Application Of Homotopy Analysis Method For Some Nonlinear Problems In Fluid Mechanics

Problems involving nonlinear differential equations usually come in the field of science and engineering. Finding the solution of these equations has great value in mathematics and strong background in applications. In general, these important subjects seem difficult to obtain analytic solutions. A method that bypasses the mathematical difficulties and also emphasizes those features of the solution is necessary. The homotopy analysis method(HAM) is an analytic tool for nonlinear problems that has been successfully applied to many nonlinear problems. Despite much progress made both theoretically and practically, it can not be state that a full understanding of HAM has been achieved. There is a need to present HAM to a wider field and more difficult problems.In this dissertation, a simple review on analytic techniques for nonlinear problems and a systematic description of HAM is given. The main contents and contributions of this dissertation may be summarized as followsFirst, an explicit analytic travelling wave solution for all possible wave speeds is given. Such kind of explicit series solutions has never been reported, to the best of author's knowledge. Also, the relationship between the steepness of the wavefront and the wave speed has been studied.Secondly, some typical flow and heat transfer problems are considered. In Chapter 3, the problem of the free convection boundary-layer flow over a semi-infinite vertical flat surface in a porous medium is investigated. This problem does not admit a similarity solution. By means of HAM, the first time, an analytic solution is given. The accuracy of this analytic solution is verified by numerical results. Chapter 4 is devoted to the boundary-layer flow of the incompressible fluid over a stretching surface. Two branches of solutions are found. Studies are performed to investigate the effects of several physical parameter. Using asymptotic analysis, the Prandtl number is embedded into the base function, and temperature profiles can be obtained in a large region of Prandtl number. In the following two chapters, two boundary-layer problems are studied mathematically. One is boundary-layer flow with algebraically decaying property. A different method for solving this problem is proposed and some remark on the mechanism of multiple solutions is given. The other is boundary-layer of a second grade fluid, which indicates that we have much larger freedom to solve nonlinear problems than we thought. The induced unsteady flow due to a stretching surface in...