Analytical approximation of Blasius' similarity solution with rigorous error bounds

Abstract

We use a recently developed method \cite{Costinetal}, \cite{Dubrovin} to find accurate analytic approximations with rigorous error bounds for the classic similarity solution of Blasius of the boundary layer equation in fluid mechanics, the two point boundary value problem $f^{\prime \prime \prime} + f f^{\prime \prime} =0$ with $f(0)=f^\prime (0)=0$ and $\lim_{x \rightarrow \infty} f^\prime (x) =1$. The approximation is given in terms of a polynomial in $[0, \frac{5}{2}]$ and in terms of the error function in $[\frac{5}{2}, \infty)$. The two representations for the solution in different domains match at $x=\frac{5}{2}$ determining all free parameters in the problem, in particular $f^{\prime \prime} (0) =0.469600 \pm 0.000022 $ at the wall The method can in principle provide approximations to any desired accuracy for this or wide classes of linear or nonlinear differential equations with initial or boundary value conditions. The analysis relies on controlling the errors in the approximation through contraction mapping arguments, using energy bounds for the Green's function of the linearized problem.