Thermal convection instability of an Oldroyd-B fluid in bidispersive porous media

This study investigates the thermal convection instability of an Oldroyd-B viscoelastic fluid in bidispersive porous media based on the Darcy model. The governing Darcy-Oldroyd momentum equations are formulated under the local thermal equilibrium assumption, and the corresponding dimensionless perturbation equations are derived. Linear stability analysis is employed to identify both stationary and oscillatory modes of instability, and the critical Rayleigh-Darcy numbers are numerically computed. The results reveal that the Oldroyd-B model admits both stationary and oscillatory instabilities, while the Newtonian limit yields only stationary instability. Increasing the permeability ratio between macropores and micropores significantly suppresses convection. In oscillatory modes, the critical Rayleigh-Darcy number is generally lower than in stationary modes, indicating enhanced instability. The relaxation time λ1 intensifies the instability and promotes heat transfer, whereas the retardation time λ2 exerts a stabilizing effect. Weakly nonlinear analysis further demonstrates the evolution of the Nusselt number with respect to key parameters: larger relaxation times amplify convective heat transfer and induce periodic oscillations in the Nusselt number, while greater retardation times inhibit convection. The consistency between linear and weakly nonlinear results validates the robustness of the analysis.