On the Use of Recursive Evaluation of Derivatives and Padé Approximation to Solve the Blasius Problem

Asai Asaithambi

doi:10.1155/2016/3698251

Abstract

The Blasius problem is one of the well-known problems in fluid mechanics in the study of boundary layers. It is described by a third-order ordinary differential equation derived from the Navier-Stokes equation by a similarity transformation. Crocco and Wang independently transformed this third-order problem further into a second-order differential equation. Classical series solutions and their Padé approximants have been computed. These solutions however require extensive algebraic manipulations and significant computational effort. In this paper, we present a computational approach using algorithmic differentiation to obtain these series solutions. Our work produces results superior to those reported previously. Additionally, using increased precision in our calculations, we have been able to extend the usefulness of the method beyond limits where previous methods have failed.

Highlights

The boundary value problem described by f󸀠󸀠󸀠 (η) + β0f (η) f󸀠󸀠 (η) = 0, f (0) = f󸀠 (0) = 0, (1)f󸀠 (∞) = 1 is called the Blasius problem [1]
We develop a shooting method that successfully obtains Taylor series expansions of arbitrarily large orders by computing exact derivatives, not approximations to the derivatives, directly by using recursive formulas derived from the differential equation itself
We begin by letting y(0) = α and obtain the solution of (5)-(6) as Taylor series expansion of degree J

Summary

Introduction

The boundary value problem described by f󸀠󸀠󸀠 (η) + β0f (η) f󸀠󸀠 (η) = 0, f (0) = f󸀠 (0) = 0, (1). F󸀠 (∞) = 1 is called the Blasius problem [1]. It is one of the well-known problems in fluid mechanics in the study of boundary layers. Blasius [1], Howarth [2], and Asaithambi [3] provide direct analytical and numerical treatments of (1). With β0 = 1/2, Blasius [1] obtained the series solution f (η) = ∞ ∑ j=0 (− 1 2 j )

Methods

Results

Conclusion