TY - JOUR

T1 - Asymptotic Expansions of Solutions for the Boltzmann Equation

AU - Caflisch, Russel E.

N1 - Funding Information:
I. Introduction For the Boitzmann equation with a small mean free path (or small Knudsen number) E, approximate solutions may be found as asymptotic expansions in powers of L. me two classical and most natura expansions are the Hilbert expansion and the Chapman-Enskog expansion, each of which has a range of validity but certain limitations. 'Ihe *Research supported in part by the Air Force Office of Scientific Research contract number AFOSR 85-001 7.

PY - 1987/6/1

Y1 - 1987/6/1

N2 - The Hilbert and Chapman-Enskog expansions approximate solutions of the Boltzmann equation, but each has some disadvantages: The Hilbert expansion, which results in nonlinear and linearized Euler equations, is invalid for weak shocks, weak boundary layers and long time asymptotics. The Chapman-Enskog expansion results in nonlinear Euler then Navier-Stokes then Burnett and super-Burnett equations. Navier-Stokes is correct for weak shocks, weak boundary layers and long time asymptotics, but the Burnett equations have spurious high order dispersive effects. In this paper a modified expansion is developed, which combines the best features of the two expansions. It results in nonlinear and linearized Navier-Stokes equations only and is valid in the above-mentioned regimes.

AB - The Hilbert and Chapman-Enskog expansions approximate solutions of the Boltzmann equation, but each has some disadvantages: The Hilbert expansion, which results in nonlinear and linearized Euler equations, is invalid for weak shocks, weak boundary layers and long time asymptotics. The Chapman-Enskog expansion results in nonlinear Euler then Navier-Stokes then Burnett and super-Burnett equations. Navier-Stokes is correct for weak shocks, weak boundary layers and long time asymptotics, but the Burnett equations have spurious high order dispersive effects. In this paper a modified expansion is developed, which combines the best features of the two expansions. It results in nonlinear and linearized Navier-Stokes equations only and is valid in the above-mentioned regimes.

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U2 - 10.1080/00411458708204310

DO - 10.1080/00411458708204310

M3 - Article

AN - SCOPUS:0040156965

SN - 0041-1450

VL - 16

SP - 701

EP - 725

JO - Transport Theory and Statistical Physics

JF - Transport Theory and Statistical Physics

IS - 4-6

ER -