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

Abstract

In this paper, the temporal stability of multiple similarity solutions (flow patterns) for the incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is analyzed. This work extends the recent results of similarity perturbations of [1] by examining the temporal stability with perturbations of general form (including similarity and non-similarity forms). 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 Corresponding author Email address: [email protected] (Ping Lin ) Preprint submitted to PHYSICS OF FLUIDS July 24, 2021 T hi s is th e au th or ’s p ee r re vi ew ed , a cc ep te d m an us cr ip t. H ow ev er , t he o nl in e ve rs io n of r ec or d w ill b e di ffe re nt fr om th is v er si on o nc e it ha s be en c op ye di te d an d ty pe se t. P L E A S E C IT E T H IS A R T IC L E A S D O I: 1 0 .1 0 6 3 /5 .0 0 5 1 8 4 6 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.