Singularities in the nonisotropic Boltzmann equation

Physica A 165 (1990) North-Holland

361-369

SINGULARITIES IN THE NONISOTROPIC BOLTZMANN EQUATION C.R. GARIBOTTI’, Centro Ardmico Bariloche,

Received

12 June

M.L. MARTIARENA’ 8400 Bariloche,

and D.H. ZANETI’E’

Argentina

1989

We consider solutions of the nonlinear Boltzmann equation (NLBE) with nonisotropic singular initial conditions. The NLBE is transformed into an integral equation which is solved iteratively. The evolution of delta and step singularities in the distribution function during the initial layer is discussed and compared with the isotropic case.

1. Introduction The nonlinear Boltzmann equation (NLBE) describes the evolution of a monoatomic dilute gas towards its equilibrium state. The difficulty in solving this equation is due to the complex mathematical structure of the nonlinear terms. The simplified linearized version has given relevant results in kinetic theory; however, nonlinear effects can become important in many cases, in particular when the system is far from equilibrium [l]. A good approximation to the NLBE is given by the fluid dynamics equations, when the gas flow variations on a macroscopic scale are much larger than the particle scale. This approximation provides the normal solution as a perturbative expansion in terms of a parameter E, of the order of the mean free path or mean free time [2]. When the distribution function has a fast variation in a time or space interval of order E, the fluid dynamics equations do not provide an appropriate description. This is the situation for a short initial layer, a narrow spatial boundary layer and internal shock layers. In these cases the distribution function must be obtained by solving the NLBE. A large number of methods has been proposed to obtain that solution [l-3]. Further to the perturbative approaches, other techniques are based upon the moment methods, where the solution is expanded in a complete set of functions, i.e., Hermite [4], Sonine and Laguerre polynomials [3]. Recently we proposed an iterative solution of the NLBE in terms of the deviation from ’ Supported

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362

C.R. Garibotti et al. I Singularities in nonisotropic Boltzmann equation

equilibrium [5]. Significant progress has been achieved in the analytic approaches for velocity isotropic and spatially homogeneous gases, when particular interaction models are considered [3]. The nonisotropic distribution f(u, t) can be expanded in an orthogonal base for Maxwell molecules [6,7] and we studied the numerical convergence for some particular conditions in Z&(0,00). A modified expansion was proposed to improve convergence properties at small times and large energies [8]. In this paper we deal with the early evolution of the solution of the space-homogeneous, velocity nonisotropic NLBE. We assume singular initial conditions which allow for an analytic expression of the structure of the layer. This initial layer problem has been formerly considered by Hendriks and Ernst [8] in the isotropic case, for some model of molecular interactions. They considered singularities in the energy distribution and found that new weaker singularities appear in the distribution as time evolves. These singularities are located at energies which are multiples of the initial position and smoothed out as the time elapses. We have to mention that delta-like singularities in the velocity distribution have also been considered for the study of spatial relaxation of strong shock waves [9]. Singular initial conditions can be considered as being interesting to deal with the physical problem of the penetration of particles in a gas. They give a simplified model for a monochromatic beam crossing over a rarefied target. This nonisotropic system represents a far from equilibrium situation and our results give its evolution in the kinetic stage, where time is of the order of the mean free time and each particle suffers few collisions.

2. Iterative solution When external forces are absent, the NLBE is [l]

$ +u -Vf=

B[f, f] .

(1)

In the initial layer aflatand the collision term are large, and the convection term can be neglected [lo]. Therefore it can be described by the spatially homogeneous NLBE [3]

g

where

=

fTf7 fl-

I[f,

fl >

C.R.

