-6) should be viewed simply as an i n t e g r a b i l i t y condition on the normal v e l o c i t y f i e l d . Use of the chain r u l e and Eq. (3-7) y i e l d s ^ f ( x , t ) = | ^ f ( x ( z , t ) , t ) (3-8) which implies that d ^ d t and 3/3z commute. Eqs. (3-1), (3-2), (3-3), (3-4)' and (3-6) w i l l be c o l l e c t i v e l y r e f e r r e d to the hydrodynamic equations, which are shown i n the following section to be equivalent to a v a r i a t i o n a l p r i n c i p l e . 3-3 Z i l s e l ' s V a r i a t i o n a l P r i n c i p l e Z i l s e l ' s Lagrangian supplemented with a Lin's c o n s t r a i n t for v" i s given by L = p [ % (l-x)V 2. + % X V 2 ] - pe(p,s, X) - a{f£- + ^ . ( p ( l - x ) V s + PXV )) - e { | ^ ^ - + v".(psV )} + p x Y J { f f ^ + V -VzJ} (3-9) n dt n dt n where the dependent v a r i a b l e s of L are p,s,V n,V s,x,z,a,3 and y a n d the independent v a r i a b l e s are ( x , t ) . The i n t e r n a l energy density e(p,s,x) i s defined by Eq. (3-5). A c t u a l l y i n Z i l s e l ' s procedure Eq. (3-5) i s not assumed; instead the v a r i a t i o n of x gives the equa-t i o n (8e/8x) = % (V -V ) 2 ; when Eq. (3-5) i s assumed i n i t i a l l y then p, s n s the v a r i a t i o n of x gives an i d e n t i t y . The v a r i a t i o n a l equations of L are given by -36-6a: ||+ V . ( p ( l - x ) V s + p XV n) = 0 (3-10) 6g: + V.(psV n) = 0 (3-11) « V V g = - Va (3-12) 6V n: V n = - Va - ^  V~3 - y '^vV (3-13) 6p: % ( 1 - X ) V 2 + % xV 2-e- - + | f +((1- X)V + XV )-Va + s = 0 (3-14) s n p o c n s at 6 s : dTT = T < 3- 1 5> 6 X: - % pV 2 + % pV 2 + % p(V -V" ) 2 + p(V -V )-(Va) = 0 (3-16) S Tl Tl s n s dt 6y: j t - = 0 (3-17) d Y 6 z : i r = - p 7 Y ( 3 - 1 8 ) Eqs. (3-1), (3-2) and (3-6) are recovered as the v a r i a t i o n a l equations of a,3 and y r e s p e c t i v e l y . When Clebsch's lemma i s used Eq. (3-12) implies ^ = - V ( | ^ + V Q 4a + % V 2 ) (3-19) dt dt s s Substitution of Eqs. (3-12), (3-15) into Eq. (3-14) y i e l d s If + + % V s " 6 + f ~ T s " *<V*s)2X 5 * ( 3" 2 0> which when combined with Eq. (3-19) gives the equation of motion of -37-the s u p e r f l u i d Eq. (3-3). The i r r o t a t i o n a l condition VxV g = 0 follows from Eq. (3-12). Clebsch's lemma applied to Eq. (3-13) y i e l d s d V d a d d 6 d y j • d„z j dP " " '<3T + * V - dt - P < t - £ - ^ df" <3-21> Eq. (3-14) can be rewritten using Eqs. (3-12), (3-15) as d a J _ + % V 2 = e + - - T s + % (1- X)(V -V ) 2 (3-22) dt n p n s Substitution of Eqs. (3-22), (3-15), (3-17), (3-18) and (3-2)' (which follows from Eqs. (3-10) and (3-11)) into Eq. (3-21) gives d V n n _ . P _ . , x ,± ± . o . s d : _ V ( e + i - Ts + h ( l - x ) ( V n - V g ) 2 ) - | VT + (f VB + y^z^) ^ - (3-23) Eqs. (3-12) and (3-13) and the i d e n t i t y I VP = V(e + - - Ts) + sVT - % (V -V )V X (3-24) P p n s shows that Eq. (3-23) i s j u s t Eq. (3-4)'. Hence a l l solutions of the v a r i a t i o n a l equations are also solutions of the hydrodynamic equations. -»-->- -*-The i r r o t a t i o n a l condition VxV = 0 implies that V can be s r s written as a gradient V g = - Va (3-25) Clebsch's lemma and Eq. (3-3) give d V d a d a T T ^ + ( 7 f - + ^ v i ) = (T|- + % V 2 - y) = 0 (3-26) dt dt s dt s or equivalently d a + % V 2 - y = 0 dt ' '* 's (a function of time can be absorbed into a) which i s identical with Eq. (3-14) provided 3 = / Tdt (the integration is carried out at 0 constant z). From Eq. (3-6) follows the identity ^ ((V j-V j) ^d) = ( ^ - ^ L2) ^ 4 + (Vk-Vk)(vJvk-vV) ^ 4 (3-27) dt n s „ l dt dt „_i n s n s „_i Substitution of Eqs. (3-3) and (3-4)' into Eq. (3-27) gives i i ( ( V j-V j) 4^) = - f- V jT 4^ + (V j-V j) -^^ 4 (3-28) d t n s 3 Z 1 X 3 Z 1 s n P X 3z X From the defin-ition of 3 given above and using Eq. (3-2)' then Eq. (3-28) can be rewritten as d . . j _E (* ( V 1 _ V J •+ JL V J S ) «L_) = o (3-29) dt s n s X . x 3z which can be integrated as V" -V = - - V g - Y J V z j (3-30) n s X where y = - (sx(z,o)/s(z,o)x)(V j(z,o)-V^(z,o)). Eqs. (3-25) and XI s -39-(3.30) are i d e n t i c a l with Eqs. (3-12) and (3-13) r e s p e c t i v e l y , furthermore i t i s easy to show that a, g and y defined above s a t i s f y Eqs. (3-14), (3-15) and (3-18) r e s p e c t i v e l y . Thus the v a r i a t i o n a l equations are completely equivalent to the hydrodynamic equations. 3-4 Symmetries and Conservation Laws As pointed out i n Sec. 1-5 the Monge p o t e n t i a l s a, 8, y and z may be subjected to "gauge" transformations which do not change the value of V and which keep the v a r i a t i o n a l equations form i n v a r i a n t , leading to a conservation law v i a Noether's theorem. For the Landau two-fluid equations the requirement that V G be unchanged and that Eq. (3-12) be form i n v a r i a n t gives V6a = 0 or equivalently a a' = a + 6 a ( t ) . I f V N i s unchanged and Eq. (3-13) i s form i n v a r i a n t then the i n f i n i t e s i m a l transformations B -*- B ' =8 + 63, y3 -* Y ' J = Y J + 6y J and z-1 z'-1 = z J + 6z J must s a t i s f y V ( - 68 + y j 6 z j ) = 68V - - 6y jVz j + 6z jVy d = 0 (3-31) The form invariance of Eqs. (3-14)-(3-18) gives inr < 6 a ) = Sr ^ - d f ( 6 l ) - d f ( 6 ? > - 0 (3-32) which have the solutions 6a = ea , 68 = 68(s,y,z), 6y = 6y(s,Y,z), 6z = 6z(s,y,z) (3-33) Substitu t i o n of Eqs. (3-33) into Eqs. (3-31) implies that the i n f i n i t e s i m a l gauge transformations have the form 68 = e 3(s/ X) 3G 6y J = - e 9G (3-34) -40-where a = constant, e i s an i n f i n i t e s i m a l constant and G i s a o function of s /x , Y and z homogeneous of degree one i n s/x and y i . e . From Eq. (1-25) the conservation law associated with t h i s symmetry transformation i s (pa + pXG) + v".(pa V + pXV G) = 0 (3-36) By choosing G = 0 the conservation of mass equation i s recovered while the choice aQ = 0, G = s/x gives the conservation of entropy equation. The G a l i l e a n transformation properties of p, s, V , V and z are known ;however the transformation properties of a, 3 and y must be deduced by re q u i r i n g that the Lagrangian be i n v a r i a n t . (i ) Under the t r a n s l a t i o n of axes x -»• x = x + ox, z -> z = z + 6z -> -»• (p,s,x»V and V are unchanged). The Monge po t e n t i a l s a, 3 and y n s transform as 6 a = e a o ' 6 e " e I(s7xT ' 6 r > = £ f l ( 3 _ 3 7 ) oZ where G = G(— , z) i s homogeneous of degree one i n s/x- Eqs. (3-37) X completely s p e c i f y the G a l i l e a n tranformation properties of a, 3 and y, From Eq. (1-25)the conservation law associated with t h i s symmetry i s j u s t the conservation of momentum equation (3-4). -41-( i i ) Under the time t r a n s l a t i o n t - » - t ' = t + 6t the i n i t i a l p o s i t i o n vector z z' = z - V(z,o)6t. Using the arguments developed i n ( i ) , the invariance of L implies y y'~* = y^ - y^ (SV 3 (z,o)/9z)6t apart from a gauge transformation