On paired decoupled quasi-linearization methods for solving nonlinear systems of differential equations that model boundary layer fluid flow problems.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.Two numerical methods, namely the spectral quasilinearization method (SQLM) and the spectral local linearization method (SLLM), have been found to be highly efficient methods for solving boundary layer flow problems that are modeled using systems of differential equations. Conclusions have been drawn that the SLLM gives highly accurate results but requires more iterations than the SQLM to converge to a consistent solution. This leads to the problem of figuring out how to improve on the rate of convergence of the SLLM while maintaining its high accuracy. The objective of this thesis is to introduce a method that makes use of quasilinearization in pairs of equations to decouple large systems of differential equations. This numerical method, hereinafter called the paired quasilinearization method (PQLM) seeks to break down a large coupled nonlinear system of differential equations into smaller linearized pairs of equations. We describe the numerical algorithm for general systems of both ordinary and partial differential equations. We also describe the implementation of spectral methods to our respective numerical algorithms. We use MATHEMATICA to carry out the numerical analysis of the PQLM throughout the thesis and MATLAB for investigating the influence of various parameters on the flow profiles in Chapters 4, 5 and 6. We begin the thesis by defining the various terminologies, processes and methods that are applied throughout the course of the study. We apply the proposed paired methods to systems of ordinary and partial differential equations that model boundary layer flow problems. A comparative study is carried out on the different possible combinations made for each example in order to determine the most suitable pairing needed to generate the most accurate solutions. We test convergence speed using the infinity norm of solution error. We also test their accuracies by using the infinity norm of the residual errors. We also compare our method to the SLLM to investigate if we have successfully improved the convergence of the SLLM while maintaining its accuracy level. Influence of various parameters on fluid flow is also investigated and the results obtained show that the paired quasilinearization method (PQLM) is an efficient and accurate method for solving boundary layer flow problems. It is also observed that a small number of grid-points are needed to produce convergent numerical solutions using the PQLM when compared to methods like the finite difference method, finite element method and finite volume method, among others. The key finding is that the PQLM improves on the rate of convergence of the SLLM in general. It is also discovered that the pairings with the most nonlinearities give the best rate of convergence and accuracy

A numerical study of entropy generation, heat and mass transfer in boundary layer flows.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface between mathematical modelling of fluid flows and numerical methods for differential equations. It is an investigation, through modelling techniques, of entropy generation in Newtonian and non-Newtonian fluid flows with special focus on nanofluids. We seek to enhance our current understanding of entropy generation mechanisms in fluid flows by investigating the impact of a range of physical and chemical parameters on entropy generation in fluid flows under different geometrical settings and various boundary conditions. We therefore seek to analyse and quantify the contribution of each source of irreversibilities on the total entropy generation. Nanofluids have gained increasing academic and practical importance with uses in many industrial and engineering applications. Entropy generation is also a key factor responsible for energy losses in thermal and engineering systems. Thus minimizing entropy generation is important in optimizing the thermodynamic performance of engineering systems. The entropy generation is analysed through modelling the flow of the fluids of interest using systems of differential equations with high nonlinearity. These equations provide an accurate mathematical description of the fluid flows with various boundary conditions and in different geometries. Due to the complexity of the systems, closed form solutions are not available, and so recent spectral schemes are used to solve the equations. The methods of interest are the spectral relaxation method, spectral quasilinearization method, spectral local linearization method and the bivariate spectral quasilinearization method. In using these methods, we also check and confirm various aspects such as the accuracy, convergence, computational burden and the ease of deployment of the method. The numerical solutions provide useful insights about the physical and chemical characteristics of nanofluids. Additionally, the numerical solutions give insights into the sources of irreversibilities that increases entropy generation and the disorder of the systems leading to energy loss and thermodynamic imperfection. In Chapters 2 and 3 we investigate entropy generation in unsteady fluid flows described by partial differential equations. The partial differential equations are reduced to ordinary differential equations and solved numerically using the spectral quasilinearization method and the bivariate spectral quasilinearization method. In the subsequent chapters we study entropy generation in steady fluid flows that are described using ordinary differential equations. The differential equations are solved numerically using the spectral quasilinearization and the spectral local linearization methods

