The Solution of the Boltzmann Equation for a Shock Wave using a Restricted Variational Principle

1954 ◽  
Vol 22 (6) ◽  
pp. 1045-1049 ◽  
Author(s):  
Philip Rosen
Keyword(s):  
Shock Wave ◽  
Boltzmann Equation ◽  
Variational Principle ◽  
The Boltzmann Equation

A note on Mott-Smith's solution of the Boltzmann equation for a shock wave

1957 ◽  
Vol 3 (3) ◽  
pp. 255-260 ◽  
Author(s):  
Akira Sakurai
Keyword(s):  
Shock Wave ◽  
Boltzmann Equation ◽  
Mach Number ◽  
Interpolation Formula ◽  
Velocity Space ◽  
Unique Determination ◽  
Finite Region ◽  
Wave Problem ◽  
The Boltzmann Equation

After a Modification, the interpolation formula of Mott-Smith (1951) for the shock wave problem is found to be a solution of the Boltzmann equation at large Mach number in a finite region of molecular velocity space. This modification gives a unique determination of the shock wave thickness, removing the ambiguity for this in Mott-Smith's formula.

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

1974 ◽  
Vol 65 (3) ◽  
pp. 603-624 ◽  
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.

High-accuracy deterministic solution of the Boltzmann equation for the shock wave structure

Shock Waves ◽  
2015 ◽  
Vol 25 (4) ◽  
pp. 387-397 ◽  
Author(s):  
E. A. Malkov ◽  
Ye. A. Bondar ◽  
A. A. Kokhanchik ◽  
S. O. Poleshkin ◽  
M. S. Ivanov
Keyword(s):  
Shock Wave ◽  
Boltzmann Equation ◽  
Wave Structure ◽  
High Accuracy ◽  
Shock Wave Structure ◽  
The Boltzmann Equation ◽  
Deterministic Solution
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