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On the approach to local equilibrium and the stability of the uniform density stationary states of a… Le, Dinh Chinh 1970

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ON THE APPROACH TO LOCAL EQUILIBRIUM AND THE STABILITY OF THE UNIFORM DENSITY STATIONARY STATES OF A VAN DER WAALS GAS BY LE DINH CHINH • B.Sc., Case Institute of Technology, 1963 M.Sc, Case Institute of Technology, 1965  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE.REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA DECEMBER, 1970  In presenting this thesis in partial  fulfilment  of the requirements for  an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it  freely available for reference and study.  I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.  It  is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of  Pwn  The University of B r i t i s h Columbia Vancouver 8, Canada  ABSTRACT  Some e q u i l i b r i u m and n o n - e q u i l i b r i u m p r o p e r t i e s o f a gas o f h a r d s p h e r e s w i t h a l o n g range a t t r a c t i v e  p o t e n t i a l a r e i n v e s t i g a t e d by  c o n s i d e r i n g t h e p r o p e r t i e s o f an e q u a t i o n , proposed  by d e S o b r i n o  (1967),  f o r a o n e - p a r t i c l e d i s t r i b u t i o n f u n c t i o n f o r t h e gas model c o n s i d e r e d . The  s o l u t i o n s o f t h i s e q u a t i o n obey an H-theorem i n d i c a t i n g  t h a t our gas  model approaches l o c a l e q u i l i b r i u m . E q u i l i b r i u m s o l u t i o n s o f t h e k i n e t i c e q u a t i o n a r e s t u d i e d ; t h e y s a t i s f y an e q u a t i o n f o r t h e d e n s i t y n ( r ) f o r w h i c h space and  dependent s o l u t i o n s e x i s t and c o r r e s p o n d t o a m i x t u r e  l i q u i d phases. The  kinetic  o f gas  .-'  e q u a t i o n i s next l i n e a r i z e d  and t h e l i n e a r i z e d  equa-  t i o n i s a p p l i e d t o t h e study o f t h e s t a b i l i t y o f t h e u n i f o r m d e n s i t y s t a t i o n a r y s t a t e s o f a Van d e r Waals gas. A b r i e f a s y m p t o t i c sound p r o p a g a t i o n an a p p r o x i m a t i o n Gross  and J a c k s o n  analysis of  i n d i l u t e gases i s p r e s e n t e d i n view o f i n t r o d u c i n g of the linearized (1959).  To f i r s t  Boltzmann c o l l i s i o n i n t e g r a l  due t o  o r d e r , t h e d i s p e r s i o n i n t h e speed o f  .sound a t low f r e q u e n c i e s i s t h e same as t h e B u r n e t t and Wang Chang-Uhl e n b e c k v a l u e s w h i l e t h e a b s o r p t i o n o f sound i s s l i g h t l y l e s s t h a n t h e B u r n e t t v a l u e and s l i g h t l y g r e a t e r t h a n t h e Wang Chang-Uhlenbeck v a l u e ; a l l t h r e e a r e i n good agreement w i t h experiment. thod developed rized  F i n a l l y , u s i n g t h e me- .  i n t h e p r e v i o u s s e c t i o n , an a p p r o x i m a t i o n  Enskog c o l l i s i o n i n t e g r a l  f o r the l i n e a -  i s obtained; a d i s p e r s i o n r e l a t i o n i s  d e r i v e d and used t o show t h a t t h e u n i f o r m d e n s i t y s t a t e s which c o r r e s pond t o l o c a l minima o f t h e f r e e energy stable, are i n fact  and t r a d i t i o n a l l y c a l l e d  meta-  stable against s u f f i c i e n t l y small perturbations.  - i i -  TABLE OF CONTENTS Abstract  i i  Table of Contents . Table of Figures  .  . .  . . .  Acknowledgement Chapter 1 - -  . '. i i i v  ; . .  vi .  Introduction  1  1.  1  Introduction  Chapter 2 - - Approach to Local Equilibrium and Equilibrium Properties  5  2. The Kinetic Equation . . . . . . .  5  3.  10  The H-Theorem  4. Equilibrium Solutions Chapter 3 - —  17  Some Linear Non-equilibrium 5.  6.  Results  21  Sound Propagation at Low Frequencies  . . . . .  23  . . ... . .  23  (a)  The GJS Model  (b)  The Dispersion Relation . . . . . . . . .  (c)  The Propagation Constant k at Low Frequencies . . . . . . . . . . . ...... . . .  (d)  Discussion  . . . . . . .  Metastable States (a)  - 31 35  ..... . . . . . .  41  . . . . . . . .  46  The GJS Model for a Gas of Hard Spheres with  a Long Range Attractive Potential .  47  (b) The Dispersion Relation . . . . . . . . .  50  (c)  Stability Criterion . . . . . .  . . .  55  (d)  Discussion  . . .  72  Chapter 4 - - Conclusion 7. Conclusion  . .  -iii-  75 75  Appendix A Appendix B  . ... .  .  78 92  Appendix C Bibliography  -iv-  .  95  . . .  97  TABLE OF FIGURES Figure 1.  Radial Distribution Function g. vs 5=>r/<r for Hard Discs at V= f-v<>/NKr= 5" ; Vo = HO-//27 for an average of Four Independent Calculations of ^, for a system of 192 discs. .  9  Comparison of Theoretical Speeds of Sound of a Maxwell Gas with Experiment . . . . . . . . . . . . .  27  3  Figure 2. Figure 3. Figure 4 . Figure 5 .  Comparison of Theoretical Absorption Coefficients of a Maxwell Gas with Experiment  .  27  Comparison between Theory and Experiment of the Dispersion i n the Speed of Sound at Low Frequencies  .  45  Comparison between Theory and Experiment of the Non-Kirchoffian Frequency Dependence of the Absorption of Sound at Low Frequencies . .  .  45  Figure 6.  Plot of f(y) vs. y . . .  64  Figure 7.a  Plot of E(w,k) for ( + "5+ 10 e + bVi^'-Cr |(i+-r<=)V Unstable Situation  71  Plot of E(w,k) for -f(»+r6)\ i+'5 + io« + bVij < o. Unstable Situation . . . .  72  Figure 7.b Figure 7.c Figure 8 .  /  Plot" of E(w,k) for Stable Situation Plot of ^>(u>,k)  1 + 5 + 10 e + bV^' >.o. 7.2  Showing Analiticity of E(w,k) on S . +  94  ACKNOWLEDGEMENT I wish to thank Dr. L. Sobrino for suggesting this problem and for the many valuable discussions I have had with him. A University of British Columbia Graduate Fellowship and research grants from the National-Research Council of Canada are gratefully acknowledged.  CHAPTER 1. 1.  INTRODUCTION  INTRODUCTION Properties of dense gases can be studied by means of f l u i d  mechanics.  The validity of this method i s restricted; the f l u i d  equations which describe the evolution i n time of the local density, velocity and internal energy density are subject to the requirement that these macroscopic quantities be slowly varying functions of space and time  i.e. that the ratio of the change in a macroscopic variable  to., that variable, over a distance of the order of the mean free path and time of the order of the collision time i s negligible. A more fundamental approach is that of kinetic theory which provides a molecular description of the gas and whose validity i s not restricted by the above conditions.  A kinetic theory of dense gases  was f i r s t introduced by Enskog (1922); this i s a direct extension of the  kinetic theory of dilute gases based on the Boltzmann equation.  In the derivation of this equation i t was assumed that only binary collisions occur and that the molecules have no extension in space. For  a dense gas these assumptions no longer hold; Enskog made a f i r s t  step i n the right direction by considering a gas of rigid spheres of diameter <r . For such a gas, collisions are instantaneous so that the  probability of multiple encounters remains negligible.  change i n the collision frequency X  However a  results from the f i n i t e volume  of - the molecules; this change i s made up of an increase i n K due to the  decrease i n the phase space of the gas and a decrease in A.  due  to molecules screening one another from the other oncoming molecules;thus the collision integral on the RHS of the Boltzmann equation i s increased by a factor of yj^ ; furthermore the one-particle distribution functions in that integral are evaluated at the center of the colliding molecules. i  Although closer to reality, Enskog's theory does not allow for the occurance of ternary and higher order collisions nor i s the spherical model a faithful description of the real intermolecular potential. A rigorous kinetic theory of dense gases based on the Liouville equation was introduced independently by Born and Green, Kirkwood, Bogoliubov, Yvon in 1946. The only assumption i n this theory i s that the interaction potential i s an additive two-particle potential.. A set of coupled equations, the so-called B-B-G-K-Y hierarchy, i s obtained by integrating the Liouville equation over the phases of N-s molecules,(N i s the total nuumber of molecules of a gas and s= 1,2,... N-l).  A method of solving this hierarchy due to Bogoliubov i s clear-  ly presented i n a review article by E.G.D.Cohen (1968). This approach to the study of dense gases takes into account ternary and higher order collisions; however, although more satisfying than the method of Enskog from a mathematical point of view, Bogoliubov's method has met with some d i f f i c u l t y ( due to the divergence of the expansion of the two-particle distribution function in terms of the one-particle distribution function) which prevents i t s use, at the present time, for the study of the problem of condensation. DeSobrino (1967) proposed an equation for the one-particle d i s t r i bution function of a Van der Waals gas obtained from the f i r s t  equation of the B-B-G-K-Y hierarchy by rewriting the two-particle distribution function ^ in terms of the one-particle distribution function £ and the pair correlation function at contact  . When  is  assumed to be velocity independent, this equation reduces to the Enskog  equation with an attractive potential taken into account; deSobri-  no further assumed that K| i s equal,to (i-«b)' which is equivalent to stating that the model under consideration is the traditional Van der Waals gas. The problem of determining the stability of uniform density stationary states was investigated using a simple relaxation time approximation for the collision integral. In this thesis the problem of approach to local equilibrium and of stability of uniform density stationary states are studied in a more accurate manner. We do not use any specific form of ^ which we simply r e s t r i c t to be a monotonically increasing,, continuous function of the density; we also obtain a dispersion relation needed in the study of stability using a more accurate approximation of the kinetic equation along the lines of a method developed by Gross and Jackson (1959) and Sirovich (1965a). This thesis i s divided into two parts; the f i r s t part to be found in chapter 2, deals with the approach to local equilibrium and the equilibrium solutions of the kinetic equation (2.10). An H function i s defined which, in the equilibrium l i m i t , yields the correct thermodynamic functions of a Van der Waals gas; the corresponding H theorem i s proved for a distribution function |. which satisfies our kinetic equation; this implies that the gas described by this equation  approaches local equilibrium.  