Computational Analysis for Nonlinear Boundary Value Problems

Abstract

Mostly fluids which are very useful in our daily life and industry do not obey the Newtonian law of viscosity. These fluids are called non-Newtonian fluids, these can be subdivided into two categories i.e. viscoelastic and viscoinelastic fluids. Viscoinelastic fluids are those in which viscosity is prominent factor regarding internal opposition of molecules. Viscoinelastic fluids are further categorized with shear thinning and shear thickening etc. The common examples of these fluids are lubricating greases, waterborne coatings, multiphase mixers, paints, suspensions, emulsions, blood, polymer sheets and biological fluids etc. The flow of electrically conducting fluids is encountered in almost every branch of science e.g. astrophysics, geophysics, mechanical engineering, aerospace engineering, nuclear engineering and bio engineering etc. Also, this concept is incorporated in many industrial processes and daily use devices such as MHD pumps, MHD power generator and electromagnetic propulsion etc. Additionally, turbulence of fluid is also handled with magnetic field. Hall current effects are noteworthy in electrically conducting fluids when applied magnetic field is strong or fluids with low density. This is an important phenomenon having many applications such as Hall accelerates, Hall senores etc. Heat transfer over stretching surfaces is encountered in many practical applications, because cooling/heating is an important factor to achieve the desired quality of product. For instance many processes/products like extrusion, paper production, fiber-glass production, hot rolling, condensation process, crystal growing, polymer sheets, artificial fibers and plastic films etc. based on heat transfer of boundary layer flows. Thermal conductivity of fluids enlarges by increasing the temperature of fluid and vice versa. Thus fluids having low thermal conductivity are used for cooling in many industrial or daily life products. Experimentally, it is observed that when fluid particles move, the viscosity of fluid alters some part of kinetic energy into thermal energy i.e. dissipate energy. Such type of dissipation is called viscous dissipation. In heat transfer of many practical problems, it plays an important role. Joule heating is an important factor in heat transfer of electrically conducting fluid flows. This phenomenon has a lot of applications such as electric stoves, electric heaters, cartridge heaters, electric fuses, electronic cigarettes, vaporizing propylene glycol and vegetable glycerin etc. Also, it is utilized in some food processing equipment. Nanofluids are modern type of fluids which are combination of base fluid and nano-sized metal particles. The principal objective of including these particles to enlarge the thermal conductivity of base fluids, because it is observed that mostly fluids (water, oil, ethylene glycol, engine oil etc.) which are traditionally utilized in thermal processes possessing low thermal conductivity. Recently, nanofluids are found very useful in bio-medicine and bio-engineering. Thus due to substantial importance of these phenomena has motivated to explore these important features of fluid flows in current work. Since Sisko fluid and Prandtl-Eyring fluid models have great importance in industry but have not been discussed for stretching cylinder so for. Thus, present work has main focus on these models. Numerical solutions are obtained through robust numerical techniques (Keller Box method, Finite element method and Shooting method). The layout of this thesis is as follows: Literature review of present work is presented in chapter 0. In chapter 1, the boundary layer flow of MHD Sisko fluid over stretching cylinder along with heat transfer is investigated. The cylindrical polar co-ordinates are used to model the physical problems. This modeling yields the nonlinear set of partial differential equations. The modelled equations are transferred to non-dimensional form after application of appropriate similarity transformations. The obtained equations are solved numerically with the aid of shooting technique in conjunction with Runge-Kutta fifth order scheme. The expressions for velocity and temperature are computed under different parametric conditions and deliberated in graphical manner. The local Nusselt number and wall friction coefficient are calculated and described in quantitative sense through graphs and tables. The contents of this chapter are published in International Journal of Numerical Methods for Heat & Fluid Flow 26(2016)1787-1801. Chapter 2 extends chapter 1 by factoring thermal conductivity effects into account. This chapter addresses the influences of variable thermal conductivity and applied magnetic field on boundary layer flow of electrically conducting Sisko fluid over stretching cylinder. The modelled partial differential equations are highly nonlinear. A set of ordinary differential equations is obtained after applying appropriate similarity transforms. The attained equations are solved with efficient numerical technique (i.e. shooting method). Impact of flow controlling parameters on velocity, temperature, coefficient of wall friction and wall heat flux is delineated via graphical and tabular manners. A comparison with literature is made to ensure the validity of computed results. The contents of this chapter are published in AIP Advances 6 (2016) 025316, DOI: 10.1063/1.4942476. Chapter 3 focuses on the physical aspects of viscous dissipation and variable thermal conductivity on non-Newtonian Sisko fluid flow over stretching cylinder under the influence of normally impinging magnetic field. The modelled partial differential