Buoyant variable viscosity fluid flow and instability in a vertical porous slab with imperfectly conducting boundaries

The study investigates the influence of temperature-dependent viscosity on the stability of buoyancy-driven flow in a vertical porous slab bounded by impermeable walls and subject to Robin thermal boundary conditions. Three viscosity–temperature relationships are considered – linear, quadratic and exponential. A normal-mode linear stability analysis is carried out for two porous flow models: Model I, representing Darcy flow with the transient velocity term neglected, and Model II, its counterpart that retains it. The resulting stability eigenvalue problem is solved numerically to obtain neutral stability curves and the critical parameters, with emphasis on the roles of the Prandtl–Darcy number, Biot number and viscosity parameters. The similarities and differences between the two models, as well as those among the viscosity–temperature laws, are examined in detail. For linear and quadratic viscosity variations, Model I predicts unconditional stability irrespective of boundary heat exchange, whereas Model II admits instability when the boundaries are thermally imperfect. In contrast, exponential viscosity variation promotes instability in both models. In Model I, instability is confined to a restricted range of Biot numbers, which depends sensitively on the exponential viscosity parameter, while the flow remains stable for all thermal boundary conditions when this parameter is less than 8.2070. Model II, however, displays qualitatively distinct behaviour characterised by mode transitions and the emergence of two instability regimes of Biot number separated by a stability window.