Stationary solutions of (2.10) are then  found to obey an equation for the density  -n-Cc) which i s identical  with the equation obtained by Van Kampen (1964) from equilibrium stat i s t i c a l mechanics; the existence of space dependent solutions of this equation has been discussed by Van Kampen (1964) and Strickfaden (1970). This equation also gives excellent agreement with experiment i n calculations of surface tension not too close to the c r i t i c a l point (Strickfaden and deSobrino, 1970). Because of the complicated expression for'the Enskog collision integral, a calculation of non equilibrium solutions of the kinetic equation (2.10) i s d i f f i c u l t .  However, for a gas i n a near e q u i l i -  brium state, this equation can be linearized and the result expanded to f i r s t order in gradients V  r  collision integral  about r . The linearized Enskog :  Sf (&) i s equal to .'Y^t(fJ) +• K{&) • €  the linearized Boltzmann operator and term.  where 2/ i s  i s the linearized non-local  Even in this form the linearized kinetic equation does not lend  i t s e l f readily 'to practical computations and an approximation must now be introduced.  In section 5 a method of approximating  2?(L) due to  Gross and Jackson i s introduced i n connection with a brief review of the problem of sound propagation in a dilute gas at low frequencies. In section 6 this method i s applied to the linearized kinetic equation of a Van der Waals gas i n order to investigate uniform density stationary states.  the stability of the  We verified the conclusion arrived  at by deSobrino i n his less accurate calculations that states of a Van der Waals gas, traditionally regarded as metastable, are indeed stable against small dynamical perturbations.  CHAPTER 2. 2.  APPROACH TO LOCAL EQUILIBRIUM AND EQUILIBRIUM PROPERTIES  THE KINETIC EQUATION In this section we briefly describe the gas model under consi-  deration and derive a kinetic equation for this model. We consider, inside a cube of side L, a Van der Waals gas.of N molecules whose interaction potential i s of the form (  t*6  \_ v(r)  r 4 <r  -for  W  r  >  r  where r i s the distance from the center of a given molecule and V(r) is a weakly attractive potential with a range A satisfying the inequality •^3 «  «  L  L i s assumed so large that wall effects are negligible.  No external  force i s present. For this gas model, the following equation for the one particle distribution function -f(.r,|,fc.) was obtained by Grad (1958) from integration of the Liouville equation over a l l coordinates except those of a given molecule.  r  For convenience, we have adopted deSobrino's notation where JL denotes a-unit vector; the svmbol fl indicates integration  over a l l values of J L such that and f i n a l l y  H.(r>0;  i s equal to  | and ^' are r e l a t e d to |_ and ||' as follows  To rewrite eq.(2.1) i n terms of the one-particle d i s t r i b u t i o n function we introduce  the p a i r c o r r e l a t i o n function  defined  by  In (2.3) we have assumed that <fr i s not a function of v e l o c i t y , i . e . that there i s no c o r r e l a t i o n between the v e l o c i t i e s of neighbouring p a r t i c l e s .  Substituting the RHS of (2.3) into the  i n t e g r a l on the LHS of (2.1) and integrating over  ^  - f c l t a ' ^ * * ' > £ * ( w W > * l % * .....  where  (?.S)  we obtain  '  M A ) ^  "The value of ^  j jf C r ^ t ) ^  i n F i g . 1 i s a reasonable approximation to the  £j, of our model. d i f f e r s appreciably  The region where the pair c o r r e l a t i o n function from unity i s very small  (. <r < l r - i - ' i ^  1.3 cr) »  Therefore the contribution to the i n t e g r a l on the RHS of (2.4) from this region i s small; so that i t makes l i t t l e ; d i f f e r e n ' c e  whether, i n that region, we write | or I . Furthermore, for \r-V|> 1.3 (T , cj,~ \ . F i n a l l y since  lr-r\<S" i s very small compared to  the range of the a t t r a c t i v e p o t e n t i a l , the integration i s extended over the sphere  \ Y - V \ <s-  Thus the  R H S of (2.4)  can be  approxi-  mated by  For a gas i n equilibrium, i t has been shown that, to f i r s t in density gradient,  (Lebowitz and Percus,  order  1963)  I t i s reasonable to expect that the non equilibrium behaviour of w i l l be s i m i l a r and we s h a l l assume that  Returning to eq.(2.1), we note that the two p a r t i c l e bution function i n this equation, therefore  i s evaluated at contact  •  ^aU.r-^U'.tJ*  Where  distri-  ^l«(r-irA,t))f ( r . i ^ ^ C r - ^ - a j ; ^ )  i s the pair c o r r e l a t i o n function o, evaluated at contact  8  (Ml)  Substituting  the RHS  of eqs. (2.6) , (2.9a) and (2.9b) into  eq. (2.1) and expanding to f i r s t derivatives  i n r about  £  we obtain the k i n e t i c equation ^  where we have used the abbreviations  Eq. (2.10) i s j u s t the Enskog equation with a long range attractive p o t e n t i a l taken into account by a s e l f consistent approximation.  field  The derivation of (2.10) from Grad's equation  (2.1) was due to deSobrino (1967).  Throughout t h i s thesis,  the pair c o r r e l a t i o n function at contact, YJ^ , w i l l be l e f t as an undefined increasing  function of the density  H(r,.  1  t  i  !  !  l  i  >  I  L  t  L  s  Figure 1. Radial d i s t r i b u t i o n function <fr vs % - r/<rfor hard discs at y - ^ V / N < T = f . Vo = H<r /fZ from an average of four independent calculations , of a for a system of 192 discs( W.W.Wood,1968). 3  :  0  3.  THE H-THEOREM Wie wish to show that the k i n e t i c equation  H-theorem. spheres.  (2.10)  obeys an  F i r s t we define an H function f o r a gas of hard  One way of doing this i s to f i n d the entropy per unit  volume of such a gas i n a uniform, equilibrium state and guessing the H-function from this entropy.  For an i n f i n i t e system of hard  spheres the pressure i s related to the free energy per unit volume p L ,*,J Bj  r  by (Ruelie,  1 9 6 3)  t  On the other hand we have the w e l l known equation ( -*)  I = /r)KT(i  3  1  4 nrj(n))  .  (We have set  | ).  Equating the RHS of ( 3 . 1 ) and ( 3 . 2 ) and integrating the r e s u l t with respect to n. we obtain an expression f o r the free energy i n terms of  where G(j*>) i s some, as yet undefined, function of the temperature.  The entropy per unit volume follows immediately  The RHS  of (3.4) i s an exact expression f o r the entropy per unit  volume of an i n f i n i t e system of hard spheres i n terms of the density and pair c o r r e l a t i o n function at contact.  (3.4)  sug-  gests an H function of the form  In f a c t i n the case of equilibrium when | i s the absolute Maxwell ian  ^  f i n d , a f t e r replacing i n (3.5)  by the RHS  of (3.6)  and  integrating, (3.7)  H =  Ti-&» - n, + Y\ f v i n  +- L n (A%m.  n  ~\) . •  From which the entropy per u n i t volume i s s -  Comparison with (3.4-) y i e l d s (3.1)  Cr (h) f  s  1 K . ( ^ ' JS— - i ) - K :  ;. • .  whence  The H-function defined  i n eq.(3.5) f o r a hard sphere gas i s  also the appropriate H-function f o r the Van der Waals gas described  by the k i n e t i c equation (2.10) since, as we s h a l l  see, the s e l f - c o n s i s t e n t f i e l d term in. this equation i s non dissipative.  >  '  ,  We shall now proceed to the proof of the H-theorem. Differentiating eq. (3.5) with respect to t we obtain  I •  H  9fc  i  where  ,  ,  From equation (2.10)  where j  "  : K  _  $5^ C H'-  »  S f & l  G-.? <rUa<A|'  l(4 l' a  ** J') n ( « W j § * ^ ( S - * > a  $r  Integrating eq. (3.11) over  §_  dr  - _  * #  "L  we obtain the conservation of  mass equation (3.i*v  d  , *  -  fte-is  In equation (3.10), replacing  by i t s value in (3.11), we  the term involving the long range potential thus V(r) i s non dissipative. We have the following identities  v(ir_-*^0  vanishes;  get  •  equation (3.13) becomes, taking into account (3.12), (3.14) and (3.15)  Tf i s a function of density  IT  Y\( >0 r  > therefore  A ~  $  —  if-  I t i s w e l l known that (3,1*)  * SM*£+0T*&  ^ I ' ^ ^ ^ ( 4 i W f ' ) ^ £ l  s  and one can readily show that  (M<0  Eqs.  J C^£ + 0  — W <*Ufc'<*  (fr-Aj  ^ 'ii  JL(.  (3.17), (3.18) and (3.19) are then substituted into eq.(3.16)  which becomes, a f t e r some rearrangement  14 With the definitions  . '  the RHS of eq. (3.20) becomes  The integrand on the RHS of (3.21) i s identical to deSobrino's eq. (3.12).  S"  To prove the H-theorem, deSobrino assumed that  are independent variables.  vary  %  From the definition of %  one must vary the functional form of ^  are also functions of ^  and  , to  ; but S~ and  ; therefore i n general 5" and  independent variables.  r  S  5" are not ^  <  We have the equality  so that P(r,tJ reduces to  Let us set  and study the sign and magnitude of unity.  4(%,S~}  as  %  approaches  We are interested in near-equilibrium states' therefore we can  write  with  IM«I  . To f i r s t order in  (M « I  Since  JU%= 4 n ( l + C U )  /  >  -  so that  «.U,<n  (3.13)  . ( l - l f )  CU*'  We recall that <Tr 1 j i " — r  ..£ ^ •£  -  i<r.n. .5£/d>- _ _  -  i<m.3£/V ^  -  In this form we see that <T i s the ration of the change i n ^ over the distance of a molecular radius to. ^. . £ vary appreciably over such a small distance so that  does not ( S\ « |  Therefore the value of <x(t,,S) given i n (3.23) i s always less or equal to zero; i t i s equal to zero when X = ) i.e. ^ - £ (see (3.22a)). P(^tJ•= 0  Returning to (3.22) we see that  when  < 0 >  Plr.t) ^ O  ;  Equation (3.20 can be rewritten as  follows  (W)  ^ = - j l . f | ( ^ - j ^ ) rff - L . 