Bulk Viscosity and Relaxation Pressure Theory

8/6/2019 Bulk Viscosity and Relaxation Pressure Theory

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Journal of Applied M athematic s and Mechanics 69 (2005) 943-954

O N T H E T H E O R Y O F B U L K V I S C O S I T Y A N DR E L A X A T I O N P R E S S U R E tV . S . G A L K I N a n d S . V . R U S A K O V

Zhukovskiiemail: f [email protected]

(Received 16 December 2003)A general theo ry of bulk v iscosity and re laxation pressure , based on the generalized Ch apma n-En skog method, is g iven, andalso s imple quali ta t ive theories of them. Problems associa ted with the defin it ion of temperatu re are d iscussed. Form ulae aregiven for the bulk viscosity coefficient and th e the rma l conductivity of a gas possessing rotatio nal degrees of freedom. The influenceof bulk v iscosity on the absorption and dispersion of sound and on the s tructure of a shock wave in n itrogen is investigated .9 2006 Elsevier Ltd. All rights reserved.

In aerodynamics, use has previously been made chiefly of Euler 's and boundary-layer equat ions in whichthe bulk viscosi ty is not o f the sam e orde r a nd is consequently omit ted. In fact , the rat io of the bulkviscosi ty gradient to the convect ive part of the equ at ion of motion will be of the o rder of the inverseReynolds number, provided that the dynamic and bulk viscosi ty coefficients are com mens urate and theveloci ty divergence wil l be of the order of the rat io of the characterist ic veloci ty to the characterist iclength. As a result, in handb ooks o n gas mechanics, th e bulk viscosity is either generally igno red or isdescribed insufficiently clearly [1], including, for example, in the problem of sound propagation [2].How ever, at present, to calculate the laminar flows of a compressible fluid, the equation s of gas dynamicsin the Navier-Stokes-Fourier approximation (or, more concisely, the Navier-Stokes equat ions) arewidely used, in which account is taken of terms of the or der o f the inverse Reynolds number, andtherefore i t is necessary to analyse the role of bulk viscosi ty problem s of gas flows around bodies.Different areas o f the kinet ic theory of bulk viscosity have been develop ed [3-11], but insufficientattention has been paid to its simple qualitative modest [8, 11].The quest ion of the defini t ion of the temperature remains debatable [5, 6, 9]. In the case of idealpolyatomic gases, the bulk viscosity occurs in the expression for the stress tensor, provided the distributionover the internal energies of the molecule s is close to local equilibriated, and the temp eratu re T is definedfrom the total internal energy of the gas (as in the therm odynamics of i rreversible processes). If thetempe rature is defined from the t ranslat ional energy of the part icles (the t ranslational tem peratu re ofthe gas Tt), then the stress tensor has the same form irrespective of the degre e of excitation of the internaldegrees of freedom of the molecules, and the bulk viscosi ty is not present , but s imilar term occurs inthe expression for the internal energy of the gas. Such a definition of the te mpe ratu re is mo re "physical",s ince the s t ress tensor is determine d by t ransfer o f the mo men tum of the part icles [5, 6]. In non-ideal(dense) mona tomic gases the bulk viscosi ty is governed by the potent ial ene rgy of interact ion of themolecules. When the temperature Tt is introduced, the expression for the bulk viscosity coefficientchanges, and a similar term occurs in the formula for the internal energy of the gas [9].A similar proble m arises with the relaxation pressu re - a fine effect which occu rs in gas flows, provide dthat som e of the internal degrees o f freedom of the molecules are close to the local ly equil ibriated s tatewhile oth ers relax [4-8, 10, 11].A conse quence of the general ized Chap man -Ens kog metho d (Sect ion 1) is a general system ofequations of physicochemical (relaxation) gas dynamics - a system of equations with level kinetics [5,6]. The lat ter describes, in particular, the regions o f gas flows where the internal degrees of freedo m

tPrikl. Mat. Mekh. Vol. 69, No. 5, pp. 1051-1064, 2005.0021-8928/S--see front m atter . 9 2006 Elsevier Ltd . All r ights reserved.doi: 10.1016/j.jappmathmech.2005.11.016


