Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

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Abstract
This paper is concerned with the derivation of asymptotic approximations to the solutions of the Orr‐Sommerfeld equation and its adjoint which are uniformly valid in a full neighborhood of the critical point. In the case of plane Couette flow, it is shown that the expansions can be expressed to all orders in terms of a restricted class of generalized Airy functions and that the slowly varying coefficients in these expansions can be determined from simple recurrence relations. The expansions can easily be related to the Laplace integral representation of the solutions but this is not essential in their derivation. The structure of the theory also suggests how it can be immediately generalized to provide “first approximations” for a more general class of velocity profiles. Detailed results are given in the special case when the wave number α → 0. In this limit, the characteristic equation can be reduced to an especially simple form which permits a convenient global description of the modal structure.

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