given by Eqs. (3-37). The conservation of energy equation associated with t h i s symmetry follows from Eq. (1-25) as ~ (% P V 2 + h p V 2 + pe) + V« (h p V 2V + h P V 2V + (pe + P)V dt n n s s n n n s s s + psT(V -V) + % p (V -V ) 2 ( V -V)) n n n s n = 0 (3-38) ( i i i ) Under the r o t a t i o n of axes x x' = x + 6Q x x, V -> V' + 66 x v , n n n ' V -> V' = V + 58 xV and z -> z' = z + 6 ^ x 1 . The invariance of L s s s s •> ->• gives y Y = Y + 68 x y apart from a gauge transformation given by Eqs. (3-37). The conservation of angular momentum equation associated with t h i s symmetry i s ^ - ( e i j k x j t p + p V k ] ) + V £ ( e i j k x j [ p V V k + p v V 3t n n s s s s s K n n n + P6 ]) = 0 (3-39) (iv) Under the Gal i l e a n boost x •+ x' = x + 6V t , V -»• V' + 6V and o n n o -*• -»• -> . ->-V -»• V = V + 6V . The invariance of L gives a -> a = a-x*6V s s s o o apart from a gauge transformation given by Eqs. (3-37). The conservation of center-of-mass associated with t h i s symmetry i s ( t [ p V * + p V*] - px 1) + V j ( t [ p V V + p V V ] + P t 6 i j dt n n s s n n n s s s - x-[p V j + p V j]) = 0 (3-40) n n s s -42-This completes the review of Herivel's v a r i a t i o n a l p r i n c i p l e for the Landau two-fluid equations. Chapter 4 w i l l extend t h i s v a r i a t i o n a l p r i n c i p l e to the two-fluid equations of rotating superfluid helium as formulated by Khalatnikov and Bekarevitch. -43-CHAPTER 4 A VARIATIONAL PRINCIPLE FOR SUPERFLUID HELIUM WITH VORTICITY 4-1 Introduction I t has been known for some time that su p e r f l u i d v o r t i c e s with c i r c u l a t i o n quantized i n u n i t s of (h/m) can e x i s t i n sup e r f l u i d 3 6 » 3 7 helium. The quantization of c i r c u l a t i o n i s connected with the multiple-valuedness of the phase of the wavefunction of the Bose 38»39 + + condensate. The superfluid v o r t i c i t y V x V g s t i l l vanishes everywhere on a microscopic scale except i n the cores of v o r t i c e s ; however when V g i s averaged over a macroscopic region which contains a f i n i t e density of v o r t i c e s then V x V g ^ 0. If the averaging i s done over a region large compared to the separation between v o r t i c e s then the normal v e l o c i t y V^ and the superfluid v e l o c i t y V g w i l l be smoothly varying functions throughout the f l u i d . 2 5 Khalatnikov and Bekarevitch (KB) have derived the equations of motion for the l a t t e r case with a phenomenological approach by allowing the i n t e r n a l energy of the f l u i d to depend on the absolute value of the s i i p e r f l u i d v o r t i c i t y . The hydrodynamic equations are 31 then derived by the standard method from G a l i l e a n invariance requirements and by manipulating the redundant conservation of energy equation. In t h i s procedure a number of phenomenological c o e f f i c i e n t s appear which can be derived from a d e t a i l e d vortex model. H a l l has examined the same problem by using a microscopic model of e x c i t a t i o n s i n t e r a c t i n g with v o r t i c e s ; the two-fluid equations he derives agree with KB. For a short review of t h i s -44-subject see the a r t i c l e by Chester. 9 hi L i n and more re c e n t l y L h u i l l i e r , Francois and Karatchentzeff (LFK) have given v a r i a t i o n a l p r i n c i p l e s incorporating a generalized Lin's constraint to describe the Landau two-fluid equations with microscopic superfluid v o r t i c i t y , i n d i s t i n c t i o n from the two-fluid equations of KB where only the macroscopic superfluid v o r t i c i t y i s non-vanishing. The purpose of Chapter 4 i s to f i n d an extension of Z i l s e l ' s v a r i a t i o n a l p r i n c i p l e which i s equivalent to the two-fluid equations of KB with zero entropy production. The hydrodynamic equations are summarized i n Sec. 4-2 and a Lagrangian for these equations i s given in Sec. 4-3. I t i s found necessary to use two constraint equations, one constraint f o r V and as shown i n Sec. 4-4, the other c o n s t r a i n t giving the superfluid vortex equations of motion. A d i s c u s s i o n of the symmetries and conservation laws i s given i n Sec. 4-5. 4-2 The Hydrodynamic Equations Following KB the fundamental assumption i s that the i n t e r n a l energy d i f f e r e n t i a l has the form de = Tds + ( P / p 2 ) d p + % (V -V ) 2 d x + ( A / p ) d c o (4-1) n s where e i s the s p e c i f i c i n t e r n a l energy, T i s the temperature, s i s the s p e c i f i c entropy, the pressure P = p(-e + Ts + % (V -V ) 2 x + u), Tl S X = (p / p ) where p i s the normal f l u i d density, u i s the chemical n n p o t e n t i a l , X i s a phenomenological c o e f f i c e n t and u> = |eo| where to = v x v . To make the notation agree with Z i l s e l the s p e c i f i c i n t e r n a l energy e d i f f e r s from that of KB (denoted ( e / p ) ) by (z/p)-e -45-= % (V -V )x» The meaning of P and y i s unchanged. From Eq. (4-1) XI s and the d e f i n i t i o n of P follows the useful i d e n t i t y % = I VP - sVT - 4 V(V -V ) 2 + - Vto (4-1)' p 2 n s p From KB the hydrodynamic equations with zero entropy production are + V(p V +p V ) = 0 (4-2) 9t s s n n | ^ - + V.(psV ) = 0 (4-3) 91 n d V ~r———• + Vy = (6 *- — ) w x (Vx(Xv)) - B'p a x(V -V ) (4-4) dt p s n s s 9(p V ^ p V 1) . . . . . .. . . , . - — 5 S n n + V J(p v V + p v V + P S 1 3 + Xo)61J -Xu)V/u>) = 0 (4-5) 9t s s s n n n where = 1,2,3 (sum repeated i n d i c e s ) , v = (D/U, ^ = — + V^*V and the su p e r f l u i d density p g = p - P r . Following Z i l s e l the factor T i s defined as — ^ + ^ . ( p n v ) = - ( ^ + V-(p V ) ) = r (4-2)' dt n n dt s s Using Eq. (4-2)* i f follows that Eqs. (4-3) and (4-5) can be rewritten r e s p e c t i v e l y as d (s/x) n (4-3)' dt PX' -46-iZlL + I fa + -2. SV"T + ^  V(V -V ) 2 + (V -V ) ^ -dt p P n 2p n s n s px 8' p - v"o» + w x (V -V - — V x(Xv)) (4-5) ' p p n s p n The phenomenological c o e f f i c i e n t s A,8' have been computed by KB from a vortex model as B' p 8' = T - ^ , A = J p i n (J) (4-6) zpp m s a s where B' = constant, m i s the mass of a He1* atom and — i s the r a t i o a of the distance between v o r t i c e s to the e f f e c t i v e radius of a vortex. 4-3 The Lagrangian Consider the Lagrangian L = L[p,s,x,V ,V ,a,8 ,y 3 > z ^»Y 3> z^1 given XI s by L = p[%(l-x)v2+%XV2i]-pe(p,s,x,v9JxVi:i)-a{|£- + v"-(p(l- X)V +PxV n) - p { | i P S l + v . ( p s ^ ) + P X y J { | f + V d t n d t n + P Y J { f r " + I(1-X)V + X V „ ] ' V £ j (4-7) o L s n where the representation for e i s given by de = Tds + (P/p 2)dp + %(V -V ) 2 d X + (A/p)d|Vyjxv'z:i | (4-8) n s From Eq. (4-8) i t i s easy to derive the following u s e f u l v a r i a t i o n a l d e r i v a t i v e s - 4 7 -6(pe) -*-~i.,->~k -v^k,., 1 — = -[V*(XVy XVZ / IVy xVz j )]-Vy J ( 4 - 9 ) 6z J — ^ = [Vx(AVy xVz /|Vy xVz |)]«VzJ ( 4 - 1 0 ) 6y3 The v a r i a t i o n a l equations of L are 6a: |£ + v " . ( p ( l - x ) ^ s + P X ^ ) = 0 ( 4 - 1 1 ) 6 3 : + v".