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944 V.S. Galkin and S. V. Rus akovof the molecules are frozen. The conc ept of the t ranslat ional temp eratur e I t is employed. T he stresstensor does not change i ts form as a funct ion of the degree of exci tat ion of the internal degrees offreed om o f the molecules. Expressions for the bulk viscosi ty and relaxat ion pressure are obtain ed bytaking the limit as "r/t, --4 0 and by redefini t ion o f temperatur e Tt to T, wh ere "~ is the time of relaxa tionof the "rapid" process of the establ ishment of a quasi-equilibrium state, and t , is the gas dynamic t ime.These quant i t ies characterize the difference of Tt from the temp erature T, defined taking the internalenergy of molecules for rapid processes into account .

Sections 2 and 3 illustrate the positions of the general theory on the well-known [8, 12] relaxationgas dynamic models without turning to kinet ic theory.Below, we consider a gas with rotat ional degrees o f freedom close to local equil ibriated, an d withthe remaining internal degrees of freedom of the molecules frozen. We emphasize that , when thetempe rature Tt is used, the systems of equat ions for scalar correct ions to the dist ribut ion funct ions aresimplified (Sect ion 4). Ou r main at tent ion is devote d to a qual i tative examinat ion. Q uest ions o f thesignificance of bulk viscosity are illustrated by simple but representative examples (Sections 6 and 7).Similar quest ions for the relaxat ion pressure are mo re complex, have not been de velope d and are notconsidered here.1. T H E G E N E R A L I Z E D C H A P M A N - E N S K O G M E T H O D

We will consider the case of ideal gases whose m olecules interact only on instantaneo us col lis ions, inthe quasi-classical approximation, w hen the translational energy of the molecule s is conside red classicallyand the internal energy is t reated q uantum-mechanical ly [3-8]. There are no chemical react ions. Thesystem of general equat ions of physiochemical gas dynamics includes equat ions of the balance ofpopulat ions no, mome ntum and internal energy (see the reviews [5, 6])Dn~ V-~ +rico . u+ V.ntoVto = rco, Kco = IJc o(f ,f)d ~ (1.1)DuP'D--7 + V . II = 0 (1.2)

H e r eD ~n-b-7 + I I : Vu + V . q = 0

D0-5 = ~+ u. V, ,, = ]~no,,o) p = m n

(1.3)

nco is the nu mbe r density of particles in qua ntum state 03, i .e. possessing internal energ y Eco, andJo~(f, f ) is the operator of inelastic collisions in the kinetic equation; summation is carried out for allvalues of the subscripts 03, ~ = 0, 1, 2, . . . , N, whe re N is the n umb er o f quantu m levels, and m is themass of a molecule.Summ ing Eq. (1.1) over 03, we ob tain the continuity e qua tion

D nO---t + nV . u = 0 (1 .4 )The general ized Chapman-Enskog method gives the series for the dis t ribut ion funct ions

foJ = f(o y+ f(a) . . . according to the Knudsen num ber Kn ~ 1 relat ive to the locally Maxwell funct ionf(O) fh ~ 3/2t~ = n ~ ) exp(-hc2) , e = ~ -u , h - m (1.5)2 k T t

where m is the mass and { is the veloci ty of the part icle. T he perturbat ion{ },~, .,~,c V~9co:(1) f~) -A ~c " V I n T t - B o cc-~Ic Vu- 9 n -G~= O(Kn)f(o)

(1.6)


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On the theory of bulk viscosity and relaxation pressure 945Here I is the unit tensor , andA , B, D and G are scalar coefficients of the order o f the Knudse n num berKn; they depend on hc 2 and the independen t of the gradients of the macropara mete rs and explici tlyof u.We will consider the version of the metho d when the following quanti t ies are def ined in terms o ff(~


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946 V.S. Galkin and S. V. Rusa kovw he r e

n ~Y a = n ' n a = E n ~ R a = y ~ K ~

and na is the rotat ional population. The quanti t ies with the superscr ipt R include only rotat ional an dtranslat ional-rotat ional exchanges, and the quanti t ies with superscr ipt V include only exchanges ofvibrational energy [10]. The term R~ )v is om itted as being of a higher order .

