Analytical study of fluid flows with slip boundary
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One of the important scientific research focuses worldwide has been on the study of behavior of materials at micro and nano-scales. Continuing theoretical and technological development in this area has led to the development of many biological and engineering devices and systems which involve fluid flow through micro-channels, referred to microflows. Typical examples include fuel cells, drug delivery systems and energy conversion devices.As the functional characteristics of micro-systems depend, to a large degree, on the behavior of fluid flow in the micro-channels, it is extremely important to study microflows. In recent years, many investigations were carried out to study various flow problems of Newtonian and non-Newtonian fluids under the steady state condition. However exact solutions to many flow problems of Newtonian fluids in micro-channels under the unsteady condition have not been developed and investigated. Hence, in this project, we study the unsteady flow of incompressible Newtonian fluids through rectangular and elliptic micro-channels with boundary slip.For the unsteady flow through micro-channels of rectangular cross-section, the governing equations are constructed and formulated in the rectangular coordinate system. Then by using Fourier series expansion and separation of variables, the governing partial differential equation for the velocity field is successfully reduced to two simpler families of boundary value problems which are then solved analytically. From the derived exact solution of the velocity field, the transient flow rate and the stress field in the fluid are subsequently derived. An investigation is then conducted to study the behaviour of microflows in rectangular channels. Various interesting results, showing the influence of boundary slip and cross-section geometry (width to depth ratio) on the flow behaviour and efficiency, have been obtained and presented in the thesis.For the unsteady flow through micro-channels of elliptic cross section, the complete set of governing equations, including the partial differential equation and the boundary conditions, are formulated in elliptic cylindrical coordinates. Then by using Fourier series expansion and separation of variables, the partial differential equation for the velocity field is successfully reduced to two families of Mathieu type equations which are then solved analytically subject to the symmetric condition and the slip boundary condition. Exact solutions for the transient flow rate and the stress field in the fluid are then derived subsequently. The solutions are expressed in terms of the Mathieu functions and the modified Mathieu functions, which are in series form and are determined by computing their characteristic numbers and the coefficients of the series. A numerical investigation is then conducted to demonstrate the flow behavior of fluids in elliptic micro-channels.
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