STUDY OF SOME NON-NEWTONIAN BOUNDARY LAYER FLOWS

Abstract

Even though the 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 boundary layer flow of non-Newtonian fluids. Almost all the fluids used in the industries are non-Newtonian and most of the flows can be numerically mod-eled by flow in rotating disk system, flow past a stretching sheet, flow near a stagnation point and flow over a flat plate. The purpose of the present research is to gain a better understanding of the behavior of different non-Newtonian fluids in the laminar boundary layer flow on aforementioned geometries. In this thesis, the constitutive equations for the non-Newtonian fluids are modeled by those for incompressible Reiner-Rivlin fluid, and few fluids of differential type, namely, second grade fluid, third grade fluid, and fourth grade fluid. These problems are not only important because of their technological sig-nificance, but also in view of the interesting mathematical features presented by the equations governing the flows. In fact, the intrinsic nonlinearity of the model of fluids of differential type and the paucity of the boundary conditions due to the presence of higher order material derivatives of the strain tensor ex-clude the possibility of an analytical solution. Moreover, there are empirical evidence that most of the non-Newtonian fluids flowing past solid boundaries Abstract exhibit partial slip boundary conditions, which has also been taken into account. This makes the boundary value problem more complicated. All the problems, studied in this thesis are generalized by incorporating the energy equation and diverse physical effects. Effective numerical schemes have been adopted to solve the resulting systems of highly nonlinear differential equations with inadequate boundary conditions. The thesis begins with the numerical investigation of the effects of partial slip on the flow and heat transfer of an electrically conducting non-Newtonian Reiner-Rivlin fluid due to a single disk in free space. The problem has been gen-eralized by considering two cases, which are respectively included in Chapter-2 and Chapter-3 of the thesis. In the first, the disk is rotating with an uniform angular speed and there is no flow far from the disk, which is often referred to as the Von Kdrz-naz-z flow; in the second, the fluid far from the disk is rotating and the disk itself is stationary, which is referred to as the Bodewadt flow. The mod-els find immediate technical applications in rotating machinery, heat and mass exchangers, biomechanics and oceanography. Finite difference method along with shooting method are used to solve the resulting system of highly non-linear dif-ferential equations. The manner in which the various material parameters affect the structure of the boundary layer is represented precisely through graphs and tables. It is observed that slip has a prominent effect on the velocity and tem-perature fields. Subsequently, the flow and heat transfer of different non-Newtonian fluids past a stretching sheet in various situations have been considered. The study has received considerable attention due to its immense applications in the industries, for example, materials manufactured by extrusion process, the boundary layer along a liquid film in condensation process and heat treated materials traveling vi Abstract between a feed roll and a wind-up roll or on conveyor belt poses the features of a moving continuous surface. In Chapter-4, the existence and uniqueness of the couple of nonlinear differential equations arising due to the flow and heat transfer of a thermodynamically compatible second grade fluid past a planar stretching sheet has been studied rigorously. Afterwards, Chapter-5, Chapter-6 and Chapter-7 of the thesis deal with the combined effects of the uniform transverse magnetic field and the partial slip on the boundary layer flows and heat transfer of second grade and third grade fluids past axisymmetric, planar and exponentially stretching sheets respectively. The issue of paucity of the boundary conditions has been addressed and effective nu-merical schemes combining the features of the finite difference method and the shooting method are adopted to solve the systems of highly nonlinear differential equations without augmenting any extra asymptotic boundary conditions. The use of Broyden's method, instead of Newton's method or secant method, as the zero-finding algorithm has enhanced the efficiency of the numerical scheme by reducing the computational time. The manner in which the emerging flow pa-rameters affect the momentum and thermal boundary layers has been discussed elaborately with physical interpretations. It is interesting to find that in the limiting case, as the slip parameter approaches infinity, the resistance between the viscous fluid and the surface is eliminated and the stretching of the sheet does no longer impose any motion on the fluid, i.e. the flow behaves as though it were inviscid. The stretching sheet problems are followed by the finite difference solution of the boundary layer flow and heat transfer of electrically conducting second grade and third grade fluids near a stagnation point. The effects of various flow parameters on the boundary layer flow and heat transfer have been computed vii Abstract and shown graphically. From a practical point of view, these flows have applica-tions in forced convection cooling processes, where a coolant is impinged on a flat plate. Chapter-8 deals with the axisymmetric stagnation point flow (Homann flow) and heat transfer of a second grade fluid. It is found that the presence of viscoelasticity in the fluid initiates the process of flattening the velocity profiles and thus, increases the momentum boundary layer thickness. Moreover, the mag-netic field has also a substantial effect on the velocity and temperature boundary layers. Subsequently, the two-dimensional stagnation point flow (Hiemenz flow) and heat transfer of a third grade fluid has been studied in Chapter-9. It is ob-served that the third grade fluid parameter increases the momentum boundary layer as well as the thermal boundary layer thicknesses. The simplest example of the application of the boundary layer equations is afforded by the flow along a flat plate, known as the Blasius flow. In Chapter-10, the Blasius flow and heat transfer of an electrically conducting third grade fluid subject to partial slip boundary conditions has been studied by using the finite difference method. Subsequently, in Chapter-11, the effects of partial slip on the Blasius flow and heat transfer of an electrically conducting fourth grade fluid has been investigated numerically by a shooting method. The manner in which the various material parameters affect the structure of the boundary layer is delineated. It is interesting to find that increasing the higher order material parameters in non-Newtonian fluids causes further thickening of the boundary layer. Each Chapter begins with a brief literature review pertaining to the con-cerned problem and the motivation behind the study. Efforts have been made to delineate the manner in which the various material and flow parameters af-fect the structure of the velocity and thermal boundary layers, and are shown viii Abstract graphically. Emphasis has been given to the physical interpretation of the new findings. Moreover, each Chapter contains a precise 'Conclusion' part followed by 'Major contributions and recommendations'. Eventually, an 'Appendix' has been added, which contains the brief descrip-tion of certain important technical terms and numerical methods used in this thesis. All the numerical methods used in the thesis are translated into FORTRAN 90 program and are run on a pentium IV personal computer, using FORTRAN Power Station 4.0 compiler. The above studies and results are novel, and to the best of author's knowledge no attention has been given to the aforementioned boundary layer flows with diverse physical effects heret