(ps^n) = 0 ( 4 - 1 2 ) 6 ^ s : ^ s = " Va - 9JVzj ( 4 - 1 3 ) 6^n: V~n = - Va - f Vg - y JVz j - y % j ( 4 - 1 4 ) 6p: % ( l - X ) V | + % xV2-e- ^ + |^+ ( ( l - x ) ^ + X \ ) -Va+s ^ + Y j { | r - + I(l-x)^ + X^]- V z j >= 0 ( 4 - 1 5 ) 6s: ^ - = T ( 4 - 1 6 ) 6 x : -%pV2+%pV2-%p (^  )2+ p(^ J . ^ + ^ V z ^ ) = 0 ( 4 - 1 7 ) . d z j 6y J: ^ — = 0 ( 4 - 1 8 ) d J i n 1 r i 6 z : d t p ^ <4-19) 6y J: + t(l - X ) v " + XV~ 1-v"^ = - - IV~x(Av) ] .Vz j ( 4 - 2 0 ) o L s n - . p -48-<5zj: | l i + I ( l - X ) V s + x \ ] ' V Y J = ^  [v"x (Xv) ]-V^ (4-21) Eqs. (4-2) and (4-3) are recovered as Eqs. (4-11) and (4-12) while Eqs. (4-13) and (4-8) give Eq. (4-1). When use i s made of Clebsch's lemma | | = _ $(dt + w 2 / 2 ) - |f vV* - ^ |f (4-22) where — = h w*V and w = -Vii - n v£ (a=l,...,m) and the vector dt 9 1 i d e n t i t y Ax(BxC) = (A-C)B - (A'B)C then Eqs. (4-11) - (4-21) imply a f t e r a lengthy but straightforward c a l c u l a t i o n d V + $ = _ 1 ^x(Vx(Xv)) - Xwx(V -V ) (4-23) dt p n s dt p p 2p n s px n + - - wx(Vx(Xv))+(l-x)wx(V -v" ) (4-24) p p n s which are Eqs. (4-4) and (4-5)' with B'=2. Note that due to the (X/ P)dw terms i n de the exact form of Eq. (4-20) i s c r u c i a l to the gauge invariance of Eqs. (4-23) and (4-24) ( i . e . that the po t e n t i a l s a and c a n be eliminated from Eqs. (4-23) and (4-24)). Hence the v a r i a t i o n a l p r i n c i p l e i s confined to the case B'=2. The preceeding arguments have shown that a l l solutions of Eqs. (4-8) - (4-21) are solutions of Eqs. (4-1) - (4-6) with B'=2. The converse statement can be proven using a ge n e r a l i z a t i o n of Weber's -49-transformation of a c l a s s i c a l one component f l u i d provided Eqs. (4-18) and (4-20) are adopted at the outset and i t i s assumed that z 1 = z ^ x . t ) and z 1 - z i(x,t) ( z 1 and z 1 denote the values of the functions z^(x,t) and z 1(x,t) respectively) possess d i f f e r e n t i a b l e inverses x 1 ( z , t ) and x 1 ( z , t ) r e s p e c t i v e l y (indices are omitted unless necessary). This means that x 1 ( z , (x, t) , t) = x 1, x ^ z t e , t) ,t) = x 1, z 1 ( x ( z , t ) , t ) = z 1 and z 1 ( x ( z , t ) , t ) = z 1 (the x 1 are the coordinates, x 1 ( z , t ) and x 1 ( z , t ) are f u n c t i o n s ) . The chain r u l e implies the r e l a t i o n s 9x X 9 z l = 9x1 $SL- i s l i i ^ - <>2!L- l i l - * i k 9z j 9x k 9Z 1 9xj 9Z 1 9xj 9z j 9x k = & (4-25) where the p a r t i a l d e r i v a t i v e s have the meaning 9x 1/9z : ] = 9x X(z, t ) / 9 z J , 9z j/9x k = 9 z j ( x , t ) / 9 x k , d^/dz1 = 9x j (z, t ) / 9 z \ 9z k/9x j = 9z k(x, t)/9x j. Def ine V L = (1-x)V g + X V n + j vx(Xv) (4-26) d L d n then by using the chain r u l e the d e r i v a t i v e s -r— and — have the at dt meaning d L 9 d n 3 dt f ( x > t ) = JE f ( x ( z , t ) , t ) , g(x,t) = g ( x ( z , t ) , t ) (4-27) d L 9 d n 9 Eqs. (4-27) implies that — and r commute as do — and r . By dt l dt l i i i i 3 z 8 z s e t t i n g f = x and g = x Eqs. (4-27) imply \ d T x 1 i = 9x (z,t) = _L_ L 9t dt K ' -50-i , . d x 1 v ± _ 3x (z,t) _ _n n 3t dt (4-29) which states that V.f and are the tangents to the particle paths J_J n x"*"(z,t) and x^(z,t) respectively. Eq. (4-28) implies the identity ^ ( V J i £ ) = (^|Is + IJ x ( t 4 ) ] j ) + (V 2/2+V -(V -V )) (4-30) dt s ~j dt L s ~J ~ i s s L s 3z J " 3z J 3z Substituting Eq. (4-23) into Eq. (4-30) yields ^ { ^ - (V j + V j [ / (y-V 2/2-V . ( V - V ))dt])} = 0 (4-31) d t „ ~ i s 0 s s L s d Z where the integration is carried out at constant z 1 . Eq. (4-31) can be integrated as V J = -yJ[ / ( y - V 2/2-V - (V T -V ) ) d ^ ] + V 1 ( z , o ) V j z 1 = - V ^ a - y V z 1 (4-32) s 0 S S L s s where a = / (y-V 2 / 2 - V ' ( V - V ))dt and y 1 = - V ± ( z , o ) . It i s easy to 0 s s L s s verify that V s , a and y 1 satisfy Eqs. (4-13), (4-15) and (4-21) respectively. Eq. (4-29) implies the identity v j _ v j ) 2*1) = ( V k - V k ) ( V j V k-vV) 4^ (4-33) d t v v n s' „ r v dt dt „_ i n s n s' „ _ i 3z 3z 3z Substituting Eq. (4-25) into Eq. (4-33) gives ^ • { ^ ^ 4 ( V j - V j + - V j B ) } = 0 (4-34) dt s i n s x d Z -51-t i where B = / Tdt (the integ r a t i o n i s c a r r i e d out at constant z ). 0 Eq. (4-34) can be integrated as V n ~ V l = - 7 v 3 g 4 l Ifcnl ( v \ z , o ) - V ^ ( z , o ) ) V j z 1 = - f- V j B-y V z 1 (4-35) n s A X s(.z,o) n s X where y 1 = -(s X(z,o)/xs(z,o))(V^(z,o)-V^(z,o)). I t i s easy to v e r i f y that V n,6 and y s a t i s f y Eqs. (4-14), (4-16) and (4-19) r e s p e c t i v e l y . This proves that Eqs. (4-1) - (4-6) with B'=2 and Eqs. (4-18) and (4-20) are completely equivalent to the v a r i a t i o n a l equations of L. Since Eqs. (4-18) and (4-20) are added to the hydrodynamic equations s u i t a b l e physical i n t e r p r e t a t i o n s must be given these equations. Just as i n c l a s s i c a l one-component hydrodynamics Eq. (4-18) may be viewed as an i n t e g r a b i l i t y constraint on the normal v e l o c i t y f i e l d . In Sec. 4-3 Eq. (4-20) i s interpreted as the st a t e -ment that the superfluid v o r t i c e s , averaged over many v o r t i c e s , ->-move with v e l o c i t y V . By following L i n or LFK and using only Eq. (4-20) as a con-s t r a i n t an i n t e r e s t i n g problem develops. The Lagrangian and v a r i a -t i o n a l equations f o r t h i s case are obtained j u s t by dropping the terms involving y and z . The equation for V o remains unchanged but the equation f o r V becomes n V ^ s - " f ^ ( 4 " 3 6 > M u l t i p l y i n g Eqs. (35) and (36) by — and taking the c u r l gives r e s p e c t i v e l y [Ui (V - V^J^^VzVV 3 d—-{^r (v"(z,o)-Vm(z,o))} (4-37) n s 3 z p s(z,o) n -52-[V x (V n - V ^ } ] 1 = 0 (4-38) i i k where e i s the Levi-Civita symbol. A necessary condition that Eq. (4-37) agree with Eq. (4-28) for a l l values of z m is £Jlmp {U^2l ( vm ( z > o ) _ ^(z.o))} = 0 (4-39) 3 zP s(z,o; n s or (x(z,o)/s(z,o)) (V^z.o) - V^z.o)) = 3 ^ ( z ) / 8 z m . But the 3 which n s appears in Eq. (35) may be subjected to a gauge transformation 3 -*• 3 ' = 3 + ^(z) (which does not alter Eq. (4-16)) which just cancels the Y^V^Z 1 terms. Hence Eq. (4-39) i s a necessary and sufficient condition that Eq. (4-35) reduce to Eq. (4-36). Note that the preceeding arguments did not depend on the presence of the X terms. Thus a necessary and sufficient condition that the variational principle given by Eq. (4-7) be equivalent to the hydrodynamic equations i s that either constraint Eqs. (4-18) and (4-20) are used with V x {(v - V )x/s> arbitrary or only Eq. (4-20) is used with the XX s i n i t i a l constraint V x {(V - V )x/s) = 0. Note that this i s the proof n s of a claim made by LFK for pure V g vortices. Since KB do not assume 7 x{(v - V* )x/s} = 0 both Eqs. (4-18) and (4-20) are needed. n s 4-4 Interpretation of z(x,t), V and the restriction B' = 2 Taking the curl of Eq. (4-23) yields an equation for the superfluid vort i c i t y - 5 3 -which can be integrated using Eq. (4-28) as " I = ^ j( z>°) *2L. (4_ 4 1) p p(z,o) 3 z J Eq. (4-41) j u s t states that w/p i s fixed r e l a t i v e to a set of coordinate axes composed of the same p a r t i c l e s x(z,t) for a l l t or equivalently that to/p i s transported with v e l o c i t y V T. Since the macroscopic v o r t i c i t y to was assumed to arise as a result of averaging over many superfluid vortices i n some small volume t h i s means i t i s consistent to interpret x(z,t) as a mean vortex path. Hence z(x,t) would be the i n i t i a l position of a mean vortex with position x at time t. Eq. (4-28) then states that the mean superfluid vortices move with v e l o c i t y V T. Apart from the A terms t h i s corresponds to ±4 the pure V g vortices of LFK. Note that the v a r i a t i o n a l p r i n c i p l e given by Eq. (4-7) i s confined to the case B' = 2. I f B' Is ar b i t r a r y then taking the c u r l of Eq. (4-4) implies that the ve l o c i t y of the vortices i s given by V , ' - 7 ( P V + P V ) + (B' - 2)V - (B' - 2)V + L p s s n n 2 p n 2 p s B'p ( 7 - - - ~ ) ^ x ^ < 4 " 4 2 > P s 2 p P s Since the A terms arise solely as a result of the averaging procedure the usual Landau two-fluid equations must be regained i n the l i m i t A 0. Hence the condition B' = 2 i s necessary and s u f f i c i e n t for the superfluid vortices to t r a v e l with the mass f l u x v e l o c i t y f = - (p f +p V ) p s s s s On the other hand the work of LFK shows that the l a t t e r condition Is equivalent to the requirement that superfluid vortices be regarded as s i n g u l a r i t i e s i n the superfluid v e l o c i t y V^.Thus the physical interpretation of the mathematically necessary r e s t r i c t i o n B' = 2 i s -54-that the superfluid vortices (before the averaging i s carried out ) be regarded as s i n g u l a r i t i e s i n the superfluid v e l o c i t y V . Note that t h i s view of superfluid vortices i s consistent with Eq. (B-6). k3 A direct measurement of B' has been made by Snyder (1963) and tk Snyder and Linekin (1966) by observing the mode s p l i t t i n g i n a rotating second-sound resonator.They found that B' was approximately 9 zero and s l i g h t l y temperature dependent. L i n (1963) has c r i t i z e d t h i s experiment on the grounds that secondary motion along the axis of rotation may occur. From Eq. (4-42),B' = 0 implies that superfluid vortices t r a v e l with v e l o c i t y V (apart from the X terms ). LFK show s that t h i s corresponds to superfluid vortices being regarded as - > - - » • -»•-»• combinations of s i n g u l a r i t i e s i n V"s and A = ( x/s) ( v n -^ s)» which i s not consistent with Eq. (B-6). Hence there appears to be a contradiction between the experimental results of Snyder and Linekin and Eq. (4-1) of the KB equations. For further discussion of the experimental results and the case B' = 0 see Appendix D. 4-5 Symmetries and Conservation Laws Just as i n Sec. 3-3 there are certain gauge transformations of the Monge potentials a, 6, z*, Y * and z^ which do not change the values of V q or V G and which keep the v a r i a t i o n a l equations form ->- -> invariant. The requirement that V N and V G be unchanged and that Eqs. (4-13) and (4-14) be form invariant under the i n f i n i t e s i m a l gauge transformations gives V(6a + Y ^ Z J ) = -Syiyz-i + 6z Jv" Y J (4-43) V(63 - + ' Y j6z j) = 63V (-) - 6 Y jVz j + 6 z j V Y j (4-44) X X -55-The form invariance of Eqs. (4-15)-(4-21) gives £ (6a) = £ (6Y J) = £ (62 3) = 0 (4-45) dn d n ->- d n -> (66) = (6 Y) = (6z) = 0 (4-46) which have the solutions 6a = 6a(y,z), 6y 3 = 6Y 3(Y,Z), 6z 3 = 6£3(y,z) (4-47) 66 = 66(s /x,Yz), 6y = 6Y(S/X,Y» z), 6Z = 6Z(S/ X,Y,Z) (4-48) Substitution of Eqs. (4-47) and (4-48) into Eqs. (4-43) and (4-44) give (e ^-r + 6 f j ) v z j + - 6Z3)V"Y3 = 0 9z 3 99 J (e T T T T - 66)Vs + ( e + 6 Y 3 ) V z 3 + (E - 6Z 3)VY 3 = 0 (4-49) 9 ( S / X ) 9z 3 9 Y 3 where EG(Y,Z) = 5a + y2&z2 and EH(S/X,Y,Z) E (S/X)66 + Y 36Z 3. Eqs. (4-49) imply that the i n f i n i t e s i m a l gauge transformations have the form 6a = EG + EY 3 , 6 y j = - e , 6z 3 = E (4-50) 9y 3 9Z 1 9y 3 66 = E , 6y 3 = — E ~ ~ r > 6z 3 = e % (4-51) 3(f) 9z 3 9YJ A where G = G(y.z) i s an arbitrary function of y2 and z 3 and H = H(—,y,z) X s - i i s homogeneous of degree one i n — and y . From Eq. (1-25) the X conservation laws which arise from these symmetries are (pG) + V-(pGVL) = 0 (4-52) -56-~ (p XH) + V-(p XH V n) = 0 (4-53) Under the Gal i l e a n transformations the transformation properties -> -> i i of p, s, Xi V , V , z and z are known while the transformation n s properties of a, 3, y 1 and yX are deduced by re q u i r i n g that the Lagrangian remain i n v a r i a n t . ( i ) Under the t r a n s l a t i o n of axes x ->• x' = x + 6x and z 1 -»• z' 1 = z 1 + Sx1* z 1 -> z' 1 = z 1 + fix1. In t h i s case the gauge transformation — s i s given by G = G(z) and H = H(— , z ) . The conservation of momentum X equation (4-5) follows from Eq. (1-25). ( i i ) Under the time t r a n s l a t i o n t t' = t + 6t and z 1 -* z' 1 = z1 - V 1 ( z , o ) 6 t , z 1 z' 1 = z 1 - V 1(z,o)At. The Monge po t e n t i a l s y1 and n L y 1 transform as ^ + y = Y + Y J(8V J(z,o)/3z )St and Y 1 Y ' 1 = Y 1 + Y J O V j ( z , o ) / 9 z 1 ) 6 t g apart from a guage transformation of the type G = G(z) and H = H(— , z ) . X From Eq. (1-25) the conservation of energy equation i s -r- P v +h P V 2 + pe) + V-(h P V 2V +h P V 2V + (pe + P)V 9t n n s s K n n n s s s + psT(V -V) + h P (V -V ) 2 ( V - V) + X - x (V_ x to)) n n n s n co L = 0 (4-54) • + - y - y - y - y - y - y ( i i i ) Under the r o t a t i o n of axes x ->• x = x + 66 x x and V -»• V = ^ ' n n V + 66 x V , V -»- V = V + 66 x V , z -v z' = z + 66 x z, n n s s s s' - 5 7 -f + z' = z + 6$ x ?. The Monge potentials transform as y Y = Y' + 6^ x y and y y' = y + <5^  x y apart from a gauge transformation of the type G = G(z) and H = H(—, z ) . From Eq. (1-25) the conservation X of angular momentum equation i s £ ( e i j k P x J V k ) + V ^ e 1 ^ * 3 [p V V + p v V + P 6 U 3t n n n s s s + X a ) 6 k £ - AwVVu)]) = 0 (4-55) (iv) Under the Galilean boost x -»• x' = x + 6 V t and V -> V' = V" + 6V , o n n n o V -> V' = V + 6 V . The Monge potential a a ' = a - x*6V apart from s s S 0 0 g a gauge transformation of the type G = G(z) and H = H(— , z ) . From X (1-25) the conservation of center-of-mass equation i s (pvS-px1) + Vd([p vV+p v V + A6o 1 J - AuV/aiJt du n n n s s s - p x V ) = 0 (4-56) In conclusion, the relaxation of the conservation of p a r t i c l e l a b e l constraint i n two-fluid hydrodynamics gives r i s e to superfluid v