Viscous heating driven convection in a Darcy porous channel with viscoelastic fluid: Effects of lower Robin boundary condition and Local thermal non-equilibrium

The paper investigates the dual influence of a convective boundary condition and Local thermal nonequilibrium (LTNE) on buoyancy-driven convection in a viscoelastic fluid, precipitated by viscous dissipation in a Darcy porous channel. The impermeable boundaries covering the porous medium are constrained by a constant temperature at the top and a Robin-type thermal condition at the bottom, with a distinct Biot number for each phase. Since the system exhibits a propensity for LTNE, a two-energy model is implemented to delineate separate solid and fluid temperature fields. The general motion of the viscoelastic fluid is modelled by incorporating the Oldroyd-B constitutive relation into Darcy’s law. Key non-dimensional parameters, including retardation and relaxation times, are utilised to capture the rheological behaviour of the working fluid, while the inter-phase heat transfer coefficient ( H ) and the thermal conductivity ratio ( γ ) govern the prevalence of either the LTNE or LTE regime. The study evaluates a spectrum of limiting cases, involving symmetric and disparate Biot numbers as well as the asymptotic limit of an infinite Péclet number ( P ) . A linear stability theory is performed to examine the basic flow response to infinitesimal disturbances, developed as normal modes. The resulting ordinary differential equations for marginally stable convective rolls are resolved numerically via a Runge–Kutta solver within the Mathematica environment. It is shown that the loss of stability is significantly advanced when the fluid Biot number is markedly lower than its solid counterpart. Once the limits of LTNE ( H → 0 or γ → 0 ) and LTE ( γ → ∞ ) are achieved, the onset of convection becomes entirely independent of the solid Biot number. Furthermore, it is proved that unlike viscoelastic fluids, which consistently favour longitudinal rolls as the most stable basic flow, Newtonian fluids generally lack a preferred disturbance orientation across most regimes; however, at certain H values, longitudinal rolls can emerge as the favoured mode for convective instability.