COMMENTS Comments on "A generalized BKW solution of the ...

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Comments on "A generalized BKW solution of the nonlinear Boltzmann equation with removal" [Phys. Fluids 27, 2599 (1984)] A. Santos and J. J. Brey Departamento de Flsica Te6rica, Universidad de Sevilla, Apdo. Co"eos 1065, Sector Sur, 41080 Sevilla, Spain (Received 29 October 1985; accepted 12 December 1985) It is shown that there is a direct relationship between the solutions of the Boltzmann equation for a homogeneous system of Maxwell molecules with and without particle removal Recently, Spiga 1 has found an exact, particular solution of the nonlinear Boltzmann equation for Maxwell molecules that includes a removal effect, making the total number of particles decrease in time. His special solution is a generalization of the formerly obtained solution in the absence of removal events (the so-called BKW mode2 ). · We show here that there· exists a simple relationship between the solutions of both kinetic equations for arbitrary initial conditions. The Boltzmann equation for Maxwell molecules considered by Spiga1 is

a

-f(v,t) = - (Cs

at

+

+ CR )n(t)f(v,t)

JJ dv1

dfi a(x)f(v',t)f(v; ,t),

( 1)

where Cs = Jdfi a(x) and CR is the removal collision frequency. The evolution equation for the number density is

d~~t) = -

CR [n(t) ] 2,

(2)

whose solution is

(5)

f(v,-r)=[nofn(t)] f(v,t).

Thus, }"is proportional to the probability density function of the system, but now the role of time is played by -r. Of course, when CR = 0, T = t andf=f. By substituting Eq. (5) into Eq. ( 1), we obtain

~ f(v,-r) =

- C5 n0 f(v,-r)

+

f f dv1

dfi

Xa(x)f(v',-r)f(v; ,-r),

(6)

which is formally the usual Boltzmann equation (i.e., with conservation of the total number of particles). Thus, given an initial condition, the solution ofEq. ( 1) is closely related to the solution ofEq. (6). In particular, there exists aBKWlike solution to Eq. ( 1):

F(v,t) = F(v,-r)/(1

+ n0 CRt),

(7)

where -r is given by Eq. (4) and F(v,-r) is the well-known BKW mode. 2 The function F is precisely the solution obtained in Ref. 1 by using a heuristic method. ACKNOWLEDGMENTS

(3)

The authors are very thankful to Professor M. H. Ernst for an enlightening discussion about this work.

where n0 is the initial density. Now, let us introduce the time-dependent parameter

,

n(s) 1 ds--=--ln(l+n 0 CRt) o n0 n0 CR and the distribution function

l

-r(t)==

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Phys. Fluids 29 (5), May 1986

(4)

1 2

0. Spiga, Phys. Fluids 27,2599 (1984). Extensive reviews are given in M. H. Ernst, Pbys. Rep. 78, 1( 1981 ); J. Stat. Pbys. 34, 1001 (1984).

0031-9171 /86/051750-01$01.90

@ 1986 American Institute of Physics

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