A note on the inviscid Orr-Sommerfeld equation

Abstract

The inviscid Orr-Sommerfeld equation for ϕ(y) in y > 0 subject to a null condition as y → ∞ is attacked by considering separately the intervals (0, y₁) and (y₁, ∞), such that the solution in (0, y₁) can be expanded in powers of the wave-number (following Heisenberg) and the solution of (y₁, ∞) regarded as real and non-singular. Complementary variational principles for the latter solution are determined to bound an appropriate parameter from above and below. It also is shown how the original differential equation may be transformed to a Riccati equation in such a way as to facilitate both the Heisenberg expansion of the solution in (0, y₁) and numerical integration in (y₁, ∞). These methods are applied to a velocity profile that is linear in (0, y₁) and asymptotically logarithmic as y → ∞, and it is found that the mean of the two variational approximations is in excellent agreement with the results of numerical integration of the Riccati equation.