A linear stability analysis of a pressure driven, incompressible, fully developed laminar Poiseuille flow of immiscible two-fluids of stratified viscosity and density in a horizontal channel bounded by a porous bottom supported by a rigid wall, with anisotropic and inhomogeneous permeability, and a rigid top is examined. The generalized Darcy model is used to describe the flow in the porous medium with the Beavers-Joseph condition at the liquid-porous interface. The formulation is within the framework of modified Orr-Sommerfeld analysis, and the resulting coupled eigenvalue problem is numerically solved using a spectral collocation method. A detailed parametric study has revealed the different active and coexisting unstable modes: porous mode (manifests as a minimum in the neutral boundary in the long wave regime), interface mode (triggered by viscosity-stratification across the liquid-liquid interface), fluid layer mode [existing in moderate or O(1) wave numbers], and shear mode at high Reynolds numbers. As a result, there is not only competition for dominance among the modes but also coalescence of the modes in some parameter regimes. In this study, the features of instability due to two-dimensional disturbances of porous and interface modes in isodense fluids are explored. The stability features are highly influenced by the directional and spatial variations in permeability for different depth ratios of the porous medium, permeability and ratio of thickness of the fluid layers, and viscosity-stratification. The two layer flow in a rigid channel which is stable to long waves when a highly viscous fluid occupies a thicker lower layer can become unstable at higher permeability (porous mode) to long waves in a channel with a homogeneous and isotropic/anisotropic porous bottom and a rigid top. The critical Reynolds number for the dominant unstable mode exhibits a nonmonotonic behaviour with respect to depth ratio. However, it increases with an increase in anisotropy parameter ξ indicating its stabilizing role. Switching of dominance of modes which arises due to variations in inhomogeneity of the porous medium is dependent on the permeability and the depth ratio. Inhomogeneity arising due to an increase in vertical variations in permeability renders short wave modes to become more unstable by enlarging the unstable region. This is in contrast to the anisotropic modulations causing stabilization by both increasing the critical Reynolds number and shrinking the unstable region. A decrease in viscosity-stratification of isodense fluids makes the configuration hosting a less viscous fluid in a thinner lower layer adjacent to a homogeneous, isotropic porous bottom to be more unstable than the one hosting a highly viscous fluid in a thicker lower layer. An increase in relative volumetric flow rate results in switching the dominant mode from the interface to fluid layer mode. It is evident from the results that it is possible to exercise more control on the stability characteristics of a two-fluid system overlying a porous medium in a confined channel by manipulating the various parameters governing the flow configurations. This feature can be effectively exploited in relevant applications by enhancing/suppressing instability where it is desirable/undesirable.

This study is motivated by the preliminary direct numerical simulations in double-diffusive (DD) core-annular flows with slip at the wall which displayed elliptical shaped instability patterns as in a rigid pipe case; however, slip at the pipe wall delays the onset of instability for a range of parameters and increases the phase speed. This increased our curiosity to have a thorough understanding of the linear stability characteristics of the miscible DD two-fluid flow in a pipe with slip at the pipe wall. The present study, therefore, addresses the linear stability of viscosity-stratified core-annular Poiseuille flow of miscible fluids with matched density in a slippery pipe in the presence of two scalars diffusing at different rates. The physical mechanisms responsible for the occurrence of instabilities in the DD system are explained through an energy budget analysis. The differences and similarities between core-annular flow in a slippery pipe and in a plane channel with velocity slip at the walls are explored. The stability characteristics are significantly affected by the presence of slip. The diffusivity effect is non-monotonic in a DD system. A striking feature of instability is that only a band of wavenumbers is destabilized in the presence of moderate to large inertial effects. Both the longwave and shortwave are stabilized at small Reynolds numbers. Slip exhibits a dual role of stabilizing or destabilizing the flow. The preliminary direct numerical simulations confirm the predictions of the linear stability analysis. The present study reveals that it may be possible to control the instabilities in core-annular pressure driven pipe flows by imposing a velocity slip at the walls.

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