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 fluid mechanics that are mathematically related by their large-distance asymptotic behaviours in semi-infinite domains. The first problem is ‘Sakiadis’ boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. second static air–liquid meniscus with surface tension intersects wall at given contact angle and limits to pool away from wall. For latter problem, solution—given as distance function height—has long known (Batchelor, G. K. (1967) An Introduction Fluid Dynamics, chapter 1: physical properties fluids. Cambridge). Here, we provide an explicit height versus elucidate structural similarities Sakiadis layer. Although readily obtainable governing equations, show that—in both problems—the diverge due non-physical complex or negative real-valued singularities. In cases, these singularities can be moved expanding exponential gauge functions motivated respective large enable convergence full this not only provides convergent Taylor (and conjectured exact) ODE, but also means evaluate shear parameter other properties) within any desired precision. nature ODEs precludes general conclusions, our results indicate useful when proposing variable transformations overcome divergence.