Hypersonic expansion of the Fokker--Planck equation
A systematic study of the hypersonic limit of a heavy species diluted in a much lighter gas is made via the Fokker--Planck equation governing its velocity distribution function. In particular, two different hypersonic expansions of the Fokker--Planck equation are considered, differing from each other in the momentum equation of the heavy gas used as the basis of the expansion: in the first of them, the pressure tensor is neglected in that equation while, in the second expansion, the pressure tensor term is retained. The expansions are valid when the light gas Mach number is O(1) or larger and the difference between the mean velocities of light and heavy components is small compared to the light gas thermal speed. They can be applied away from regions where the spatial gradient of the distribution function is very large, but it is not restricted with respect to the temporal derivative of the distribution function. The hydrodynamic equations corresponding to the lowest order of both expansions constitute two different hypersonic closures of the moment equations. For the subsequent orders in the expansions, closed sets of moment equations (hydrodynamic equations) are given. Special emphasis is made on the order of magnitude of the errors of the lowest-order hydrodynamic quantities. It is shown that if the heat flux vanishes initially, these errors are smaller than one might have expected from the ordinary scaling of the hypersonic closure. Also it is found that the normal solution of both expansions is a Gaussian distribution at the lowest order.
- Research Organization:
- Department of Mechanical Engineering, Yale University, New Haven, Connecticut 06520
- OSTI ID:
- 6554603
- Journal Information:
- Phys. Fluids A; (United States), Journal Name: Phys. Fluids A; (United States) Vol. 1:2; ISSN PFADE
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
DENSITY
DIFFERENTIAL EQUATIONS
DILUTION
DISTRIBUTION FUNCTIONS
EQUATIONS
FLUID FLOW
FLUIDS
FOKKER-PLANCK EQUATION
FUNCTIONS
GASES
HYPERSONIC FLOW
MACH NUMBER
PARTIAL DIFFERENTIAL EQUATIONS
PHYSICAL PROPERTIES
PRESSURE GRADIENTS
SERIES EXPANSION
TENSORS
VELOCITY