Unconditional nonlinear stability for double-diffusive convection in a porous medium with temperature-dependent viscosity and density

In this study, fluid flow in a porous medium is analyzed using a Forchheimer model. The problem of double-diffusive convection is addressed in such a porous medium. We utilize a higher-order approximation for viscosity-temperature and density-temperature, such that the perturbation equations contain more nonlinear terms. For unconditional stability, nonlinear stability has been achieved for all initial data by utilizing the L3 or L4 norms. It also shows that the theory of L2 is not sufficient for such unconditional stability. Both linear instability and nonlinear energy stability thresholds are tested using three-dimensional (3D) simlations. If the layer is salted above and salted below then stationary convection is dominant. Thus the critical value of the linear instability thresholds occurs at a real eigenvalue sigma, and our results show that the linear theory produces the actual threshold. Moreover, it is known that with the increase of the salt Rayleigh number, R-c, the onset of convection is more likely to be via oscillatory convection as opposed to steady convection. The 3D simulation results show that as the value of R-c increases, the actual threshold moves towards the nonlinear stability threshold, and the behavior of the perturbation of the solutions becomes more oscillatory.