In dimensionless form, the term R~ )n - 1/Kn, and the rem aining terms o f Eq. (1.12) a re of the ord erof unity. We will seek a solution in the form_(o) . (1) 0) . (o) O(Kn)Ya = Ya + Ya , Yc~ /Ya =

In the zeroth app roximation we haveR(o)R = 0 ~ y(aO) = y~a(rt) (1.13)

where the Bol tzmann func t ionyeaq(T, Sa= ~ e x p ( - E a ) , E ~Q = ] ~Sa e xp( - e = ) , e , = kT---'~, (1.14)

O~and Sa is the statistical weight of the rotational level a.The system of equations for the slow variables, i .e . for the relat ive vibrational populations

y[~ = nl~, n13 = E n ~n Otis obta ined by summ ing Eqs (1.1) ov er a taking Eq. (1.4) into accou nt

= ~ n ' ( ~ ( 1 . 1 5 )DY~+ V.nfj Vf~ = R~~ nf~Vfj = ~.,,,oV,o, ~ ~.,--,oDt Ot OtThe r ight-hand side of Eq. (1.15) results f rom the vibrational exchanges and is calculated forf i~ Withthe assumptions made, the quanti ty R[ 1)V i s of the order o f Kn com pared wi th R[~ and is there foreomitted.In the expressions for 11, V~ and q, with an error O(Kn) com pared with unity, the p opulation nab iseq (0)replaced by ya (Tt)n~. Af ter linearization with respe ct to y , the righ t-han d side of Eq. (1.12) will belinear in y0) and contain, generally speaking, a free term ( indep end ent of n0)) . The lef t-hand side ofEq. (1.12) must be set equal to

Oy ~ ) dyeaq(T,)OT,n----~- = n dT, Dt (1.16)

w he r eDTt - C-o1 kTtV.U+ nZ.d,~f ~,,fj ), C v = r2k + c oDt f~ (1.17)

The "rotat ional" specific heat at constant volum eR d eqC v ---- -~ t~ R( Tt ), ~eRq = EE ay ~( Tt ) (1.18)


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On the theo ry of bulk viscosity and relaxation pressure 947Form ula (1.17) was obtained neglecting terms of the o rder of Kn in Eqs (1.3) and (1.15). Theseequations take the form

D 3 %~q + %v (1.19)b5 kT, + + kr , v. u = o, =

Equa lity (1.17) follows from re lations (1.19).Taking into acco unt eq ualities (1.16) and (1.17), for y~ ) we find a sy stem of linear alge braic equ ation swith an inhom ogen eous par t that is l inear in V 9 . As a resulty~l) = as + ba V . u

whe re the coeff icients as and b~ are proportional to Kn.Consequent ly~kT , + %R 3 eq ]F Ea(ae~ + ba V. u) (1.20)=2kT,+%R(Tt)+Q, Q =

The quanti ty Q is the non-equil ibr ium correction to the internal energ y of the gas, which is govern edby translat ional, rotat ional and vibrational degrees of freedom of the molecules. Instead of equali ty(1.20), in the ther mo dyn am ics of irreversible processes, th e following expression is used

~kT, + %R = ~ kT + %~q(T) (1.21)whe re T is the translat ional-rotat ional tempe rature. In o rder to switch from equali ty (1.20) to (1.21),we will per turb th e translat ional tempe rature

T, = T+T ~), T


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948 V.S. Galkin and S. V. Rus akov2. M O D E L O F T H E R E L A X A T I O N O F R O T A T I O N A L E N E R G Y