Ternary ion exchange in fixed beds : equilibrium and dynamics

A model was developed to simulate ion exchange within fixed beds for ternary systems. Models of the fluid phase material balance, phase equilibria, and diffusion of ions through the film and within the resin phase incorporated the latest advances in ion exchange theory. The separate model elements were combined after testing into an overall general model. Non-linear regression support programs were developed to estimate equilibrium parameters and resin phase diffusion coefficients. A computer program was developed to estimate axial dispersion coefficients from experimental data. General correlations derived from literature sources were tested, and axial dispersion terms were included in the electrolyte phase material balance equations. The rational thermodynamic equilibrium constant, utilizing resin phase activity coefficients based on the 3 suffix Redlich-Kister equation and the Bromley equation for electrolyte phase activity coefficients, was selected. The Wilson and NRTL equations were tested but were not as good. This model was used to correlate published data on 13 binary systems and to predict ternary compositions for comparison to published data on 4 ternary systems. Average root mean square of % normalized difference was about 3% on the binary systems and 4-12% on data predicted for the ternary systems. The Nernst-Planck equation was used to model resin phase diffusion. An integrated form of the Nernst-Hartley equation, based on the Bromley equation, was developed and tested to predict the effect of concentration on electrolyte phase diffusion coefficients. These coefficients were used in a pseudo electric field model which was developed and tested to approximate the electric field effect on diffusion of ions in the film. The overall ternary system model resulted in four coupled non-linear second order parabolic partial differential equations, with appropriate boundary conditions. The equations were reduced to a set of algebraic equations by finite difference approximations and solved by the implicit Crank-Nicholson method. Non-linear terms were quasilinearized. The resulting five diagonal coefficient matrix describing the fluid phase, coupled with the 7 diagonal coefficient matrices describing the resin phase, were inverted with algorithms developed in this work. An iterative procedure resolved all nonlinear terms at each time step. Comparison of concentration histories generated by the model with experimental results obtained by previous researchers showed that the ternary model could be used in practice to optimize process design applications with a bed in a condition of partial presaturation, and for favorable or unfavorable ion exchange. Resin phase activity coefficients developed in correlation of the equilibrium data were used to test chemical potential as a driving force in the systems simulated. Indication that use of chemical potential would obviate the need for ion pair specific diffusion coefficients in the Nernst-Planck model, or the use of ion pair corrector coefficients (Stefan-Maxwell), is shown by comparison of results on seven binary systems. Implications for industrial application and directions for further research are discussed

Slip-flow and heat transfer of chemically reacting micropolar fluid through expanding or contracting walls with Hall and ion slip currents

AbstractThe present article deals with the effects of velocity slip, chemical reaction on heat and mass transfer of micropolar fluid in expanding or contracting walls with Hall and ion slip currents. Assume that there is symmetric suction or injection along the channel walls, which are maintained at nonuniform constant temperatures and concentrations. The governing Navier–Stokes equations are reduced to nonlinear ordinary differential equations by using similarity transformations then solved numerically by quasilinearization technique. The effects of various parameters such as wall expansion ratio, chemical reaction parameter, Prandtl number, Schmidt number, slip parameter, Hall and ion slip parameters on nondimensional velocity components, microrotation, temperature and concentration are discussed in detail through graphs. It is observed that the concentration of the fluid is enhanced with viscosity. Further, the temperature and concentration of the fluid are increased whereas the microrotation is decreased for an expansion or contraction of the walls

Numerical solution of the inverse Gardner equation

In this paper, the numerical solution of the inverse Gardner equation will be considered. The Haar wavelet collocation method (HWCM) will be used to determine the unknown boundary condition which is estimated from an over-specified condition at a boundary. In this regard, we apply the HWCM for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. It is proved that the proposed method has the order of convergence O(∆x). The efficiency and robustness of the proposed approach for solving the inverse Gardner equation are demonstrated by one numerical example.Publisher's Versio

Spectral relaxation method and spectral quasilinearization method for solving unsteady boundary layer flow problems

Nonlinear partial differential equations (PDEs) modelling unsteady boundary-layer flows are solved by the spectral relaxation method (SRM) and the spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral based methods that have been successfully used to solve nonlinear boundary layer flow problems described by systems of ordinary differential equations. In this paper application of these methods is extended, for the first time, to systems of nonlinear PDEs that model unsteady boundary layer flow. The new extension is tested on two problems: boundary layer flow caused by an impulsively stretching plate and a coupled four-equation system that models the problem of unsteady MHD flow and mass transfer in a porous space. Numerous simulation experiments are conducted to determine the accuracy and compare the computational performance of the proposed methods against the popular Keller-box finite difference scheme which is widely accepted as being one of the ideal tools for solving nonlinear PDEs that model boundary layer flow problems. The results indicate that the methods are more efficient in terms of computational accuracy and speed compared with the Keller-box

(R2030) Generalized Quasilinearization Method for a Initial Value Problem on Time Scales

We have investigated that the generalized quasilinearization method under some convenient conditions for nonlinear initial value problem (IVP) of dynamic equation on time scale constructed by monotone sequences of function by using comparison theorem that is the solution of linear IVP of dynamic equation on time scale which converge uniformly and monotonically to the unique solution of the original problem, and the convergence is quadratic