2- , J j / ^ * +  The physical meaning of this equation is clear.  The increase in  entropy per.unit time in a fixed unit volume located at 2" ^ time a  "t  , drffit, i s equal to the flux of entropy into the unit volume due  to.fluid flow ( f i r s t integral on the RHS of (3.24)) plus the flux  of entropy due to the fact molecules are not points but occupy a f i n i t e volume (2nd integral) plus increase in entropy per unit time per unit volume due to binary collisions,  f ( h t ) .  We note that the  long, range potential does not contribute to the increase in entropy. Integrating eq. (3.24) over the volume of the container we obtain  H i s bounded from below (Chapman & Cowling, 1958) and for a container of f i n i t e volume H  o  is also bounded from below, therefore H cannot ' o  decrease indefinitely but must tend to a limit corresponding to a state of the gas i n which  dHoM*- zO but i f olHo/«U = o , then  the distribution i s the local Maxwellian ~  ZTTKTCr.t) '  %. = 1 and  4.  EQUILIBRIUM SOLUTIONS The H-theorem of section 3 gives a strong indication that the  gas described by the kinetic equation (2.10) w i l l approach equilibrium.  It is natural to seek the equilibrium (i.e. time independent)  solutions of (2.ID) among the local Maxwellian -id  I  - ~(  To find the functional dependence of n ,T and for £ the value of £  solutions of the form I"  < ««-0*"  c on £ we substitute  given in (4.1) into (2.10) and obtain using  U )  the summation convention  where  ^ =  |-£  ;  ^""s  */%r"-  •  J  £  jdr'^r-r'ij ( 'j  Each coefficient of the powers of <y must vanish separately. coefficient of wV w r  1  "  ft'  A  gives immediately  = ^—  The temperature is uniform.  = constant .  M  r  The  .  From the coefficient of W*w we obtain the following equation M  For  For  ( i - |  JAJ: V  jj_ -  v  n  ) ( d V . 3 V )  ^ ^ V s - ^  ;  0  (4.4) reduces to  (AIJV^I.Z^)  we get from (4.4), adding the results  The general solution of (4.5) and (4.6) i s C =  (k.l)  u>_x r-  • +. Co  where UJ and c are constant. 0  stationary state  The general motion of a gas i n a  i s a uniform rotation and a constant translation  One example of such a motion i s a circular helix whose axis i s along the  i-direction; then  c = (-w^w^c) For a motion described i n  (4.7) the shear stress tensor i s zero to f i r s t order in the velocity gradient. Since f'= constant and using eq. (4.6) we obtain for the coefficient of the zeroth order i n (M;  c*-a%  _  o  This means that n. remains constant on the flow line. From the coefficient of vv* in eq. (4.2) we have  when  c =o  and using the identity  we reduce eq. (4.9) to  or  This equation was derived by Van Kampen (1964). for an extremum of a function that  ^YvVv^r  = r-i .  5(*(tl)  It i s the condition  subject to the requirement  When this extremum is a minimum -5/VT can  be identified as the free energy of the gas and the constant on the RHS of (4.11a) is then the chemical potential divided by KT constant = where ^  1- ( M- - 3. KT fa 021  )  i s the part of the chemical potential due to the potential  energy of the gas and  -IKT^"--—  i s the contribution to the chemical  potential from the kinetic energy. Strickfaden (1970), using for  the Pade approximant of Ree  and Hoover (1964) and for V(r) the (12,6)Lennard-Jones potential, showed that for T<T , space dependent solutions of (4.11a) exist c  which correspond to a mixture of liquid and vapor phases. For an equilibrium state, eq. (4.11a) becomes  -a.-  - » r ^ > .  ^ - P 5 ? r  from which one easily deduces the free energy per unit volume at _r i  (f».»0  - HJ!L£!1:  =  M r ) W r ) + «{£)  - U l r ) [vHv^d*  -  The thermodynamic functions of a gas of hard spheres with a long range attractive t a i l are derived from the definition of the H function given in (3.5).  Substituting in (3.5) £  by the local Maxwellian  20  4^= ^ ^ ~ ^ T ) £ **T^1 W  and carrying out the integration  V  T  over lg we get the entropy per unit volume  This i s the same equation as (3.8) except that now w  i s a function  of jr . The internal energy i s  =  IfiKT  +  1 J | vi (v;) v«<vj)  where V i s the long range part of  r-v'|)Mjrd f'  U  The free energy i s  >,r)  A  = V - T S  •  From (4.15) i t follows that the local free energy per unit volume i s  ^(Mt))  £ w ( t ) j * ( r M V ( \ r - r i ) d r ' +- K.r(^Ij)^w{r> ,  This i s the same as equation (4.12) which was obtained directly from the kinetic equation (2.10). be readily obtained  For a uniform density the pressure can  21  CHAPTER 3.  SOME LINEAR NON-EQUILIBRIUM RESULTS  We have been c o n s i d e r i n g t h e approach t o l o c a l e q u i l i b r i u m the e q u i l i b r i u m s o l u t i o n s of the k i n e t i c  equation (2.10).  We  now  i n v e s t i g a t e some n o n - e q u i l i b r i u m p r o p e r t i e s o f t h i s e q u a t i o n . c o m p l i c a t e d e x p r e s s i o n on i t s RHS  The  r e s t r i c t s us from t h e s t a r t t o t h e  c a s e o f n e a r - e q u i l i b r i u m s t a t e s where e q u a t i o n (2.10) can be f i e d by  and  simpli-  linearization.  A problem connected  with n e a r - e q u i l i b r i u m s t a t e s i s t h a t o f  determining the s t a b i l i t y of the uniform d e n s i t y s t a t i o n a r y states o f a Van  der Waals gas a g a i n s t s m a l l p e r t u r b a t i o n s ; t h e s e  d e n s i t y s t a t e s correspond o f t h e f r e e energy  (1959) and  are t r a d i t i o n a l l y c a l l e d  l a t e r extended  F o r t h i s problem t h e  e q u a t i o n i s s t i l l unmanageable and  f o r i t must be found.  linearized kinetic  and  states respectively.  of the l i n e a r i z e d k i n e t i c approximation  t o t h e a b s o l u t e minimum and t h e l o c a l minima  (Van Kampen, 1964)  s t a b l e and m e t a s t a b l e  A method due t o Gross  by S i r o v i c h (1965a) i s u s e d ;  RHS  an  and  Jackson  (the modified  Boltzmann c o l l i s i o n  L ( h ) i s expanded i n terms o f t h e e i g e n f u n c t i o n s o f t h e o p e r a t o r o f a Maxwell gas; t h e f i r s t  i n g t h a t , as f a r as t h e s e terms a r e concerned,  integral  linearized  few terms i n t h e expan-  s i o n a r e r e t a i n e d w h i l e t h e r e m a i n i n g terms a r e approximated  and  the  e q u a t i o n w i l l be c a l l e d t h e GJS model f o r c o n v e n i e n c e ) .  A c c o r d i n g t o t h i s method, t h e l i n e a r i z e d  collision  uniform  t h e gas  by assum-  i s Maxwellian  f u r t h e r m o r e , t h a t a l l e i g e n f u n c t i o n s have t h e same e i g e n v a l u e .  The GJS model i s the only one which yields results on sound propagation in good agreement with experiment over a wide frequency range (Sirovich and Thurber, 1965b);of particular importance i s the good agreement with experiment at high frequencies.  This implies  that this model i s valid at high frequencies as well as at low frequencies so that i t i s particularly suitable to the study of the stability of the uniform density states which, from a mathematical point of view, i s closely related to the problem of sound propagation and consists i n deriving a dispersion relation  E (UD, k V, n.) =o  for a Van der Waals gas and evaluating E as a function of u> increases from  - ««» to  co  1>  as u;  5.. SOUND PROPAGATION AT LOW FREQUENCIES In this section, we present the GJS model i n connection with an asymptotic analysis of sound propagation in rarefied monatonic gases in order to acquaint" the reader with this model which w i l l be applied to the more complicated problem of determing the stability of the uniform density states of a Van der Waals gas.  The results  of this section w i l l also serve as a means of checking the calculations in the stability problem. An expansion of the wave number k for a Maxwell and hard sphere gas up to 3rd power i n <x>  i s derived using the GJS model i n  which the f i r s t five terms in the expansion of L(h) are kept intact. This value of k i s compared to those obtained from the Navier-Stokes, .Burnett, Super-Burnett, 13-moments and recently Wang Chang - Uhlenbeck approximations.  (For these values of k, see Greenspan, 1965;  also Foch and Uhlenbeck, 1967). (a)  The GJS Model From the Boltzmann equation the GJS equation for % = 5 (Tl i s the  number of non-approximated terms i n the expansion of L(h) in eigenfunctions of L ; •',',) is now derived.  For a GJS model, the collision frequency X- i s velocity independent and i t has been shown (Sirovich and Thurber, 1969) that in this case the expansion of k i n powers of oo does not converge; However, though i t i s not convergent, the series i s asymptotic.  The Boltzmann equation for a one dimensional flow i s  where  - .  H  (rs  .  •  j's  etc.  Eq. (5.1) i s linearized by writing  3'i  where  eq. (5.1) becomes, ignoring terms quadratic in ^ teii)  (L  +  (If 4„' t i l  UU.G-^fcUJU'  -  where  •  W  ^  Dimensionless variables are introduced  where  A.  i s an undefined constant frequency.  new variables (5.4) becomes  In terms of these  where  - v-V (n)  1_  =  '^(rj'jr, t")  , e.  i s expanded in terms of the eigenfunctions  linearized collision operator, index i n \ ^ 1965a).  -  , of a Maxwell gas.  can be reduced to a single index  of the The double  (Sirovich and Thurber.  •  Then  where  •.  '  Substituting (5.8) into (5.6) and dropping the prime superscript on t' >  i  ,  x ,6- and  (!T.«o)  g> f L  +  V ^_)^ 3  « . Lit,)  =  ?.er,L*t»«  5  Q.'Aij^j  where (C.i.)  Aij *  J-ur  ^[t^oW  The following approximation of the RHS of (5.10) i s due to Sirovich and Thurber (1965a)  N  06  This approximation of L ( h ) i s substituted into the RHS of (5.10); One obtains  This model differs from the one of Wang Chang and Uhlenbeck in that the streaming term which becomes important at high frequencies is not truncated; this may account for the better agreement of this model with experiment at those frequencies (Figures 2 and 3 ). Furthermore the expansion of L(h ) is not truncated; an approximation for those terms which, in the Wang Chang - Uhlenbeck model, have been neglected i s now provided and taken into account.  (For a more  detailed discussion of the properties of the Wang Chang - Uhlenbeck and GJS models at high frequencies,  See Sirovich and Thurber (1967),  (1969)). For Ti= 5 and for a plane wave perturbation a dispersion relation i s derived from eq. (5.13).  The wave number k i s solved in terms  of <JO The following quantities which are needed in the derivation of the dispersion relation are now written down. The eigenfunctions of  c /c 0  Figure 2.  Comparison of theoretical speeds of sound of a Maxwell gas with experiment.  oO  63  -.4 -.1 > PeKei-is er |_  -.or  Q  WftVv'e\r-STofe.«s - .0? -.01  O  .OOi  —1  .oof  :  ,Figure 3.  exper, n,e«t,i values o f C - r e t o s ^  .01  L_  '  L - i _ l _J  l—L_l_X___l  1-  I I I  |  Comparison of theoretical absorption coefficients' of Maxwell gas with experiment.  t  of the linearized collision operator of a Maxwell gas,  where  Sp,.,  and  respectively.  (5.15-)  are the Laguerre and Legendre polynomials  The f i r s t five eigenfunctions are  ^  =  •+©» =  ^  s  t,o  £  --  Substituting these values of the corresponding coefficients  ({T.K)  a,  (5".I7)  °-z  2  cue  c '<9«i  -  3  (  Ulv ) 1  <^,-  i)  into (5.9) one obtains  c\;  ».  =  V and r are the dimensionless deviations in density and temperature U  S  5 fv.y», and  5  3  are the dimensionless velocity, stress tensor and  heat vector respectively.  Using the following correspondence between the two ways of indexing i  . M<'/  <(')  I  o  o  Z  o  I  I  i we rewrite the matrix  /AM  An Aij  Am  in the form  ''AooOO  Air\  An Ait A i j  AiV A t f  Kovjoo  A.3) K31 A n  M>+  AK  \\OOO  Ai<5 Am*  Aur  Xt(i Am  \A\\.oc  \Y«<V*'  AO«;0>  Ao»,tO  AoO'Oi.  A«»o.\\  AM.«»  AQUVO  No\^ox  Aoijiv  NlO;lo  X>»;Ot.  K\Oj\\  Au;ot  Ai\;\o  The normalized values of the matrix elements  Ai\;0*.  Art;*'*'  Avxjll^  for a gas of  hard spheres are given by Sirovich and Thurber (1965a);it i s found that the above matrix for a hard sphere gas i s diagonal, thus f o r ] l = /  eq. (5.13) i s identical for both Maxwell and hard sphere gases. The matrix elements of t£(h)  defined in (5.4) are written down  explicitly; to f i r s t order in perturbation they are the same for both Maxwell and hard sphere gases (see Grad, 1949)  - o  OVO"S  )  »  30  where for a Maxwell gas  \4is the strength of the potential (S.2.U)  V(v).=  V»/r  r  b i s related to 9 by • P.  and for a hard sphere gas  For the definition 'of the dimensionless  i n (5.5) we take  Consequently the eigenvalues of the dimensionless operator L are A«>0;«o  s.^oi.el  -  Aiojlo  o  3  It i s now assumed that h i s a plane wave perturbation  where x  and t' are the dimensionless length and time defined i n  (5.5) and k. and u>' the dimensionless wave number and frequency  A  From now on a l l prime superscripts w i l l be dropped. (b)  The Dispersion Relation In eq. (5.13) the value of  5 ^  a; ,  and  ^  given  in eqs. (5.16) - ( 5 . 2 0 ) , ( 5 . 2 9 ) , ( 5 . 1 5 ) , (5.30) are substituted; the  result, multiplied by  . .( 5,33)  is  (u> +i - kv-jH  In eq. (5.33) , ' V  4  VJ^  TC • ^ )  Vi  leaving a relation between  and  eqs.  -un.v-}-vr^  w i l l be successively eliminated  t-o and t=o .  reexpressed In terms of -v" , by  S3  First  IK- , and x:  .,'integrating the result over  ( 5 . 1 9 ) , (5.18) and (5.16) one obtains  .  |\,yj and  S  3  are  Multiplying eq. (5.33) f\r and making use of  32  8  •*•  ^ «  +^  ^ U X ^ ^ ) 3  -  K ^-iz  ,  -  + V r r)  solving for  Eq. (5.33) i s multiplied by 'wt-v-iv'' > the result integrated over 1  'V*  yields an equation for S 3  Eq. (5.33) i s multiplied by  •VJ-W)  ; the result i s integrated over  y\r ; one gets the continuity, equation (531)  OJV  = feu 3  The RHS of eq. (5.35) - (5.37) are,substituted the result i s  into eq. (5.33);  Eq. (5.38) i s multiplied by  ;  the result i s inte-  (U>+-0-fciTj)  grated over  AT  ; one finds  + -L. S i ( 3 c - tv)(V|_  To eliminate  r  f i r s t the integral eq. (5.38) by over  31^)  4-V  +- i i v i r +• 1 (*£-») "C 3  we evaluate the integral J aukv£")djr  W(v)(v,V^~)  j  J^^vMjr  i s evaluated; multiplying a n d  integrating the result  v- we obtain, making use of (5.39) Pi -«Tz/  z  Prom eq. (5.34)  Adding (5.40) and (5.41)  ;  i • "5 ( r - t - v ) 4.  = solving for 77  -1  t 3-  k  S  3  \  ^  8  V  i t A - id'^B  3  3  R  where  Replacing i n (5.39) the value of T  just obtained we get the  dispersion relation  3  -fi  T  B  where  This i s the dispersion relation from which one can extract numerically the curves labelled  S T ( U * S ) i n , .figures. 2.  and  3.  ^ ^ . ^ . . ^ v , ^ , ?  35  (c)  The Propagation Constant  k  At Low Frequencies  From the dispersion relation, eq. (5.45), which i s of the form tA.  =. o .  w  e  calculate the propagation constant  low frequency limit. powers of  uj  .  To this purpose,  \r(vjj,k)  h, in the i s expanded i n  The f i r s t step i s to obtain a series expansion  of the integrals A,B, C and D for. small values of UJ . We note that in the limit  u> —>• o  —  for the dimensionless values of U J and k (5.32) limit  ~  -  \J r / 3  .  Co = J-5"fcr/3 . therefore. 0  M  defined i n (5.31) and  so that when w i s small, k i s  also small and of the same order of magnitude; the integral A can be expanded as follows:  where  .carrying out the integration up to the 6th power in' u)  one finds  where ^ •=  aj/i  In a similar manner we find  1 (ir*,+ r  iSo"*. * 3  lo5\x )  -V:  r  Zoovc?  X  Substituting these values of A,,B, C and D into (5.45) and after tedious but straightforward  simplification and rearrangement, one  finds (5:5-2)  ^  -j^(20X*-u)  + ?<\U ') + 1  +- ^ (LIT-G-^-ktr*. *) 1  536 +  I04gx*  +  _i»VJ x * - ^  « - ^lox - . 8  r  The expression i n the square bracket i s of the form  Z ) 6  ]  =»  0  r  ) }  37  where  '  A solution X of  :  must be of the form  substituting this value of % into  (5.53)  and expanding i n Taylor  series  (£5Tj  £(*) =  =  For  £(-*.) - 0  separately, thus  Eo (x*  +1 x ) t~- + 1 E (5^ + - ^ ^ ; S  3  3  2  K) + ... + i.  , each coefficient of V  +1**,  (  n e0/  + ** f  ( y  %i  .. 3 ) -  Xo  m u s  +  *j  t vanish  <r ) lhto  38  fate)  £tM  . (srs)  =*  x E * (OU) ^ i i c."(*0 + r  W(*0 +  Substituting i n (5.56) the value of Eo for Xi  x  ; with  0  x  »  0  given i n (5.53), we solve  known, we proceed to (5.57) and solve for  0  and so on; in this manner we find  VJ7r~ 100  (SlCo)  -  , floo  140  5  therefore  v  k  -\  and i n terms of the real uj  Co  •  too  *0  % '  and k.  X  •?& U -  goo  VA '  g  '  io /K i s reexpressed in terms of  defined by Greenspan. From  (5.28) and (5.23)  A = 9« B ^  (r.cs)  1  )  ^ f ^  =  r  (  0  )  ^  !  P  ^ )  (for a Maxwell gas)  hence  In.  Z(o)(.^)i  1  On the other hand (Greenspan, 1965)  3  (Co  =  5 " K r / 3M 6  P^rnn  ;  '  p(i)  0  ;  <f = e  L  '  For a Maxwell gas the exact value of jx.  y  so that  AX -  ^ / r r )  "  2P(l)(ifC(o)/jr)  I  ( ^ T . / i / / " ' •  viscosity)  i s (Chapman and Cowling,  1961)  (s-.U)  ; ( / i r coefficient of  "  40  from (5.64) and (5.67) we find  T " Furthermore Greenspan's definition of the propagation constant (1965) is  (5".69)  kn -  ( U, i s the absorption coefficient and  Y- UJ/C )  (cj  c/,  while in terms of (£7o)  k  and  X"  our  k  is  &  so that  (i^ ') 7  ko = —-—-— d I UJ.|  (  #•  denotes complex conjugation)  ^ Co-  In (6.71), replacing  k  by the RHS of (5.62) and  uo/X, by -i/s^,  we  find, for a Maxwell gas,  A.  for a gas of hard spheres i s , taking into account eq. (5.27),  whence to  (r.7«  ^  -  • 5"  The fourth oz-der approximation of the coefficient of viscosity of a gas of hard spheres i s (Chapman and Cowling^1961)  yV  fa?)  Substituting  rVKT  1.016  W  into (5.65) we find  5"( i.ois) w From (5.74) and (5.76) the following relation results  3so that f o r a hard sphere gas the propagation constant i s  117  • (d)  Discussion We note that for both the hard sphere and Maxwell gases the  coefficients of powers of  ^  ^  , ^=0,^1,1  i n the expansion of  are independent of temperature.  gas, the coefficients of  J- , - i - • ;  .  in  For a hard sphere  are 1.6%, 3.2% and 4.8%  smaller than the corresponding coefficients for a Maxwell gas; since  these models are the limits of the soft and hard atoms one may deduce that velocity and absorption of sound are almost independent of the nature of the intermolecular potential, however i t was found (Sirovich and Thurber,1965b)that in the moderate and high frequency ranges the hard sphere results agree slightly better with experiment. h^.  The various expansions of order i n ~r  for a Maxwell gas up to 3rd  are  Navier-Stokes _  c  i5Tg  I'M  Burnett +  J L  ?,0 0 0  'T-a  Super-Burnett l  +  iJLL  _7_ io  t<  . 1*6  -7. io?5T3 1,3  *  4  13-Moments _T_ l O  JJlL 4-0  Foch - Uhlenbeck  10 3/ Ho  H-00  *  Sirovich - Thurber  The f i r s t terra in the expansion i s the constant speed; the second term i s the classical attenuation; the third term i s the dispersion in the speed.  K '&  All  agree up to  •  N-S  ; a l l k'i>  a  value agree up to  different.  J- V  except the  '  ; a l l coefficients of  i  are  They are 0.779, 2.101, 2.872, 1.318, 2.578 and 2.185  respectively. The Navier-Stokes equations which are a result of the f i r s t order Chapman - Enskog method of expansion i n powers of a parameter proportion to order i n  i I a r e  therefore valid only up to f i r s t  i / i ^ , so that the coefficients of  in the Navier-Stokes expansion of In figures and  (\ - co/c)  an(  ^ lA^s  cannot be trusted.  4. and 5 are plotted the quantities (—*<£)/.!*' vs  . *t  z  for the different theoretical models  and the corresponding experimental values of Greenspan for Neon. Recalling that  to  and  we see that (i-c/c) and ( . - ^ - t ' - ^ ^ / i t third therm of  are proportional to the  1  and the fourth term of  respectively; thus a l l values of (i-c./c)  K<^,  multiplied by c^_ r  except the Navier-Stokes  value are equal and a l l values of  are different.  The Navier-Stokes and 13-Moments values of in figure 4 are certainly too small.  (lir'-I^Wli  1  The slope of the B line seems  slightly smaller than the slope of the experimental curve at the origin since the B line l i e s below the 2nd and 3rd experimental points from the origin. As expected, the Burnett value of ( i — c/c) in figure 5 agrees well with experiment while the Navier-Stokes result i s too small. In..conclusion, we note that in the expansion of of  i /T_  v  the coefficient of  The coefficient of  n.^  fj^  k^,  i n powers  i s the same for a l l models.  i s sensitive to the method -of approxima-  tion and i s different for each model.  Comparison with experimental  data for Neon does not reveal which coefficient of  TJ"^  i s the  most reliable;-however, i t can be f a i r l y safely deduced that the experimental value l i e s somewhere between 2.101 and 2.872 and, as far as agreement with experiment i s concerned, any -of the four coefficients of b a l l park.  iT^  of B, SB, ST and FU i s within the right  45 .3 r-  Figure 4.  Comparison between theory and experiment of the dispersion in the speed of sound at low frequencies. SB = Super Burnett; FU = Foch-Uhlenbeck; ST = Sirovich-Thurber; B = Burnett; 13M = 13-Moments; dots are experimental values for Neon.  .07  Figure 5.  r  i-Co/c  Comparison between theory and experiment of the nonKirchoffian frequency dependence of the absorption of sound at low frequencies.  6.  METASTABLE STATES *  . The existence of superheated liquid and supercooled vapor states i s well known (see for instance Landau and Lifshitz, 1965); they correspond to the sections on the  curve with a positive slope  below and above the parallel to the \r-axis which divides this curve into two parts of equal areas.  Van Kampen (1964) showed that the den-  s i t i e s YL of these states are the homogeneous solutions of eq.(4.11a) which correspond to local minima of the free energy; a more general treatment of this topic was later given by Lebowitz and Penrose (1966). The correspondence of these states to local minima of the free energy implies that they are thermodynamically stable i.e. they are stable against small perturbations which vary so slowly that they can be considered as a succession of equilibrium states. One cannot find out from equilibrium s t a t i s t i c a l mechanics theories whether these states are stable against non quasi-static perturbations; for such an investigation a kinetic theory approach i s more appropriate; deSobrino (1967) was able to show, using certain approximations for the RHS of the linearized kinetic equation (6.19), that superheated liquid and supercooled vapor states are stable against sufficiently small perturbations.  In this section, the pro-  blem of determining the stability of these states i s reconsidered making fewer and more accurate approximations. From a mathematical standpoint the s  problem of determining the  s t a b i l i t y of a state i s closely related to the problem of sound propagation; i t requires deriving the dispersion relation  (<f.l)  E(  u > > ,  V(k)  ;  m.)  =  O  for the Van der Waals gas described in section 2 and studying the properties of the roots of this equation.  The main problem is again  to approximate the linearized collision integral, T(&) gas. J(&.) can be s p l i t into two parts  where L(*0  , for such a  (appendix A).  i s the familiar linearized collision integral of a dilute  gas and K(&) , a non local term.  In deSobrino's paper U£) is approx-  imated by the Krook model which i s a special case of the GJS model with 11 = 3 ;  KM  by  K(l -itJ J  . and r| by  ( i-ntO"'  The Krook model does not yield correct results for sound propagation (Sirovich and Thurber, 1965b)and though i t might be suitable for a qualitative analysis such as the study of stability, a certain uneasiness about the adequacy of this model persists. In this thesis the GJS model of the previous section (TT. = 5), is used to approximate LU-i) ;  K(*i) i s approximated by  K(S-^v' Vi') : v  furthermore, except for the requirement that i t be a monotonically increasing function of density, which i s physically plausible, no assumption i s made on the functional form. (a)  The GJS Model For a Gas of Hard Spheres with a Long Range Attractive Potential In this section the kinetic equation of a gas of hard spheres  with an attractive t a i l , eq. (2.10),  is linearized and a GJS model  of this linearized equation i s derived for a plane wave perturbation .  On the LHS of the kinetic equation  one writes  (6.*)  *x  (6.0 ^-jj  »  % H  U  l  u  ,  t  V)  =  •V  ?  k  =  v(v-)  e  ;  -  V ^  and obtains, to f i r s t order in h and V, =  (a).  l k){> + '"0/00 » 3  k  v  l  where •(ci)  v(k)  •=  J V ( 0 c.  - 4ir . _  The collision integral for a gas of hard spheres  where  2 =is linearized and expanded in Taylor series up to f i r s t order; the result i s  where  It i s shown i n appendix A that Kl&) can be approximated by  l<(&«).  For a plane -wave the RHS of (6.12) and (6.13) become  (*••*•)  2°U)= ALU}  _ A| i „ ( ^ - l ; +|S VJ(^-I) - [v i-u,^ 1  r  N  + J. ( ^ - 3 ) r + X ji„  3  ^  v  (  v*_,) -j ft } +  (i- i vj - i ^ ) u _ JL (- 1| + | yV | ^ ,r ^ 1  3  3  .  «,ir, -. 4  where  Vj' =  <iv|/d( ,b) h t  the dimensionless collision frequency X i s defined in (5.5); the dimensionless  collision integral  L(£) i s defined in (5.6); i t s value,  the RHS of (6.15), has already been derived i n section 5. luation of K(JUy i s carried out i n appendix A. In the linearized kinetic equation  The eva-  50 replacing  by the RHS of (6.8), til) and K(«) by the RHS of t  (6.15) and (6.16) and expressing the result in terms of the dimensionless variables  and  AT  defined in (6.17), one finds, after multiplication by L  53  1  iso  j  The prime superscripts on u) and fe. have been dropped.  Eq. (6.22)  i s the GJS model for a gas of hard spheres with an attractive potent i a l and for a plane wave perturbation. (b)  The Dispersion Relation As i n section 5, we proceed to eliminate successively [ H 3 , S u r 3>  and  V  from eq. (6.22).  We multiply (6.22) by VJ(V)IT  3  3)  integrate  the result and taking into account eqs.. (5.16) - (5.20) solve for |t  fan)  >33  - W ? - .  S-^H-^V) -<'- "n.) b  I  t  lb ^ . n  L  33  where (f.?«»)  S  To find 5  3  H  < T  0  urK)V  , we multiply (6.22) by  3,  and integrate the result;  we obtain ju  S  ix  (  - 2V(  1+  Multiplying (6.22) by  -*\))  1 n1  <ur(v) and integrating one finds the continuity  equation tuV - ku.3 = o  In (6.22), replacing  ,S - and u by their values in (6.23), (6.25) 3  3  and (6.26) we obtain (fi,-?!)  ( to  • U  LZ  k<Vj)£ =  '  (6.27) i s multiplied by  wt^)  +  b ^ (inn  _!  i_f  J J_ ( ^ 14"  £ ) 3  and the result integrated over <£  52  +. - ^  o  ^ ^ (-  ^  + fVj  j  + V ( feir ( 5 + 2^n^  ::  3  we must evaluate the integral  we multiply (6.27) by ^ V " )  +)  V  bVrj' + ^) 1  JJr3\<v'W ,  .  ei+CvJ"))  JurWM^*T«  and integrate the result.  + Jt b ^ u ; V  .The integral f ur^^d-ir  +  L-^^™-- {"(4 1  * V  +  ^  X (2.1 + |  kbm^V- ) j. 3  has already been evaluated when  reexpressed in terms of V  and Z  Adding (6.29) and (6.30) we find  L  First  Making use of (6.28), we find  (^)  +  ^^t^,)}  X ( i ( ^ ) 4  +  To eliminate T  )  (eq. (6.23)); i t i s  was  35  •And from (5.16) and (5.18) we also have (<S.3z)  ^TJT^V^av- =  3(X + V)  Equating the RHS of (6.31) and (6.32) and solving for x  Substituting this value of x, into (6.28) we obtain the dispersion relation  I = - (it^ ^ ^ ^  (6.*)  .4  l ' - ^  V  | b  jl(c- ) B  +  n  i L >_^) + feth»(-6 rt- +5-f).L- I • + feB( 5  -  (I  5"  - Cft-<\C  +  *T)~T"> =  ) •  + i  (c-IA) +  -4  0  B  <  >  M  10  ( Bb  l  J 0 ) U  J  where A, B, C, D are defined in (5.44) and (5.46) 04  -ATs/t,  and  In the limit as  bin-* o ,  o  and  I , equations (6.33) and  (6.34) reduce to equations (5.43) and (5.45) of section 5 which are respectively the equation for u a dilute gas.  and the dispersion relation for  (c)  Stability Criterion . To study the stability of uniform density states we consider  the behaviour of the perturbation - ^ v R ^ ^  in time. The  space dependence of %\ i s kept fixed. The dispersion relation, eq. (6.34) i s of the form  k, VlW) and w are parameters and ou i s variable and complex. The values of ui( k^O),n) for which oscillation of the f l u i d .  E=o  are the frequencies of  A zero of EE  in the upper half plane S  +  corresponds to an exponential growth indicating that the unperturbed state i s unstable. A zero of E on the real axis or on the lower half plane S_ corresponds to a stationary or damped perturbation. The problem i s to determine whether E has zeros on the 5+ plane. If  One has the following theorem, (see for example Wylie.,1960): ^(-j) i s analytic within and on a closed curve C  has no zeros on C  then the number of zeros of  and i f within C  is  The RHS of this equation i s just the net number of times ^ty) encloses the origin counterclockwise on a complex C  ^-plane as  moves along  once. To find the number of zeros of E  is analytic on S  +  on S-*. , we f i r s t show that E  , (this i s done i n appendix B) then take for con-  tour C the segment of the real axis between - R  and R. and the  "'semicircle on 6-*- with radius R. and centered at the origin.  In the  limit as  00  ,C encloses  . Next  we plot £U>) on the  complex E-plane as the complex variable u> moves along  C  ; this  plot w i l l show the number of times E(.w) encloses the origin counterclockwise.  It w i l l be shown that in the limit as R->> °o > E (u>)  remains constant on the semicircle and i t i s sufficient to plot Etw) as u> increases from  -«6  tt «o  on the real axis.  In fact we  need only find the zeros of the imaginary part of E (OJ) , the corresponding values of the real part of EUu)  Etuo)  and the direction in which  crosses the real axis at these points. As i t stands, the function  B(UL>,fe,V(,^,v\)  defined i n (6.34),  does not readily lend i t s e l f to analysis. We now derive approximations for this function for various frequencies. We have mentioned, in the beginning of this subsection that the dimensionless wave number fe. i s kept fixed; we w i l l now estimate the order of magnitude of k for which linearized kinetic equation  *^'T^ ^feV  (6.22), due  the attractive potential, i s  non negligible compared to the collision integral. V(r)  The Fourier transform of V{k)  =  j v M ef -- dr L  can be rewritten as  and since V(r) has a f i n i t e range c L  , the term in the  We assume that for o< t-< A. there exist an M and an m such that _ M < Y-  VO) < - *r\  then we have the following inequality Mir fcd-i)  ( cos  (i)  W«l  When  in powers of  <  kd  •  <  v« U o s . k d - \ )  then  ^=  »4  and  cosW  can be expanded  ; the above inequality becomes  f o r a slowly varying potential,M i s not much larger than m and M and m are of the order of  rV  where  0  of the potential; i t follows that interparticle distance.  