Her e and in the following section, the general theo ry of bulk viscosity and relaxation pressure d escr ibedabove will be i l lustrated on gas dynamic models, avoiding kinetic theory. As stated in Section 1, thebalance equations in the Euler approximation are used. The trace of the stress tensorP = nk T t (2.1)

We will consider a gas with relaxing rotat ional degrees of freed om o f the molecules, and the vibrationsfrozen. The energy equation (1.3) takes th e formD% % 3+ kT tV . u = O, = ~k T, + %R (2.2)D t

whe re % is the internal en ergy of the gas, referred to th e nu mb er of par t icles per un it volume.For the rotat ional ene rgy %n, we will use the simplest relaxation equ ation [8, 12]D~; R _ I [ % ~q ( T , ) _ % R ] ( 2 . 3 )

Dt "cR

wh ere "oR is the rota tiona l relax ation tim e, an d %~q is the kn own equilib rium fu nctio n Tt ( see formulae(1.14) and (1.18)). E qua tion (2.3) is obtained fro m Eq. (1.12), if we om it R( a (the vibrations are frozen),assume the term R~ )R to be negligibly small , mult iply by Ea, s um in te rms of a and approxim ate th er ight-hand side with the simplest relaxation expression.Local equil ibr ium ( i.e. comp letely excited rotations) occurs in the l imit as xn/ t . ~ O. We will writethe solution of Eq. (2.3) for this case in the form o f a ser ies power of us

% , = ~ o ) + ~ , , ~ ] , ) + . . .

In the zero th approximation, we have the equ il ibr ium solution(2.4)

%(R~ = %~q(T,) (2 .5)In the following approximation, taking (1.4) and (2.2)-(2.5) into account, we obtain

(2.6)_~(RI) _ O~eRq(Tt) RD Tt R -1D t - c v ' - ~ = - k T t c v C ~ V 9The quantities Cu and c~ are given by formulae (1.7) and (1.18). Using relations (2.6), we reduce thesecond form ula of system (2.2) to the form

(2.7)= ~kT3 + %~q(Tt) + A ( T , ) V 9A( Tt ) R -1= "CRcokTtC v (2.8)

We om it quan tities prop ortion al to x~, n _> 2 everyw here.We emphasize that the term A V 9 is the relaxation contr ibution to the energy of the gas. Here ,according to form ula (2.1), the hydro static pressure n k T t is calculated from th e translat ional tem peratu reTt. We will introduce the translat ional-rotational temp eratu re T by the equali ty

% = ~ k T + ~ R q ( T ) (2.9)wh ere for %, form ula (2.7) also holds.We will per turb [5] the translat ional temp eratu re T t = T + ~RT 1).Linearizing the r ight-ha nd side offormu la (2.7) with respect to xR, and equating i t to the r ight-hand side o f formula (2.9) , we f ind

Cv'CRT(O +A V. u = 0


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On the theory of bulk viscosity and relaxation pressure 949such that

T t - T = "cRT l ) = -A ( T) c -o l v 9 - - l q v , u (2.10)Substituting the expressions (2.8) and (2.10) into the right-hand side of equality (2.1), we "convert"the relaxation term into the stress tensor such that i ts t race

P = nkT t = p - gV .u , p = nkT (2.11)wh ere the pres surep is calculated from the translat ional-rotat ional temp eratu re T. Taking into accountform ula e (2.8) and (2.10), we write the exp ression for the bulk viscosity coefficient

2 R 3nk T'cRco(T )(~k R -2= + c o ( T ) ) (2.12)For the coeff icientA, in the non-equil ibr ium correction to the specific internal energy (2.7) , we ob tainthe expression

Co(Tt )g (T , )A ( T t ) = (2.13)The quant i ty g(Tt) is given by form ula (2.12) with T rep laced b y Tt.By vir tue of relat ions (2.10), the d ifference between the translat ional and translat ional-rotat ionaltem pe ratu res is prop ortio nal [11] to the bulk viscosity: gV 9 .

