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 2.ooo 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) -H2.ck(r-fc*--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 (.R.it-) 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 e.tr ; 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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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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Title | On the approach to local equilibrium and the stability of the uniform density stationary states of a Van der Waals gas |
Creator |
Le, Dinh Chinh |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | Some equilibrium and non-equilibrium properties of a gas of hard spheres with a long range attractive potential are investigated by considering the properties of an equation, proposed by deSobrino (1967), for a one-particle distribution function for the gas model considered. The solutions of this equation obey an H-theorem indicating that our gas model approaches local equilibrium. Equilibrium solutions of the kinetic equation are studied; they satisfy an equation for the density η(r) for which space dependent solutions exist and correspond to a mixture of gas and liquid phases. The kinetic equation is next linearized and the linearized equation is applied to the study of the stability of the uniform density stationary states of a Van der Waals gas. A brief asymptotic analysis of sound propagation in dilute gases is presented in view of introducing an approximation of the linearized Boltzmann collision integral due to Gross and Jackson (1959). To first order, the dispersion in the speed of sound at low frequencies is the same as the Burnett and Wang Chang-Uhlenbeck values while the absorption of sound is slightly less than the Burnett value and slightly greater than the Wang Chang-Uhlenbeck value; all three are in good agreement with experiment. Finally, using the method developed in the previous section, an approximation for the linearized Enskog collision integral is obtained; a dispersion relation is derived and used to show that the uniform density states which correspond to local minima of the free energy and traditionally called metastable, are in fact stable against sufficiently small perturbations. |
Subject |
Gases Van der Waals forces |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0302448 |
URI | http://hdl.handle.net/2429/34204 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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