On Convection Flow of Viscous Fluid Along a Surface with and without Radiation Effect

Abstract

In this thesis, the mathematical model, for steady natural and mixed convection boundary layer ows of incompressible uid, is developed. The ows are induced over inclined horizontal surface, vertical surface and horizontal circular disk. The important physical quantities such as thermal radiation and magneto hydrodynamics are incorporated for physical and experimental considerations. Some important physical features like heat transfer and mass transfer are added for engineering processes. The fundamental governing equations and the theory of boundary layer are discussed in detail in Chapter 2. This chapter contains the elementary eld equations of uid mechanics as partial di erential equations in terms of physically important unknown parameters such as velocity, pressure and energy and concentration variables. The boundary layer equations for momentum, thermal and mass transport are nally developed for ready reference in physical models considered subsequently. In Chapter 3, investigation has been made on the natural convection boundary layer ow of a viscous incompressible uid over a semi in - nite at plate inclined at a small angle to the horizontal. The e ects of internal heat generation and thermal radiation are taken into ac- count. In both cases viscosity of the uid is taken as exponential function of temperature. The e ect of important parameters are seen on local skin friction coe cient and local Nusselt number. The veloc- ity and temperature distributions are obtained at the separation point and discussed physically. The dimensionless boundary layer equations are transformed into the suitable nonlinear equations with the help of stream function formulation (SFF) and primitive variable formula- tion (PVF) which are respectively solved by using iterative schemes namely; implicit nite di erence Keller-box method and implicit nite di erence method along with Gaussian elimination method. Compar- ison with the previously published results is made and an excellent agreement has been found between the two. Chapter 4 contains the study of MHD natural convection ow of an electrically conducting and optically dense gray uid above a heated vertical surface. Two cases of periodic and non periodic boundary layer ows are considered together with the interaction of thermal ra- diation. It is worth mentioning that the obtained results are for the low Prandtl number uids known as liquid metals. Solutions of the governing equations are obtained for the entire range of local Hart- mann parameter. For the constant magnetic eld (non periodic case) asymptotic solutions are also obtained for small and large values of locally varying parameter ξ . The numerical values of skin friction co- e cient, rate of heat transfer, velocity and temperature distributions are discussed for various values of physical parameters. Conjugate e ects of heat and mass transfer on the natural convection ow of an electrically conducting uid along a semi-in nite vertical at plate is examined in Chapter 5. It contains two case studies: (a) When the e ects of uniform heat and mass ux are absent (b) When both are present. The problem is particularly investigated un- der the in uence of strong cross magnetic eld for liquid metals. For entire range of local Hartmann parameter, ξ , the reduced governing equations are integrated with the help of the implicit nite di er- ence Keller-box scheme. However, for slightly small values of local Hartmann parameter, ξ , problem is tackled with regular perturbation method whereas asymptotic solutions are obtained for larger values of ξ by using matched asymptotic technique. The discussion, for several physically important parameters, has been carried out for the numeri- cal results of shear stress, τw , heat transfer rate, Qw and mass transfer rate, mw . The velocity, temperature and species concentration pro les are plotted and critically analyzed in the presence of strong magnetic eld. In Chapter 6, the conduction-radiation interaction on the laminar two-dimensional steady state mixed convection ow of a viscous in- compressible uid over a semi-in nite vertical porous plate has been studied. In this chapter the solution of the problem corresponds to the situation where density of the uid varies exponentially with temper- ature. Therefore, the underlying problem deals with the solutions for high temperature di erence between the surface and the uid, which in turn provide more accurate results. Taking into account primitive variable formulation (PVF), the governing boundary layer equations are reduced to parabolic equations, which are solved numerically using implicit nite di erence method together with Gaussian elimination method (see Appendix B). The numerical results are discussed for the emerging parameters appearing in the analysis of the problem. Chapter 7 contains the in uence of conduction-radiation on the nat- ural convection ow over the horizontal circular disk. Two numerical techniques are employed to solve the boundary layer problem, namely; (i) implicit nite di erence Keller-box method (see Appendix A) and (ii) implicit nite di erence scheme along with Gaussian elimination technique (see Appendix B). The numerical results are compared graphically showing a good compatibility between the two methods. The results are presented for the whole range 0 < R < 1 of the radius of the horizontal circular disk when the Prandtl number is consider- ably small. Discussion has been carried out on the basis of numerical results obtained in terms of local skin friction coe cient and local Nusselt number. Finally in Chapter 8, the important ndings of the physical models investigated in this thesis are highlighted and the valid conclusions are drawn.