Study of Non-Newtonian Swirling Flows Near Rotating Disks

Abstract

Swirling, whirling and rotating flows have fascinated people for centuries and the fascination continues to this day. Atmospheric or oceanic flows, typhoons and tornadoes, and closer to our daily lives, bathtub vortices and stirring tea in a teacup, are all examples of the ubiquity of swirling flows at all scales in nature. Moreover, rotating flows are of crucial importance in a wide range of scientific and engineering applications. Description of such flows, in more than just observational details had to wait for the Navier-Stokes equations. Even than the equations were so difficult to solve, that it had to wait for the advent of computers and with them many numerical, analytical or semi-analytical techniques.

This thesis is devoted to investigation of a few problems of convective heat and mass transfer related to rotating disk systems. Rotating disk systems are widely used to model the flow and heat transfer associated with the internal-air systems of gas turbines, where disks rotate close to a rotating or a stationary surface. In addition, rotating-disk systems are used in electro-chemistry (rotating-disk electrodes), bio and chemical reactors, transport engineering (automobile brakes), rotating-disk cleaners, etc. These flows are among those few problems in fluid dynamics for which Navier-Stokes equations admit an exact solution.
In view of such theoretical and practical importance of rotating disk flows, this thesis focuses on obtaining approximate analytical solutions of equations governing the flow problems arising in five different configurations of the rotating disk systems, viz. (i) flow over single rotating disk (under the influence of partial slip), (ii) flow between two rotating disks, (iii) fluid impinging over a stretchable rotating disk, (iv) flow between two stretchable rotating disks, and (v) flow over a stretchable rotating disk (in a rotating frame of reference). In addition, two different non-Newtonian fluid models, namely, visco-inelastic Reiner-Rivlin fluid and viscoelastic second grade fluid, are considered and the corresponding steady, laminar flows are investigated in the above mentioned configurations. The equations governing these flows are fully coupled and highly nonlinear, which provides a level of complexity that it becomes difficult to get a closed form analytic approximation.

In this thesis, we have adopted a popular and promising non-perturbation technique, called, the ‘Homotopy Analysis Method (HAM)’ to obtain approximate analytical solutions of the considered problems. HAM is a general approximate analytic approach used to obtain series solutions of nonlinear equations of different types, such as algebraic equations, ordinary differential equations, partial differential equations and differential integral equations. The reasons why HAM is used are, first, this method is valid no matter whether or not a nonlinear problem contains small or large physical parameters, which is an essential requirement for perturbation techniques; second, unlike other perturbation or non-perturbation methods, the HAM provides us a simple way to guarantee the convergence of solution series by means of ~-curves. In addition, HAM provides us with the freedom to choose proper base functions to approximate a given nonlinear problem.

In the consequent chapters, results in the form of a convergent Taylor series are obtained using HAM. In some cases the results obtained by using HAM are compared with those obtained by using commonly used semi analytical techniques, namely, Adomian decomposition method (ADM) and Homotopy perturbation method (HPM) to demonstrate the efficiency of HAM. In addition, convergence of the obtained series solutions are also analyzed. The effects of various parameters, such as, Reynolds number, stretching parameter, non-Newtonian parameter etc. on the flow fields are discussed in detail. Moreover, streamlines are drawn to better visualize and understand the flow problems under consideration.