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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.

Hydraulic fracturing is a process for the extraction of hydrocarbons from underground formations. It involves pumping a specialized fluid into the wellbore under high pressures to form and support fractures in the rock. Fracturing stimulates the well to increase the production of oil and the natural gas which are the pillars of the energy economy. Key to this process is the use of proppants, which are solid materials used to keep the fractures open. Understanding the transport of proppant particles through a fluid is important to improve the efficiency and reduce environmental impact of fracturing. An increase of the settling velocity for instance, will impede the hydraulic fracturing process by reducing well productivity, or necessitate use of chemical additives. This thesis presents a theoretical investigation of the settling velocity of proppant particles. The effect of different parameters on the settling velocity were studied by manipulating the main factors that can influence particle transport. These include size of the particle (300 μm- 2000 μm), sphericity, density (1200 kg/m3-3500 kg/m3) and concentration. These typical values were obtained from commercially available proppants currently used in industry. Various correlations were investigated, assuming the carrier (fracturing) fluid to be an ideal Newtonian and as a power law (non-Newtonian) fluid. This will help predict the settling velocity for proppant particles in order to increase well productivity, and improve hydraulic fracturing efficiency. The models show that changing the carrier fluid viscosity and particle properties such as diameter, density, sphericity, and concentration leads to a significant change in the proppant settling velocity. For instance, reduction in particle size, density, and sphericity tend to reduce the settling velocity, while increasing the concentration of the particles and the fluid viscosity reduce the settling velocity.

With the depletion of oil reserves and increase in oil price, the enhanced oil recovery methods such as polymer flooding to increase oil production from water flooded fields are becoming more attractive. Effective design of these processes is challenging because the polymer chemistry has a strong effect on reaction and fluid rheology, which in turn has a strong effect on fluid transport. We have implemented a well-established polymer model within the Implicit Parallel Accurate Reservoir Simulator (IPARS), which enables parallel simulation of non-Newtonian fluid flow through porous media. The following properties of polymer solution are modeled in this work: 1) polymer adsorption; 2) polymer viscosity as a function of salinity, hardness, polymer concentration, and shear rate; 3) permeability reduction; 4) inaccessible pore volume. IPARS enables field-scale polymer flooding simulation with its parallel computation capability. In this thesis, several numerical examples are presented. The result of polymer module is verified by UTCHEM, a three-dimensional chemical flood simulator developed at the University of Texas at Austin. The parallel capability is also tested. The influence of different shear rate calculations is investigated in homogeneous and heterogeneous reservoirs. We observed that the wellbore velocity calculation instead of Darcy velocity reduces the grid effect for coarse mesh. We noted that the injection bottom hole pressure is very sensitive to the shear rate calculation. However, cumulative oil recovery and overall oil saturation appear to not be sensitive to grid and shear rate calculation for same reservoir. There are two models to model the ground water infiltration in vadose zone. One is Richard’s Equation (RE) model. And the other is two-phase flow model. In this work, we compare the two-phase model with an RE model to ascertain, under common scenarios such as infiltration or injection of water into initially dry soils, the similarities and differences in solutions behaviors, the ability of each model to simulate such infiltration processes under realistic scenarios, and to investigate the numerical efficiencies and difficulties which arise in these models. Six different data sets were assembled as benchmark infiltration problems in the unsaturated zone. The comparison shows that two-phase model holds for general porous media and is not limited by several assumptions that must be made for the RE formulation, while RE is applicable only for shallow regions (vadose) that are only several meters in depth and a fully saturated bottom boundary condition must be assumed. / text

Fluid dispensing is a method by which fluid materials are delivered to the targeted boards in a controlled manner and has been extensively applied in various packaging processes in the electronics assembly industry. In these processes, the flow rate of the fluid dispensed and/or the fluid amount transferred onto a board are two important performance indexes. Due to the involvement of the compressibility and non-Newtonian behaviour of the fluid being dispensed, modeling the fluid dispensing process has proven to be a challenging task. This thesis presents a study on the modeling of the positive displacement dispensing process, in which the linear displacement of a piston is used to dispense fluid. Also, this thesis presents an evaluation of different designs of the fluid dispensing system based on the axiomatic design principles. <p>At first, the characterization of the flow behaviour of fluids used in the electronic packaging industry is addressed. Based on the previous experiments conducted in the authors lab, a 3-parameter Carreau model for the fluid Hysol FP4451 is derived for use in the present study. Then, taking into account fluid compressibility and flow behaviour, a model is developed to represent the dynamics of the flow rate of the fluid dispensed. The resulting model suggests that the dynamics of the flow rate in the positive displacement dispensing process is equivalent to that of a second order system. Based on the model developed, the influences of the fluid compressibility and the process parameters such as the dispensing time and needle temperature are investigated by simulations. <p>In the positive dispensing process, it is noticed that the fluid amount dispensed out of needle is different from the fluid amount finally transferred to the board, if the fluid amount dispensed is very small. This difference is considered one major problem affecting dispensing performance. In order to determine the fluid amount transferred to the board, a 3-step method is developed in the present study, based on existing theories of liquid bridges and Laplaces equation. Simulations are conducted based on the developed method to study the influence of surface tension and initial fluid amount on the final fluid amount transferred onto the board. <p> Finally, this thesis presents a new approach to evaluate and compare different designs of the fluid dispensing system, namely air-pressure, rotary-crew, and positive- displacement. In this approach, the axiomatic design principles, i.e., the Independence Axiom and the Information Axiom, are employed. This approach can be used not only to evaluate existing dispensing systems, but also to design new dispensing systems.

