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

Mathematical models for heat and mass transfer in nanofluid flows.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.The behaviour and evolution of most physical phenomena is often best described using mathematical models in the form of systems of ordinary and partial differential equations. A typical example of such phenomena is the flow of a viscous impressible fluid which is described by the Navier-Stokes equations, first derived in the nineteenth century using physical approximations and the principles of mass and momentum conservation. The flow of fluids, and the growth of flow instabilities has been the subject of many investigations because fluids have wide uses in engineering and science, including as carriers of heat, solutes and aggregates. Conventional heat transfer fluids used in engineering applications include air, water and oil. However, each of these fluids has an inherently low thermal conductivity that severely limit heat exchange efficiency. Suspension of nanosized solid particles in traditional heat transfer fluids significantly increases the thermophysical properties of such fluids leading to better heat transfer performance. In this study we present theoretical models to investigate the flow of unsteady nanofluids, heat and mass transport in porous media. Different flow configurations are assumed including an inclined cylinder, a moving surface, a stretching cone and the flow of a polymer nanocomposite modeled as an Oldroyd-B fluid. The nanoparticles assumed include copper, silver and titanium dioxide with water as the base fluid. Most recent boundary-layer nanofluid flow studies assume that the nanoparticle volume fraction can be actively controlled at a bounding solid surface, similar to temperature controls. However, in practice, such controls present significant challenges, and may, in practice, not be possible. In this study the nanoparticle flux at the boundary surface is assumed to be zero. Unsteadiness in fluid flows leads to complex system of partial differential equations. These transport equations are often highly nonlinear and cannot be solved to find exact solutions that describe the evolution of the physical phenomena modeled. A large number of numerical or semi-numerical techniques exist in the literature for finding solutions of nonlinear systems of equations. Some of these methods may, however be subject to certain limitations including slow convergence rates and a small radius of convergence. In recent years, innovative linearization techniques used together with spectral methods have been suggested as suitable tools for solving systems of ordinary and partial differential equations. The techniques which include the spectral local linearization method, spectral relaxation method and the spectral quasiliearization method are used in this study to solve the transport equations, and to determine how the flow characteristics are impacted by changes in certain important physical and fluid parameters. The findings show that these methods give accurate solutions and that the speed of convergence of solutions is comparable with methods such as the Keller-box, Galerkin, and other finite difference or finite element methods. The study gives new insights, and result on the influence of certain events, such as internal heat generation, velocity slip, nanoparticle thermophoresis and random motion on the flow structure, heat and mass transfer rates and the fluid properties in the case of a nanofluid

Free Convection Fluid Flow from a Spinning Sphere with Temperature-Dependent Physical Properties

Conference ProceedingsNatural convection from a spinning sphere with temperature dependent viscosity, thermal conductivity and viscous dissipation was studied. A unique system of non-similar partial differential equations was solved using the bivariate local-linearization method (BLLM). This method use Chebyshev spectral collocation method applied in both the η and ξ directions. Similar equations in the literature are normally solved by inaccurate time-consuming finite difference methods. This work introduces a robust method for solving partial differential equations arising in heat and mass transfer. The numerical method was validated by comparison to the results previously published in the literature. The method is fully described in this article and can be used as an alternative method in solving boundary value problems. This work also presents rarely reported results of the effect of selected parameters on spin-velocity profiles g(η)

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

A numerical study of heat and mass transfer in non-Newtonian nanofluid models.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.A theoretical study of boundary layer flow, heat and mass transport in non-Newtonian nanofluids is presented. Because of the diversity in the physical structure and properties of non-Newtonian fluids, it is not possible to describe their behaviour using a single constitutive model. In the literature, several constitutive models have been proposed to predict the behaviour and rheological properties of non-Newtonian fluids. The question of interest is how the fluid physical parameters affect the boundary layer flow, and heat and mass transfer in various nanofluids. In this thesis, nanofluid models in various geometries and subject to different boundary conditions are constructed and analyzed. A range of fluid models from simple to complex are studied, leading to highly nonlinear and coupled differential equations, which require advanced numerical methods for their solution. This thesis is a conjoin between mathematical modeling of non-Newtonian nanofluid flows and numerical methods for solving differential equations. Some recent spectral techniques for finding numerical solutions of nonlinear systems of differential equations that model fluid flow problems are used. The numerical methods of primary interest are spectral quasilinearization, local linearization and bivariate local linearization methods. Consequently, one of the objectives of this thesis is to test the accuracy, robustness and general validity of these methods. The dependency of heat and mass transfer, and skin friction coefficients on the physical parameters is quantified and discussed. Results show that nanofluids and physical parameters have an important and significant impact on boundary layer flows, and on heat and mass transfer processes.The year on the title page reflects as 2019 on the thesis and differs from that on pages ii to iv which indicates the year 2020

Fully-coupled pressure-based finite-volume framework for the simulation of fluid flows at all speeds in complex geometries

