Onset of instability in Hadley–Prats flow in a weakly heterogeneous porous layer with viscous dissipation

Abstract

The stability of a flow subjected to an inclined temperature gradient (Hadley-type flow) in a horizontal porous media is studied in the presence of a basic horizontal mass flow (Prats flow). Therefore, the basic flow is called the Hadley–Prats flow. A weak vertical heterogeneity in permeability and conductivity is considered. The effect of viscous dissipation is taken to be non-negligible. The Rayleigh number corresponding to the vertical thermal gradient RaC is considered as an eigenvalue. Other parameters are the Péclet number (Pe) associated with the horizontal through flow, horizontal Rayleigh number (RaH) associated with the horizontal temperature gradient, Gebhart number (Ge) associated with viscous dissipation; parameters γ1 and γ2 represent the changes in permeability and conductivity, respectively. A linear stability analysis is done and the governing equations are solved numerically to obtain the critical Rayleigh number and wave number. Longitudinal and transverse rolls are discussed. Longitudinal rolls are the preferred modes for instability in most scenarios. It is found that when throughflow is present, the heterogeneity in permeability can show a stabilizing effect for longitudinal rolls but destabilizing effect for transverse rolls and vice versa depending on the direction of the throughflow. Increase in conductivity also may stabilize or destabilize the flow depending on the mass flow and viscous heating. The horizontal thermal gradient shows interesting effects in the presence of weak heterogeneity and horizontal throughflow. Significant change in the critical Rayleigh number is observed even for small values of the horizontal Rayleigh number.