Group theoretical and compatibility approaches to some nonlinear PDEs arising in the study of non-Newtonian fluid mechanics

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015.This thesis is primarily concerned with the analysis of some nonlinear problems arising in the study of non-Newtonian fluid mechanics by employing group theoretic and compatibility approaches. It is well known now that many manufacturing processes in industry involve non-Newtonian fluids. Examples of such fluids include polymer solutions and melts, paints, blood, ketchup, pharmaceuticals and many others. The mathematical and physical behaviour of non-Newtonian fluids is intermediate between that of purely viscous fluid and that of a perfectly elastic solid. These fluids cannot be described by the classical Navier–Stokes theory. Striking manifestations of non-Newtonian fluids have been observed experimentally such as the Weissenberg or rod-climbing effect, extrudate swell or vortex growth in a contraction flow. Due to diverse physical structure of non-Newtonian fluids, many constitutive equations have been developed mainly under the classification of differential type, rate type and integral type. Amongst the many non-Newtonian fluid models, the fluids of differential type have received much attention in order to explain features such as normal stress effects, rod climbing, shear thinning and shear thickening. Most physical phenomena dealing with the study of non-Newtonian fluids are modelled in the form of nonlinear partial differential equations (PDEs). It is easier to solve a linear problem due to its extensive study as well due t

Unsteady MHD Convective Heat And Mass Transfer Past An Infinite Vertical Plate Embedded In A Porous Medium With Radiation And Chemical Reaction Under The Influence Of Dufour And Soret Effects

A two-dimensional unsteady MHD free convective heat and mass transfer flow past a semi-infinite vertical porous plate in a porous medium in presence of thermal radiation and chemical reaction has been studied numerically including the Dufour and Soret effects. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. The resulting equations are then solved numerically using shooting method along with Runge-Kutta fourth order integration scheme. The numerical results are displayed graphically showing the effects of various parameters entering into the problem. Finally, the local values of the skin-friction coefficient, Nusselt number and Sherwood number are also shown in tabular form

Self-Similar Unsteady Flow of a Sisko Fluid in a Cylindrical Tube Undergoing Translation.

The governing nonlinear equation for unidirectional flow of a Sisko fluid in a cylindrical tube due to translation of the tube wall is modelled in cylindrical polar coordinates.The exact steady-state solution for the nonlinear problem is obtained.Thereduction of the nonlinear initial value problem is carried out by using a similarity transformation.The partial differential equation is transformed into an ordinary differential equation, which is integrated numerically taking into account the influence of the exponent n and the material parameter b of the Sisko fluid. The initial approximation for the fluid velocity on the axis of the cylinder is obtained by matching inner and outer expansions for the fluid velocity. A comparison of the velocity, vorticity, and shear stress of Newtonian and Sisko fluids is presented

Closed-Form Solutions for a Nonlinear Partial Differential Equation Arising in the Study of a Fourth Grade Fluid Model

The unsteady unidirectional flow of an incompressible fourth grade fluid bounded by a suddenly moved rigid plate is studied. The governing nonlinear higher order partial differential equation for this flow in a semiinfinite domain is modelled. Translational symmetries in variables and are employed to construct two different classes of closed-form travelling wave solutions of the model equation. A conditional symmetry solution of the model equation is also obtained. The physical behavior and the properties of various interesting flow parameters on the structure of the velocity are presented and discussed. In particular, the significance of the rheological effects are mentioned

Lie Group Analysis of Natural Convection Heat and Mass Transfer in an Inclined Surface

Natural convection heat transfer fluid flow past an inclined semiinfinite surface in the presence of solute concentration is investigated by Lie group analysis. The governing partial differential equations are reduced to a system of ordinary differential equations by the translation and scaling symmetries. An exact solution is obtained for translation symmetry and numerical solutions for scaling symmetry. It is found that the velocity increases and temperature and concentration of the fluid decrease with an increase in the thermal and solutal Grashof numbers. The velocity and concentration of the fluid decrease and temperature increases with increase in the Schmidt number

Symmetry analysis for steady boundary-layer stagnation-point flow of Rivlin–Ericksen fluid of second grade subject to suction

An analysis for the steady two-dimensional boundary-layer stagnation-point flow of Rivlin–Ericksen fluid of second grade with a uniform suction is carried out via symmetry analysis. By employing Lie-group method to the given system of nonlinear partial differential equations, the symmetries of the equations are determined. Using these symmetries, the solution of the given equations is found. The effect of the viscoelastic parameter k and the suction parameter R on the tangential and normal velocities, temperature profiles, heat transfer coefficient and the wall shear stress, have been studied. Also, the effect of the Prandtl number Pr on the temperature and the heat transfer coefficient has been studied

