Depth-dependent permeability effects on the oscillatory onset of Oldroyd-B porous convection

The oscillatory onset of buoyancy-driven convection in an Oldroyd-B fluid-saturated porous layer with linearly depth-dependent permeability is investigated. The viscoelastic rheology introduces elastic memory into the momentum balance through an extended Darcy law. Both increasing and decreasing permeability gradients are considered to examine how permeability stratification modifies elastic instability mechanisms. The resulting linear stability eigenvalue problem is solved using a Galerkin method to determine the critical conditions for instability over a wide range of relaxation, retardation, and variable permeability parameters. The onset of oscillatory convection is advanced by elastic relaxation and by permeability decreasing with depth, whereas retardation effects and permeability increasing with depth delay the instability. The retardation threshold at which the instability switches from oscillatory to stationary mode increases with the relaxation parameter and is only weakly influenced by the variable permeability parameter, irrespective of its sign. In the Maxwell limit, removal of stress retardation promotes instability and favors shorter-wavelength modes, highlighting the qualitative distinction between Maxwell and Oldroyd-B viscoelastic behavior. In the Newtonian limit, the classical results for convection with depth-dependent permeability are recovered.