Nonlinear Stability and Linear Instability of Double-Diffusive Convection in a Rotating with LTNE Effects and Symmetric Properties: Brinkmann-Forchheimer Model

Ali Hasan Ali,Omar Bazighifan,Jan Awrejcewicz,Khalil S. Al-Ghafri,Ghazi Abed Meften

doi:10.3390/sym14030565

Abstract

The major finding of this paper is studying the stability of a double diffusive convection using the so-called local thermal non-equilibrium (LTNE) effects. A new combined model that we call it a Brinkmann-Forchheimer model was considered in this inquiry. Using both linear and non-linear stability analysis, a double diffusive convection is used in a saturated rotating porous layer when fluid and solid phases are not in the state of local thermal non-equilibrium. In addition, we discussed several related topics such as the effect of solute Rayleigh number, symmetric properties, Brinkman coefficient, Taylor number, inter-phase heat transfer coefficient on the stability of the system, and porosity modified conductivity ratio. Moreover, two cases were investigated in non-linear theory, the case of the Forchheimer coefficient F=0 and the case of the Taylor-Darcy number τ=0. For the validation of this work, some numerical experiments were made in the non-linear energy stability and the linear instability theories.

Highlights

Due to their wide range of applications, from the unification of binary mixtures to melting runoff in saturated soil, double diffusive convection problems in fluid and porous media have received a lot of attention in the last few decades, where symmetric properties played an important role in solving these problems
For the validation of the proposed work, we present some experimental examples and focus on the numerical solutions of the instability of the linear case and the stability of the nonlinear case
The stability of a double diffusive convection problem using the local thermal nonequilibrium (LTNE) effects has been considered in this recent work

Summary

Introduction

Due to their wide range of applications, from the unification of binary mixtures to melting runoff in saturated soil, double diffusive convection problems in fluid and porous media have received a lot of attention in the last few decades, where symmetric properties played an important role in solving these problems. The study of double diffusive-convection in a rotating porous media has been covered and supported theoretically in many studies as well as through practical applications in engineering. An essential study of the effects of rotation on linear and non-linear double diffusive convection in a sparsely packed porous medium can be found in [3]. Important examples and experiments that contain geophysical framework, electrochemistry, and some other applications with an explanation of the non-linear energy stability of double diffusive convection problems, resulting in many future investment, are those in [4,5]. The same study was performed by [15], where the linear stability of thermal convection was analyzed using the Darcy–Brinkman model Another important category of studies that are related to the problems of double diffusive-convection and the melting of permafrost under the sea can be seen in [16–19]. Further studies with many different applications such as solidify and centrifugal casting of minerals, bio-mechanics, petroleum manufacture, chemical operations and food, rotating machinery, and geophysical problem are found in [21–35]

Basic Equations

Linear Instability

Nonlinear Stability Analysis with Forchheimer Coefficient F = 0

Nonlinear Stability Analysis with Taylor-Darcy Number τ = 0 (F 6= 0)

Findings

Discussion of Results

Conclusions and Future Direction