On the asymptotic expansion in three-dimensional compressible viscous flow

Abstract

An exact solution of the set of equations (of motion, energy, continuity, and state) of a three-dimensional flow of a compressible viscous fluid (in which pressure, density, coefficients of viscosity and heat conductivity are functions of position) represents insurmountable difficulties and cannot be achieved. Consequently, only approximation methods may be considered. Goldstein (1, 2) 2 proposed an asymptotic expansion for the two-dimensional steady flow of an incompressible viscous fluid behind a solid body by means of exponential functions.That type of expansion was applied by the author to a few cases of a two-dimensional steady flow of a compressible viscous flow ( 3, 4, 5). In the present paper the author extends this method to a case of a three dimensional flow of a compressible fluid.Not to obscure the problem by many items of a simple algebraic nature, the paper presents only a general outline of the method of attack, applied to the simplest possible case, that is, to a flow along a yawed flat plate whose plane is parallel to the direction of the undisturbed flow.From two requirements: ( a) to solve exactly or approximately the equations, ( b) to satisfy exactly or approximately the boundary conditions, in a case of a compressible viscous fluid (when no use is made of velocity potential—or stream—functions) it is possible to satisfy the requirement ( a) approximately and ( b) exactly. This was shown by the author in the present as well as in previous papers.The presented method may be easily adjusted to three-dimensional wakes and jets similarly as it was done in plane flows.