SOME PROBLEMS OF HEAT TRANSFER IN FLUIDS

Abstract

This thesis, running into eight chapters, comprises of an analysis of heat transfer (by forced or free convection) in certain flows of viscous and second order fluids. The effects of a magnetic field on heat tra nsfer have also been examined in some cases. The first chapter is introductory and deals with the fundamental concepts of flow and heat transfer in fluids. The governing equations (consisting of the consti tutive equation and the equations of conservation of mass, momentum and energy) for the viscous flow heat transfer as well as those governing it in second-order or electri cally conducting incompressible viscous fluids have been given in tensorial form. The author's contribution forms the subject matter of the seven chapters that follow. They are classified into parts A and B, dealing respectively with the heat transfer in viscous (electrically conducting or non-conduct ing) and second-order fluids. In case of second-order fluids, interpretation of results has been based, wherever possible, on the experimental values of material constants (Markovitz,H., and Brown, D.R.; Proceedings, Int. Symp. on Second-order Effects in Elasticity, Plasticity and Fluid Dynamics,1962,585). ii The detailed coverage of parts A (Chapters II-V) and B (Chapters VI-VIII) is as follows:- Part A: Viscous Flow Heat Transfer Chapter II deals with a study of heat transfer from an enclosed rotating disc. The effects of a transverse mag netic field, when the fluid is electrically conducting, have also been considered. The solution has been affected by expanding the temperature in ascending powers of Reynolds number, assumed small. The effects of radial outflow and inflow on the temperature profile and the Nusselt numbers on the rotor and the stator have been investigated in regions of no-recirculation and recirculation. In the magnetic case the effects of magnetic field on temperature and velocity 1/2 . are governed by the Hartmann number S f= u. H (—) Z , i_ | e o yx.' oj ' ZQ being the gap length, HQ the applied magnetic field, i^ the magnetic permeability, a- the electrical conductivity, Q. the density and ~j) the kinematic coefficient of viscosity of the fluid. Some of the results have been illustrated graphically. Chapter III studies the effects of a transverse weak magnetic field on heat transfer from an infinite rotat ing disc, maintained at a constant temperature (or else insulated), to an electrically conducting incompressible viscous fluid occupying the semi-infinite space over the disc. Fourth degree polynomials in terms of the dimensionless distance along Z-axis have been assumed for temperature functions. The main parameter governing the temperature is iii the magnetic number Km (-^ fi2 ^ ^ where ^ ig thQ angular velocity of the disc. A thermal boundary layer depending on the magnetic field, has been determined. For both the cases, viz., those of disc at constant temper ature and when the disc is insulated, it is found that the thermal boundary layer as well as the temperature therein increases with an increase in the magnetic number. Chapter IV takes into account the effects of a magnetic field on fully developed natural convection flow between two parallel porous plates maintained at constant temperatures. A method of successive approximations has been developed to solve the non-linear differential equation involved. The velocity and temperature are both affected by the presence of the magnetic field through the Hartmann number S[- flQ HQ h («• /p. )1/2]. The effects of natural convection are taken care of by dimensionless group }\(» PrGp fx h/Cv), where Pr is the Prandtl number, G the Grashof's number, p the coefficient of volumetric expansion, fx the body force, h the distance between the plates and Cy the specific heat at constant volume . Besides other results it is found that the velocity as well as the temper ature decrease with an increase in S and so does also the Nusselt number (its numerical value) along the walls. The problem of heat transfer to a viscous fluid occupying the space above a torsionally oscillating plane forms the subject matter of Chapter IV. Boundary layer type solution of the energy equation has been attempted by assum ing the amplitude of rotational oscillations to be small. IV Solution correct to 0(e), where 6 is the small amplitude of oscillations, has been discussed. The temperature, like velocity, is found to have a steady component and a component with frequency twice that of the plate. As happens for the velocity field, the steady part of temperature, based on series expansion too does not satisfy the condition at inf inity. A re-examination by Ka'rman Pohlhausen type method, however, reveals that no part of the steady temperature will in fact survive outside a secondary thermal boundary layer. The thermal and velocity boundary layersare found to be in the usual ratio l:"fP » ?r being the Prandtl number. One of the interesting revelations is that the secondary thermal and velocity boundary layers, calculated on the basis of Ka'rman Pohlhausen type method, also tend to be in the same ratio at high Prandtl numbers. Part B: Heat Transfer to Second-order Fluids. Chapter VI deals with the problem of heat transfer in the flow of a second-order fluid near a stagnation point. The study covers both the two-dimensional and axi-symnetric cases. Solution valid in a thermal boundary layer is developed in terms of a sextic polynomial of the dimensionless distance from the plate. The effects of elastico-viscosity and crossviscosity of the fluid are found to depend upon the dimensionless numbers K ( - a^/v )and Kr(=a Vg/l> ) respectively, 'a' being a fluid flow parameter with dimensions T" related to the potential flow andV1»^>2 are *he kinematic coefficients V of elastico-viscosity and cross-viscosity respectively. It is concluded that in both the cases the plane of maximum temperature shifts nearer to the plate with an increase in Kg. The temperature decreases in the same situation. That the Nusselt number is negative and decreases (numerically) with an increase in K is another conclusion. In Chapter VII the effects of uniform suction (or injection) on the natural convection flow of a second order fluid from a vertical porous flat plate have been studied. As is usual for all two-dimensional flows, the flow is inde pendent of the cross-viscosity of the fluid. A momentum integral method similar to that of Ka'rman Pohlhausen has been employed. The effects of the elastico-viscosity of the fluid depend upon the dimensio.nless parameter K while those of suction on S f» _ (Ngp )-1/4 (j))-1'2 y -1 where U L W J N is a constant of proportionality, while g and v represent respectively the acceleration due to gravity and the constant velocity of injection. A negative sign with v, denotes suction (Su>0 for suction, <0 for injection). For a fixed value of K , the boundary layer thickness is found an to decrease withAincrease in suction while the behaviour gets reversed for increase in injection. For a constant suc tion or injection, the boundary layer thickness decreases with an increase in elastico-viscous parameter. The concluding chapter VIII is devoted to the study of flow of a second-order fluid in a channel with porous walls. The heat transfer analysis is also given. The fluid is assumed to be uniformly blowing in or blowing out of the walls. The problem of heat transfer is examined when the w vi walls are at different temperatures. Both the velocity and temperature functions are expanded in powers of the suction parameter Su(» h vw/^> ). The effects of elasticoviscosity are governed by the dimensionless parameter Rt(= Ke/Su), where KQ(-^ vw/h ). The behaviour of the coefficients of skin friction and heat transfer at different values of R, and S„ has been studied in x u detail. Almost the entire numerical work has been carried out on I.B.M. 1620 Computer installed at Structural Engineering Research Cent re, Roorkee. '^e results of invest igations are summarized at the end of each chapter.