26 The Eigenvalue Spectrum for Small Reynolds Numbers

For sufficiently small values of the Reynolds number we expect viscosity to have a purely stabilizing effect. Since c has been made dimensionless with respect to V we would expect the damping rate ?c _i to become large (and negative) like 1/ R as R ?0 so that the product ?Rc is independent of V in this limit. Thus, as R ?0 it is clear that we must recover Rayleigh s (1892a) results for the small oscillations of a fluid at rest, and this is confirmed by the present discussion.

The Orr-Sommerfeld equation has been studied in detail for small values of the Reynolds number by Southwell & Chitty (1930) for the particular case of plane Couette flow. A more general discussion was given later by Pekeris (1936), who also gave detailed results for both plane Couette and plane Poiseuille flow. The problem has been considered again independently by Birikh, Gershuni & Zhukhovitskii (1965). These results all show that there are important qualitative differences in the spectrum depending upon whether the basic flow is odd or not.

26.1 A perturbation expansion

Consider then, following Pekeris, an expansion of the solution in powers of i ?R of the form

For simplicity we will restrict the present discussion to the determination of c ⁽⁰⁾ and c ⁽¹⁾ which give the limiting values of the damping rate and wave speed...