SOME FLOW PROBLEMS IN NON-NEWTONIAN FLUIDS

Abstract

The thesis is an attempt to study certain problems of flow of incompressible Second-order fluids. It runs into eight chapters. The first chapter is introductory and covers in the first instance, the development of fundamental concepts of classical hydrodynamics so as to embrane generalization leading to the theory of Second-order fluids. It also pro vides the transformations to different coordinate systems (given in the Appendix and used in the thesis) of the funda mental equations of motion and the constitutive equation of the Second-order fluids. The subsequent content of the thesis which forms the main contribution, is classified under parts Aand B, which deal respectively the steady flows with and without suction and/or injection and unsteady flows with suction and/or injection. Part Aconsists of four chapters and Part B consists of three more chapters. , The equations involved in the thesis have been non-dimensionalized and the interpretation of the results is based wherever possible on the experimental values of the material constants provided by the experiments conducted by Markovitz for the solution of Poly-iso-butylene in Cetane at 30°C (see Truesdell1)). ii The brief outlines of the Chapters in Part A (Chapters II-V) and Part B (Chapters VI-VIII) are as follows : Chapter II comprises the study of the steady forced flow of a Second-order fluid against a rotating disk with uniform high suction. The effects of uniform high suction on the flow produced by the rotation of a disk in an infi nite fluid which is otherwise at rest and flow near a stagna tion point occurring on a flat plate are particular cases of this problem. Assuming series expansions for the velocity components in descending powers of the suction parameter, it has bean possible to obtain the solution in ascending powers of the cross-viscous parameter. Thus the aolution is valid for small values of dimensionless cross-viscous para meter and large values of the suction parameter. Numerical computations have been carried out for the cases when one of these two parameters is kept fixed and the other is varied. The effects of elastico-viscous and cross-viscous forces in the flow and those due to suction on the radial, transverse and axial velocity components have been studied in detail and illustrated graphically. Chapter III deals with a study of the flow of an incompressible Second.order fluid through a slightly curved pipe of circular cross-section. The curvature of the pipe has been assumed small, that is, the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of cross-section. Asolution iii is developed by the method of successive approximations. The streamlines in the plane of symmetry and the projection of the streamlines on a normal section have been obtained and illus trated through figures. Expressions for normal stresses, rate of out flow and the axial drag on the pipe etc.have also bean obtained. Chapter IV is devoted to a theoretical investigation of aflow of an incompressible Second-order fluid in a curvedannulus of circular cross-section. The flow parameters are evaluated by the method of successive approximations and a particular example has been solved numerically, considering that the radius of curvature of the annulus is large, implying that the ratio of the radius of the outer curved pipe to that of the circle in which the common axis of the two curved pipes is coiled is sufficiently small. The projection of the stream lines on a normal section and rate of out flow have been obtained and illustrated graphically. The Chapter V deals with a study of the steady flow of an incompressible Second -order fluid between two infinite porous rotating disks with the assumption that the rate of injection of the fluid at one disk is equal to the rate of suction at the other. The velocity components have been expre ssed in terms of three dimensionless-functions, which in turn are obtained in ascending powers of the Reynolds number (taken to be small) , defined in terms of the angular velocities of Iv the disks for the three cases viz (i) „h8n the disks rotate in the same sense (11) when the disks rotate in the opposite sense and (ill) when one disk rotates and the other Is at rest. Effects of variation in the ela.tlco-visccus and cross-viscous forces in the now and those due to porosity on the radial, transverse and axial velocity components have been studied and depicted graphically. Chapters VI, VII and VIII deal with the problems of the flow of a Second-order fluid due to unsteady rotations of two infinite porous disks assuming (1) injection at one disk, (2) suction at one disk and (3) suction at one and the ingestion at the other disk. The rotatory vibrations of a rotating disk about a constant non-zero mean over the other disk have been considered. It is ass»ad that the angular velocity of the disk vd.th unsteady rotation consists of a basic steady distribution together «Uh a week-time varying distribution. The amplitude of the super imposed vibration is supposed to be »all. a.A„ solution is sought by expanding the parameters in powers of the rotational Taylor number, assumed small. The effects of the .l.sticoviscosity, cross-viscosity and porosity have been discussed in detail and shown graphically for following three cases : (1) When the disks vibrate in the same sense, Ot) When the disks vibrate in the opposite sens, and (HI) When one disk vibrates and the other is at rest. The entire „»,erlcal work has be.n carried out on I.B.M 1620 Computer installed at structural peering Research Centre, Roorke.. The results are s^ariz.* ,t ^ enfl „ ^ chapter.