equations are transfigured into ordinary differential equations with the aid of scaling group of transforms. The highly nonlinear ordinary differential equations are tackled with numerical scheme Runge-Kutta-Fehlberg method. The focused physical quantities (velocity, temperature, coefficient of wall friction and wall heat flux) are calculated and variations in these quantities are displayed graphically as well as tabular by choosing feasible values of involving physical parameters. The accuracy of adapted method is certified by comparing present results with reported literature. The contents of this chapter are published in Results in Physics 7(2017)139-146. Chapter 4 explicates the combined effects of viscous dissipation and Joule heating on boundary layer flow of MHD Sisko nanofluid over stretching cylinder. The flow governing equations are highly nonlinear set of partial differential equations with appropriate boundary conditions. These equations are converted into system of ordinary differential equations by using suitable similarity transformations and then solved the resulting boundary value problem with the help of shooting technique. The velocity, temperature and nanoparticle concentration are computed numerically in dimensionless form. The effects of pertinent flow parameters on these quantities are displayed with the help of graphs. Physical phenomena in vicinity of stretching surface are explained with the help of skin friction coefficient, local Nusselt number and local Sherwood number. Also, effects of physical parameters are depicted with the assistance of graphs and tables. The comparison of present and previous results exhibits good agreement which leads to validation of the presented model. The contents of this chapter are published in Journal of Molecular Liquids 231(2017)341-352. In chapter 5, a computational study has been established to explore the physical aspects of non-Newtonian Prandtl-Eyring fluid flow over stretching sheet under the influence of normally applied magnetic field. The mathematical formulation of this flow configuration produces highly nonlinear system of simultaneous partial differential equations. The formulated problem is transformed to non-dimensional model by applying suitable scaling variables. The numerical solution of resulting nonlinear boundary value problem is computed with the assistance of finite difference scheme Keller-Box method. The concerned physical quantities i.e. velocity and coefficient of wall friction are computed and discussed under different parametric conditions. The graphs and tables are developed to deliver the effects of involved physical parameters on focused physical quantities. A comparative study has been made of obtained results with the literature. The contents of this chapter are published in Neural Computing & Applications 31(2019)425-433. Applications of generalized Fourier law of heat conduction on MHD Prandtl- Eyring fluid flow over stretching sheet with temperature dependent thermal conductivity has been made in chapter 6. Modelled system of nonlinear partial differential equations is transformed to system of ordinary differential equations by using similarity transformations after simplifying through boundary layer assumptions. Shooting method is applied to compute numerical results for the obtained nonlinear system of ordinary differential equations. The graphs are adorned to investigate the influences of governing parameters on fluid velocity and temperature. Local wall friction coefficient and local Nusselt number are computed to explore physical aspects near the stretched surface. The impacts of controlling parameters on these physical quantities are revealed through tables. Additionally, a correlation between present and previous results is presented to justify the validation of present results. The contents of this chapter are published in International Journal of Numerical Methods for Heat & Fluid Flow (2019), DOI: 10.1108/HFF-02-2019-0161. Chapter 7 spotlights the effects of Hall current on Christove-Cattaneo heat flux model for viscous fluid flow over stretching sheet with variable thickness. The modelled problem comprises highly nonlinear partial differential equations with presubscribed boundary conditions. To facilitate the computation process, an appropriate group of similar variables is utilized to transfigure governing flow equations into dimensionless form. To find more compatible and realistic solution of obtained non-dimensional boundary value problem well-known numerical scheme shooting method is used. Numerical results are accomplished and interesting aspects of axial velocity, normal velocity and temperature are visualized via graphs by varying involving physical parameters. A comprehensive discussion is presented to delineate the effects of flow parameters on axial wall friction, transverse wall friction and wall heat flux in tabular form. A comparison of computed results (in limiting case) is presented to authenticate the present computations. The contents of this chapter are under review in peer-reviewed journal. Chapter 8 narrates the physical features of gravity-driven swimming microorganisms in the boundary layer regime of MHD viscous nanofluid flow over stretching surface. The mathematical configuration of flow problem yields the nonlinear simultaneous partial differential equations. These equations are then transformed to non-dimensional form by applying scaling variables on it. The numerical solution of resulting system of non-dimensional partial differential equations is computed with finite element method. The deviations in interesting physical quantities are demonstrated through graphs by varying the flow governing parameters.