o r t i c i t y dependent contributions to the in t e r n a l energy. The hydro-dynamic equations describe superfluid helium with a f i n i t e density of superfluid vortices where a l l hydrodynamic variables are averaged over many superfluid vortices. -58-CHAPTER 5 LIN'S CONSTRAINT GENERALIZED TO INCLUDE HIGHER ORDER DERIVATIVES 5-1 An Extension of the Results of Chapter 2 The v a r i a t i o n a l p r i n c i p l e given i n Chapter 2, which depends on derivatives no higher than f i r s t order, can ea s i l y be extended to arbitrary order derivatives. The fundamental assumption i s that the i n t e r n a l energy of the f l u i d contains an additional contribution pgCco^V^co^jV^k* " "a)*) which depends on the v o r t i c i t y and gradients of the v o r t i c i t y . Just as i n Chapter 2, this theory may be interpreted as describing a f l u i d with a f i n i t e density of vortices where the hydro--y -y dynamic variables V, p , s, and z have been averaged over a region containing many vortices. The additional V ' V terms arise from considering interactions between neighboring vortices. The Lagrangian for this case i s given by L = h PV2 - pe(p,s) - p g C c / . v V . V ^ ' - ' c o 1 ) - »{|^+ V-(pV)} + <* f + ^ # «"» The dependent variables are {V,p,s,z,a,3,y} and co = -V$ x Vs - Vy 3 x v"zJ -y (only after the v a r i a t i o n of V i s carried out can the i d e n t i f i c a t i o n -y -y -y OJ = V x v be made). The v a r i a t i o n a l derivatives of pg with respect to -y -y 3, s, y and z are straightforward to compute: (-pg) = vs-V x [± ( p g ) ] , A (_pg) = 43.^ x [5 ( p g ) ] ( 5_2) O P oo) o s 6o) ->-with si m i l a r expressions for Y a n d z. The v a r i a t i o n a l equations of L are given by -59-6V: V = -Va - 3Vs - y JVz J (5-3) 6a: I 6 - + V-(pV) = 0 (5-4) „ da , 0 ds , j dz-1 . i „? P n /-c o 6p: — + 3 — + Y J — + Js V 2 - e - - - g = 0 (5-5) 63: p | = -^s-V x [A (pg)] (5-6) 6s: - -V3-^ x [A (pg)] - PT (5-7) dt Soa 6 Y j: p ^ ^vV-v" x [A (pg)] (5-8) d t 6a> 6 z j : p = - V * ^ * [4: (P8) ] (5"9> a c 6a) Clebsch's lemma yields the identity | - - * (f + '^2> - f *• " ft - ^ " A £ (5-10) Substitution of Eqs. (5-3)-(5-9) into Eq. (5-10) gives p • — + V 1? = -pvS + V3-V x [A ( p g ) ] V i s d t 5oo - vs-^  x [A (pg)]vi3 + vVv x [A ( P g)]vV 6co 6o) - vV-V x [A (pg)]V iy j (5-11) 00) By using the vector identity A x (B x C) = (A»C)B - (A«B)C Eq. (5-11) becomes P + VP = -Vg + V x [A (pg) ] x w (5-12) a t 6a) or equivalently p — - + V j(P6 i j + T l j) = 0 (5-13) -60-Th e symmetric stress tensor i s given by T i j = 6 ( p g ) _ J « ( p g ) + V V 3 ( p g ) 6o> 6o> 3VJo) + 2V 1 V (pg) - V* [ V 1 ^ — i — (pg)] + ... (5-14) Multiplication of Eq. (5-8) by 9z-,/3x'L yields the equivalent form v = l*£tl_^x f 6 ( p g ) ] ( 5 _ 1 5 ) which states that V' = V + ^- v" x [ — ( p g ) ] ±s the tangent to the P So) particle paths x = x(z,t). The equivalence of the hydrodynamic equations (5-4), (5-6) and (5-12) with the variational equations (5-3)-(5-9) follows from a straightforward extension of Weber's transformation as developed in Sec. 2-2. In addition the conservation law associated with the gauge invariance of L is unchanged from Eq. (2-30). 5-2 A Negative Result for Viscous Fluids The variational principle given in Chapter 2 adds terms to the momentum equation which depend on second derivatives of the velocity. A natural question arises: Can the velocity terms p - 1 v " x v" x v which occur in the momentum equation for incompressible, viscous fluids be derived solely from a generalized Lin's constraint of the type given in Chapter 2? Note that there is no chance of obtaining the compressible terms p _ IV(V*V) since these terms involve a which would alter the conservation of mass equation. The variational principle of Chapter 2 uses the fact that the Lin's constraint may be generalized from dz-'/dt = 0 to dz J/dt = p - 1 (VXA) • V^z3 -61-without a f f e c t i n g the conservation of mass equation and without destroying the gauge invariance of the momentum equation. Consider the Lagrangian L = h PV 2 - pe(p.s) - a {||- + V-(pV)} + p3 { % + ^ x l 4 } ot dt p + py j {^r~ + - v" x A • Vz 3} ( 5 - 1 6 ) ' dt p ->• -+ ->-where A = A ( V z ) . The v a r i a t i o n a l equations are ( 5 - 1 7 ) ( 5 - 1 8 ) 6V: V = -Va - 3Vs - y j ; 6a: ^ ( p V ) = 0 5 p : dz3 dt 63: p £ . jBj x at ->-A 6s: p £ = jfe-S x -»• A -6y 3: dz3 $ i ± p d t = -Vz 3-V '-»• x A 6z 3: dy 3 A i A p dir = ~V y v ->-x A e - J = 0 ( 5 - 1 9 ) P ( 5 - 2 0 ) ( 5 - 2 1 ) ( 5 - 2 2 ) ( 5 - 2 3 ) 6z 3 Clebsch's lemma combined with the vector i d e n t i t y A x (B x C) = (A*C)B - (A*B)C y i e l d s the momentum equation d t ~ + ^" v ± p = T t (^ x A ) x (V x ft]* + (t-V" x ( 5 - 2 4 ) dx oz By expanding the v a r i a t i o n a l d e r i v a t i v e f o r z 3 Eq. ( 5 - 2 4 ) can be written as -62-iz l + 1 vS - - 1 dt p p ,kmn 3A dz 3 3vV Sx 1 3A* 3vV 3z J Bx 1 r V V + - [(VxA) x ( V x v ) ] 1 (5-25) P A necessary condition to obtain (y/p)v" x v" x v on the R.H.S. of M . Eq. (5-25) i s ([y] LT ) kn(m 9A 3z J _ iP(m &)Pn „ s . . ye e S V ^ V 3 X 1 (5-26) i r Mu l t i p l y i n g Eq. (5-26) by 3x /3z and summing over m and H gives kmn 3A k „ 3x n £ r = 2y r 3 V V (5-27) 3z-D i f f e r e n t i a t i n g Eq. (5-27) by 3/3(V nz J) gives 0 = 2 y 9 ( 9 x n / 8 z J ) E _2y 4^ t 0 3(3z J/3x n) 3z J 3 Z 3 (5-28) where the de r i v a t i v e s of 3x n/3z J are computed from Eqs. (1-8). Eq. (5-28) shows that no s o l u t i o n f o r A e x i s t s , hence the following theorem: The v e l o c i t y terms p-1 v"(V«V) and p-1V x v" x v which occur i n the momentum equation f o r viscous f l u i d s cannot be derived s o l e l y from a generalized Lin's constraint of the form dz^/dt = p-1 (V~ x A) • v"z J . ->-The theorem can e a s i l y be extended to include A = A(p,s,Vz). I t i s possible that a d i f f e r e n t modification of the Lin's constraint could generate the incompressible terms p-1V" x v~ x v, but t h i s would give no in s i g h t i n t o the compressible terms p-1v"(V«V) or the entropy production equation. This suggests that a generalized Lin's constraint plays no ro l e i n describing viscous fluids,whose v a r i a t i o n a l formulation remains an unsolved problem. -63-5-3 Conclusion This thesis has presented a new extension of Clebsch's v a r i a t i o n a l p r i n c i p l e for perfect f l u i d s , based on a generalized version of the conservation of p a r t i c l e l a b e l constraint. For the one-component case the v a r i a t i o n a l , p r i n c i p l e gave a macroscopic description of a f l u i d with a f i n i t e density of v o r t i c e s , for the two-fluid case i t yielded the Khalatnikov equations for rapidly rotating superfluid ^He. To the author's knowledge th i s v a r i a t i o n a l p r i n c i p l e has not been described i n the l i t e r a t u r e and represents o r i g i n a l research. The