Hydrodynamic limits with shock waves of the Boltzmann equation

Boltzmann brought a fundamental contribution to the understanding of the notion of entropy, by giving a microscopic formulation of the second principle of thermodynamics. His ingenious idea, motivated by the works of his contemporaries on the atomic nature of matter, consists of describing gases as huge systems of identical and indistinguishable elementary particles. The state of a gas can therefore be described in a statistical way. The evolution, which introduces couplings, loses part of the information, which is expressed by the decay of the so-called mathematical entropy (the opposite of physical entropy!).

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Shock waves and the Boltzmann equation

Nonlinear partial differential equations - Contemporary Mathematics ◽

10.1090/conm/017/706084 ◽

1983 ◽

pp. 35-44 ◽

Cited By ~ 5

Author(s):

Russel Caflisch ◽

Basil Nicolaenko

Keyword(s):

Boltzmann Equation ◽

Shock Waves ◽

The Boltzmann Equation

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Solution of the Boltzmann equation at the upstream and downstream singular points in a shock wave

Journal of Fluid Mechanics ◽

10.1017/s002211207400156x ◽

1974 ◽

Vol 65 (3) ◽

pp. 603-624 ◽

Cited By ~ 8

Author(s):

J. P. Elliott ◽

D. Baganoff

Keyword(s):

Shock Wave ◽

Boltzmann Equation ◽

Mach Number ◽

Singular Point ◽

Shock Waves ◽

Strong Shock ◽

Singular Points ◽

The Boltzmann Equation ◽

Closure Relations ◽

Strong Shock Waves

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocityu, rather than the spatial co-ordinatex, is used as the independent variable, and an equation for ∂f/∂uat a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.

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