Even though fluid mechanics is well developed as a science, there are many physical phenomena that we do not yet fully understand. One of these is the deformation rates and fluid stresses generated in a boundary layer for a non-Newtonian fluid. One such non-Newtonian fluid would be a waxy crude oil flowing in a centrifugal pump. This type of flow can be numerically modeled by a rotating disk system, in combination with an appropriate constitutive equation, such as the relation for a Bingham fluid. A Bingham fluid does not begin to flow until the stress magnitude exceeds the yield stress. However, experimental measurements are also required to serve as a database against which the results of the numerical simulation can be interpreted and validated. The purpose of the present research is to gain a better understanding of the behavior of a Bingham fluid in the laminar boundary layer on a rotating disk. For this project, two different techniques were employed: numerical simulation, and laboratory investigations using Particle Image Velocimetry (PIV) and flow visualization. Both methods were applied to the flow of a Bingham fluid over a rotating disk. In the numerical investigations, the flow was characterized by the dimensionless yield stress Bingham number, By, which is the ratio of the yield and viscous stresses. Using von Kármáns similarity transformation, and introducing the rheological behavior of the fluid into the conservation equations, the corresponding nonlinear two-point boundary value problem was formulated. A solution to the problem under investigation was obtained by numerical integration of the set of Ordinary Differential Equations (ODEs) using a multiple shooting method. The influence of the Bingham number on the flow behavior was identified. It decreases the magnitude of the radial and axial velocity components, and increases the magnitude of the tangential velocity component, which has a pronounced effect on the moment coefficient, CM, and the volume flow rate, Q. In the laboratory investigations, since the waxy crude oils are naturally opaque, an ambitious experimental plan to create a transparent oil that was rheologically similar to the Amna waxy crude oil from Libya was developed. The simulant was used for flow visualization experiments, where a transparent fluid was required. To fulfill the demand of the PIV system for a higher degree of visibility, a second Bingham fluid was created and rheologically investigated. The PIV measurements were carried out for both filtered tap water and the Bingham fluid in the same rotating disk apparatus that was used for the flow visualization experiments. Both the axial and radial velocity components in the (r-z) plane were measured for various rotational speeds. Comparison between the numerical and experimental results for the axial and radial velocity profiles for water was found to be satisfactory. Significant discrepancies were found between numerical results and measured values for the Bingham fluid, especially at low rotational speeds, mostly relating to the formation of a yield surface within the tank. Even though the flow in a pump is in some ways different from that of a disk rotating in a tank, some insight about the behavior of the pump flow can be drawn. One conclusion is that the key difference between the flow of a Bingham fluid in rotating equipment from that of a Newtonian fluid such as water relates to the yield surface introduced by the yield stress of the material, which causes an adverse effect on the performance and efficiency of such equipment.

Tissue engineering is a multidisciplinary subject aimed at producing the immunologically tolerant artificial tissues/organs to repair or replace damaged ones. In this field, tissue scaffold plays a key role to support cell growth and new tissue regeneration. For fabrication of tissue scaffolds with individual external geometry and predefined inner structure, rapid prototyping (RP) systems based on fluid dispensing techniques have proved to be very promising. The present research conducted a comprehensive study on the dispensing-based fabrication process. <p>First of all, the scaffold materials are characterized in terms of their biocompatibility and flow behaviour. The biocompatibility of biomaterials of PLLA, PCL, collagen, chitosan, and gelatine is evaluated in terms of supporting neuron cells adhesion and outgrowth. Chitosan solution (2% w/v) in acetic acid is shown to be the most promising among the examined biomaterials for the fabrication of nerve tissue scaffolds. Its non-Newtonian flow behaviour is identified by using a commercial rheometer. <p>In the fabrication process, the flow rate of biomaterials dispensed, the profile of strand cross-sections, and the scaffold porosity are very important and must be precisely controlled. A model is developed to represent the flow rate of biomaterials dispensed under the assumptions that the flow is incompressible, steady, laminar, and axisymmetric. Also, the profile and size of line strands at different layers and portions are modeled based on the Young-Laplace equation. Thus the dispensing-based fabrication process can be predicted in terms of the flow rate and the scaffold porosity. <p>The effects of operation conditions on the fabrication result are identified theoretically and experimentally. Simulation result shows that a higher driving pressure, a higher temperature, and a larger needle diameter will result in a larger size of the strand cross-sections and lower scaffold porosity. The change pattern, however, is nonlinear, which is affected by the fluid surface tension and non-Newtonian flow behaviour of scaffold biomaterials. <p>To verify the effectiveness of the developed models, experiments were carried out on a commercial dispensing system (C-720, Asymtek, USA). To avoid the possible error derived from the temperature difference between the dispensing system and the rheometer, a new method is presented to characterize the fluid properties used for model predictions. Experimental results illustrate that the developed models, combined with the new identification method, are very promising to predict the dispensing-based fabrication process.

