Abstract
The spatial stability of similarity solutions for an incompressible fluid flowing along a channel with porous walls and driven by constant uniform suction along the walls is analyzed. This work extends the results of Durlofsky and Brady [Phys. Fluids 27, 1068 (1984)] to a wider class of similarity solutions, and examines the spatial stability of small amplitude perturbations of arbitrary shape, generated at the entrance of the channel. It is found that antisymmetric perturbations are the best candidates to destabilize the solutions. Temporally stable asymmetric solutions with flow reversal presented by Zaturska, Drazin, and Banks [Fluid Dyn. Res. 4, 151 (1988)] are found to be spatially unstable. The perturbed similarity solutions are also compared with fully bidimensional ones obtained with a finite difference code. The results confirm the importance of similarity solutions and the validity of the stability analysis in a region whose distance to the center of the channel is more than three times the channel half-width.
Highlights
The Navier–Stokes equation for an incompressible viscous flow along a channel with porous walls, driven by constant uniform suction, admits a similarity solution
The lowest branch, and the one that determines the instability of the solution, corresponds to an antisymmetric eigenfunction. This branch crosses the line ϭ1 at R ϭRI, which means that these solutions lose their temporal and spatial stability at the same value of the Reynolds number
Solutions of type I and Ia are the only branches of solutions of Berman’s equation which are spatially stable for a certain range of Reynolds numbers
Summary
The Navier–Stokes equation for an incompressible viscous flow along a channel with porous walls, driven by constant uniform suction, admits a similarity solution. This solution was studied in 1953 by Berman, who reduced the bidimensional Navier–Stokes equation to a fourth order nonlinear ordinary differential equation with two boundary conditions at each wall. This equation depends on a sole nondimensional parameter, the transversal Reynolds number, R, defined in terms of the channel width and the suction velocity.
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