NUMERICAL SOLUTIONS OF SOME PROBLEMS OF FLOW AND HEAT TRANSFER

Abstract

The thesis deals with numerical solutions to some problems of flow and heat transfer in viscous fluid through different geometries. The thesis runs into nine chapters. The first chapter is of introduction and presents the basic equations of fluid dynamics. The rest of the eight chapters form the main contribution of the author. In all, eleven problems have been discussed in these eight chapters. In the end bibliography is given. According to the nature of the methods used, the chapters two to nine are divided into two parts viz: Part I ; Solution by non-iterative methods. Part II : Solution by iterative methods. In Part I, the problems have been solved by following methods: (i) Parameter differentiation (ii) One parameter imbedding technique (iii) Method of characteristics (iv) Perturbation Part II consists of the application of following methods: (i) Finite difference (ii) Quasilinearization (iii) Newton Raphson (iv) Method of Continuation. \i A chapterwise survey of the two parts is as follOWB8- PART I The second chapter presents an application of the method of parameter differentiation to the problem concerning the flow of a viscous fluid between two stationary infinite porous discs. The boundary value problem representing the phenomenon, is first converted into an initial value problem, with a parameter as the independent variable. Through step by step integration with respect to the parameter, the solution has been sought for successive values of Reynolds number. The results have been shown graphically. The method may be used to seek results for a wide range of the parameter. The third chapter contains the discussion of one parameter imbedding technique, applied to the problem of laminar source flow between rotating coaxial porous discs. The general form of the procedure is given and the resulting algorithms are classified as one loop and multi loop. The results have been discussed for various values of the three parameter involved, namely Reynolds numbers based on suction velocity, rotation and strength of the source. The results for one loop and multi-loop algorithms are compared. In the fourth chapter the method of characteristics has been employed to solve the partial differential equations 111 governing the unsteady flow in open chonnel with slope when friction is present. Sudden closure is applied at one end of the channel. The partial differential equations are converted into total differential equations and are solved. The stability condition and convergence criterion has been satisfied. The velocity and pressure distribution in the transient condition has been analysed for various values of friction coefficient and slope factor. The fifth chapter comprises of the solution of two problems by perturbation method: a. Heat transfer near a stagnation point between porous discs with uniform blowing: Here the temperature function has been expanded as a series in perturbation parameter R, the Reynolds number for injection, and terms up to third order have been considered. The effect of Reynolds number upon temperature distribution has been discussed. b. Fully developed free convection flow in a circular pipe: In this problem a non-linear density temperature relationship has been taken to express the body force term as buoyancy term. A perturbation solution for velocity and temperature in terms of Reynolds numbers, has been developed, which works only for small values of the parameter. A com parison of these results with corresponding ones for quadratic density temperature variation has been made. w PART II The sixth chapter deals with the following three problems: a. Unsteady flow past a flat plate with suction: The governing partial differential equation has been converted into a set of ordinary algebraic equations using an explicit finite difference scheme. The solution has been sought in view of the stability criterion, using ordinary iterative procedure correct to four places of decimals. The shearing stress at the wall has been obtained. b. Natural convection in a channel with porous walls under transverse magnetic field: An iterative approach has been employed to solve the set of non-linear algebraic equations obtained through an implicit finite difference scheme applied to the differential equations governing this problem. The non linear density temperature relationship has been considered. The results for a wide range of suction Reynolds number have been found and compared with those for linear density temperature relation. Skin friction and Nusselt number has also been calculated. c. The extension of the problem (a) in fifth chapter, for large injection Reynolds number is made. The governing differential equations are converted into algebraic equations and are solved by Gauss Seidel Iterative method. The seventh chapter again contains the solution of two problems namely, a. flow between porous walls of different permeability. b. natural convection flow between vertical parallel plates, by the method of quasilinearization. The first problem reduces to the solution of single non-linear differential equation while second one to that of solving a set of simultaneous non-linear differential equations. Instead of solving non-linear problem, a sequence of linear two point boundary value problem, is solved. The numerical results correct up to six places of decimal have been computed and compared with those obtained earlier in the literature. The eighth chapter is devoted to the study of flow and heat transfer from an enclosed rotating disc with suction and injection. The governing differential equations which are highly non-linear, are transformed into algebraic equations using an implicit finite difference scheme. These equations are then solved iteratively by Newton Raphson method. The radial and transverse velocity profiles have been plotted for various values of Reynolds number in the region of no-recirculation. The effect of net radial inflow and outflow on temperature has been discussed. The problem of magnetohydro dynamic flow between two porous discs-one rotating and the other stationary, VI forms the subject matter of the last ninth chapter. The problem for moderately large values of R± (Reynolds number due to rotation) and R2 (Reynolds number due to suction) is solved by Newton Raphson method but for very large value of R2 the method fails. This case has been treated by the method of continuation. The solution is first obtained by shooting method for a short interval which is then extended to a bigger interval. The procedure is continued, each time the terminal conditions shifted on the end point of the new interval till the final point is reached. The results have been shown graphically. The entire numerical work has been carried out on IBM 16 20 and IBM 360 computers.