fore  W=-jv(r)dir  e  " ( > . o f V° ti*) ; v  k  r  &  i s the  In order for the so-called metastable states  to exist, the temperature where  o<r<d and \/ i s the strength  and  T  must satisfy the condition b=  £~-.>"^jf »  ; but - j v ( o d r ~ v  there-  0  v\.VQ) ^ o((£")^-) • The term, in the linearized kinetic equa3  tion, due to the attractive potential is- ^.V^) sionless wave number : k  k'v^ y  ; the dimen-  i s the ratio of the mean free path  the perturbation wavelength  JL \ k'= —  to  ;^ =  for a near equilibrium state the velocity distribution i s almost Gaussian so that [ A ] ^* ^"  "O^ ~ I  v,fe'v ~< 0 ( g ) £ ^-vj  . Therefore  3  / For a Van-der Waals gas  V  ("Zc) ~"  W = 3b C  hence  so that, for a dense gas, ^ ) ~ 0(»)  ?  The mean free path ^ ^ 0(^-) -^ 0 (') 2  a  i s of the order of  = (o^k  5 then  ; the mean free path,, for a dense gas, i s  of the order of the interparticle distance.  In section 2 we have  assumed  that  and  ^therefore  dyy t\ , then  The r a t e  o f change  sionless  variables,  lengths  -lyydyxx  the  attractive (ii)  Jin v-  of A  ;  OL->>OL  .  Also,  since  therefore  due t o c o l l i s i o n s ,  o f the order o f  &  (<3-K/^t) , i s , i n d i m e n c  . ; therefore,  ~ v>  , t h e term i n the l i n e a r i z e d k i n e t i c potential  When  is negligible then  k,d^\  o<r<c(  1 si_ ^  f o r wave-  equation  due t o  compared t o t h e c o l l i s i o n d  ; our inequality  term,  becomes  and  In t h i s to  case t h e c o n t r i b u t i o n t o the l i n e a r i z e d k i n e t i c  the potential  (ii)  <A y> o_  become  t e r m i s much l a r g e r .  nonn e g l i g i b l e  (iii)  When  I f we c h o o s e  T  than  in (i)  e q u a t i o n due  wUre -f»<b?4  s m a l l enough  this  while i n  term w i l l  compared t o t h e c o l l i s i o n t e r m .  Vd >11  then  i=.l  .  For  c\  V  we h a v e  The potential term is much smaller than i t s value i n ( i i ) since i t i s proportional to ^ j * " (iv)  When  while in ( i i ) i t is proportional to  kdL»l and  ^.  , then I  *V(k) \?y\j KT  i s smaller than i t s value i n ( i i i ) .  3  In conclusion, we see that the values of wavelength L for ^Y^.  which  V* k V  i s largest are those of the order of the range  3  of the potential: t~A >>a-  so that  \i'g. -j- <r< |  . For these  values of Ji , when the temperature i s sufficiently small, the contribution due to the attractive potential i s non negligible compared to the c o l l i s i o n term. For wavelengths too large or too small ( l*i>l a n d ^ c i ), HJ^Jil V h.V i negligible. We shall choose / KT such that 3  "'(6.3*)  k'~  «  s  '  In the above discussion on the magnitude of the wave number we have •used the original notation where k is the real wave number with L  dimension  and  k  is the dimensionless wave number defined i n  (6.20). We return now to the problem of approximating the function ^(U^R  1  V(k ),n) ,  and for the remaining part of this chapter, w i l l be  dealing only with the dimensionless wave number k  ; the prime super-  script on k.' is again dropped. As u) increases from - <o ( i. 11)  leu | »  k  to o  on the real axis, the inequality  i s always satisfied except for values of u> i n the vicinity of the origin.  This i s a consequence of condition (6.38).  When (6.39) holds,  the integral A can be expanded as follows  A= i - - J  e  ^  -  J _ i J.  Similarly  Substituting these values of A,B,C,D and F into (6.34), expanding in powers of k retaining only the zeroth order i nfe.we find  where  (CM)  RtE  0 =  ]  +6,^4- I H 6 ^  4- 141 ^ + Hit*  cy + CK;  + l  6 ] f u. - _ i ?  ,6  ?' <^ £-HUfc" +6(  3 + m.4  3 +I4.4fr + 2ce  __:  . g + t ?  '  4 4-  6  i  .  € +-nrfc' -4. l  uA  6  T  — 4o36  '  s  ( l+loe + si^faofe*) -1 . . ' — J ?6 4- \<?<Jfe + 4 Ho 3 fc 3  I /f y/1 J  /  L  i&16  +  ui •nsV'f Va3fc  . de«o>vii^<tfoi-of r  3  b 1 • K^EoV J  and  For a function E(m,k) of the form  We can show, using Newton's well known method of successive approximation (this method i s given in most books of Mathematical Functions; see, for instance, Abramowitz and Stegun, 1965), that the zeros of E are of the form  The zeroth order approximation of E order in k  , a l l the zeros of E  expansion of E  ,E  > contains to zeroth  Q  so that retaining more terms in the  does not yield more zeros but only more accurate values  of the zeros of E  .  Unless we want to determine these zeros accurate-  l y , i t is sufficient, in a qualitative analysis such as the study of s t a b i l i t y , to keep the f i r s t term, Eo , of the expansion of E Since i t was assumed that K « I mations of E  0  and u) are very good approxip  and u> .  IvrtEo  has a zero at  X w v U  IVY,  M036*)j'{  E  R« E ( M j 6  'UJ- oc =  6  . Furthermore i^vCC  - ( U+isle + mr€* +  ^ ( l t ? e ) ( l + ? 6 , 5 - ^ t-a?76 '+n<6 +l<a©fc ')j. c  so that as OJ increases from upward at  . E  l  - °o , E-(u>,fc)  3  l  crosses the real axis  which i s  X-vwt"o (oujt) has also a t r i p l e zero at the origin.  This t r i p l e zero  w i l l be examined later using an approximation of E (ojjK) values of  .  [ | << \ . U)  valid only for | j » k w  valid for  (Recall that E given in (6.45) - (6.47) i s and cannot be trusted for U J near the origin).  The other zeros of TMEc^ujjfr)  are those of the expression  63  ^(^feU  (C.fa)  ^  +• - J  L _ _  +21 6 f i l e *  ^ ( » +lo€ +  3o6 ) 3  which i s of the form (<^4)  ^ v  . ^(^fe)=  J  ^ c o i^-t- ^ct)i|.  t-  Affc)  where \  W  2  V  and \u >o  We see that  j  •= ) fe 0 }  < o  6  and  =  <0  _ ©o  crosses the real axis at least once.  s o that ^ c ^ e )  The derivative of ^ ( ^ t j  .with respect to ^_  has, for a l l values of & , a positive and a negative root. _  *  >  0  64  This implies that ^C^> of ^  fe)  has a local maximum at a negative value  and a local minimum at a positive value of  ^(0^61= IC^J  <o fe)  .  .  Finally  From a l l these informations, one deduces that  must have one of the'three following forms:  Figure 6.  Plot of f(y) vs. y.  Whether -^('^e) has two complex conjugate roots, a double root or two unequal real roots i s irrelevant since these roots are either complex or negative and hence unphysical (recall that ^ - u>  v  the other hand, root,  (^fe) •  fc)  ; <*> real).  On  always has one and only one real, positive  This root is physically meaningful.  At this point we introduce explicitly the assumption that the pair correlation function at contact increasing function of the density n_ (6.48), i s a monotonically  increasing  . n,  i s a monotonically  ; so that  increasing function of  does not depend on the density  EaCujje.)  through £  ^(bvt)  n.  & , defined in KL  .  explicitly but  Because' of the above assumption, increasing .  e  implies  We now return to the discussion of the double root i V^Tu) - l^,(e) of  TJMEO  . For a dilute gas,  £=o  ,  has a positive real root at  to which corresponds +  ULM  (O)  =  +  0,5"^  We wish to show that the root .^.\Cfc) monotonically  increasing function of £  dix C) that the functions are monotonically  decreasing functions of  h,' =  Now  since  we have  1  4t  - ae  etc-  \.  .  is a  . It can be shown (appen-  , o^ce) and  where  of  AO)  €• :  defined i n ( 6 . 5 3 )  From figure 6 i t i s found that Therefore  u,( ) 0  and cu, fo)  Let us take for example for  ihOho and  [^)  >o hence  are the lower bounds of  bn =|  , a very high density.  ^  ^vU)  so  .  >d  ar  If we take  the Pade approximant  I f o.OOJnb + o. o 113 n*b*  Then to nb-=l corresponds  and J ( «J, 0.IH6) -  +  6  ' *  r  ^ - 0.16 8  0.0 35"  has a positive real root at  whence + t o , ( o . U 16 ) ± O. In recapitulation, we have shown that, in addition to the roots s  at infinity and at the origin, iv^Eo (.cu, e) . has always a double root  iuo^t.) n,  whose absolute value i s an increasing function of &  and for densities  between o and i  hence  , bounded by  We determine the direction in which "E crosses the real axis on the complex E - plane at  R«E(+iu fc} . The imaginary part of E i s Xj  of the form  where  Recalling that  w o.i  v  =  and that  that the derivative of T ^ £  0  [Hl-i)  >o (figure 6 ) , we find ;  with respect to U J at ± UJ, i s  negative:  A ^j7~'±^r  V  thus as to increases from  -u»-£  M  to -oj| + t , TUA E?  0  decreases from a positive value to zero at -u^ value at - u>( v d R*E ( - U J j., € )  ; £ (.u>j  • Similarly  at the same point as UJ  Next we evaluate  ( c u , £ )  to a negative  crosses the real axis downward at E(ujjO  crosses the real axis downward  increases from .  V.? to ( ±u>i,  fcj  u i | - £  . f?eHo  to' <J-»I+ E  CUJ,  €r)  , defined in  68  (6.46)  i s of the form  lM M<\.l*'>),, tfe,  lA^al.f^Wj  a  n  d  Hi-1**'  are increasing functions of. € 0  to  0.416  and  0.542  to  0.639  written out explicitly in . As € and  <JJ» C&)  (6.46)  increase from  respectively the expression in the  square bracket in the numerator becomes less negative while the expression in the square bracket in the denominator is positive and increases.  Compared to the rate of change of these quantities, the  rate of decrease of Therefore .(^.5"1)  —L-,  —  £-e£o(uj k) w  s  irr i s greater.  increases, and i s bounded by .  KzSLo (t O.Zklj o)  =  Rf B (± (jj^ ) ^  -6,121 N<  0  fe  -0.6ttl|.7 = R e E o f + O . ' O T , - o . t t l O  We now investigate the zeros of I^E where both  \LL>(  and  k.  at the origin.  are small, the integrals A, B, C and D  have already been evaluated (eq.  (5.48)  -  (5.51));  their expansions  up to fourth order in k and in terms of  v  1  *  k  x.  can readily be deduced from  In the case  (5.48)  -  (5.51).  They are  '(.6.40  A -t  + ^  &  -^('^^V'^h+V  fu^(  -I ^ )  •:  i-  »  (*.