3. M O D E L O F R O T A T I O N A L - V I B R A T I O N A L R E L A X A T I O NSuppose that , apar t f rom rotat ions, vibrations of the m olecules are also excited in the gas. Th e en ergyequa tion is given by the first form ula of system (2.2), but vibrational en ergy %v is added to the expressionfor th e specific intern al en ergy of the gas, i.e.

3= ~ k T t + ~ R + ~ v (3.1)Th e rotation al ene rgy %R satisfies Eq. (2.3), and for the energ y %vwe co nsider the relaxation equation

D % v _ 1 eqDt Zv [%v (Tt) - %v] (3.2)to hold , where Xv is the vibrational relaxation time.Equ atio n (3.2) is obta ined fro m Eq. (1.15) in th e Eul er app roxim ation (w hen V[~ = 0) by multiplyingby E[~ and summ ing over [3 for a mo del o f the molecules - harmo nic oscil lators with single-quantumtransitions [4].Supp ose, as in Section 2, that "OR~t, ---) O. We will consider th e case of slowed ex changes of vibrationalenergy of the mo lecules: "Cv/t, = O(1). Again, we expand the solution of Eq. (2.3) in series (2.4). Asabove, we obtain

%(0) eq __~(1) RDT,= %R (T, ) , = cv D tUsin g the en ergy equatio n (the first form ula of system (2.2)), expression (3.1) for the specific intern alenergy of the gas an d relaxation equation (3.2) , we f ind

( ~ )T , = _ Co l k T t V . u + % v ( T t ) - % vDt z vTh e q uantity Cv is define d by formu lae (1.17) and (1.18).


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950 V.S. Galkin and S. V. Rusa kovPerturbing Tt with respect to the rotat ional-t ranslational temp eratur e T, in much the sam e way as inSect ion 2, we o btain formula (1.24) for the specific internal ene rgy of the gas, and instead of formula(2.11) we ob tain the expression

P = P + P r e l - q V ' u , p = n k TThe gas flow is described by the system of equat ions shown at the end o f Sect ion 1. The bulk viscosi ty

coefficient q is given by form ula (2.12).The stress tensor contained the relaxat ion pressureeq


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On the theory of bulk viscosity and relaxation pressure 951and cons equently the simpler approximation

r,~ = y3(% ~(T,) - E a) (4.5)We recall that th e coefficients 7k are co nsta nt qu antities (k = 1, 2, 3). By virtue of cond ition (4.4),the pe r turbation (4.1) and (4.5) m akes a contr ibu tion only to the expression for the specific internal

ene rgy (2.7). By using eq uatio ns given in [3], it is easy to pro ve th e validity of formu la (2.13).The computing advantages of introducing Tt increase on changing to m ore com plex cases (for example,a mixtu re of gases with qua si-stationary vibrational s tates [7]).

5. T H E B U LK V I S C O S I T Y C O E F F I C I E N T A N D T H E R M A LC O N D U C T I V I T YConsider the t r anspor t proper t ies of a gas with rota t iona l degrees of f reedo m of the molecules c loseto local equil ibr ium ( the case of rapid exchanges of rotat ional energy of the molecules) . The vibrationsof the mo lecules are frozen, and th ere a re no ch emical reactions. The stress tensor is given by formula(1.23), whe re, in t he given case, Prel = 0. We will ignore the depe nden ce of the dynam ic viscositycoeff icient 11 on the rotat ional degrees of fre edo m [3] , and for the function r l(T) th ere are rel iableexperimen tal data.We will write form ula (2.12) for th e bu lk viscosity coefficien t g in the form

1 n (3 n~ a xn _rc 'qq = 7 ~n kc on Z sk + c o ) ; Z = - - s x , - 4 7, (5.1)w he r e xt is the relaxation t ime of the translat ional de grees of freedo m of the molecules.The same express ion for g is obta ined by the Ch apman -Enskog metho d in the main approximat ionin terms o f Sonine and Wald mann -Triibenbach er polynomials in [8] . Below, the specific heat capacityRc,, governed by the rotat ional degrees of freedom of the molecules, is assumed to be equal to k (adiatomic gas; here, the specific-heat ratio y = 7/5, %~q = kT). For the ratio of the rotational andtranslat ional relaxation t imes Z, Parker 's approximation form ula

