| The boundary-layer flow is an important topic of fluid mechanics for its wide ap-plications in science and engineering. Though it has been developed over one hundredyears, many researchers have been invested in this field for all kinds of new applica-tions. On the other hand, the nanofluid, as a new subject, has attracted more and moreresearchers’ interests for its potential prospects of applications. Both the boundary-layer flow problems and the nanofluid flow problems, in the view of computation, canbe classified as solving certain nonlinear differential equations. How to effectivelysolving these problems is a huge challenge for theoretical researchers.The study of analytical solutions to nonlinear differential equations plays a veryimportant role in penetrating the inner structure, in analyzing the relationship of thingsas well as in interpreting various physical phenomena. However, for most nonlinearproblems, it’s difficult to get the analytical solutions, in this case, the analytical ap-proximations of nonlinear problems play an important role to help us to understandthe behaviors those physical problems. The Homotopy Analysis Method (HAM) isan effective approach to get analytical approximate solutions for nonlinear equations.Although great progress has been done, the method still need to develop and improvefurther.In this dissertation, some new problems of boundary layer and nanofluids field-s with potential applications in science and engineering have been studied by meansof HAM. The content is composed of three aspects, they are, respectively, bound-ary layer flow problems, nanofluids flow problem and the optimization calculation ofHAM. For boundary layer flow, we firstly investigate the unsteady stagnation flow andheat transfer towards a stretching or shrinking sheet. Then we study a steady con-vection thermal boundary-layer flow over a semi-infinite permeable flat plate driven by a power-law shear with asymptotic velocity profile and provide the valid existenceregions and conditions for multiple solutions of temperature distribution for the firsttime. For nanofluids flow, firstly we study the steady, three dimensional mixed convec-tion flow of a nanofluid past a stretching sheet in the presence of an applied magneticfield. Then we study the fully developed mixed convection flow of vertical or hori-zontal channel filled with nanofluids. We offer the explicit approximate formula ofsome important physical quantities in these problems for the first time, such as theskin friction coefficient, the Nusselt number, which is better than the traditional lin-ear regress approximation formula. For the optimization calculation of HAM, takethe Bonhoeffer-van der Pol model as an example, we define some intermediate vari-ables to avoid the repeated computation and introduce a new approximation skill, thecomputational efficiency is improved greatly.Through studies on these nonlinear problems, we provide highly precision analyt-ical approximate solutions, which are uniformly valid in the whole region and furtherverified the effectiveness of the HAM. In addition, according to the features of thesenonlinear problems, we give some advanced techniques to calculate the equation er-rors and determine all sorts of auxiliary parameters in HAM, which further improvethe HAM.The present work shows the capability of HAM for solving nonlinear problemsin new fields and exhibits superior features of HAM, such as independent on smallparameters, convenient to control the convergence of approximation series, wide ap-plication range, high flexibility and so on. It is expected that the HAM could be appliedto resolve nonlinear problems in new fields of science and engineering. |
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