Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T13:54:53.472Z Has data issue: false hasContentIssue false

Brownian motion far from equilibrium: a hypersonic approach

Published online by Cambridge University Press:  26 April 2006

P. Riesco-Chueca
Affiliation:
Yale University, Department of Mechanical Engineering, PO Box 2159 Y.S., New Haven, CT 06520, USA
J. Fernández de la Mora
Affiliation:
Yale University, Department of Mechanical Engineering, PO Box 2159 Y.S., New Haven, CT 06520, USA

Abstract

An investigation is carried out on the effects of Brownian agitation in the motion of small particles in a carrier gas in situations far from equilibrium. Although the standard near-equilibrium closure of the hydrodynamic equations is not valid for the heavy particles, the smallness of their speed of thermal agitation allows an alternative systematic hypersonic closure. The hypersonic equations are solved for two known instances where the kinetic Fokker–Planck equation describing the non-equilibrium particle distribution function admits exact solutions. These problems are characterized by a null or a spatially constant value for the gradient of the velocity field in the carrier gas, both being free from boundary surfaces. In the first case, where the background velocity is uniform, a fundamental solution (expressed as an integral) is obtained for the steady flow of particles from a point source; this result has obvious applications for the description of the Brownian broadening of particle streamlines. An asymptotic integration of the fundamental solution yields analytical expressions for the particle hydrodynamic properties valid everywhere except near the source, where a direct integration of the Vlasov equation completes the description. The exact solution for the second example, where the background velocity field gradient is uniform, is taken from the literature. Once these reference solutions have been established, the hypersonic equations are attacked by a variety of methods. In particular, for the uniform steady flow, a boundary-layer analysis yields analytical results identical to those obtained from the asymptotic evaluation of the kinetic fundamental solution. In both problems, the agreement found between kinetic and hydrodynamic solutions is excellent even for values of order one of the inverse particle Mach number, the expansion parameter of the hypersonic theory.

Type
Research Article
Copyright
© 1990 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)