3/2 2 -1

is of ten us ed an d also a revised formula [8] differ ing from (5.2) by the presence in square b rackets ofthe addit ional term/~3/203/2.For nitrogen , T, = 91.5 K and Z~ = 18.2.Figure 1 shows graphs of the rat io g/r l against T, calculated for nitrogen from fo rmula (5.1) usingParker 's formula (5.2) ( the continuou s curve) and using the revised formula [8] ( the das hed curve) .Th e limit value (as T ---) ~ ) of this ratio is equal to 2.23. For T > 900 K, the relative con tribu tion o fthe vibrational degrees of freed om to the specif ic heat becom es substantial .The heat f lux vector is given by formula (1.25). We will write the therma l conductivity in th e fo rm[131= gnA, A = + n = k_ (5 .3 )

m

whe re ~ and 9~ are du e to the translational and rotat ional degrees of freedom respectively. In practice,one is usually l imited to the m odif ied Eucken approximation [3] . Her e)~* = 1 )~ = al_7.~[~t - a~ 1.328, [~t = p '~ (5.4)' . , l . , 11

where ~ is the self-diffusion coeff icient of the gas, ignoring rotat ional degrees o f freedom. T he quanti ty~t depen ds slightly on the intermolecular potentials, and therefo re i ts average value, given in the secondform ula of system (5.4), is used [3]. In this app roxim ation, th e Pran dtl n um be r Pr = 7C~I]/9~ = const


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95 2 V. S. Galkin and S. V. Rusa kov1 / 1 1 1

"~ . . ~/rl ~ ~ ~ '0.75 \ , " ~ ~ j '

/ / P - - r - - " : - ,0"2500/ 4 1 700 10010 T

Fig. 1

(recall that ~, and C~ are c onsta nt quan tities). A t low T, the accu racy of approximation (5.4) is inadequate ,and the Mason-Monchic approximation [8] is used, namely~.* = 1 A ~ P~ R-~ , ~,* = ( I +A ), 13n= ~]

5 - 2 ~ n V l + 2 ( 5 +13R O = - - -A = n Z t_ '

= 1 3 , q , ( r )(5.5)

wher e ~n is the self-diffusion coefficient of the gas, taking into account the rotational degrees of freedom.The rat io ~ n / ~ will be est imated using Sandler 's approximate formula [8]~ n l ~ - ~ (T) = 1 + 0.27Z -I - 0.44 Z -2 - 0.90Z -3 (5.6)

In approximation (5.5), (5.6), the Prandtl number depends on the temperature T.

6. T H E A B S O R P T I O N A N D D I S P E R S I O N O F S O U N DThe bulk viscosity is important if it is of the ord er of the dy namic viscosity and must be taken into accou ntmainly by the Kn < 1 approximation. An example is the propa gat ion of small perturbat ions in amolecular gas with exci ted rotat ional degree s of freedom. Th e dimensionless absorpt ion coefficient

= ~xlKn+O(K n3), Kn = O~rllp~. 1 (6.1)whe re co is the frequency . We emph asize that all the qu antities (i.e. p, 11, g, etc.) occ urring in form ula(6.1) and formulae (6.2)-(6.4) given below are equal to their values in a stationary gas.The coefficient cq is well known [2], since it is given by the Navier-Stokes approximation

2B 8 ~ 3_g (6.2)~ 1 = ~ + A ( T - 1 ) 2 , B = l + 4 r l

Much less famil iar is the fact that dispersion of sound should b e invest igated within the framew ork ofBurnet t 's approximation, s ince the Navier-S tokes and Burnet t terms of the t ransport propert ies ma kea contribut ion of the same or der of magnitude to the deviat ion of the dispersion coefficient from thelimit value as Kn -~ 0 (see, for exam ple, [13]).The dimensionless dispersion coefficient[3 = 1 - ~2Kn 2 + O(K n4), [3 2 = K1 + K2 (6.3)