considerable d i f f i c u l t i e s i n finding a v a r i a t i o n a l p r i n c i p l e for f l u i d s are connected with the presence of f i r s t order derivatives and n o n - l i n e a r i t i e s i n the equations of motion. Clebsch's solution of this problem sharply r e s t r i c t s the form of the constraint equations for the mass density, the entropy and the p a r t i c l e l a b e l . The author has shown that the conservation of p a r t i c l e l a b e l constraint may be generalized to Bz^/St + (V + p - 1 ^ x A) • v"zJ = 0 without destroying the "gauge" invariance of the conservation of momentum equation. A review of C.C. Lin's Lagrangian for the adiabatic case was given i n Chapter 1. The consequences of relaxing Lin's constraint for a one-component f l u i d were examined i n Chapter 2 and yielded v o r t i c i t y dependent contributions to the i n t e r n a l energy and the stress tensor of the f l u i d . In Appendix B, the hydrodynamic equations were interpreted as describing a f l u i d with a f i n i t e density of vor t i c e s , where a l l hydrodynamic variables have been averaged over regions containing many vortices. As background material, Chapter 3 reviewed Herivel's v a r i a t i o n a l -64-p r i n c i p l e for the Landau two-fluid equations. In Chapter 4 a new v a r i a t i o n a l p r i n c i p l e for the Khalatnikov equations of rapidly rotating superfluid ^He was presented. I t was found necessary to use two Lin's constraints, one constraint for the normal v e l o c i t y f i e l d and the other constraint expressing the p o s s i b i l i t y of l a b e l l i n g a super-f l u i d vortex, averaged over many vortices. Chapter 5 concluded with an extension of the v a r i a t i o n a l p r i n c i p l e to arbitrary orders of derivatives i n the v o r t i c i t y . In addition i t was shown that a generalized Lin's constraint cannot be used to describe viscous f l u i d s . -65-REFERENCES 1. L. Euler, Hist. Acad. B e r l i n , Opera Omnia I I J.2, p. 54, (1755). 2. J.L. Lagrange, Nouv. M6m. Acad. S c i . B e r l i n , Oeuvres 4_, p. 706, (1781) 3. A. Clebsch, J. Reine angew, Math 56, 1 (1859). 4. G. Monge, Mem. Acad. S c i . Paris 1784, 502 (1787). 5. H. Bateman, Proc. Roy. Soc. A 125, 598 (1929). 6. H. Lamb, Hydrodynamics (Cambridge University Press, 1916), p. 239, 7. A.H. Taub, Proc. Symp. Appl. Math 1, 148 (1949). 8. J.W. H e r i v e l , Proc. Camb. P h i l . Soc. 51, 344 (1955). 9. C.C. Lin (unpublished, 1959); C.C. L i n , Liquid Helium, Proc. Int. School of Physics, Course XXI (Academic Press, 1963) p. 93. 10. J. Serrin, Handbuch der Physik, v o l . 8 (Springer-Verlag, 1959), p. 123. 11. C. Eckart, Phys. Fluids 3^ , 421 (1960). 12. P. Penfield, Phys. Fluids 9^, 1184 (1966). 13. R.L. Seliger and G.B. Whitham, Proc. Roy. Soc. 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Donnelly, Experimental S u p e r f l u i d i t y (The U n i v e r s i t y of Chicago Press, 1967) p. 130. - 6 7 -APPENDIX A PROOF OF CLEBSCH'S LEMMA Define V(x,t) = VA(x,t) + B a ( x , t ) V C a ( x , t ) (A-l) where a = l,...,m. Straightforward d i f f e r e n t i a t i o n y i e l d s the r e s u l t s || . S(|A> + | £ ^ + ^ ( A . 2 ) while a lengthy algebraic manipulation gives the i d e n t i t y (V-V)V + V(h V 2) = V(V-VA) + (V-VB a)VC a + B aV(V-v"c a) (A-3) Addition of Eqs. (A-2) and (A-3) gives Clebsch's lemma where d/dt E 3/8t + (V«V). - 6 8 -APPENDIX B INTERPRETATION OF THE EQUATIONS OF MOTION F i r s t consider the case of the rotating superfluid as discussed -»• -»• i n Chapter 4. To maintain the i r r o t a t i o n a l condition 7 * V = 0 i n a s rotating superfluid, superfluid vortices are formed (possibly at the boundary) which rotate r i g i d l y with the container, i . e . Figure 1. Rotating Superfluid H^e Integrating around a closed path enclosing a l l the vortices gives a r e l a t i o n between the angular rotation m and the t o t a l number of vortices N. If the vortices each have a strength h/2nm then or N/A = 2a)m/h. Hence for a rapidly rotating superfluid many superfluid vortices are formed to maintain the i r r o t a t i o n a l condition. Consider an array of such vortices each with core size a and some mean separation b. Since the v e l o c i t y f i e l d of each vortex i s 2iuA = 7* x V «dA = <& V -dZ = Nh/m (B-l) s I s -69-V g = (h/2Timr)e (B-2) then the energy per un i t length of a vortex i s E' = (% p V 2)2Trr dr = (p h 2/4irm 2)ln(b/a) (B-3) s s s a Now define an average v e l o c i t y f i e l d <V g> by averaging over a region containing many su p e r f l u i d v o r t i c e s i n such a way that the c i r c u l a -t i o n i s due to the enclosed v o r t i c e s . I f <Vs> does not vary appreciably over the area enclosed then |V x <v >|A = V x <v >«dA = <V >'dl = Nh/m (B-4) S J S I S Hence the number of v o r t i c e s per un i t area i s Y = |V x V |m/h (B-5) A S (dropping the average symbol < >) and the energy per unit volume due to the v o r t i c e s i s e = (p gh/4irm)ln(b/a) |V x | j = A|v x V | (B-6) The t o t a l i n t e r n a l energy i s given by the generalized Gibbs r e l a t i o n de(p,s,x,oj) = T ds + (P/p 2)dp + %(V - V ) 2 d X + (A/p)du> (B-7) XI s - > • - » • - » -where X = p /P and to = V x V . The Khalatnikov equations follow n s from Eq. (B-7) by the standard technique i n hydrodynamics, namely manipulating the redundancy of the conservation of t o t a l energy equation. The conservation of mass equation remains unchanged -70-|f i . + v". j = 0 (B-8) d C - > - - » - ->• where J = p V + p V , p = p + p . The conservation of momentum and n n s s n s t o t a l energy equations become | i l + V j ( n i : i + T T I J ) = 0 (B-9) ot If + V-(Q + q) = 0 (B-10) where = p v V + p v V + P6 i : i , n n n s s s P = p(-e + T + %(v - V ) 2 X + y) , t> n s E = pe + h P V 2 + h P V 2 , n n s s 5 = H pnVl\ + h p s V f s + ( p e + P ) ^ / p + p T s ( ^ n " V + h P(vn - v s) 2(vn - J/p) and TT , q remain to be determined. The entropy equation remains unchanged. A ( p s ) + v.(psV n) = 0 ( B - l l ) while the sup e r f l u i d equations become oV + (V -V)V + Vy = f (B-12) dt S S where f i s to be determined. Eqs. (B-8)-(B-12) give nine equations i n eight unknowns V n, V g, s, ii •*• -»• p; the self-consistency conditions determine TT , q and f. From the d e f i n i t i o n of E and using Eqs. (B-7), (B-9) and (B-12) i t follows that -71-| i . _v".Q - V 1 ( T T i j v i ) + T [ ^ r (ps) + V.