3 research outputs found
From the Boltzmann Equation to the Euler Equations in the Presence of Boundaries
The fluid dynamic limit of the Boltzmann equation leading to the Euler
equations for an incompressible fluid with constant density in the presence of
material boundaries shares some important features with the better known
inviscid limit of the Navier-Stokes equations. The present paper slightly
extends recent results from [C. Bardos, F. Golse, L. Paillard, Comm. Math.
Sci., 10 (2012), 159--190] to the case of boundary conditions for the Boltzmann
equation more general than Maxwell's accomodation condition.Comment: 22 pages, work presented at the Eighth International Conference for
Mesoscopic Methods in Engineering and Science (ICMMES-2011), Lyon, July 4-8
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The Steady Boltzmann and Navier-Stokes Equations
The paper discusses the similarities and the differences in the mathematical
theories of the steady Boltzmann and incompressible Navier-Stokes equations
posed in a bounded domain. First we discuss two different scaling limits in
which solutions of the steady Boltzmann equation have an asymptotic behavior
described by the steady Navier-Stokes Fourier system. Whether this system
includes the viscous heating term depends on the ratio of the Froude number to
the Mach number of the gas flow. While the steady Navier-Stokes equations with
smooth divergence-free external force always have at least one smooth
solutions, the Boltzmann equation with the same external force set in the
torus, or in a bounded domain with specular reflection of gas molecules at the
boundary may fail to have any solution, unless the force field is identically
zero. Viscous heating seems to be of key importance in this situation. The
nonexistence of any steady solution of the Boltzmann equation in this context
seems related to the increase of temperature for the evolution problem, a
phenomenon that we have established with the help of numerical simulations on
the Boltzmann equation and the BGK model.Comment: 55 pages, 4 multiple figure
Second-order Knudsen-layer analysis for the generalized slip-flow theory II: Curvature effects
Numerical analyses of the second-order Knudsen layer are carried out on the basis of the linearized Boltzmann equation for hard-sphere molecules under the diffuse reflection boundary condition. The effects of the boundary curvature have been clarified in details, thereby completing the numerical data required up to the second order of the Knudsen number for the asymptotic theory of the Boltzmann equation (the generalized slip-flow theory). A local singularity appears as a result of the expansion at the level of the velocity distribution function, when the curvature exists