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On the theory of bulk viscosity and relaxation pressure 953character izes the change in the phase velocity as Kn increases. The coeff icient

2B 2 25 B. . )2 225 2.!r I = -- 1 + 1)3( 3~/- 7)3),2 + "~T3At) - 12874 A tT - (6.4)

is given by the Nav ier-St okes approximation, while the coeff icient] ( - l ( 7 r ~ - Q 3 ) + 2Q23),

is governed by the Burnett terms of the transpor t proper t ie s [13], where1 5 o (~,2 5-- .2"~ 8, ~(n _l )( ~, t , 1~,~)QI = 7"(i+~ )~. t + ~aA'R ) - ffkt -

5 .Q2 = I + 2 ( 1 + 6 ) ( B - 1 ) 2, 03 = ~'t*+~(~'t-(Y~,~)(B-1)

(6.5)

(6.6)

The quanti t ies in expressions~6.2) and (6.4)-(6.6) wil l be calculated using the formula e f rom theprevious section for nitrogen (c~ = k, ~, = 7/5) . The qu anti ty B is def ined by the second formula ofsystem (6.2).We will denote by

0~* = al ( ~ = 0)/ (l l(~ :;/:0) , ~* ---- ~2(~---- O)/~2(g :g: O)the rat ios def ining the contr ibution of the bulk viscosity to the ab sorption and dispersion coeff icients .The values of the quanti t ies in the numera tors are obta ined if we put B = 1 in formula e (6.2) , (6.4) and(6.6).The results of calculations of these ratios are given in Fig. 1. The bulk viscosity has a particul arlystrong e ffect on the dispersion coefficie nt [32; he qu antity Q2 is quad ratic in ~. We emph asize , howev er,that , when the bulk viscosity has a considerable inf luence, i t might be necessary to change to a relaxationdescription [8], i.e. xR/t. must not be assumed to be a small quanti ty .

7. T H E S H O C K - W A V E S T R U C T U R EAs noted in the introductory par t of this paper , the relat ive magnitude of the bulk viscosity gradient inthe equation of motion of a compressible f luid in general is of the ord er of Re -1. Howe ver , this is validif the gas te mpe rat ure is sufficiently high, such that q - 11, and if the velo city diverg ence is not small.An up per est imate of this influence is obtaine d if we conside r the question of the inf luence of the bulkviscosity on the shock wave structure, since here R e - 1, q ~ rl and V 9 ~ u /L . The prob le m of thestructure of a wea k shock wave in a molecula r gas was solved ear l ier in [13]. We will denote by u* andT* the reduc ed valves of the velocity and tem perat ure in a shock wave

u* u(~) - u( -~ ) T* T(~) - T(-**) (7.1)= u ( o o ) - u ( - o o ) ' = T ( o . ) - T ( - o o )where { is the dimensionless s treamwise coo rdinate [13]. The f irs t approximation with respect to theparam eter of the shock wave intensity is given by the Nav ier-Stok es approximation ( in the followingapproximation, al lowance for the Burnet t t ranspo r t p roper t ies is necessary [13]). Here

u * = T* = ( l + th (~ / b ) ) /2 , b = 8"~2~1(~t+1) (7.2)The absorption coeff icient ~1 is def ined by the f irst formula of system (6.2) and is calculated f rom datafor unper turb ed f low. Wh en the rat io c.Jrl increases, the coeff icient b increase. C onsequently, the regionof per turbed f low expands in terms of { .In applicat ions, the question of the inf luence of bulk viscosity on the T and p prof i les in shock wavesof mode rate and high intensity is of greater interest . The equations

A r = T ( q - 0 ) / T ( q # 0 ) , Ap = p ( { = 0 ) / p ( q # 0 )