A generalized finite-volume framework for the solution of fluid flows at all speeds in complex geometries and on unstructured meshes is presented. Starting from an existing pressure-based and fully-coupled formulation for the solution of incompressible flow equations, the additional implementation of pressure–density–energy coupling as well as shock-capturing leads to a novel solver framework which is capable of handling flows at all speeds, including quasi-incompressible, subsonic, transonic and supersonic flows. The proposed numerical framework features an implicit coupling of pressure and velocity, which improves the numerical stability in the presence of complex sources and/or equations of state, as well as an energy equation discretized in conservative form that ensures an accurate prediction of temperature and Mach number across strong shocks. The framework is verified and validated by a large number of test cases, demonstrating the accurate and robust prediction of steady-state and transient flows in the quasi-incompressible as well as subsonic, transonic and supersonic speed regimes on structured and unstructured meshes as well as in complex domains

Convective heat and mass transfer in boundary layer flow through porous media saturated with nanofluids.

Doctor of Philosophy in Mathematics. University of KwaZulu-Natal, Pietermaritzburg 2016.The thesis is devoted to the study of flow, heat and mass transfer processes, and crossdiffusion effects in convective boundary layer flows through porous media saturated with nanofluids. Of particular interest is how nanofluids perform as heat transfer fluids compared to traditional fluids such as oil and water. Flow in different geometries and subject to various source terms is investigated. An important aspect of the study and understanding of transport processes is the solution of the highly non-linear coupled differential equations that model both the flow and the heat transportation. In the literature, various analytical and numerical methods are available for finding solutions to fluid flow equations. However, not all these methods give accurate solutions, are stable, or are computationally efficient. For these reasons, it is important to constantly devise numerical schemes that work more efficiently, including improving the performance of existing schemes, to achieve accuracy with less computational effort. In this thesis the systems of differential equations that describe the fluid flow and other transport processes were solved numerically using both established and recent numerical schemes such as the spectral relaxation method and the spectral quasilinearization method. These spectral methods have been used only in a limited number of studies. There is therefore the need to test and prove the accuracy and general application of the methods in a wider class of boundary value problems. The accuracy, convergence, and validity of the solutions obtained using spectral methods, have been established by careful comparison with solutions for limiting cases in the published literature, or by use of a different solution method. In terms of understanding the physically important variables that impact the flow, we have inter alia, investigated the significance of different fluid and physical parameters, and how changes in these parameters affect the skin friction coefficient, the heat and mass transfer rates and the fluid properties. Some system parameters of interest in this study include the nanoparticle volume fraction, the Hartmann number, thermal radiation, Brownian motion, the heat generation, the Soret and Dufour effects, and the Prandtl and Schmidt number. The dependency of the heat, mass transfer and skin friction coefficients on these parameters has been quantified and discussed. In this thesis, we show that nanofluids have a significant impact on heat and mass transfer processes compared with traditional heat transfer fluids

Numerical Solutions of Free Convective Flow from a Vertical Cone with Mass Transfer under the Influence of Chemical Reaction and Heat Generation/Absorption in the Presence of UWT/UWC

The purpose of this paper is to present a mathematical model for the combined effects of chemical reaction and heat generation/absorption on unsteady laminar free convective flow with heat and mass transfer over an incompressible viscous fluid past a vertical permeable cone with uniform wall temperature and concentration (UWT/UWC).The dimensionless governing boundary layer equations of the flow that are transient, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicholson type. The velocity, temperature, and concentration profiles have been studied for various parameters viz., chemical reaction parameter , the heat generation and absorption parameter , Schmidt number Sc , Prandtl number Pr , buoyancy ratio parameter N . The local as well as average skin friction, Nusselt number, Sherwood number, are discussed and analyzed graphically. The present results are compared with available results in open literature and are found to be in excellent agreemen

Double-diffusive convection flow in a porous medium saturated with a nanofluid.

In this work, we studied heat and mass transfer in a nanofluid flow over a stretching sheet. Fluid flow in different flow geometries was studied and a co-ordinate transformation was used to transform the governing equations into non-dimensional non-similar boundary layer equations. These equations were then solved numerically using both established and recent techniques such as the spectral relaxation and spectral quasi-linearization methods. Numerical solutions for the heat transfer, mass transfer and skin friction coefficients have been presented for different system parameters, such as heat generation, Soret and Dufour effects, chemical reaction, thermal radiation influence, the local Grashof number, Prandtl number, Eckert number, Hartmann number and the Schmidt number. The dependency of the skin friction, heat and mass transfer coefficients on these parameters has been quantified and discussed. The accuracy, and validity of the spectral relaxation and spectral quasi-linearization methods has been established

VIM Solution for Mixed Convection over Horizontal Moving Porous Flat Plate

The non-viscous, laminar mixed convection boundary-layer flow over a horizontal moving porous flat plate, with chemical reaction, is considered. The governing equations are expressed in non-dimensional form and the series solutions of coupled system of equations are constructed for velocity, temperature and concentration functions using variational iteration method. The investigated parameters are: buoyancy parameter, chemical reaction parameter, order of chemical reaction, Prandtl number and Schmidt number