Stability of a plane Poiseuille flow in a channel bounded by anisotropic porous walls

Abstract

Linear stability of a plane Poiseuille flow in a channel bounded by anisotropic permeable walls supported by rigid walls is studied. Characteristic instability features due to two-dimensional infinitesimal disturbances of the most unstable wall mode are investigated in detail. A detailed parametric study displays the existence of wall modes, porous modes, and center modes in both the presence and absence of inertial effects. The results reveal that an increase in mean permeability decreases the critical Reynolds number, destabilizing smaller wavenumbers. Although anisotropy has no significant effect on the growth rate at smaller wavenumbers, the impact is substantial at larger wavenumbers, particularly destabilizing short-wave modes and enlarging the bandwidth of unstable wavenumbers. Furthermore, in relation to the configuration with isotropic permeability, the one with larger (smaller) relative wall-normal permeability is more (less) unstable with a large bandwidth of unstable wavenumbers covering short-wave lengths when mean permeability is high and when the fluid channel thickness is the same as the thickness of each of the porous walls. The critical Reynolds number increases with an increase in anisotropic permeability, while the critical wavenumber decreases with an increase in anisotropic permeability. This demonstrates the possibility of enhancing (suppressing) instability by designing the channel walls as one with anisotropic permeability and appropriately tuning the relative wall-normal permeability to be higher (lower). Furthermore, anisotropic permeability can be used to control instabilities for any arbitrary relative thickness of the porous medium beyond a minimum relative thickness that depends on the relative magnitude of wall normal anisotropic permeability.