Double-diffusive convection flow in porous media with cross-diffusion.
In this thesis we study double-diffusive convection and cross-diffusion effects in flow through porous media. Fluid flows in various flow geometries are investigated and the governing equations are solved analytically and numerically using established and recent techniques such as the Keller-box method, the spectral-homotopy analysis method and the successive linearisation method. The effects of the governing parameters such as the Soret, Dufour, Lewis, Rayleigh and the Peclet numbers and the buoyancy ratio on the fluid properties, and heat and mass transfer at the surface are determined. The accuracy, computational efficiency and validity of the new methods is established. This study consists of five published and one submitted paper whose central theme is the study of double-diffusive convection in porous media. A secondary theme is the application of recent numerical semi-numerical methods in the solution of nonlinear boundary value problems, particularly those that arise in the study of fluid flow problems. Paper 1. An investigation of the quiescent state in a Maxwell fluid with double-diffusive convection in porous media using linear stability analysis is presented. The fluid motion is modeled using the modified Darcy-Brinkman law. The critical Darcy- Rayleigh numbers for the onset of convection are obtained and numerical simulations carried out to show the effects of the Soret and Dufour parameters on the critical Darcy-Rayleigh numbers. For some limiting cases, known results in the literature are recovered. Paper 2. We present an investigation of heat and mass transfer in a micropolar fluid with cross-diffusion effects. Approximate series solutions of the governing non-linear differential equations are obtained using the homotopy analysis method (HAM). A comparison is made between the results obtained using the HAM and the numerical results obtained using the Matlab bvp4c numerical routine. Paper 3. The spectral homotopy analysis method (SHAM) as a new improved version of the homotopy analysis method is introduced. The new technique is used to solve the MHD Jeffery-Hamel problem for a convergent or divergent channel. We show that the SHAM improves the applicability of the HAM by removing the restrictions associated with the HAM as well as accelerating the convergence rate. Paper 4. We present a study of free and forced convection from an inverted cone in porous media with diffusion-thermo and thermo-diffusion effects. The highly nonlinear governing equations are solved using a novel successive linearisation method (SLM). This method combines a non-perturbation technique with the Chebyshev spectral collection method to produce an algorithm with accelerated and assured convergence. Comparison of the results obtained using the SLM, the Runge-Kutta together with a shooting method and the Matlab bvp4c numerical routine show the accuracy and computational efficiency of the SLM. Paper 5. Here we study cross-diffusion effects and convection from inverted smooth and wavy cones. In the case of a smooth cone, the highly non-linear governing equations are solved using the successive linearisation method (SLM), a shooting method together with a Runge-Kutta of order four and the Matlab bvp4c numerical routine. In the case of the wavy cone the governing equations are solved using the Keller-box method. Paper 6. We examine the problem of mixed convection, heat and mass transfer along a semi-infinite plate in a fluid saturated porous medium subject to cross-diffusion and radiative heat transfer. The governing equations for the conservation of momentum, heat and solute concentration transfer are solved using the successive linearisation method, the Keller-box technique and the Matlab bvp4c numerical routine.