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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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