On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs

Abstract

We examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviors in semi-infinite domains. The first problem is that of the "Sakiadis" boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air-liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution -- given as distance from the wall as function of meniscus height -- has long been known (Batchelor, 1967). Here, we provide an explicit solution as meniscus height vs. distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that -- in both problems -- the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviors to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviors can be useful when proposing variable transformations to overcome power series divergence.