On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales

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Abstract
The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.

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