12/12

954 V.S . Galkin and S. V. Rusakov

1.0

0.5

\Ap

- 4 - 2 0 x *Fig. 2

denote the ratios of the values of temperature and density calculated using the Navier-S tokes equationsby the me tho d pro pose d in [14] for g = 0 and g ~ 0 (we recall that, in a shock wave , p - 1/u). The datain Fig. 2 were obtained for T(--~) = 100 K, the Mach number upstream of the wave M = 5 (the dashedcurves) and M = 11 (the continuous curves), the streamwise coordinate x* is referred to the mean freepath upstream of the wave, and the value of the reduced density p* = 1/2 corresponds to the valuex* = 0 [141.The formulae from Section 4 are used, and, in particular, formula (5.2) for Z.The bulk viscosity changes the density and temperature profiles considerably, especially in the frontzone of the shock wave. Its influence increases as the Mach num ber increases.

We wish to thank G . A. Tirskii for helpful comments.This research was supported financially by the Russian Foun dation for Basic Res earch (02-0 1-00501),by the "State Support for Leading S cientific Schools" program me (N Sh-1984.2003 .1), and by the RussianMinistry of Edu cation (E02-40-52).

R E F E R E N C E S1. EM AN UE L, G., Bulk viscosity of a dilute polyatomic gas. Phys. Fluids, 1990, 2, 12, 2252-2254.2. LAN DA U, L. D. and LIFSHIT Z, E. M., Fluid Dynamics. Pergamon Press, Oxford, 1987.3 . FERZIGER, J . H. and KAPER, H. G., Mathematical Theory of Transport Processes in G ases. North-Holland, Amsterdam-London, 1972.4. KOGAN, M. N., Rarefied Gas Dynamics. Nauka, Moscow, 1967.5 . KOGAN, M. N., GALKIN, V. S. and MAKASHEV, N. K., Generalized Chapman-Enskog method: derivation of t h enonequilibrium gas dynamic equations. In Rarefied Gas Dynamics. Papres of ll th International Symposium on Rarefied GasDynamics, Cannes, 1978. CEA, Paris, 1979, Vol. 2, 693-734.6. KOG AN , M. N., Kinetic theory in aerothermodynamics. Prog. Aerospace Sci., 1992, 29, 4, 2771-354.7 . NAG NIBED A, Ye. A. and KUSTOVA, Ye. V., Kinetic Theory of Transport and Relaxation Processes in Non-equilibrium Flowsof Reacting Gases. Izd. SPbGU, St Petersberg, 2004.8 . ZHDA NOV, V. M. and ALIYEV SKII , M. Ya., Transport and Relaxa tion Processes in Molecular Gases, Nauka, Moscow, 1989.9. ERNST, M. N., Transport coefficients and tem perature definition. Physica, 1966, 32, 2, 252-272.10. KU ZNE TSOV , V. M., Theory of the bulk viscosity coefficient. Izv. Akad. Na uk SSSR. MZh G, 1967, 6, 89-93.11. WALDMANN, L., Transportersscheinungen in G asen von mittlerem Druc k Hand buck der Physik. Ban d XI I (Ed. S. Flugge).Springer, Berlin, 1958, 295.12. CH ER NY I, G. G. and LOSEV, S. A. (Eds), Physicochemical P rocesses in Gas Dynamics. Hand book. Vol. 2. Physiocochem icalKinetics and Thermodynamics. Nauchno-Izd. Tsentr Mekhaniki, Moscow, 2002.13. GA LKIN , V. S. and ZHAR OV, V. A., Solution of problems of the propagation of sound and the structure of a weak shock

w a v e in a polyatomic gas using Burnett 's equations. Prikl. Mat. M ekh., 2001, 65, 3, 467-476.14. GA LKIN , V. S. and RUSA KOV, S. V., Burnett 's mo del of the structure of a shock wave in a molecular gas. Prikl. Mat. M ekh.,2005, 69, 3, 419-426.Translated by P.S.C.

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Bulk Viscosity and Relaxation Pressure Theory

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