An Exact Analytic Solution for the Sakiadis Boundary Layer

We examine the classical problem of the Sakiadis Boundary Layer. Though a power series solution can be found, it has finite--and qualitatively small--radius of convergence, while the problem's domain is infinite. Barlow et al [Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] developed a solution combining an asymptotic series around infinity with the Taylor series via the method of asymptotic approximants. While this series solution was shown to converge over the physical domain, its representation as an exact solution to the ODE was not demonstrated. We demonstrate that the asymptotic series may be viewed as a Taylor series analytic on a disc larger than one containing the domain after suitable transformation of the relevant ODE. Furthermore, we provide explicit recursions that allow efficient computation of higher-order coefficients. We demonstrate that this exact expression converges uniformly, with the infinity norm of the error on the whole domain decreasing exponentially with the number of terms. Moreover, we verify the location of the singularity limiting the radius of convergence for the untransformed problem. We discuss conditions under which this methodology may be adapted to similar problems.