Supplementary material from "Stability of buoyant viscoelastic fluid flow in a vertical porous layer with horizontal throughflow"

The stability of buoyancy-driven convection in a viscoelastic fluid saturating a vertical porous layer under the influence of a horizontal throughflow is studied. The viscoelastic behaviour is displayed by means of the Oldroyd-B type fluid, and its flow is represented through a suitable extension of Darcy's Law. The basic velocity and temperature fields turn out to be independent of viscoelastic rheology but are significantly influenced by throughflow. A linear stability analysis leads to a differential eigenvalue problem, which is numerically solved to obtain the neutral stability curves and the critical Darcy–Rayleigh number marking the onset of instability. The transition to instability is governed by the Péclet number and viscoelastic parameters, both of which influence the position of the neutral stability curve and the critical Darcy–Rayleigh number. The elasticity of the fluid primarily drives instability, with stress relaxation and strain retardation parameters exerting opposing effects. Though throughflow by itself does not render the system unstable, it profoundly influences the onset of instability once it arises—regardless of direction—by inducing both stabilizing and destabilizing effects.