On the solution for steady state and time dependent flows in certain channels with small wall curvature

An asymptotic scheme is presented for the solution of the steady-state and time-dependent stream functions for flows in symmetric curved walled channels. In this scheme a class of nonlinear Jeffery-Hamel solutions appear at 0(1), and thus provide the first approximation to the steady-state stream function. This class of Jeffery-Hamel solutions are evaluated by using a simple perturbation about Poiseuille flow. The classic Orr-Sommerfeld eigenproblem appears at 0(1) in the asymptotic development of the time-dependent stream function, but here there is a slow streamwise dependence. This eigenvalue problem, for a complex wave number, is solved using an algorithm which automatically provides an initial guess which is then used to iterate to the correct eigenvalue. Higher-order terms in the asymptotic development, for both the steady-state and time-dependent stream functions, are evaluated to provide a solution for the total stream function.