On the asymptotic solution of a high–order nonlinear ordinary differential equation

We examine Berman's similarity solution for steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls. We convert the two–point boundary–value problem for the similarity function into an initial–value problem in which the position of the upper channel wall and the Reynolds number are determined as part of the solution, and consider two cases. In the first case equal suction or injection takes place at each wall and solutions are sought that are symmetrical about the mid–line of the channel. In the second case fluid can pass through only the upper channel wall. In each case analysis of a single initial–value problem yields two distinct solutions, one corresponding to injection, the other to suction, and our analysis is valid in the limit of large suction or injection. The injection solutions are well known. For the first (symmetrical) case the suction solution has previously been given only incompletely, while for the second case the asymptotic structure of the solution for large suction has been an open problem. We complement our asymptotics with numerical integrations of the ordinary differential equation governing the similarity function and find excellent agreement between the two.