Some flow and heat transfer problems with special reference to numerical procedures

Abstract

This thesis deals with elegant application of certain known mathematical methods and with certain numerical procedures devised to obtain solutions of the differential equations arising in the flow and heat transfer problems of viscous and visco-elastic fluids. The techniques devised here replace the usual expansion methods in terms of small parameters. These methods are applied to physical problems to shov/ their applicability and efficiency. Satisfactory solutions have been obtained either in the closed forms or in terras of simple quadratures or without using the brute-force trial and error method specially when the two-point boundary value and eigen-value problems are involved. Thus the solutions of the physical problems as well as the methods developed here are of interest. The thesis consists of seven chapters. Chapter I gives a general introduction to the matter discussed in the succeeding chapters. Here we have stated in brief and in general teims the main types of problems that we have investigated in the thesis. We have also recorded here the constitutive equations of the fluidj and the equations of motion relating to some of the geometries that we have employed in our studies. Each of the subsequent chapters deals exclusively with one particular type of problem. - 1 - Each of these chapters starts with a short introduction to the problem followed by the author's contribution* Every care has been taken to present the contribution in a connected manner to avoid repetition as far as possible. Wherever necessary a chapter is split up into different parts to present the related work. All the equations and formulae etc. of chapter I have been numbered separately. Each of the other chapters has its own section numbers and the numbers of the equations etc. Chapter II deals with time-dependent heat transfer in the presence of a heat source in a Newtonian fluid adopting the Laplace Transform technique and is based on the following papers: (i) A note on the temperature distribution in a channel bounded by two co-axial cylinders (in collaboration with Prof.P.L.Bhatnagar), Proc.Gamb.Phil,Soc62, (1966), 301-302. (ii) Heat transfer in a flow of viscous incompressible fluid in a cylinder and cylindrical anrailus, Jour.Ind.Inst.Sci., 49, (1967), 1-9. Chapter III deals with the steady laminar flow and heat transfer in Noll fluid for which formal solutions have been obtained and is based on the following paper : Laminar flow of elastico-viscous fluid inside a cylinder, Ind.Jour.Pure and Appl.Phys., 5, (196 7), 246-247. Chapter IV deals with the steady laminar flow and heat transfer in Rivlin-Ericksen fluid between two infinite parallel plates - one stationary and other rotating - and with an iterative procedure of solving two-point boundary value problems associated with coupled nonlinear ordinary differential equations. These investigations are reported for the first time in this thesis. Chapter V deals with steady laminar flows and heat transfer with suction and injection for which the solutions of two-point boundary value problems associated with ordinary differential equations are obtained without trial and error. This chapter is based on the following papers : (i) Plane Gouette flow with suction or injection and heat transfer for Rivlin-Sricksen fluid, Jour.Ind.Inst.Sci., 50, (1968), 244-257. (ii) Steady laminar flow with suction or injection and heat transfer through porous pipe "to appear in Proc .Ind.Acad.Sci . Chapter VI deals with a new numerical procedure for initial value problem associated with second order non-linear ordinary differential equation: and is based on the following paper : Numerical procedure for second order nonlinear ordinary differential equations and application to heat transfer problem (in collaboration with Miss Swarnalata Prabhii) [to appear in Proc.Ind. Acad .Sci.] Chapter VII deals with the application of Galerkin’s method to the study of stability of laminar flow of a Maxwell fluid inside a circular cylinder. This study is reported for the first time in this thesis.