<u)  P=  .-*k-cife\  r+.ay)*--jV M  We can find F i n a similar way  The RHS of (6.61) - (6.65) are substituted into (6.34) which becomes after tedious but straightforward calculations  +  I •<\  i+  .(i+w)(i+?6) (it?e)  /  •• / i > 7 £ i  1  15-  i (i+?+i6t4l)Vy l  €  +  1  1  !»JMeV ii2£« ,  15"  315-  31?  tr  tro  5T  1  a +  1 1  j(r  uLe  +  7  1  .roe  e  4  63oe ) • r  +  3 15-  J  to  J  J  :  +13 + 1  5  L  €  3ir  5o  In the dilute gas limit as bvi-? o  and  5 ~> o  (recalling that -j =-L  the RHS of (6.66) reduces to the f i r s t two terms of the LHS of (5.52) which i s the dispersion relation of a rarefied gas at low frequencies. From (6.66) we see that  rwE  has a zero at 1g. = °j  crosses the real axis downward at  0+z«)(u-*fe )"(n-re)^  7  1  7  '  7  .  ( if "5 ttofe 4-b n ^ ) .  R-tE (k,6jfe) has the sign of the expression XwvG  »  v  v  1  has a double zero at  (j>.(>8)  | ^  14- -<=>  =  +  l 0  e +  fcVy1£  ( H-s^y " 1  1  There are three possibilities (a)  For sufficiently large negative values of H  and the zeros  ±  ^  4- |  + 16 t +  of XVA E  rit-?6)  V  are imaginary.  S  •< O  It follows from (6.69)  that  (:.b',) For the zeros of  ~ | ( i + r e ) < !+ 5 v  4-106+  fc>Vvj'<o  TnruEr '-y-'+T^t.^ are. real and i t can readily be seen  from (6.66) that  &*(fe i>,*) <foe(M,eJ <o ;  (c)  I-f'5 + 10 f + fcVy > o  For  the coefficients of a l l powers of (\+~s +\ot+ bVv^ ) are negative hence 1  From (6.66) i t can be seen that (a>  T w t € (  b^jfe)  )  >o  so that when the zeros ± ^ T _ of IWIP axis upward both times at Rt£  are real, E crosses the real  $•».«-fe)•  To the three cases discussed above correspond the following hodographs  Figure 7.a. Plot of E(w,k) for 1+ 5+- ioe + bV/<-ifi+^/. Unstable situation. ' 3  Plot of E(w,k) for - | 0 + f * ) \ (*• S + ' O f + tV^'<o. Unstable situation.  Figure 7.b.  \ — <o  OJX,  /  Jo  Figure 7.c.  (d)  Plot of E(w,k) for Stable situation.  V(r). -  of  RtE  1+- 5 + 10 € +- t>Vvj'> o . . ,  Discussion Let us consider the quantity |+  Since  00  (4 W/W) l  e"  dr  410  fr +•  b vi*"K|' v  .  For  , ' " 5 ( 0 i s of the form 5 { I 0 »'^CoJ+0 ( k ) • v  k « I , one can in general perform a Taylor series expansion about  h = e> and keep only the f i r s t term, *5(o)  , so that  73  But the equation of state of a gas of hard spheres with an attractive t a i l i s  from which one finds that  1  therefore, to zeroth order i n _L_  , ' 1+  U O HOfe-+ ^ ^ v ^  times the compressibility of this gas.  1  If we take  i s just V| (v- bv\) i =  5  |+"S(o) •Hoe+bV'v| reduces to ^[o'JA-r^ which is proportional to the coml  pressibility of the Van der Waals gas as was pointed out by deSobrino. Figures(7.a) and(7.b) show that when the compressibility i s negative, the gas i s i n an unstable state; figure(7.c) shows that the states traditionally regarded as metastable are stable against sufficiently small perturbations. Qualitatively, figures(7.a),(7.b) and(7.c) are identical to the corresponding figures of deSobrino. the approximations of L(£)  This i s as we expect since i n  and k(£) the difference l i e s in the  accuracy but not i n the method of approximation.  Furthermore this  agreement of our findings with deSobrino's was already strongly hinted  at by the calculations of Sirovich and Thurber on sound pro-  pagation as i t was found that the Krook model, though not quantitatively correct, has the same qualitative properties as the highermoment models.  74  Finally, two remarks on the validity of the approximations made i n this section: On the study of sound propagation, the agreement with experiment at high frequencies which Sirovich and Thurber obtained, indicatesthat the GJS model seems most suitable to the study of stability of metastable states which requires evaluating the linearized kinetic equation at frequencies ranging from  _ <>o  to • »o  .  Some doubt was raised as to the validity of approximating hy  K(&e>) at high frequencies. We have argued that (appendix A) the  non zero terms i n the expansion, of L(,£\) are much larger than the corresponding terms of k!(£\) so that the latter may be neglected. In conclusion, we have investigated the stability of metastable states using a more accurate approximation of the Enskog collision integral and a more general pair correlation function firmed deSobrino's results.  and con-  CHAPTER 4. 7.  CONCLUSION  CONCLUSION By not restricting ourselves to a specific form of the pair  correlation function at contact  , we have shown that the results  obtained by deSobrino for the traditional Van der Waals gas are valid for a more general gas of hard spheres with an attractive long range potential. Using the method of Sirovich and Thurber we find that, to f i r s t order, the dispersion of sound at low frequencies i s the same as that obtained by previous calculations (except for the Navier-Stokes  value  which i s incorrect); the absorption of sound, up to 3rd order in to , is slightly less than the Burnett value and slightly greater than the Wang Chang - Uhlenbeck value; a l l three are i n f a i r l y good agreement with the experimental result of Greenspan. Qualitatively, the results shown in figures 7.a,7.b,7.c , concerning the stability of uniform density stationary states are "identical to those of deSobrino; here again the conclusion is valid for any Van der Waals gas.  This consistency with previous results  along with careful checks indicate that calculations are free of errors and that the approximations used are adequate.  We have shown  that a l l uniform density states are stable against- small perturbations these include those states at temperatures compressibility small.  T<Tc  (-§fe)- Kr( it- £(o) + Vo 6 + ^ k l j ) r  and for which the i s positive and  On the other hand i t was shown numerically by Strickfaden and  deSobrino (1970) that these same states are unstable against s u f f i -  ciently large perturbations therefore these states are metastable. In this thesis we have assumed the following dependence of  r  on  and  t  In analogy to equilibrium.theory where ^ i s a function of separation, density and temperature, we may assume that ^ i s also a function of local temperature (we use here Chapman and Cowling's definition of local temperature). faO  \=  \{ ^(^>bV T(i >t) <r) )  a  ;  this may be more r e a l i s t i c and the calculations do not seem much more complicated.  Dymond and Alder (1966) implicitly took into account  the dependence of  on temperature through <T(T); they obtained  values for the transport coefficients for rare gases at voru^  T>T < C  and  which agree to within 10% with experiment.  A more d i f f i c u l t problem i s that of using a velocity dependent frequency  X{%) model to approximate  gas, the spectrum of the  . Except for the Maxwell  -operator of a gas with a f i n i t e range  has a continuous part (Grad, 1963) so that the method of «L(4\.)  in eigenfunctions of ^"Maxwell  m a  y  expanding  he a very faithful  reproduction of the spectrum of the ^-operator of a Van der Waals -X-  gas.  For a cut off potential, o£(-k) takes the form  where W\ has a complete discrete spectrum (7.3)  NHi* =  M£)^C-  so that  i=i  -  (f,g).is the inner product in an five value  L  space.  Noting.that the f i r s t  eigenfunctions are just the summational invariants with eigenA' =1, L  i = 1,  5,  Cercignani (1966) introduced the  following model  which, when A i s velocity independent, becomes the BGK model.  The  next step would be to use the approximation on the RHS of (7.5) to investigate the problems studied i n this thesis.  * However Grad (1963) showed that for a hard potential( V= ^ 6 ) 5>r) with an angular cut off the Gross and Jackson approximation may be used. !  APPENDIX A In this appendix we shall present the calculations leading to an approximation of the linearized Enskog collision integral. The Enskog collision integral  is linearized by writing  (M  4 = ^o(»t^)  The result i s , to f i r s t order i n  where ( .3a) n  K'-  •MJ^tr+I-r^)-»\(t-i^)}^ct.xi^g.«l| >  /  and  4-  < U H e/(r-«L} ) The integral  K i s independent of  the eigenfunctions  A»  ) -  iin£Z.)(  ^ <r,^ d a d i s expanded i n terms of  *YC of the dimensionless linearized collision  operator of a Maxwell gas L  (A.4)  . K  <TUT.)  <i* + A i  M  where  £  MM)  H 0 =  2.  and cx»  Z  A=  (A. « * b )  X  Substituting into (A.3) we get ( K ' + Z (i )  ^fe(Vi=  +  l  D  Let us consider i n detail the integral  -Sf'f&Os  (M)  (j  ^ ( ^ 4 - 1 ^ )  (hit)  + V(^'BVU)  )  - y}(y-i<r_a)(  expanding in Taylor series, keeping f i r s t derivatives and f i r s t order in perturbation (A.l)  i'Cli,)-  t (l ) t  +-^'/^j  where  and  c£  i s the linearized Boltzmann collision operator defined i n  . section 5. . For a plane wave perturbation  Substituting into (A.7) we note that since the f i r s t three eigenfunctions  and  )  4.  al t w o *  (M)  are summational invariant,  If Hf' + ^.•-^•'-^O^'^'f Js<*|'*o (t=i,^)  while  However, for coefficient of  t>5 H'S'il^o and ( r ^ , ' - ^ ; ) ^  ^I4i)jf0 ; furthermore, the  s\la « (J-  in the integral i n (A.9) i s  one while the coefficient of the same quantity in the integral in (A.10) i s of the order of a-/([(k)«\ so that unless l t " * ? ] ) ~ 0 ( x - " t i l )  fe. (which i s not the case for ^  and Mr  defined in (5.15) and, we  hope, in general) we can assume that for l°£  I  Ih.H)  i s much larger than  \ K"(4i»^\  ^(4.)  <t'(V) =  ' and therefore  ~ *\&tl>)  Eq. (A.5) becomes (  a^Uo-  = £  2  (  & (  ,<' + K* +  + sc'^,)  v^li*}  +V|K"M  )  + >\£(U)  81  Expanding cr  , defined i n (A.3), i n Taylor series to f i r s t order in  and substituting the result into the definition of K  we obtain for  i n (A.12)  Mho)  Substituting into (A.13) the values ,  .  i (  - wt)  we get  '" " •  n  where  An approximation of the linearized Boltzmann c o l l i s i o n integral <L{k) in eq. (A.12) has already been given i n section 5 i n connection with the study of sound propagation. the integral  k.(£e)  We now turn to the evaluation of  on the RHS of (A. 14); to be consistent with r  the . 5-Moments approximation of  we take  & =. 