(psV ) dt n d t n + X | ^ + X v V c o + i r i j v j V j + ( J - p V ) . ( l + ux (V -V ) ) (B-13) 9t n n n n s Taking the c u r l of Eq. (B-12) gives X | ^ = Xv • Vx[f + co x (V - V )] - Xv-Vx(<o" x V ) (B-14) 3t n s n where v = <o/<o. Substitution of Eqs. ( B - l l ) and (B-14) in t o Eq. (B-13) gives | ^ + V ^ Q 1 + / V + X{v x (f + t x (V - V ) } ± 91 n n s = ( T T I J - Xco61:i + X c o V / c o ) V : i V 1 n + [f + co x (V - V ) ] • [J - pV + V x ( X v ) ] (B-15) H' s n. Comparison of Eqs. (B-10) and (B-15) shows that q 1 = 7 r i J V j + X{v x (f + t x (V - V ) } 1 (B-16) n n. s T T 1 j = XcofS1^ - XcoVVco (B-17) [f + t x (V - V ) ] • [ J - pV + V x ( X v ) ] = 0 (B-18) n s n The lowest order s o l u t i o n of Eq. (B-18) which holds f or a l l values of ->- •+ p, V and V i s ' n s f = - w x (V - V ) + aco x ( J - pV + Vx (Xv ) ) (B-19) n s n In order that the Landau equations be regained i n the l i m i t X -*- 0, the parameter a = p - 1 . Substitution of Eqs. (B-17) and (B-19) s int o Eqs. (B-9) and (B-12) give the Khalatnikov equations as presented i n Chapter 4, j u s t i f y i n g t h e i r i n t e r p r e t a t i o n as a macroscopic theory of -72-s u p e r f l u i d helium. Requiring that there be many v o r t i c e s per unit area and that the s u p e r f l u i d v e l o c i t y be less than the c r i t i c a l v e l o c i t y l i m i t s the a p p l i c a b i l i t y of the Khalatnikov equations to the region (h/mL2) < OJ < (h/mL 2)ln( — ) where a i s the core radius and L i s some - - a length c h a r a c t e r i s t i c of the flow. A s i m i l a r i n t e r p r e t a t i o n can be given the one-component f l u i d s discussed i n Chapter 2. Consider a perfect f l u i d with an array of v o r t i c e s present, each vortex separated by a mean distance b, with a v e l o c i t y f i e l d V = ( Y/r)6 (B-20) and a core s i z e a. The energy of each vortex per unit length i s rb E' = Qi pV 2)27rr dr = irpy 2 ln(b/a) (B-21) Now define an average v e l o c i t y f i e l d <V> by r e q u i r i n g that the c i r c u l a -t i o n of <V> around some closed path be equal to the c i r c u l a t i o n due to the enclosed v o r t i c e s . I f <V> does not vary appreciably over the enclosed area then V x <V> A = x <V>'dA = (b <V>«d£. = 2TTNY (B-22) Thus the number of v o r t i c e s / u n i t area i s 0) N =  A 2TTY (B-23) where u> •= V x v (the average symbol i s omitted). The energy/unit volume due to the v o r t i c e s i s e = h py ln(b/a)ai (B-24) -73-I f the vortex strength y i s allowed to depend on the number of v o r t i c e s / u n i t area then Eq. (B-23) defines an i m p l i c i t function f o r N/A = g'(oi) and hence Eq. (B-24) can be written as e = Pg(o>) (B-25) The t o t a l i n t e r n a l energy of the f l u i d i s j u s t pe(p,s,o>) = pe(p,s) + pg(oi) where e(p,s) i s the usual expression f or the i n t e r n a l energy i . e . , de(p,s) = T ds + (P/p2)dp (B-26) The hydrodynamic equations follow from the redundancy of the t o t a l energy equation. The conservation of mass equation remains unchanged. |£- + V- ( PV) = 0 (B-27) d t while the conservation of momentum, energy and entropy equations take the form ~ (pV 1) + V j ( n i j + T r i j ) = 0 (B-28) | | + V.(Q + q) = 0 (B-29) ^ - (ps) + V-(psV) + R = 0 (B-30) where n l j = p v V + P 6 l j , E = h pV 2 + pe + pg, Q = [h pV 2+pe + pg]V and I T 1 " ' , q and R remain to be determined. From the d e f i n i t i o n of E and Eqs. (B-27), (B-29) i t follows that |f + V^Q 1) - T < £ (ps ) + V.(psv))-V j v V J + p | A V-Vo)) (B-31) d t d t dOO d t -74-Taking the c u r l of Eq. (B-28) gives 9oi i j k i _ i r -£Nk , m „ k k2., .„ . o s — = e J v V J[-(ai x v ) + TV s - p V TT ] (B-32) O t where v = u/u. Eqs. (B-31), (B-32) y i e l d — + V (Q + P T ^ E J V j [-(u) x V) + TV s — p V TT ] ) O t OO) = T ^ - (ps) + V.(psV) + vs-V x ( p | £ v)) d t oO) + [ - p ^ V ^ - (u x V ) 1 ] ^ 1 + V x ( p f - v ) 1 ] (B-33) 00) Comparison of Eqs. (B-29), (B-30) and (B-33) shows that R = Vs«V x ( p |& v) (B-34) 00) [ - p - ^ ^ T r 1 ^ - ( u x V l I p V 1 + V x ( p A ^ ) 1 ] = o (B-35) 00) The lowest order s o l u t i o n of Eq. (B-35) i s y V j = a e i j kw j [V k + p-!v x ( p |& v~) k] - p " l x V ) 1 (B-36) 00) The requirement that the usual equations be regained i n the l i m i t g -*• 0 f i x e s a = p _ 1 . Eq. (B-36) becomes V \ i j = V 1 ( p | - [a) 26 i j - o i V ] ) (B-37) 00) and hence Eqs. (B-27)-(B-30) reduce to the hydrodynamic equations of Sec. 2-2, j u s t i f y i n g t h e i r i n t e r p r e t a t i o n as a macroscopic d e s c r i p t i o n of a perfect f l u i d with a density of v o r t i c e s present. -75-APPENDIX C TURBULENT SOLUTIONS OF THE EQUATIONS OF MOTION The i r r e g u l a r and disordered flow of a f l u i d known as turbulence i s characterized by v o r t i c i t y i n three dimensions and by energy transfer from the large scales of motion to the small scales, ending i n d i s s i p a t i o n . I t i s usually assumed that turbulent f l u i d flow i s described by the Navier-Stokes equations and experimentally i t i s found that turbulence occurs when the Reynolds number R > 20,000. Since the hydrodynamic equations of Sec. 2-2 describe a f l u i d with a d i s t r i b u t i o n of v o r t i c i t y , i t i s worthwhile to check whether they can provide a model f o r turbulence. A u s e f u l concept i n the s t a t i s t i c a l d e s c r i p t i o n of turbulence i s the two-point v e l o c i t y c o r r e l a t i o n function defined as the time average <V 1(x,t)V J(x',t )> = LIM ^ V 1 ( x , t + t ' ) V J ( x ' , t + t , ) d t ' (C-l) T + 0 0 •'0 which r e l a t e s adjacent f l u c t u a t i o n s i n V 1(x,t) and V J ( x , t ) . Turbulence i s u s u a l l y assumed to be an incompressible flow, where the s t a t i s t i c a l properties are homogeneous, i s o t r o p i c and time-independent. Since space and time d e r i v a t i v e s are assumed to commute with the averaging process, homogeneity implies the r e l a t i o n s <lY_ix)_ v k ( x . ) > = _JL_ < v 1( x)V k(x*)> = ~ - <V ±(x)V k(x + r)> (C-2) 9x J 3x J 3r J where x = x + r. Isotropy implies that ^ ( x + r)P(x)> = 0 (C-3) fo r any s c a l a r function P(x). -76-The following r e l a t i o n s i n v o l v i n g two-point c o r r e l a t i o n s of the v o r t i c i t y w i l l prove u s e f u l <oi i(x)a) ; i(x ,)> = - E ^ E 3 " f <V A(x)V n(ac + r)> (C-4) 3 r k 3 r m Contraction over i and j gives <w(x)«o)(x + r)> = ¥—r <V(x)»V(x + r)> S r ^ r 1 = h —\ r <[V(x) - V(x + r ) ] 2 > (C-5) 3r 3 r x In Kolmogoroffs theory of turbulence the f l u i d flow i s pictured as a superposition of eddies of various s i z e s with energy being transferred from l a r g e r to smaller eddies at a constant rate e o [ L 2 T - 3 ] . The energy i s ult i m a t e l y d i s s i p a t e d by viscous e f f e c t s i n the smallest eddies of length scales (e v - 3 ) - " 2 * (v i s the kinematic v i s c o s i t y ) . I t i s assumed that i n the time-independent regime the s t a t i s t i c s are completely determined by e Q and v. Furthermore f o r those eddies smaller than the larges t scales but larger than viscous sca l e s , the i n e r t i a l subrange, the s t a t i s t i c s are completely determined by e . J o By dimensional analysis Kolmogoroff's theory then predicts the form