Garibotti et al. / Singularities in nonisotropic

fTf?fl = Z[fTfl =

1dw du’ dw’ W(u, wlu’, w’) f(u’,

1dw du’ dw’ W(u’, w’(u, w) f(u,

Boltzmann equation

t) f(w), t) ,

t) f(w, t)

363

(3)

(4)

are the gain and loss terms, respectively. W(u’, w’lu, w) is the transition probability per unit of time for the binary collision (u, w)-+ (u’, w’), and it is obtained from the differential cross section a( g, .$ - A) by W(u’, W’IU, w) = p2”g3-dC(g,

g* li) SCd’(u + w - u’ - w’)

x S(v2 + w* - Uf2 - w’*) ) where we consider a d-dimensional velocity space and a deterministic is the spatial gas density. Velocities are related by

(5)

model; p

u’=gu+w)+$ig, W’ =

$(u + w) - $rig )

(6)

g=u-w,

with h being the unit vector in the direction of the relative final velocity g’ = u’ - w’. We denote by P[r, S] the quadratic functional of two functions T(U, t) and s(u, t) defined as in eq. (3). Eq. (1) can be written as $ = P[fY fl -f(u,

4 Ufl

7

(7)

where Ufl=

PZd

dw f(w, t)

dri gcT(g, g-2).

(8)

We will consider interaction models such that L[f] does not depend on time; by symmetry, it only depends on u = ]u/, i.e., L[f] = L(U). This condition is satisfied for pseudo-Maxwellian particles [3], for which

gc+(g,d * 4 = 4x) and

364

C. R. Garibotti

et al.

I Singularities

and for very hard particles (VHP),

in nonisotropic

Boltzmann

equation

for which

and

where

is the mean collision frequency. In that case we find that eq. (7) can be written in an integral form, f

f(u, t) =f(u,

0) emL(“)’ +

I

e-L(“)(r-r’)P[f,

f] dt’ ,

(9)

0

which can also be obtained from the general integral equation written by Grad [ll]. We propose an iterative solution of eq. (9),

f(l)(u, t) = f(u,

0) epL(“)’ ,

f’“‘@,,

0)

t)

=f(u,

e-L(‘J)’

+ e-L(U)’

I

e+L(U)r’f,[f(“-‘),

f’“-“1

dt’

.

We note that successive iterations introduce an increasing number of binary collisions. Therefore, that number can be considered as the implicit expansion parameter. Eq. (9) exhibits a clear separation of the nonlinear gain contribution, which decays with an exponential loss factor. The convergence of this approach has been shown in the isotropic case for the interaction models here considered [5].

3. Singular initial conditions First, we introduce write f(u, 0) =fi =

an initial condition with two delta peak singularities. We

;p[qu -

uo) + 6(u + uo)] .

(11)

C.R.

Garibotti et al. I Singularities in nonisotropic

365

Boltzmann equation

This can be interpreted as the collision of two space extended monoenergetic beams. Because a single delta function can be considered as a limit of a Maxwellian, eq. (11) gives a far from equilibrium condition. In this case, we obtain for the first two iterations in eq. (10)

f(“(v,

t) = f; epL(“)’ + A (u, t) P[fi, _fJ ,

fc3)(u,

t) =f,

I

(12) emL(“jt + A,(u,

t) I’[$, A] + 2A,(u, f> p[_t, p[_t,

fill

+ A,(u, f) P[P[A, &I, p[&>AlI. The coefficients

determine

the temporal evolution of each term and are given

bY

A,=D,

A, = (D, - 20, + D3)lL2(u),

A, = (D, - D,)lUu),

9

(13) with D,(u, t) = (epL(“)’ - e-‘“+*‘L(“o)r)l[(n

+ l)L(u,)

- L(u)] .

(14)

These are the first terms of the general expansion obtained by Barrachina [12] for pseudo-Maxwell particles. The initial condition (11) does not belong to L&(0, CQ)and the expansion (12) should be interpreted as a generalized function. For d = 2, we substitute eq. (11) in eq. (3) and obtain

m3 Al =

P3bW,>6)

+ (r(2u,, 7r - S)] 6(u - U”) )

054

where 8 is the angle between u and u,,, and

J,(u, 8) =

vu, -

x;> +&T,d&, x2) 4%?

