Stability of an inviscid flow through a tube with porous wall

TL;DR

This paper analyzes the stability of inviscid flow in cylindrical tubes with porous walls, deriving conditions for instability and revealing new unstable modes caused by the porous interface, unlike non-porous flows.

Contribution

It provides the first theoretical analysis of both axisymmetric and non-axisymmetric instabilities in flows with porous walls, including explicit stability conditions and energy exchange mechanisms.

Findings

Instability exists for both axisymmetric and non-axisymmetric modes.

Hagen-Poiseuille flow with non-porous walls remains stable.

Porous walls induce instability through energy exchange between layers.

Abstract

The temporal stability of an inviscid flow through cylindrical geometries with a porous wall subjected to non-axisymmetric perturbations is investigated in the present work using an unsteady Darcy equation for the porous layer. An expression for the perturbation energy exchange term between the fluid and porous layers is derived by integrating the perturbation equation for the porous layer which is then used to prove the propositions. The necessary and sufficient condition for the existence of the instability in flows through cylindrical geometries is shown to be $\frac{d}{dr} \left( \frac{r \frac{dV}{dr}}{n^2+k^2 r^2} \right) (V-V_I)<0$ somewhere in the flow where $V$ and $V_I$ are the axisymmetric base-state velocity profile and velocity at the fluid-porous layer interface, respectively. The parameters, $k$ and $n$ are the axial and azimuthal wavenumbers, respectively. Additionally, the bounds on the phase speed ($c_r$) and growth rate ($k c_i$) are derived. The derived propositions are used to determine the stability of the sliding Couette, annular Poiseuille, and Hagen-Poiseuille flows with a porous wall. The analysis reveals, for the first time, the existence of both axisymmetric ($n=0$) and non-axisymmetric ($n\neq 0$) modes of instability in all three flows. The Hagen--Poiseuille flow through a tube with a non-porous wall is linearly stable. Therefore, the instability predicted here is solely due to the porous nature of the wall. The mechanism responsible for the predicted instability is the exchange of the perturbation energy between the fluid and porous layers.