Recent numerical techniques for differential equations arising in fluid flow problems

Abstract

The work presented in this thesis is the application of the recently introduced numerical techniques, namely the spectral quasi-linearization method (SQLM) and the bivariate spectral quasi-linearization method (BSQLM), in solving problems arising in fluid flow. Firstly, we use the SQLM to solve the highly non-linear one dimensional Bratu problem. The results obtained are compared with exact solution and previously published results using the B-spline method, Picard’s Green’s Embedded Method and the iterative finite difference method. The results obtained show that the SQLM is highly accurate and computationally efficient. Secondly, we use the bivariate spectral quasi-linearization method to solve the two dimensional Bratu problem. Since the exact solution of the two-dimensional Bratu problem is unknown, the results obtained are compared with those previously published results using the finite difference method and the weighted residual method. Thirdly, we use the BSQLM to study numerically the boundary layer flow of a third grade non-Newtonian fluid past a vertical porous plate. We use the Jeffrey fluid as a typical fluid which shows non-Newtonian characteristics. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations which are then solved using BSQLM. The influence of some thermo-physical parameters namely, the ratio relaxation to retardation times parameter, Prandtl number, Schmidt number and the Deborah number is investigated. Also investigated is the influence of the ratio of relaxation to retardation times, Schmidt number and the Prandtl number on the skin friction, heat transfer rate and the mass transfer rate. The results obtained show that increasing the Schmidt number decelerates the fluid flow, reduces the skin friction, heat and mass transfer rates and strongly depresses the fluid concentration whilst the temperature is increased. The fluid velocity, the skin friction, heat and mass transfer rates are increased with increasing values of the relaxation to retardation parameter whilst the fluid temperature and concentration are reduced. Using the the solution based errors, it was shown that the BSQLM converges to the solution only after 5 iterations. The residual error infinity norms showed that BSQLM is very accurate by giving an error of order of 10−4 within 5 iterations. Lastly we propose a model of the non-Newtonian fluid flow past a vertical porous plate in the presence of thermal radiation and chemical reaction. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations. The BSQLM is used to solve the system of equations. We investigate the influence of the ratio of relaxation to retardation parameter, Schmidt number, Prandtl number, thermal radiation parameter, chemical reaction iv parameter, Nusselt number, Sherwood number, local skin fiction coefficient on the fluid concentration, fluid temperature as well as the fluid velocity. From the study, it is noted that the fluid flow velocity, the local skin friction coefficient, heat and mass transfer rate are increased with increasing ratio of relaxation to retardation times parameter whilst the fluid concentration is depressed. Increasing the Prandtl number causes a reduction in the velocity and temperature of the fluid whilst the concentration is increased. Also, the local skin friction coefficient and the mass transfer rates are depressed with an increase in the Prandtl number. An increase in the chemical reaction parameter decreases the fluid velocity, temperature and the concentration. Increasing the thermal radiation parameter has an effect of decelerating the fluid flow whilst the temperature and the concentration are slightly enhanced. The infinity norms were used to show that the method converges fast. The method converges to the solution within 5 iterations. The accuracy of the solution is checked using residual errors of the functions f, and . The errors show that the BSQLM is accurate, giving errors of less than 10−4, 10−7 and 10−8 for f, and , respectively, within 5 iterations.