%)[&4g,, Xl) ~c%l?

x;)l (16)

with

366

C.R. Garibotti et al. I Singularities in nonisotropic Boltzmann equation

cos

x’,,* =

$

{cos

~Y[ui+ h(u* - u’,)“‘] -

cos x1,* = {2[u2, + A(u2 - u’,)““] 2 g1,2

=

u

,*

+

u;

-

u,

sin S[(u” - ui)l’* + A]} ,

- 2uu, co.5 6 - u’2 + II’,} /g,,, )

2u’uo cos x;,, )

(17)

urn= u(u - u. cos 6)/(u2 + u; - 2uu, cos 8) , A = (/

- u;)1’2 .

The subscripts 1,2 correspond to the -+ signs; in the present case we have u’ = uO. The .I2 is obtained from .I, by replacing x’,,, by T - x;,~, and Y(u) denotes the step function. We obtain that, after a first collision, particles in the beam are spread in a singular circle, given by S(u - uJ, with an angular weight determined by the cross section. A second collision introduces weaker singularities, first as a cusp in a neighborhood of u = uO, and secondly by a jump at u, = u,, given by the step function. Depending on the considered interaction model, some reinforcement of the beam could be produced by the other terms in eq. (11). The evolution of the initial beam is given by the first term in eq. (10) and is characterized by a mean decay time t, = 1 /L(u,). For u = uO, the coefficients A,(u, t) (n = 1,2,3) have maxima at t, = ln(n + 1)/L(u). This corresponds to times for which the number of particles available for single, double and triple collisions begin to decrease. Therefore the population of the circle u = uO, coming from the first scattering grows up to t = In 2/L(u,), then decays. A second collision populates the region u, =SuO, reaching a maximum for t = In 3/L(u,). This gives for VHP a longer growth for small velocities. The fourth term in the right-hand side of eq. (12) represents the collision of two singular distributions in the shell u = u0 and gives a contribution which has a step singularity at u = 2u,, which grows up to t = In 4/L(2u,). The large velocity region is slowly populated by the next terms. This indicates that the terms in the iterative expansion build up the final Maxwellian distribution centered at u = 0 as the time elapses. Next we consider an initial distribution given by a beam superposed to a distribution p(u) E Z2(0, m), f(u, 0) =A = q(u) + A6(u - uo) . For Maxwell interaction

models we can rewrite eq. (12) with

Dn(u, t) = (e-@ - e-(“-‘)p”‘)

lnp

.

(18)

C. R. Garibotti

Substituting

et al.

I Singularities

in nonisotropic

Boltzmann

equation

367

eq. (18) in eq. (3) we obtain

The first term is not singular; the second is zero by momentum and the third can be written as P[cp, 6]= 4p

conservation,

dv’ g~( g, g - ii) 6(2(v - u’) . (u - u,,)) cp(u’) cc

x [g,dg,,

Xl) (P(U’Yx’,> + g*(+(g27 x2> (P(u’,

A413

(20)

where the variables u,, g, x and x’ are related to the distribution variables u and 6 by eqs. (17). For smooth cp and cross sections, the integral is a non-singular function and P[cp, 61 has an inverse square root singularity at u = uO. Let us consider cp(u’) = (pOY(U- u’). This represents a weaker singularity than the two S-functions formerly considered. For pseudo-Maxwellian particles g(+ = LY(X)and eq. (20) gives

P[cp, aI=

In particular,

P[cp,61=

,u 1”0,,

qourn u’ du’ k’* - dJ1’*b(xJ I

+ 4x211

.

(21)

for an isotropic scattering model CX(X)= (Ye, and

,,“-“,,,

(p,Y(U - urn) (U’ - u;)1’2”0 .

