Stability of non-isothermal Poiseuille flow in a fluid overlying an anisotropic and inhomogeneous porous domain

A two-domain approach is used to investigate the thermal convection of Poiseuille flow in an anisotropic and inhomogeneous porous domain underlying a fluid domain. The flow of the Newtonian fluid is regulated by Darcy's law in the porous domain with the implementation of the Beavers–Joseph condition at the interface. The impact of medium anisotropy and inhomogeneity is inspected by virtue of linear stability analysis along with other governing parameters such as depth ratio (ratio of depth of fluid domain to porous domain), Darcy number, Reynolds number and Prandtl number concerning the stability of the fluid–porous system. The neutral curves are found to be bimodal and unimodal in nature with the anisotropy and inhomogeneity leaving its imprint on parametric variation. An increase in anisotropy or decrease in the inhomogeneity parameter follows the modal change from unimodal (porous) to bimodal (both porous and fluid). Also, it has been identified that, irrespective of the considered variations in anisotropy and inhomogeneity, the least stable mode for the depth ratio ${}0.16$ is fluid. Furthermore, energy budget analysis is carried out to classify the type of instability and validate the type of mode. The instability is found to be thermal–buoyant in nature with omission of low Reynolds numbers along with very low values of the ratio of permeability in the horizontal to vertical direction, where thermal–shear instability is witnessed. Likewise, secondary flow patterns in the context of the streamfunction and temperature contour are analysed to validate the least stable mode and the type of prevailing instability in the fluid–porous system. The presented numerical results find good experimental support from the results of Chen & Chen (J. Fluid Mech., vol. 207, 1989, pp. 311–321) in the limit of natural convection with an isotropic and homogeneous porous domain.