Even though fluid mechanics is well developed as a science, there are many physical phenomena that we do not yet fully understand. One of these is the deformation rates and fluid stresses generated in a boundary layer for a non-Newtonian fluid. One such non-Newtonian fluid would be a waxy crude oil flowing in a centrifugal pump. This type of flow can be numerically modeled by a rotating disk system, in combination with an appropriate constitutive equation, such as the relation for a Bingham fluid. A Bingham fluid does not begin to flow until the stress magnitude exceeds the yield stress. However, experimental measurements are also required to serve as a database against which the results of the numerical simulation can be interpreted and validated. The purpose of the present research is to gain a better understanding of the behavior of a Bingham fluid in the laminar boundary layer on a rotating disk. For this project, two different techniques were employed: numerical simulation, and laboratory investigations using Particle Image Velocimetry (PIV) and flow visualization. Both methods were applied to the flow of a Bingham fluid over a rotating disk. In the numerical investigations, the flow was characterized by the dimensionless yield stress Bingham number, By, which is the ratio of the yield and viscous stresses. Using von Kármáns similarity transformation, and introducing the rheological behavior of the fluid into the conservation equations, the corresponding nonlinear two-point boundary value problem was formulated. A solution to the problem under investigation was obtained by numerical integration of the set of Ordinary Differential Equations (ODEs) using a multiple shooting method. The influence of the Bingham number on the flow behavior was identified. It decreases the magnitude of the radial and axial velocity components, and increases the magnitude of the tangential velocity component, which has a pronounced effect on the moment coefficient, CM, and the volume flow rate, Q. In the laboratory investigations, since the waxy crude oils are naturally opaque, an ambitious experimental plan to create a transparent oil that was rheologically similar to the Amna waxy crude oil from Libya was developed. The simulant was used for flow visualization experiments, where a transparent fluid was required. To fulfill the demand of the PIV system for a higher degree of visibility, a second Bingham fluid was created and rheologically investigated. The PIV measurements were carried out for both filtered tap water and the Bingham fluid in the same rotating disk apparatus that was used for the flow visualization experiments. Both the axial and radial velocity components in the (r-z) plane were measured for various rotational speeds. Comparison between the numerical and experimental results for the axial and radial velocity profiles for water was found to be satisfactory. Significant discrepancies were found between numerical results and measured values for the Bingham fluid, especially at low rotational speeds, mostly relating to the formation of a yield surface within the tank. Even though the flow in a pump is in some ways different from that of a disk rotating in a tank, some insight about the behavior of the pump flow can be drawn. One conclusion is that the key difference between the flow of a Bingham fluid in rotating equipment from that of a Newtonian fluid such as water relates to the yield surface introduced by the yield stress of the material, which causes an adverse effect on the performance and efficiency of such equipment.

Tissue engineering is a multidisciplinary subject aimed at producing the immunologically tolerant artificial tissues/organs to repair or replace damaged ones. In this field, tissue scaffold plays a key role to support cell growth and new tissue regeneration. For fabrication of tissue scaffolds with individual external geometry and predefined inner structure, rapid prototyping (RP) systems based on fluid dispensing techniques have proved to be very promising. The present research conducted a comprehensive study on the dispensing-based fabrication process. <p>First of all, the scaffold materials are characterized in terms of their biocompatibility and flow behaviour. The biocompatibility of biomaterials of PLLA, PCL, collagen, chitosan, and gelatine is evaluated in terms of supporting neuron cells adhesion and outgrowth. Chitosan solution (2% w/v) in acetic acid is shown to be the most promising among the examined biomaterials for the fabrication of nerve tissue scaffolds. Its non-Newtonian flow behaviour is identified by using a commercial rheometer. <p>In the fabrication process, the flow rate of biomaterials dispensed, the profile of strand cross-sections, and the scaffold porosity are very important and must be precisely controlled. A model is developed to represent the flow rate of biomaterials dispensed under the assumptions that the flow is incompressible, steady, laminar, and axisymmetric. Also, the profile and size of line strands at different layers and portions are modeled based on the Young-Laplace equation. Thus the dispensing-based fabrication process can be predicted in terms of the flow rate and the scaffold porosity. <p>The effects of operation conditions on the fabrication result are identified theoretically and experimentally. Simulation result shows that a higher driving pressure, a higher temperature, and a larger needle diameter will result in a larger size of the strand cross-sections and lower scaffold porosity. The change pattern, however, is nonlinear, which is affected by the fluid surface tension and non-Newtonian flow behaviour of scaffold biomaterials. <p>To verify the effectiveness of the developed models, experiments were carried out on a commercial dispensing system (C-720, Asymtek, USA). To avoid the possible error derived from the temperature difference between the dispensing system and the rheometer, a new method is presented to characterize the fluid properties used for model predictions. Experimental results illustrate that the developed models, combined with the new identification method, are very promising to predict the dispensing-based fabrication process.