Graduate Thesis Or Dissertation
 

Generalized deformation-rates in secondary flows of viscoelastic fluids between rotating spheres

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  • By various experiments, it has been found that the response of real materials to external forces is, in general, nonlinear in character. In classical continuum mechanics, the use of ordinary measures of strain have forced the constitutive equations to take complex forms and since the orders of these measures are not fixed, many unknown response coefficients have to be introduced into the constitutive equations. In general, there is no basis of choosing these coefficients. Seth attempted to resolve this difficulty by introducing generalized measures in continuum mechanics and Narasimhan and Sra extended these measures in such a way as to adequately explain some rheological behavior of materials. The constitutive equation of Narasimhan and Sra essentially contains two terms and four rheological constants and, unlike some previous theories, it does not contain any unknown functions of the invariants of kinematic matrices while at the same time explains many viscoelastic phenomena. In the present investigation, a theorem has been proved establishing certain criteria for fixing the orders of generalized measures suitably so as to predict different types of viscoelastic phenomena, such as dilatancy. We have found during the course of this investigation that the constitutive equation of Narasimhan and Sra does not adequately explain such physical phenomena as pseudoplasticity. However, in order to construct a constitutive equation so that it does explain such phenomena, we have found it necessary to construct combinations of sets of generalized measures. The resulting constitutive equation is found to be quite general and is able to explain a vast range of physical behavior of fluids. To illustrate the use of this constitutive equation based upon combined generalized measures, we have investigated the important problem of secondary flows for fluids in the presence of moving boundaries. This problem is very important, since the investigation of secondary flows allows us to obtain a clearer picture of the actual motion of the fluid. For the problem of flow of a fluid in the annulus of two rotating spheres, we have obtained the solution for the velocity and pressure fields. In order to investigate the secondary flow pattern more thoroughly, we have obtained the streamline function of the flow. The streamlines in meridian planes containing the axis of rotation are found to be closed loops and the nature of the closed loops is found to be strongly dependent upon the viscoelastic parameter S. For S less than critical value, the flow is found to be very much like that of a Newtonian fluid, with the fluid advancing toward the inner sphere along the pole and outward along the equation. At the critical value the flow region is found to split into two subregions, each containing closed loops of streamlines. As S is increased further, another critical value is reached whereby the streamlines again become one set of closed loops, with the sense of rotation reversed from that of Newtonian fluid.
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