Magnetic Field Effect on Stability of Convection in Fluid and Porous Media

We investigate the linear instability and nonlinear stability for some convection models, and present results and details of their computation in each case. The convection models we consider are: convection in a variable gravity field with magnetic field effect; magnetic effect on instability and nonlinear stability in a reacting fluid; magnetic effect on instability and nonlinear stability of double diffusive convection in a reacting fluid; Poiseuille flow in a porous medium with slip boundary conditions. The structural stability for these convection models is studied. A priori bounds are derived. With the aid of these a priori bounds we are able to demonstrate continuous dependence of solutions on some coefficients. We further show that the solution depends continuously on a change in the coefficients. Chebyshev collection, finite element, finite difference, high order finite difference methods are also developed for the evaluation of eigenvalues and eigenfunctions inherent in stability analysis in fluid and porous media, drawing on the experience of the implementation of the well established techniques in the previous work. These generate sparse matrices, where the standard homogeneous boundary conditions for both porous and fluid media problems are contained within the method. When the difference between the linear (which predicts instability) and nonlinear (which predicts stability) thresholds is very large, the validity of the linear instability threshold to capture the onset of the instability is unclear. Thus, we develop a three dimensional simulation to test the validity of these thresholds. To achieve this we transform the problem into a velocity-vorticity formulation and utilise second order finite difference schemes. We use both implicit and explicit schemes to enforce the free divergence equation

Simulation of three dimensional double-diffusive throughflow in internally heated anisotropic porous media

A model for double-diffusive convection in an anisotropic porous layer with a constant throughflow is explored, with penetrative convection being simulated via an internal heat source. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the steady state throughflow. However, the required time to arrive at the steady state increases significantly as the Rayleigh number tends to the linear threshold. © 2014 Elsevier Ltd. All rights reserved

Perturbation-Iteration Algorithm for Solving Heat and Mass Transfer in the Unsteady Squeezing Flow between Parallel Plates

In this paper, heat and mass transfer in the unsteady squeezing flow between parallel plates is analyzed using a perturbation-iteration algorithm. The similarity transformation is used to transform the governing partial differential equations into ordinary differential equations, before being solved. The solutions of the velocity, temperature and concentration are derived and sketched to explain the influence of various physical parameters. The convergence of these solutions is also discussed. The numerical results of skin friction coefficient, Nusselt number and Sherwood number are compared with previous works. The results show that the method which has been used, in this paper, gives convergent solutions with good accuracy

Nonhomogeneous porosity and thermal diffusivity effects on stability and instability of double-diffusive convection in a porous medium layer: Brinkman Model

A model for double-diffusive convection in anisotropic and inhomogeneous porous media has been analysed. In particular, the effect of variable permeability and thermal diffusivity has been studied using the Brinkman model. Moreover, we analyse the effect of slip boundary conditions on the stability of the model. Due to numerous applications in micro-electro-mechanical-systems (MEMS) and other microfluidic devices, such a study is essential to have. Both linear instability analysis and nonlinear stability analysis are employed. We accurately analyse when stability and instability will commence and determine the critical Rayleigh number as a function of the slip coefficient