Temporal stability of multiple similarity solutions for porous channel flows with expanding or contracting walls

In this paper, the temporal stability of multiple similarity solutions (flow pat-terns) for the incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is analyzed. This work extends the recen-t results of similarity perturbations of [1] by examining the temporal stability with perturbations of general form (including similarity and non-similarity form-s). Based on the linear stability theory, two-dimensional eigenvalue problems associated with the flow equations are formulated and numerically solved by a finite difference method on staggered grids. The linear stability analysis reveals that the stability of the solutions is same with that under perturbations of a similarity form within the range of wall expansion ratio α (−5 ≤ α ≤ 3 as in [1]). Further, it is found that the expansion ratio α has a great influence on the stability of type I flows: in the case of wall contraction (α < 0), the stability region of the cross-flow Reynolds number (R) increases as the contraction ratio (|α|) increases; in the case of wall expansion and 0 < α ≤ 1, the stability region increases as the expansion ratio (α) increases; in the case of 1 ≤ α ≤ 3, type I flows are stable for all R where they exist. The flows of other types (types II and III with −5 ≤ α ≤ 3 and type IV with α = 3) are always unstable. As a nonlinear stability analysis or a validation of the linear stability analysis, the original nonlinear two-dimensional time dependent problem with an initial perturbation of general form over those flow patterns is solved directly. It is found that the stability with the non-linear analysis is consistent to the linear stability analysis.