Lie group analysis of equations arising in non-Newtonian fluids

It is known now that the Navier-Stokes equations cannot describe the behaviour of fluids having high molecular weights. Due to the variety of such fluids it is very difficult to suggest a single constitutive equation which can describe the properties of all non-Newtonian fluids. Therefore many models of non-Newtonian fluids have been proposed. The flow of non-Newtonian fluids offer special challenges to the engineers, modellers, mathematicians, numerical simulists, computer scientists and physicists alike. In general the equations of non-Newtonian fluids are of higher order and much more complicated than the Newtonian fluids. The adherence boundary conditions are insufficient and one requires additional conditions for a unique solution. Also the flow characteristics of non-Newtonian fluids are quite different from those of the Newtonian fluids. Therefore, in practical applications, one cannot replace the behaviour of non-Newtonian fluids with Newtonian fluids and it is necessary to examine the flow behaviour of non-Newtonian fluids in order to obtain a thorough understanding and improve the utilization in various manufactures. Although the non-Newtonian behaviour of many fluids has been recognized for a long time, the science of rheology is, in many respects, still in its infancy, and new phenomena are constantly being discovered and new theories proposed. Analysis of fluid flow operations is typically performed by examining local conservation relations, conservation of mass, momentum and energy. This analysis gives rise to highly non-linear relationships given in terms of differential equations, which are solved using special non-linear techniques. Advancements in computational techniques are making easier the derivation of solutions to linear problems. However, it is still difficult to solve non-linear problems analytically. Engineers, chemists, physicists, and mathematicians are actively developing non-linear analytical techniques, and one such method which is known for systematically searching for exact solutions of differential equations is the Lie symmetry approach for differential equations. Lie theory of differential equations originated in the 1870s and was introduced by the Norwegian mathematician Marius Sophus Lie (1842 - 1899). However it was the Russian scientist Ovsyannikov by his work of 1958 who awakened interest in modern group analysis. Today, the Lie group approach to differential equations is widely applied in various fields of mathematics, mechanics, and theoretical physics and many results published in these area demonstrates that Lie’s theory is an efficient tool for solving intricate problems formulated in terms of differential equations. The conditional symmetry approach or what is called the non-classical symmetry approach is an extension of the Lie approach. It was proposed by Bluman and Cole 1969. Many equations arising in applications have a paucity of Lie symmetries but have conditional symmetries. Thus this method is powerful in obtaining exact solutions of such equations. Numerical methods for the solutions of non-linear differential equations are important and nowadays there several software packages to obtain such solutions. Some of the common ones are included in Maple, Mathematica and Matlab. This thesis is divided into six chapters and an introduction and conclusion. The first chapter deals with basic concepts of fluids dynamics and an introduction to symmetry approaches to differential equations. In Chapter 2 we investigate the influence of a time-dependentmagnetic field on the flow of an incompressible third grade fluid bounded by a rigid plate. Chapter 3 describes the modelling of a fourth grade flow caused by a rigid plate moving in its own plane. The resulting fifth order partial differential equation is reduced using symmetries and conditional symmetries. In Chapter 4 we present a Lie group analysis of the third oder PDE obtained by investigating the unsteady flow of third grade fluid using the modified Darcy’s law. Chapter 5 looks at the magnetohydrodynamic (MHD) flow of a Sisko fluid over a moving plate. The flow of a fourth grade fluid in a porous medium is analyzed in Chapter 6. The flow is induced by a moving plate. Several graphs are included in the ensuing discussions. Chapters 2 to 6 have been published or submitted for publication. Details are given in the references at the end of the thesis

Challenges and progress on the modelling of entropy generation in porous media: a review

Depending upon the ultimate design, the use of porous media in thermal and chemical systems can provide significant operational advantages, including helping to maintain a uniform temperature distribution, increasing the heat transfer rate, controlling reaction rates, and improving heat flux absorption. For this reason, numerous experimental and numerical investigations have been performed on thermal and chemical systems that utilize various types of porous materials. Recently, previous thermal analyses of porous materials embedded in channels or cavities have been re-evaluated using a local thermal non-equilibrium (LTNE) modelling technique. Consequently, the second law analyses of these systems using the LTNE method have been a point of focus in a number of more recent investigations. This has resulted in a series of investigations in various porous systems, and comparisons of the results obtained from traditional local thermal equilibrium (LTE) and the more recent LTNE modelling approach. Moreover, the rapid development and deployment of micro-manufacturing techniques have resulted in an increase in manufacturing flexibility that has made the use of these materials much easier for many micro-thermal and chemical system applications, including emerging energy-related fields such as micro-reactors, micro-combustors, solar thermal collectors and many others. The result is a renewed interest in the thermal performance and the exergetic analysis of these porous thermochemical systems. This current investigation reviews the recent developments of the second law investigations and analyses in thermal and chemical problems in porous media. The effects of various parameters on the entropy generation in these systems are discussed, with particular attention given to the influence of local thermodynamic equilibrium and non-equilibrium upon the second law performance of these systems. This discussion is then followed by a review of the mathematical methods that have been used for simulations. Finally, conclusions and recommendations regarding the unexplored systems and the areas in the greatest need of further investigations are summarized