Asymptotic analysis of the steady-state and time-dependent Berman problem

Abstract

The Berman problem for two-dimensional flow of a viscous fluid through an infinite channel is studied. Fluid motion is driven by uniform suction (or injection) of fluid through the upper channel wall, and is characterised by a Reynolds number R; the lower wall is impermeable. A similarity solution in which the streamfunction takes the form ψ = −xF(y, t) is examined, where x and y are coordinates parallel to and normal to the channel walls, respectively. The function F satisfies the Riabouchinsky-Proudman-Johnson equation, a partial differential equation in y and t; steady flows satisfy an ordinary differential equation in y. The steady states are computed numerically and the asymptotics of these solutions described in the limits of small wall suction or injection, large wall injection and large wall suction, the last of these being given more concisely and more accurately than in previous treatments. In the time-dependent problem, the solution appears to be attracted to a limit cycle when R ≫ 1 (large wall suction). This solution has been computed numerically for ε = 1/R down to 0·011, but the structure of the solution makes further numerical progress currently infeasible. The limit cycle consists of several phases, some with slow and others with very rapid evolution. During one of the rapid phases, the solution achieves a large amplitude, and this feature of the solution lies behind the practical difficulties encountered in numerical simulations. The profile of the solution is plotted during the various phases and corresponding asymptotic descriptions are given. An exact solution to the Riabouchinsky-Proudman-Johnson equation covers most of the phases, although separate discussion is required of the boundary layers near the two walls and an interior layer near a zero of F. Particular consideration is required when this zero approaches the upper channel wall.