The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Abstract

The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semi-infinite flat plate. The Blasius function is the solution to $2 f_{xxx} + f f_{xx} = 0$ on $x \in [0, \infty]$ subject to $f(0)=f_{x}(0)=0, f_{x}(\infty) = 1$. We use this famous problem to illustrate several themes. First, although the flow solves a nonlinear partial differential equation (PDE), Toepfer successfully computed highly accurate numerical solutions in 1912. His secret was to combine a Runge-Kutta method for integrating an ordinary differential equation (ODE) initial value problem with some symmetry principles and similarity reductions, which collapse the PDE system to the ODE shown above. This shows that PDE numerical studies were possible even in the precomputer age. The truth, both a hundred years ago and now, is that mathematical theorems and insights are an arithmurgist's best friend, and they can vastly reduce the computational burden. Second, we show that special tricks, applicable only to a given problem, can be as useful as the broad, general methods that are the fabric of most applied mathematics courses: the importance of “particularity.” In spite of these triumphs, many properties of the Blasius function $f(x)$ are unknown. We give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician.