Wavelets and the numerical solution of heat transfer and Newtonian/non-Newtonian fluid flow problems

Wavelet-based methods have demonstrated great potential for solving partial differential equations of various types. In this work, the capabilities of wavelet-based methods are explored by solving various heat transfer and fluid flow problems. Inherent to these techniques are difficulties encountered in implementing the boundary conditions. Two methods are therefore studied to implement Dirichlet boundary conditions. These methods are the Fictitious Boundary Approach and the Fictitious Domain/Penalty Formulation. These methods are evaluated by solving a number of one- and two-dimensional problems. A detailed description of the procedure used is provided for each method. It is found that both methods provide an efficient and good approximation to the solution of differential equations. The Fictitious Domain/Penalty method is, however, found to be superior because the resulting system of equations is block circulant and can be inverted easily. The Fictitious Domain/Penalty method also exhibits less errors in most of the considered examples. In the Fictitious Domain/Penalty Formulation, the resulting large system of equation is solved iteratively via the Conjugate Gradient Method and the Preconditioned Conjugate Gradient Method. The problems modified with the Fictitious Boundary Approach are solved using simple Gaussian elimination. The Fictitious Domain/Penalty formulation along with the Preconditioned Conjugate Gradient Method are then used to solve heat conduction problems, as well as Newtonian and non-Newtonian fluid flow problems. The fluid flow problems in the present study are formulated in such a manner that the solution of the continuity and momentum equations are turned to solutions of a series of Poisson equations. This is achieved by using methods known as the Conjugate Gradient and Steepest Descent Methods and a segregation procedure of the dependent variables. In addition, standard methods can be used to solve the nonlinear problems associated with non-Newtonian fluid. The Picard iterative method was used here. This formulation provide accurate results rapidly using personal computers, regardless of the geometry of the problem. The fluid flow problems studied consist of the lid-driven cavity box and rotating concentric cylinders. These problems are solved for both Newtonian and non-Newtonian fluids. It was therefore shown that wavelets can be used to solve nonlinear non-Newtonian fluid flow problems using Galerkin/Fictitious Domain formulation combined with more established techniques.