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 Sun et al. [“Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls,” Appl. Math. Modell. 77, 738–755 (2020)] by examining the temporal stability with perturbations of general form (including similarity and nonsimilarity 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 the staggered grids. The linear stability analysis reveals that the stability of the solutions is same with that under the perturbations of a similarity form within the range of the wall expansion ratio α. [−5≤α≤3 as in Sun et al., “Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls,” Appl. Math. Modell. 77, 738–755 (2020)]. 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 nonlinear analysis is consistent with the linear stability analysis.
Highlights
The laminar flow in a porous channel with expanding or contracting walls has attracted much attention due to its wide applications in engineering and biomedicine, including transpiration cooling, phase sublimation, propellant burning, filtration, and blood transport in organisms
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
For α < 0, we find that the critical cross-flow Reynolds number decreases and the stability region increases as the contraction ratio (|α|) increases; for α > 0, the critical R decreases and the stability region increases as the expansion ratio (α) increases
Summary
The laminar flow in a porous channel with expanding or contracting walls has attracted much attention due to its wide applications in engineering and biomedicine, including transpiration cooling, phase sublimation, propellant burning, filtration, and blood transport in organisms. Dauenhauer and Majdalani [2] obtained a self-similar solution for a porous channel flow with expanding or contracting walls They assumed that the wall expansion ratio α was a constant and reduced the Navier-Stokes equations to a boundary value problem of a fourth-order nonlinear ordinary differential equation that could be solved by a shooting method. For type IV solutions, the profiles (shown in Fig. 9(b)) are characterized by a rapid increase in the centerline velocity and the wall velocity gradient (F ′′(1)) as R increases, and the development of reverse flow near the wall of the channel. It is not difficult to verify that all steady state similarity solutions satisfy the proposed condition (25) at the artificial boundary
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