0  Z. 4i W,A  82  The integrand of (A.14) i s made up of three terms, one involving <ko —i a second  and the third^ v  evaluated readily.  •  The  Integration  bI*jy\ VtCT.jvn 0  . The f i r s t integration can be over J l yields  (<^'^yfey l o u r ' *  >  4JT S \>  hence  M  fe^J>V«^i*»(^3 -  z-axis. has been chosen along the k vector;  VT ) S  In a similar way  the third integral i s evaluatedj the result i s  (A.14)  H^M^ft*!^  r  T' The.5-Moments approximation of *  is 5"  In the 2nd integral, replacing (A,I"U)  Ko' by the RHS of (A. 17) we find  J-bi»j vu/iaih* |(dr'd^. -a^-ft-k { v + E  3)  + -y-'u  This integral i s the sum of five integrals, the f i r s t one involving V , the 2nd one involving V " ' ; we have  "C etc....  v ' -  i s rewritten in terms of  (A. zc) (A.xi)  -  (^) ^-; ?  These values are substituted into the expression (A.17a); the f i r s t three integrals of (A. 17a) involving  v,z  and "tt  readily; as an example the integral involving  can be evaluated i s evaluated.  The third integral i s  Integration over SL i s f i r s t carried out. the direction'of easily  <j.  be shown that  then  dsi-  Sv*6d.6  We take as ; jj.j ^t se =  0  -direction > ^  c a n  After integration over xz. , (A. 22) becomes  now we take the z-axis parallel to k ; carrying out the integration we get  In a similar manner, we evaluate i , and 1^, the results are  •r  '^1 b vi«, V K r d ^  (1 - J -vr"") -v-j x  Hi^ the integral i n (A.17a) involving  1  , i s written out explicitly  using eq. (A.9)  The f i r s t integral of (A.30) can be evaluated readily; the second and third integrals are identical and are also easily evaluated with the  help of eqs. (A.23) - (A.25).  The results are  ts  The fourth integral must be evaluated with some care (A.*?*)  I\u. ^  | | >vj WoV^uT II 4"'dafi-fa>k i ( - . I J W X I J n  ^-4?V-'  In the coordinate system in which the z-axis is paralled to <^ which we shall now c a l l the <^ -system, the SL unit vector i s  The 3rd rank tensor  J7|\fij  .fLfe  >  'uz\ = ' S ^ f l s i i ^ y _C2?^  t-o&H  ; Sl\£l\  s  f  e  n  distinct elements  >  • ^-^-j^  $^&-t6iV»ti^Y  =  to^G-coS^  States Q it>C4  •SZ,Sl^~  non zero contribution after integration over  from zero to ; they are jvtj  a  >  . J^H^*  Of these,only three give  n  , _fl£_fL  i  i s a symmetric tensor (see definition of  and H^Si-  3  ; also  , eq. (5.19)).  Taking into account these informations, eq. (A.33) becomes  Z  fe,  [7  * ^ 3 bvx» ) *  fe  t  *  ^  l,<Uj- '  F i r s t , the integration over JT£ i s carried out; the result i s  Before integration over V*'  i s carried out, k ^ y j  and  fecfv^  must be rewritten i n the coordinate system in which the z-axis parallel to k  , which we c a l l the  ^ -system.  Thus i n the ^.-system (A.Vi)  fe  . in the (A.^a) in the  (ft.V?)  =  3  -  -syste' m  . i . .fe  »  ^  k  -coordinate system  (Jt.t)Z  ^ b :U  = |v «.  IA  in the  k -coordinate system  •it  Therefore the quantity which, in the  ^^(--lhn^|H^)/f  i n  t h e  -  ^ -system, i s  -  s y s t e m  fe  3  is  -  Similarly (f\.^>) in the (A,4t)  =_ (v. : l * i  fv^k; k. -coordinate system t Kfc ='  (IT,  ^  The RHS of (A.38), (A.40) and (A.42) are substituted into eq. (A.36); integration over V  i s carried out; the result i s  •The integral of (A.7a) which contains the factor  $i  ,  _ I  is the sum of two integrals: the f i r s t one, X<r\ , involving Sv<vfi and the second one, X$-  7  , involving  C; ^'(v' ^ 1  .  Substituting  for W> the RHS of (A.18), making use of (A.13) and (A.14), we obtain for ~r y\  The integral  Xrv  i S  written out;explicitly  The f i r s t four integrals of Zrz.  c a n  help of eqs. (A.23) - (A.25):  (A.u«)  Ij^bs  be readily evaluated with the  They are  | ^ i v | v i l ^ ^ ( f % - M r i - r i ^ -a-fc s  2TT  a  AT;  Va^y*? w 1  ir) v\ Virr ^' /J^^'^B- •ft J--9"k C ' S - <$•£>• v-i^uf' ,  v  6  a  and l £ " 2 ^ are more involved and w i l l be evaluated in more detail let us f i r s t consider X ^ e .  zir  Replacing, i n (ft.ri) j  «  o  by the RHS of (A.34) and integrating the  result over si. we find  k^S^ and  fe.s  .  ,  • "  are reexpressed i n the JR -coordinate system  in the k -coordinate system (ft.?*)'  —  ( s. i)(fe-i)=  ' lls,k  Substitution of the RHS of (A.53) and (A.54) into (A.52) and integration over  AT' yields  Next, we evaluate  The integration over -Q. i s f i r s t carried out following the same method used i n evaluating the integral I v ^ , (A.33).  4-  ,i_-V-'k, ) ULT  Before integrating over rewritten in the  1  AT' , the quantities i n parentheses are  K_-coordinate system  •k.l  1  S . i  a  ks  z  AT'- \  \ J= -  Substitution into (A.57) and integration over  W  AT yields 1  i s obtained by adding the RHS of (A.15), (A.16), (A.27), (A.28), (A.29), (A.32), (A.43), (A.45), (A.47) - (A.50), (A.55) and (A.59)  91  The RHS of (A.60) i s the linearized value of the plane wave. equal to  K -integral for a  The sum of the f i r s t three terms on the RHS of (A.60) i s  k($ ) y%s  where sf ^ i s the local Maxwellian; L  approximation of K(£) used by deSobrino.  X^ ) lrt  i s the  92 APPENDIX B We w i l l now show that the function persion relation 6+- . E  E(u^ k, V({0,ri) in the dis-  E = 0, (eq.(6. 34)),is regular on the upper half plane  i s an expression involving the integrals A,B,C,D and F  which can be rewritten in the form  where  "5 -  depending on whether X stands for A,B,C,D or F.  (It can be easily  shown that B, C, D and F are simply related to A).  The RHS of (B.l)  i s a Cauchy integral defined and regular on the S  plane; (see for  instance R. Balescu, 1963).  +  If follows that the poles of E , i f any,  must come from the zeros of the denominator of & defined i n (6.33) and (6.36) and the problem of proving the analyticity of E on Sireduces to showing that the denominator of Q (jj.s)  9 U ' * + '±!±^L.(..i f\ + £ kb»« 8 -H) - J L ! _ ( f 05 + .  has no zeros on  . Of course tS i s a sum.of A's and B's and  therefore i s regular on —L  ,  f  Hll±d^^  H  S  +  so that  93 is the number of zeros of 5S° inside  . An analysis similar to  the one carried out i n section 6 i s given here; the function ^ i s plotted on a complex $-plane as u? increases from - *o  to °o  on the  real axis of the complex tu-plane.^ , the integrals A and B can  For values of oj such that be expanded i n powers of  k/tto+l)  results substituted into (B.2).  The imaginary part of $  ?  (eqs. (6.40) and (6.41)) and the  To zeroth power i n ^ , t$ i s equal to  , has a zero at io =t  0 0  and  t)  crosses the real axis upward at  From (B.3) i t i s apparent that t v n ^ also has a zero at the origin. .For uu near the origin, the expansions (6.61) and (6.62) for A and B are substituted into (B.2) the result i s , to f i r s t order i n oo ,  l»nS(ofe)=0 and $ :  l  1  crosses the real axis downward at Rz$>(o, e)* *• t H^?6)  The hodograph of -5 , figure 2j S$  has no zero on S  +  does not enclose the origin hence  which i s what we .set out to prove.  .  Figure 8.  Plot of 5f((o,-6) E(w,k) on  showing a n a l i t i c i t y of  95  APPENDIX C  j I^C^U  The expressions - JvO)  a n  d |/4CO|  defined i n  (6.53) are of the form 9> (6) =  J  aw  where  then We wish to determine the sign of  F ^ ) ^ ) -  fi!(t)P(tj  1  (4, b  A  f « i t | - eiibi) + 3 (^ i> _ 3  Ay,  0  dob ) 3  Q-ib, _ a,b,)  3  sufficient condition for  dwibr,-  do ki)  0  =  "  -  0  are  -i( 4*b _ dob,.)  n-l  w  which i s the sign of  . The coefficients of p'ft-ft'P  H= O  n  S(tJ  o  sVfc) >  (Qvwbk,-dvib^<o)  for  YA  0 (?'(*)•<. o j > h_  i s that  because then a l l the  coefficients are positive (negative); i f flw>o and b ^ > 0 m's then the condition i s equivalent to (Cl)  Q±?L bw>  >  b«  'vn>rf  for a l l  96  (This i s readily generalizeable to the case where P and Q are polynomials of degree N). The coefficients of)u«-)<0  a  ,  n  {  i IMOl  <r'ce;<o  a*  and bw  of  |T.(fe) satisfy (C.2) and those  satisfy (C.I); therefore and  A(*)<o.  97  BIBLIOGRAPHY  Abramowitz, M. and Stegun, I.A. 1965. Handbook of mathematical functions ( Dover Publications, New York ). . . . •i  Balescu, R. 1963. Statistical mechanics of charged particles ( John Wiley & Sons Ltd., London). Bogoliubov,N.N. 1946. J. Phys. U.S.S.R. 10,265. Born, M. and Green, H.S. 1946. Proc. Royal Soc. A188, 10. Cercignani, C. 1966. Annals of Phys. 40, 454. Chapman, S. and Cowling, T.G. 1961. The mathematical theory of nonuniform gases (Cambridge University Press, Cambridge). Cohen, E.G.D. 1968. Fundamental problems in s t a t i s t i c a l mechanics ?II (North-Holland Publishing Co., Amsterdam). DeSobrino, L. 1967. Can. J. Phys. 45, 363. Dymond, J.H. and Alder, B.J. 1966. J . Chem. Phys. 45, 2061. Enskog, D. 1922. Kungl. Svenska Vetenskap Akademier.s Handl. 4, 63. Foch, J. and Uhlenbeck, G.E. 1967. Phys. Rev. Letters 19, 1025. Grad, H. 1949. Comm. Pure and Applied Math. 2, 331. 1958. Principles of the kinetic theory of gases, Handbuch . der Physik, Vol. XII (Springer Verlag, Berlin). 1963. In Rarefied gas dynamics, edited by J. Laurmann ( Academic Press, New York). Greenspan, M. 1965. In Physical acoustics, edited by W.P.Mason ( Academic Press, New" York). Gross, E.P. and Jackson, E.A. 1959. Phys. Fluids 2, 432. Kirkwood, J.G. 1946. J , Chem. Phys. 14, 180. Lebowitz, J.L. and Penrose, 0. 1966. J. Math. Phys. 7, 98. Lebowitz, J.L. and Percus, J.K. 1963. J . Math. Phys. 4, 116. Ree, F.H. and Hoover, W.G. 1964. J. Chem. Phys. 41, 1635. Ruelle, D. 1963. Helv. Phys. Acta 36, 183.  98  Sirovich, L. and Thurber, J.K. 1965a. In rarefied gas dynamics, edited by J.H. de Leeuw (Academic Press, New York). 1965b. J. Acoust. Soc. Am. 37, 329. 1967. J. Math. Phys. 8, 888. 1969. J. Math. Phys. 10, 239. Strickfaden, W.B. 1970. Ph.D. Thesis, University of British Columbia, Vancouver, B.C. Strickfaden, W.B. and deSobrino, L. 1970. Can. J . Phys. 48,in press. Van Kampen, N.G. 1964. Phys. Rev. 135A, 362. Wood, W.W. 1968. In Physics of simple liquids, edited by Temperley, Rowlinson and Rushbrooke (North-Holland Publishing Co., Amsterdam). Wylie Jr. , CR. 1960. Advanced engineering mathematics (McGraw-Hill Co., New York). Yvon, J. 1935. La theorie statistique des fluides et 1'equation d'etat (Hermann & Cie., Paris).  


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