From eqs. (12) and (21) results that the initial jump singularity relaxes with mean decay time r = 1 /CL; meanwhile a second jump grows at u, = U until t = In 2/p. In fig. 1 we display the curve u, = U in the velocity space. The dashed area indicates the zone where P[cp, 61 is non-zero; a forbidden region remains between u0 and U. The )u - q,I -I singularity is indicated by an arrow in the neighbourhood of the point u = uO. When q(u) is a jump function the first term in eq. (19) should be considered and presents a step discontinuity at u=2u.

368

C.R.

Garibotti et al. I Singularities in nonisotropic

Boltzmann equation

Fig. 1. The dashed area indicates the region in the two-dimensional P[ cp, S], for the collision of a delta peak with a distribution bounded the cusp.

velocity space of nonvanishing to u s U. The arrows indicate

4. Conclusions

There are relevant differences between the two initial conditions considered here and the behaviour of singularities in the isotropic case [S]. In our first case, the strength of the colliding deltas decreases with a rate of L(u,) as the particles are spread out in a shell u = uO. This is a singular shell as shown by eq. (15a) and the population in each angle is determined by the cross section. One further collision scatters particles inside the surface u, = uO, with a 1u - uo( -’ cusp transversal to the beams, which can be interpreted as the original deltas widening at small times. This effect is independent of the cross section. In the second example the beam particles collide with an arbitrary distribution p(v), producing the 1u - u,,I -’ divergence, as a consequence of the diffusion of the beam in the background gas. When q(u) has a step at u = U a first collision introduces a new jump for larger velocities, as shown in fig. 1. No particle travels behind the beam when ug > U. We note that in the isotropic case with a singular initial condition no new delta-like singularities result after the collision has started, and the original singularity keeps its shape [3]; this follows from the isotropic character of the initial condition. Our results can give an interpretation of the propagation of a beam in a medium. We consider the beam extended at the whole space; at t = 0 we start interactions and observe the evolution. This is similar to the stationary scattering theory, where the initial state is described by a spatially infinite plane wave. The first term in eq. (12),

C.R. Garibotti et al. I Singularities in nonisotropic Boltzmann equation

A6(u - q,) e-L(“)t ,

369

(22)

shows the relaxation of the monochromatic beam with a mean decay time l/L(v,). When we relate the propagation distance x with the time by x = u,t, the decay of the beam can be obtained from eq. (22), (23) This is the usual attenuation law for penetration of particles in a host medium. The following terms in eq. (12) come from the nonlinear contribution giving a shell singularity in first order and a forward direction cusp in the second one. These contributions grow and decay with characteristic times in the course of relaxation of the initial layer. Therefore, in the initial layer the beam decays more slowly than given by the attenuation formula.

References [l] C. Cercignani, Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh, 1975). [2] J.H. Ferziger and H.G. Kaper, Mathematical Theory of Transport Process in Gases (NorthHolland, Amsterdam, 1972). [3] M.H. Ernst, Phys. Rep. 78 (1981) 1. [4] H. Grad, Comm. Pure Appl. Math. 4 (1949) 331. [5] D.H. Fujii, R.O. Barrachina and C.R. Garibotti, J. Stat. Phys. 44 (1986) 95. (61 E.M. Hendriks and T.M. Nieuwenhuizen, J. Stat. Phys. 29 (1982) 591. [7] R.O. Barrachina and CR. Garibotti, J. Stat. Phys. 45 (1986) 647. [8] E. Hendriks and M.H. Ernst, Physica 120 (1986) 545. [9] M.H. Ernst, in: Fundamental Problems in Statistical Physics, E.G. Cohen, ed. (NorthHolland, Amsterdam, 1970), p. 249. [lo] R.E. Caflisch, in: The Boltzmann Equation, J. Lebowitz and E. Montroll, eds. (NorthHolland, Amsterdam, 1983). [ll] H. Grad, in: Handbuch der Physik, vol. 12, S. Fhigge, ed. (Springer, Berlin, 1958). [12] R. Barrachina, J. Stat. Phys. 52 (1988) 371.