By Daniel Evan Jones

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Thus ord / is in the overlap domain if ord / < ord 1 and at the same time lim r is a sufficiently slow limit so that lim/ q* = 1. Hence we obtain the fundamental matching principle lim/ (i — u 0) = 0 in some overlap domain. (20) Now u 0was defined as a Stokes solution satisfying the inner boundary conditions. The matching condition (20) furnishes the missing boundary condition and determines u 0to order unity. Extending the above ideas to higher order approximations we replace the Stokes expansion by an intermediate expansion th q*~ E«/(Re)u, (21) We require the n partial sum to be valid to order € n(Re) in some convex domain ord Re ^ ord / < ord /„ where the limit corresponding to /„(Re) is sufficiently slow so that the domain just described overlaps with the corresponding domain of validity of the Oseen expansion.

K A P L U N , The Role of the Coordinate System in Boundary Layer Theory. V I I I International Congress for Theoretical and Applied Mechanics (Istanbul, 1952). [ 7 ] L . , New York, 1946). [8] S. , Modern Developments in Fluid Dynamics, Vol. I (Oxford, 1938). [9] J. A . L E W I S and G . F . C A R R I E R , Some Remarks on the Flat Plate Boundary Layer, Quart, appl. Math. 7 ( 1 9 4 9 ) . S. , Ν . E. K O C H I N , Sobranie Sochinenii, V o l . I I , Collected Works (published by U. S. R. Academy of Sciences, Moscow and Leningrad, 1949).

An approximation for the con tinuous transition across the boundary layer may be found by the application of another limit process, described in the next paragraph. Second limit process. Let ε be a small parameter which, in a sense, measures the effective thickness of the transition region. In the present case ε will be taken proportional to Vv. More generally, ν may be a function of ε such that lim = const Φ 0 . (4 b) A system of coordinates (f, η) will be chosen with the only restriction that η be zero at the wall.

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