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

Abstract

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 σ, 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, Rc, 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 Rc increases, the actual threshold moves towards the nonlinear stability threshold, and the behavior of the perturbation of the solutions becomes more oscillatory.

Chemical Reaction Effect on Convection in Bidispersive Porous Medium

“…In this section, the Chebyshev collection method is employed to solve the eigenvalue systems (3.4) and (4.11). For more details see Harfash (2014a, b, c), Harfash (2015, 2016a, Harfash and Nashmi (2017), Challoob (2018, 2019); Harfash and Meften (2020), Meften (2018, 2019), Hameed and Harfash (2019) and Challoob et al (2020). In this method, system (3.4) is rewritten in terms of second-order derivatives only.…”

Slip boundary condition effect on double‐diffusive convection in a porous medium: Brinkman Model

“…Now, we utilize the Chebyshev collocation method to approximate system (20) to (22), cf. [32][33][34] To use this method, system (20) to (22) transfer to a system of second and thirdorder derivatives only. Let DW Ξ = , then Equation (20) to (22) become four equations of second-order derivatives.…”

Stability of Darcy thermosolutal convection in bidispersive porous medium with reaction

“…Double diffusive convection is a problem that is being researched, and the scientific community is paying attention to it 9,17–32 . Stability analyses in double diffusive convection were introduced in the highly influential papers of Nield 26,27 and from an unconditional energy stability point of view by Joseph 33,34 .…”