Hagen–Poiseuille flow in the pipe layered by porous medium is linearly unstable

Abstract

The long-standing linearly stable Hagen–Poiseuille flow is shown to become unstable when a low-permeable porous medium layers the inner surface of the pipe. The analysis indicates that depending upon the media permeability, a threshold value of the fluid layer thickness exists below which the onset of instability occurs under axisymmetric disturbances, whereas above the threshold value, the same occurs under non-axisymmetric disturbance. In the former case, the instability is induced due to the interaction of the dynamics of base flow with the porous layer and leads to the porous mode of instability. The latter case is due to the combined effect of Reynolds stress in the fluid regime and slip porous boundary at the interface, and gives rise to the fluid mode of instability. For instance, when the Darcy number ($Da$) and Beavers-Joseph slip coefficient (${\alpha }_{{}_{{}_{BJ}}}$) are fixed at $10^{-6}$ and 0.1, respectively, the threshold value of thickness ratio, $\hat{t}$ is around 0.0336. Our results show that the threshold value of $\hat{t}$ increases monotonically with an increase in $Da$. In the fluid mode, energy production due to Reynolds stress is balanced by energy loss via viscous dissipation, whereas in porous mode, the same is balanced mainly by combined energy loss via surface drag and work done at the interface. In addition, keeping the thickness of the porous region fixed, the fluid layer thickness for which almost similar instability characteristics are found varies directly as the square root of media permeability. Our rigorous analysis also shows that ${\alpha }_{{}_{{}_{BJ}}}$ destabilizes the flow, and the onset of instability takes place at Reynolds number as small as 695, when $Da=10^{-6},{\alpha }_{{}_{{}_{BJ}}}=0.3$ and $\hat{t}=0.016$. Furthermore, an increase in ${\alpha }_{{}_{{}_{BJ}}}$ invites the fluid mode for relatively low values of fluid layer thickness.