THREE-DIMENSIONAL FLOW IN A POROUS CHANNEL

This paper describes three-dimensional flow of a viscous incompressible fluid driven along a channel by uniform suction through parallel porous walls, generalizing recent work on two-dimensional flow. The Navier-Stokes equations are reduced to two nonlinear diffusion equations with time and the coordinate normal to the walls as independent variables, by use of a generalization of the Hiemenz similarity solution. These equations and the boundary conditions are parametrized in dimensionless form by R, a Reynolds number, and μ, a measure of the three-dimensionality. First the steady solutions of this nonlinear boundary-value problem are described, then their linear stability; particular attention is given to the case when μ = 0 corresponding to axisymmetric flow. Asymptotic results for small and large values of R are presented. In particular, new stable steady three-dimensional solutions are found such that R(μ − 1) remains finite as R → ∞, where μ = 1 corresponds to two-dimensional flow, and we analyse the non-commutability of the limits as R → ∞ and μ↓1. Finally, results of numerical integration of the initial-value problem are reported. Pitchfork bifurcations, turning points, Hopf bifurcations, chaos and the return of stable steady solutions are found as R increases.