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Revisiting Blasius Flow by Fixed Point Method

DOI: 10.1155/2014/953151

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Abstract:

The well-known Blasius flow is governed by a third-order nonlinear ordinary differential equation with two-point boundary value. Specially, one of the boundary conditions is asymptotically assigned on the first derivative at infinity, which is the main challenge on handling this problem. Through introducing two transformations not only for independent variable bur also for function, the difficulty originated from the semi-infinite interval and asymptotic boundary condition is overcome. The deduced nonlinear differential equation is subsequently investigated with the fixed point method, so the original complex nonlinear equation is replaced by a series of integrable linear equations. Meanwhile, in order to improve the convergence and stability of iteration procedure, a sequence of relaxation factors is introduced in the framework of fixed point method and determined by the steepest descent seeking algorithm in a convenient manner. 1. Introduction The Navier-Stokes equations are the fundamental governing equations of fluid flow. Usually, this set of nonlinear partial differential equations has no general solution, and analytical solutions are very rare only for some simple fluid flows. However, in some certain flows, the Navier-Stokes equations may be reduced to a set of nonlinear ordinary differential equations under a similarity transform [1, 2]. These similarity solutions could not only provide some physical significance to the complex Navier-Stokes equations but also act as a benchmarking for numerical method. The well-known Blasius flow [3–5] is possibly the simplest example among these similarity solutions. It describes the idealized incompressible laminar flow past an semi-infinite flat plate at high Reynolds numbers, which is mathematically a third-order nonlinear two-point boundary value problem: subject to the boundary conditions: where the prime denotes differentiation to the variable and is the nondimensional stream function related to the stream function as follows: is the similarity variable, where is the free stream velocity, is the kinematic viscosity coefficient, and and are the two independent coordinates. The two velocity components are then determined: According to (1) and (2) the solution is defined on a semi-infinite interval , and one of the boundary conditions is asymptotically assigned on the first derivative of function at infinity, which are the main challenges on solving the Blasius flow. The solution to this problem has the following asymptotic property [6, 7]: where is a constant and the benchmarking value provided by Boyd

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