On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods

Numerical investigation of Von Karman swirling bioconvective nanofluid transport from a rotating disk in a porous medium with Stefan blowing and anisotropic slip effects

In recent years, significant progress has been made in modern micro- and nanotechnologies related to applications in micro/nano-electronic devices. These technologies are increasingly utilizing sophisticated fluent media to enhance performance. Among the new trends is the simultaneous adoption of nanofluids and biological micro-organisms. Motivated by bio-nanofluid rotating disk oxygenators in medical engineering, in the current work, a mathematical model is developed for steady convective Von Karman swirling flow from an impermeable power-law radially stretched disk rotating in a Darcy porous medium saturated with nanofluid doped with gyrotactic micro-organisms. Anisotropic slip at the wall and blowing effects due to concentration are incorporated. The nano-bio transport model is formulated using non-linear partial differential equations (NPDEs), which are transformed to a set of similarity ordinary differential equations (SODEs) by appropriate transformations. The transformed boundary value problem is solved by a Chebyshev collocation method. The impact of key parameters on dimensionless velocity components, concentration, temperature and motile microorganism density distributions are computed and visualized graphically. Validation with previous studies is included. It is found that that the effects of suction provide a better enhancement of the heat, mass and microorganisms transfer in comparison to blowing. Moreover, physical quantities decrease with higher slip parameters irrespective of the existence of blowing. Temperature is suppressed with increasing thermal slip whereas nanoparticle concentration is suppressed with increasing wall mass slip. Micro-organism density number increases with the greater microorganism slip. Radial skin friction is boosted with positive values of the power law stretching parameter whereas it is decreased with negative values. The converse response is computed for circumferential skin friction, nanoparticle mass transfer rate and motile micro-organism density number gradient. Results from this study are relevant to novel bioreactors, membrane oxygenators, food processing and bio-chromatography

A New Approach for the Solution of Three-Dimensional Magnetohydrodynamic Rotating Flow over a Shrinking Sheet

The numerical solution of magnetohydrodynamic (MHD) and rotating flow over a porous shrinking sheet is obtained by the new approach known as spectral homotopy analysis method (SHAM). Using a similarity transformation, the governing equations for the momentum are reduced to a set of ordinary differential equations and are solved by the SHAM approach to determine velocity distributions and shear stress variations for different governing parameters. The SHAM results are analysed and validated against numerical results obtained using MATLAB's built-in bvp4c routine, and good agreement is observed

Successive Linearization Analysis of the Effects of Partial Slip, Thermal Diffusion, and Diffusion-Thermo on Steady MHD Convective Flow due to a Rotating Disk

We proposed a general formulation of the successive linearization method for solving highly nonlinear boundary value problem arising in rotating disk flow. The problem was studied under the effects of partial slip, thermal diffusion, and diffusion-thermo. The governing fundamental conservation equations of mass, momentum, angular momentum, energy, and concentration are transformed into a system of ordinary differential equations by means of similarity transformations. A parametric study illustrating the influence of the magnetic field, slip factor, Eckert number, Dufour and Soret numbers was carried out

Numerical study of nonlinear heat transfer from a wavy surface to a high permeability medium with pseudo-spectral and smoothed particle methods

Motivated by petro-chemical geological systems, we consider the natural convection boundary layer flow from a vertical isothermal wavy surface adjacent to a saturated non-Darcian high permeability porous medium. High permeability is considered to represent geologically sparsely packed porous media. Both Darcian drag and Forchheimer inertial drag terms are included in the velocity boundary layer equation. A high permeability medium is considered. We employ a sinusoidal relation for the wavy surface. Using a set of transformations, the momentum and heat conservation equations are converted from an (x, y) coordinate system to an (x,η) dimensionless system. The two-point boundary value problem is then solved numerically with a pseudo-spectral method based on combining the Bellman–Kalaba quasi linearization method with the Chebyschev spectral collocation technique (SQLM). The SQLM computations are demonstrated to achieve excellent correlation with smoothed particle hydrodynamic (SPH) Lagrangian solutions. We study the effect of Darcy number (Da), Forchheimer number (Fs), amplitude wavelength (A) and Prandtl number (Pr) on the velocity and temperature distributions in the regime. Local Nusselt number is also computed for selected cases. The study finds important applications in petroleum engineering and also energy systems exploiting porous media and undulating (wavy) surface geometry. The SQLM algorithm is shown to be exceptionally robust and achieves fast convergence and excellent accuracy in nonlinear heat transfer simulations

Adomian decomposition solution for propulsion of dissipative magnetic Jeffrey biofluid in a ciliated channel containing a porous medium with forced convection heat transfer

Physiological transport phenomena often feature ciliated internal walls. Heat, momentum and multi-species mass transfer may arise and additionally non-Newtonian biofluid characteristics are common in smaller vessels. Blood (containing hemoglobin) or other physiological fluids containing ionic constituents in the human body respond to magnetic body forces when subjected to external (extra-corporeal) magnetic fields. Inspired by such applications, in the present work we consider the forced convective flow of an electrically-conducting viscoelastic physiological fluid through a ciliated channel under the action of a transverse magnetic field. The flow is generated by a metachronal wave formed by the tips of cilia which move to and fro in a synchronized fashion. The presence of deposits (fats, cholesterol etc) in the channel is mimicked with a Darcy porous medium drag force model. The two-dimensional unsteady momentum equation and energy equation are simplified with a stream function and small Reynolds' number approximation. The effect of energy loss is simulated via the inclusion of viscous dissipation in the energy conservation (heat) equation. The non-dimensional, transformed moving boundary value problem is solved with appropriate wall conditions via the semi-numerical Adomian decomposition method (ADM). The velocity, temperature and pressure distribution are computed in the form of infinite series constructed by ADM and numerically evaluated in a symbolic software (MATHEMATICA). Streamline distributions are also presented. The influence of Hartmann number (magnetic parameter), Jeffrey first and second viscoelastic parameters, permeability parameter (modified Darcy number), and Brinkman number (viscous heating parameter) on velocity, temperature, pressure gradient and bolus dynamics is visualized graphically. The flow is decelerated with increasing with increasing Hartmann number and Jeffery first parameter in the core flow whereas it is accelerated in the vicinity of the walls. Increasing permeability and Jeffery second parameter are observed to accelerate the core flow and decelerate the peripheral flow near the ciliated walls. Increasing Hartmann number elevates pressure gradient whereas it is reduced with permeability parameter. Temperatures are elevated with increasing magnetic parameter, Brinkman number and Jeffery second parameter. Increasing magnetic field is also observed to reduce the quantity of trapped boluses. Increasing permeability parameter suppresses streamline amplitudes. Both the magnitude and quantity of trapped boluses is elevated with an increase in both first and second Jeffery parameters

Application of bivariate spectral quasilinearization method to second grade fluid flow equations.

Masters Degree. University of KwaZulu-Natal, Pietermaritzburg.In this study, the steady flow of a second grade magnetohydrodynamic fluid in a porous channel is investigated. We further investigate the hydromagnetic flow of a second grade fluid over a stretching sheet. The partial differential equations that describe the flows are solved numerically using the bivariate spectral quasilinearization method. The method is extended to a system of non-similar partial differential equations that model the steady two dimensional flow of Falkner-Skan flow of an incompressible second grade nano fluid. The work is also concerned with heat and the mass transfer from the electrically conducting second grade magnetohydrodynamic fluid over a stretching sheet. The sensitivity of the flow characteristics with respect to the second grade fluid parameter, magnetic field parameter, thermal radiation parameter, and the chemical reaction parameter are investigated. The accuracy of the numerical method is determined using the residual error analysis

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