of the two-point c o r r e l a t i o n function i n the i n e r t i a l subrange as <[V(x) - V(x + r ) ] 2 > - 4K e 2 / 3 r 2 / 3 (C-6) where K - .5 i s Kolmogoroff's constant. Eq. (C-6) has been w e l l -v e r i f i e d by experiment and any successful model of turbulence must reproduce t h i s r e s u l t . Adopting Kolmogoroff's assumption that only e Q be used i n the i n e r t i a l subrange implies by dimensional analysis that g(w) = - K ' e ^ - 1 where K' i s a dimensionless constant. In the incompressible, i s e n t r o p i c case the hydrodynamic equations of Sec. 2-2 reduce to V-V = 0 (C-7) •+ ?r- + (V«V)V = V (- - - g) - w x V x (K'e wuT3) (C-8) 3t p o v = 3x(z,t) _ + x ( K , e ^ - 3 ) . (C-9) Now assume that 3x/3t i s i d e n t i f i e d with the mean flow <V> and that -V x (K'e uiuT 3) i s the turbulent part of the v e l o c i t y f i e l d . I f <V> = 0 then Eq. (C-7) reduces to an i d e n t i t y and Eqs. (C-8) and (C-9) become # = V"(- 1 - g) (C-10) 3t p & V = -V x (K'e wo)"3) ( C - l l ) o M u l t i p l y i n g Eq. (C-10) by V 1 ( x + r) and averaging gives the time independent condition A <V i(x)V ±(x + r)> = - - \ <V i(x + r ) ( - + g)> 5 0 (C-12) 9 t 3 r X P Taking the c u r l of Eq. ( C - l l ) gives a)1 = -V^K'e v".(wuT3)) + V 2(K'e (A ) - 3 ) (C-13) 0 o M u l t i p l y i n g Eq. (C-13) by w 1(x + r) and averaging gives <w(x)'"5(x+r)> = K'e — 9 r <u>(x + r) • U( X)ID(x)~ 3> (C-14) ° S r ^ r 1 -78-To close Eq. (C-14) i t i s s u f f i c i e n t to assume the closure r e l a t i o n <u(x + r ) - o j ( x ) o j ( x ) _ 3 > = K"<w(x + r)4(x)>~1'5 (C-15) Note that Eq. (C-15) does not follow from the Lagrangian.Eq,(C-14) gives <w(x)-u>(x + r)> = K'K"e — ¥ — r <w(x + r) 4(x)> _ l £ (C-16) 3r 8r which has the s o l u t i o n <u(x)«u(x + r)> = (10 K'K " e o / 9 ) 2 / 3 r~k/3 (C-17) The i d e n t i t y Eq. (C-5) implies that the two-point v e l o c i t y c o r r e l a t i o n function i s given by <[V(x) - V(x + r ) ] 2 > = 9/2(10 K'K"e o/9) 2 / 3 r 2 / 3 (C-18) which agrees with Eq. (C-6) provided Kolmogoroff 1s constant i s i d e n t i f i e d as K = 9/8(10 K'K"/9) 2 / 3 (C-19) Hence the hydrodynamic equations of Sec. 2-2 provide a model of turbulence which i s consistent with Kolmogoroff's theory i n the i n e r t i a l subrange. From Eqs. (B-23)-(B-25) i t follows that t h i s model of turbulence consists of a f l u i d with a density of v o r t i c e s , whose i n d i v i d u a l vortex strengths depend in v e r s e l y on the vortex density to the two th i r d s power, i . e . Y a ( N / A ) " 2 / 3 (C-20) -79-APPENDIX D MEASUREMENT OF B' FROM SECOND SOUND IN ROTATING ^He The parameter B' has been measured by Snyder , Snyder and Linekin 4 5 and Lucas . The experiment of Snyder and Li n e k i n i s described b r i e f l y as follows: a standing wave of second sound with frequency a was excited i n a second sound resonator, r o t a t i n g with angular v e l o c i t y fiz; the C o r i o l i s force and the B' term removed the degeneracy of the two lowest second sound normal modes', the frequency of these normal modes was determined by observing the resonant frequencies of the temperature f l u c t u a t i o n on the resonator w a l l . Using a c y l i n d r i c a l resonator Lucas found B' = .08 ± .08 at 1.603 K and B' = .2 ± .25 at 1.426 K,the r e s u l t s of Snyder and Linekin using a square resonator are given i n Figure 2. Figure 2. B' versus T 1.0 0.8 -< B' 0.6 . 0.4 0.2 0 ' » j ' I 1.25 1.50 1.75 2.00 T (°K) Both these experiments suggest that B' i s much less than 2 and temperature dependent. - 8 0 -46 Donnelly has given a d e t a i l e d d e r i v a t i o n of the frequency s p l i t t i n g of the degenerate second sound modes; a b r i e f summary of t h i s d e r i v a t i o n i s given as follows. L i n e a r i z a t i o n of Eqs. ( 4 - 2 ) -( 4 - 5 ) y i e l d s -*- 1 P B ' p > s -> s ->• -> (D-l) V - = n P VP sVT + r 0 )x q P n 2 P • -»• 1 B ' p -»• n (D -2 ) V = s P VP + sVT - - r — wxq 2p p + p V-V + p V-V = 0 (D -3) s s n n ps + sp + psV-V = 0 (D -4) n where q = V n - V g and i t i s assumed that OJ = 2ttz. Subtracting Eq. (D -2) from Eq. (D-l) and transforming to the r o t a t i n g frame gives q + ( 2 - B ' ) f i x q = _ VT (D -5) n Using s = CT and combining Eqs. (D -3) and (D -4) y i e l d s P s T = - cp~ V-q (D -6) where C i s the s p e c i f i c heat. D i f f e r e n t i a t i o n of Eq. (D -5) combined with Eq. (D -6) gives a wave equation f o r q q + ( 2 - B T ) Q x q = U 2 ^ . ^ ) ( D _ ? ) 2 3 P r " Comparison with the case of f i r s t sound propagation i n a c l a s s i c a l r o t a t i n g f l u i d then gives the second sound normal mode frequencies as a ± . a ± M2zBji£ ( D _ 8 ) •'.•m - om o r> m T\ P s S where u = — — i s the speed of second sound £. P Li n -81-where m Is an odd interger and a i s the degenerate normal mode om frequency i n the non-rotating case corresponding to the intergers ±m. Hence the C o r i o l i s force and the B 1 term cause a s p l i t i n the 8 (2— B') Q normal mode frequencies Ao = — 7 — 7 . Note that gradients i n ^ m m u p and p would contribute a term — q to the LHS of Eq. (D-7), n s p n which causes attenuation but does not a l t e r the s p l i t t i n g of the normal modes. I f B* = 2 then no s p l i t t i n g occurs, i n co n t r a d i c t i o n 9 with the r e s u l t s of Snyder and L i n e k i n . L i n has suggested that some sort of macroscopic secondary motion may occur.supported by the r o t a t i o n , i n c o n t r a d i c t i o n to the assumptions leading to Eq. (D-8). I f B' = 0 then Eq. (4-42) states that the su p e r f l u i d v o r t i c e s t r a v e l with v e l o c i t y V* (apart from the A terms). According to s LFK t h i s corresponds to the su p e r f l u i d v o r t i c e s being regarded as combined A and V g s i n g u l a r i t i e s . This suggests that normal f l u i d v o r t i c e s may be formed i n the experiment of Snyder and -y -y -y -y L i n e k i n . I f t h i s i s the case then both VxV and VxV must be s n treated as thermodynamic v a r i a b l e s and Eq. (4-1) w i l l have an -y -y extra c o n t r i b u t i o n due to v"xV • This poses an i n t e r e s t i n g question dt -y -y -y for future consideration:Can the contributions of VxV and VxV s n to Eq.(4-1) be appropriately adjusted such that a v a r i a t i o n a l p r i n c i p l e can be found and such that the temperature dependence B' agrees with the r e s u l t s of Snyder and Linekin ? This remains an open question at present. Note that f or the experiment of Snyder and Linekin the Reynolds number of the normal f l u i d at 1.8 \ i s given by R = r 2 ^ / ^ * * (2 cm ) 2 ( 2 7r rad /sec)(.05 g/cta 3)(10~ 5 g cm / s e c ) " 1 -13,000 which suggests that normal f l u i d v o r t i c e s may have an important e f f e c t on the measurement of B'. 

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