Series Solutions Of Non-similarity Boundary-layer Flows By Means Of The Homotopy Analysis Method

Although the boundary-layer ?ows has been a topic of scientific research during thepast century, still in recent years it has received much attention due to its appli-cations in many fields of science. The study of boundary-layer ?ow along surfacesembedded in ?uid saturated porous media has received considerable interest recently,especially in the enhanced recovery of petroleum resources, packed bed reactors andgeothermal industries. The boundary-layer behavior over a moving continuous solidsurface is an important type of ?ow occurring in several engineering processes. Forexample, the thermal processing of sheet-like materials is a necessary operation inthe production of paper, metal spinning, roofing shingles, and insulating materials.Investigations of the boundary-layer ?ows of viscous ?uids due to a stretching sheethave been the interest of many researchers owing to its important applications inchemical and metallurgical industries, such as, continuous stretching of plastic filmsand artificial fibers, cooling process of metallic plate in a cooling bath, glass andpolymer industries, and metal extrusion. Due to study of heat and mass transfer inmoving ?uids, the applications of boundary-layers extended to di?erent engineeringbranches. Examples include boundary layer control on airfoils, lubrication of ceramicmachine parts, food processing, the extraction of geothermal energy, nuclear reactorcooling system, filtration process.In spite of the many applications of boundary-layer ?ows in di?erent fields ofscience, less attention has been given to the non-similarity solutions. Early stud-ies were focused on seeking similarity solutions because the similarity variables cangive great physical insight with minimal e?orts. For similarity boundary-layer ?ows, velocity profiles at di?erent x are similar. However, such kind of similarity is lostfor non-similarity ?ows. Obviously, the non-similarity boundary-layer ?ows are moregeneral in nature and more important not only in the theory but also in applications.In the investigation of non-similarity boundary layer ?ows numerical methods arewidely applied. However, by using numerical methods some additional errors anduncertainty can be found in the results. For example, for numerical computation onehad to replace the infinite domain with finite one. The partial di?erential equations(PDEs) can be solved in the infinite domain by using the analytic methods. But thetraditional analytic techniques depend on the small/large physical parameters. Thus,it is still necessary to develop new analytic method for non-similarity boundary-layer?ows.The homotopy analysis method (HAM) is a general analytic approach to get seriessolutions of various types of nonlinear equations, including algebraic equations, ordi-nary di?erential equations, partial di?erential equations, di?erential-integral equa-tions, di?erential-di?erence equations, and coupled equations of them. Di?erentfrom all perturbation and non-perturbation methods, the HAM is independent ofsmall/large physical parameters. Besides, it provides us a simple way to ensure theconvergence of solution series and great freedom to choose base functions to approx-imate a nonlinear problem. Therefore, the HAM is valid even for strongly nonlinearproblems. Researchers have successfully applied this method to various nonlinearproblems in science and engineering. Motivated by these facts, series solutions ofnon-similarity boundary-layer ?ows by means of the homotopy analysis method istaken into account in this dissertation.The objective of Chapter 1 is to provide background for constitutive equations ofboundary-layer ?ows and the homotopy analysis method.The aim of Chapter 2 is to provide the series solutions of the non-similarityboundary-layer ?ows over a porous wedge by means of homotopy analysis method.Besides, the so-called homotopy-Pade′technique is used to accelerate the convergence. The e?ect of the physical parameters on the skin friction coe?cient and the displace-ment thickness is investigated.Chapter 3 concerns the analytic solutions for the unsteady non-similarity boundary-layer ?ows caused by stretching ?at sheets. The governing partial di?erential equa-tions have been solved analytically by means of homotopy analysis method. Theconvergent series solution uniformly valid for all dimensionless time in the wholespatial region 0≤x <∞and 0≤y <∞are obtained.In Chapter 4, an analysis is performed to the ?ow and heat transfer of steadytwo-dimensional laminar boundary-layer ?ow from a permeable vertical surface in anisothermal surroundings. The analytic solutions obtained by HAM are then repre-sented in the form of the local skin friction coe?cient and the local Nusselt numberfor di?erent physical parameters.Conclusion are given in Chapter 5.