By Paco A. Lagerstrom
Fluid Mechanics and Singular Perturbations
summary: Fluid Mechanics and Singular Perturbations
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The results of the present paper apply, of course, equally well to the Oseen equations as to the Navier-Stokes equations. The Oseen equation for ψ is: 2 (vV*-U-^)v y = 0. (31a) The boundary conditions for a flat plate are ψ ^= ^y ~^ ^ ' Ihc ~^ ^ 0==at y = 0, x>0 as | y | -> oo , or as (31b) χ -> —oo . (31c) The stream function ψβis obtained from the equation resulting by the first limit process, which is j ~ V* Ψ* = 0 , 34 (32a) FLUID MECHANICS A N D SINGULAR 128 SAUL and from the boundary conditions : w, = 0 at ν -= 0, χ > 0; p PERTURBATIONS KAPLUN d ZAMP e d -> U, ^ -> 0 as j ν I -> σο, χ -> - σ ο (32b) (cf.
The problem of matching various expansions will therefore have to be reconsidered; Ref. 2 contains a detailed investigation of this problem. It is found that the matching principle used in boundary layer theory may be generalized so as to apply to the present case (this is applied in Ref. 3, p. ). A brief review of the reasoning developed in Ref. 2 is given immediately below. Intermediate Limits. In order to find the connection between the outer and inner limits (Eqs. 8 and 9) we consider other limits which in a certain sense are intermediate.
6) one obtains the Oseen expansion of q * (or other flow quantities) q* - Σ e, (Re) g,Gr<) 44 (10a) FLUID M E C H A N I C S A N D SINGULAR NAVIER-STOKES PERTURBATIONS SOLUTIONS 587 where g 0 = lirrio q*, q* - (10b) Σ *,g, g n 1+ = lirrio (10c) Similarly, a repeated application of the Stokes limit gives a Stokes expansion Ee,(Re)h,(x1). (ID By insertion of these expansions into Eq. 2 (written in Oseen and Stokes variables respectively) one obtains the equations for the g, and the h, . For a large class of bodies, including all finite bodies, (cf.