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 -