  The stability of two-dimensional buoyancy-driven convection in a vertical
porous slot, wherein a plane Couette flow is additionally present, is studied.
This complex fluid flow scenario is examined under the influence of Robin-type
boundary conditions, which are applied to perturbations in both velocity and
temperature. The inclusion of a time-derivative velocity term within the Darcy
momentum equation notably introduces intricacies to the study. The stability of
the basic natural convection flow is primarily governed by several key
parameters namely, the P\'eclet number, the Prandtl-Darcy number, the Biot
number and a non-negative parameter that dictates the nature of the vertical
boundaries. Through numerical analysis, the stability eigenvalue problem is
solved for a variety of combinations of boundary conditions. The outcomes of
this analysis reveal the critical threshold values that signify the onset of
instability. Furthermore, a detailed examination of the stability of the system
has provided insights into both its commonalities and distinctions under
different conditions. It is observed that, except for the scenario featuring
impermeable-isothermal boundaries, the underlying base flow exhibits
instability when subjected to various other configurations of perturbed
velocity and temperature boundary conditions. This underscores the notion that
the presence of Couette flow alone does not suffice to induce instability
within the system. The plots depicting neutral stability curves show either
bi-modal or uni-modal characteristics, contingent upon specific parameter
values that influence the onset of instability.
