Flow and Heat Transfer Analysis for Arbitrary and Hyperbolic Stretching Surfaces

Abstract

The main purpose of this thesis is to discuss the fluid flow driven by stretching of the sheet. In fluid dynamics, there are many mechanisms to drive the fluid flow: like the motion of the boundaries, the pressure gradient and the buoyancy force. Of these mechanisms, the motion of the boundaries remains the most important. And among many fluid flows driven by the motion of the boundaries, the flow induced by stretching of the surface has been of greatest value and importance. In fact its importance lies in the application of stretching sheets in industry and engineering. Such flows are generally generated in extrusion of polymers, fibers spinning, hot rolling, manufacturing of plastic and rubber sheet, continuous casting and glass blowing. The stretching of the sheets have been initially undertaken in Newtonian fluid and later on extended to non-Newtonian fluids-- because of their application in polymer industry. The heat transfer through a fluid in the presence of fluid flow is also a subject of immense importance both for understanding of fluid flow and its applications in vast areas of industrial problems. Keeping in view the importance of the flow and heat transfer by stretching sheet a huge amount of work has been published for linear, power law and exponential stretching of the sheets in Newtonian and non-Newtonian fluids. However, the bulk of this work describes the flow and heat transfer past a continuous stretching surface taking linear, polynomial, power law and exponential stretching velocities and temperature distributions. However, this has always been a challenge for the scientists and engineers to introduce new stretching velocities for the solutions of non-linear equations on the one hand and its industrial applications on the other hand. The present thesis extends the class of stretching problems by introducing a hyperbolic stretching velocity and temperature distribution on the sheet, for the first time. The appropriate similarity transformations are introduced to reduce PDEs into ODEs. Arbitrary stretching and non- Newtonian fluids are also taken into consideration. These investigations will go a step forward in understanding the fluid flow and heat transfer for so far unaccounted stretching of the sheet and its possible industrial applications both in Newtonian and non- Newtonian fluids. The first chapter of this thesis contains the history and literature related to stretching sheet problems and states the basic definitions and equations to be used in later chapters. In second chapter we introduce the concept of hyperbolic stretching of the sheet for the first time in this thesis. The boundary layer flow and heat transfer analysis of an incompressible viscous ixfluid for a hyperbolically stretching sheet is investigated. The analytical and numerical results are obtained using series expansion method and Local Non-Similarity (LNS) methods respectively. Analytical and numerical results for skin friction and Nusselt number are calculated and compared with each other. The significant observation is that the momentum and thermal boundary layer thicknesses decrease as the distance from the leading edge increases. The well- known solution of linear stretching is found as the leading order solution for the hyperbolic stretching. The contents of this chapter have been published in Applied Mathematics and Mechanics (English Edition), 33(4), 445–454 (2012). Flow and heat transfer of an electrically conducting viscous fluid over a hyperbolic stretching sheet with viscous dissipation and internal heat generation is investigated in third chapter. The suitable transformations reduce the governing equations in a tractable form for the analytical and numerical solutions. The same analytical and numerical methods, as in chapter two, are used to obtain the results. The essence of this paper is to examine the effects of viscous dissipation, magnetic field and heat generation in a recent paper of hyperbolic stretching sheet presented in last chapter. This work has been submitted in International Journal of Numerical Methods for Heat and Fluid Flow for publication. Fourth chapter deals with the study of mixed convection flow and heat transfer of a viscous fluid along a vertical hyperbolic stretching wall. The results have been obtained considering the effect of heat generation/absorption. The solutions for forced convection flow over a linear stretching surface with linear temperature distribution in the presence of heat source/sink are found to be the leading order solutions of mixed convection past a hyperbolic stretching wall. In next chapter, the boundary layer flow and heat transfer analysis of an incompressible nanofluid for a hyperbolically stretching sheet is presented. The model used considers the effects of Brownian diffusion and the thermophoresis. Analytical and numerical results for skin friction, Nusselt number and Sharwood number are calculated and compared with each other. The effects of different physical parameter on velocity, temperature and concentration of nanoparticles are also analyzed. The leading order solution of this problem represents the flow, heat and mass transfer of an incompressible viscous fluid over a linearly stretching surface. The boundary layer flow of second grade fluid over a permeable stretching surface with arbitrary velocity and appropriate wall transpiration is investigated in fifth chapter. The fluid is electrically conducting in the presence of constant applied magnetic field. Exact solution to the nonlinear xflow problem is presented. The contents of this chapter are published in Applied Mathematical Letters, 24 (2011), 1905–1909. The last chapter presents an exact analytical solution of magneto hydrodynamic (MHD) viscous flow over a permeable sheet with partial slip boundary conditions. The flow is induced by an arbitrary stretching of the surface. The exact analytical solution of the problem becomes possible by taking an appropriate wall transpiration velocity. An existing solution for flow generated by arbitrary stretching surface with no slip condition class of the exact solutions of the Navier Stokes equations for stretching surface.