a Similarity Solution of the Navier-Stokes Equations for Two-Dimensional Flow in a Porous-Walled Channel.

Available from UMI in association with The British Library. We describe an exact solution of the Navier-Stokes equations, of similarity form, for two-dimensional laminar flow of a viscous incompressible fluid in a channel with parallel, rigid, porous walls driven by uniform, steady suction or injection at the walls. This problem has been studied extensively when the rates of suction (or injection) through the two walls are equal, i.e. in the presence of symmetry. We consider the problem of unequal rates of suction (or injection) through the two walls. For general rates of fluid suction (or injection) we find steady solutions by numerically integrating an ordinary differential equation. These numerical results guide us in determining matched asymptotic expansions for large positive and negative values of the Reynolds number, R, for a range of ratios of suction (or injection) at the top and bottom walls. We interpret our numerical results through bifurcation theory. We analyse a Hopf bifurcation and a pair of saddle-node bifurcations by weakly nonlinear theory, and find good agreement with our numerical calculations. When only one wall is porous, and the other is impermeable, we determine analytically the number and character of the steady solutions, again with good agreement between analysis and numerical calculations. We numerically integrate a partial differential equation to find unsteady solutions: we describe and analyse these solutions for large R; near a homoclinic bifurcation we unfold the abrupt transition to chaos which is observed, at low Reynolds number, when there is equal suction at the two walls. We compare direct numerical integration of the partial differential equation which governs the streamfunction with the predictions of a theory for "Lorenz -like" dynamical systems.