Application Of Homotopy Analysis Method In Fluid Mechanics And Time-delay Dynamics System

Non-linearity is encountered in many areas of mathematics, physics, engi-neering, science, even social science. The search for methods to solve non-linearproblems is extremely important. In 1992, Prof. Liao proposed the HomotopyAnalysis Method (HAM) for solving non-linear equations using a series solution,based on the concept of'homotopy'taken from algebraic topography. In imple-menting HAM, zero- and higher-order deformation equations are constructed ofthe original non-linear problem, which is then transformed into a series of linearsub-problems. HAM has the notable feature that it does not require the per-tubation parameter to be small, and so applies not only to weakly non-linearproblems, but also to strongly non-linear problems. The method provides con-siderable freedom of choice in the selection of the perturbation parameter itself,according to the problem under consideration. Prof Liao and his co-workers haveundertaken a great deal of pioneering work in the application of HAM to physicsand engineering. In recent years, many researchers worldwide have recognised thepotential of HAM for solving the non-linear di?erential equations encountered indi?erent ?elds of application. At present, researchers are trying to develop newnumerical methods for solving non-linear problems that incorporate the HAMmethodology.The present thesis describes the use of HAM to solve several signi?cantproblems in science and engineering. These include fundamental problems in ?uiddynamics, the vibrations of o?shore structures, Newtonian and non-Newtonianboundary-layer ?ows, unsteady non-linear heat transfer, and strongly non-lineartime-delay dynamical systems. In the latter case, multiply periodic motions canoccur due to Hopf bifurcation.Chapters 2 and 3 considers the application of homotopy analysis to solve non-similar boundary-layer ?ows of Newtonian and non-Newtonian ?uids. In1908, Blasius used a similarity transformation to convert the partial di?eren-tial equations for boundary-layer ?ow into ordinary di?erential equations, andproduced a semi-exact solution for the boundary-layer ?ow over a ?at plate. Al-though the similarity solution can prove simple and e?ective in certain cases, inthe majority of applications where the thermodynamic processes are non-similar,the boundary-layer is also non-similar. In 1968, Chen and Libby carried outan extensive study of such a boundary layer for ?ow over a wedge, and foundthat the resulting ?ow could be characterized by a pair of Falkner-Skan typeself-similar boundary-layers. For cases where the boundary layer is no longerself-similar, it will be shown in the present thesis that the boundary-layer partialdi?erential equations can still be solved by the homotopy analysis method, oncea non-similarity transformation has been applied.The second major theme is the solution of various coupled or unsteady ?owand heat transfer problems using HAM. Chapter 4 considers unsteady heat trans-fer in flow started impulsively from rest, along a symmetric wedge. In this case,due to the non-linearity transformation of the boundary-layer approximation,HAM is applied to the resulting partial di?erential equations and boundary con-ditions. The analysis provides coherent and e?ective series solutions in bothtime and space, leading to explicit formulae for the local friction coe?cient. Theanalytical results are compared against alternative numerical model predictions.Chapter 5 describes the Optimal Homotopy Analysis Method (OHAM),which improves computational e?ciency when applied to complicated partialdi?erential equations, such as the boundary-layer equations. The cases consid-ered in Chapters 2 to 5 are recomputed using OHAM and the results assessed interms of relative accuracy and computational e?ciency.Finally, Chapter 6 describes the application of HAM to the time-delay vi-bration of o?shore structures, in particular cases involving strong time-delayresponses where multiply periodic motions can arise due to Hopf bifurcation.In summary, the present thesis focuses on solving partial di?erential equa-tions encountered in the ?uid dynamics of non-similar boundary-layer ?ows, un-steady boundary-layer ?ows, and the vibrations that arise in time-delay dynam-ical systems. It is found that HAM has a broad scope of application; HAM is a universal and e?ective tool for solving non-linear problems, and is potentiallyvery useful in the ?elds of engineering and science.