Study Of Several Type Of Boundary Layer Problems Based On The Homotopy Analysis Method

In fluid mechanics,the boundary-layer flow is one of the most important topics.It(?)greatly simplifies the mathematical problems in fluid motion,and provides the possibility for theoretical study.Navier-stokes equation is the basic equation of viscous fluid movement,and the boundary layer problem is based on the N-S equation with the boundary layer approximation.In the view of mathematics,both N-S equation and the boundary-layer problems can be classified as the problems of solving nonlinear differential equations.How to solve these problems effectively is a huge challenge for the theoretical researchers.Liao proposed the homotopy analysis method(HAM)to solve the complex nonlinear problems and the practical problems.HAM is a solving method which don't depend on any physical parameters and base on the theory of homotopy topology.The main idea of this method is that the original nonlinear problem transforms into infinite number of linear problems by introducing the so-called embedded variable q?[0,1].Through continuous development and improvement on homotopy analysis method,it has formed a relatively complete theoretical framework.In order to get more facts about fluid characteristic and expand the practical applications of the homotopy analysis method,in this paper,the homotopy analysis method and the improved homotopy analysis method are applied for solving mathematical equations which describe characteristics of the fluid flow.It also analyzes the problems of boundary-layer flow and the problems of heat and mass transfer in a nanofluid.Firstly,the problem of unsteady stagnation-point flow towards permeable stretching sheet in a porous medium is investigated.Using a similarity transformation,the partial differential equations which are based on the movement laws of fluids are transformed into a system of nonlinear ordinary differential equations.In the end,we used the homotopy analysis method to solve these equations.Secondly,we study the problem of magnetohydrodynamic(MHD)mixed convection slip flow near a stagnation-point on a nonlinearly vertical stretching sheet.The resulting system is solved by the homotopy analysis method which contains a rational basis function and obtain the similar solution.And graphics is used to describe the flow and heat transfer process.Finally,the nanofluid,as a new subject in recent years,has attracted many researchers'interests for its potential prospects of applications.we deal with the problem of stagnation-point flow of a nanofluid with chemical reaction past a permeable cylinder subjected to the surface heat and mass fluxes.Based on the characteristic of nanofluid,the mainly work is talking about the heat transfer of fluid and the effects of the key physical parameters on flow.In the study of these nonlinear problems,through the experiment to choose the appropriate basis function,initial guess solution,linear operator and auxiliary parameter,homotopy analysis method is successfully applied to solve the above problems.In order to verify the validity of the homotopy analysis method,we compared the series solutions of high accuracy with its numerical solutions,the results are in good agreement.