Forgotten Merits of the Analytic Viewpoint

Abstract

The early twentieth‐century German scientist Ludwig Prandtl solved a problem that at first sight seems impossible to solve without a computer: What are the shear forces exerted on the wing of an airplane during flight? The governing equations for this problem are the Navier‐Stokes equations, nonlinear partial differential equations for which it is not possible to obtain a general analytic solution. Prandtl solved the problem through strong physical insight, by realizing that frictional effects of the air on the wing are localized within a thin boundary layer, with airflow far from the wing behaving as if it were an ideal fluid, that is, one of zero viscosity. By separating the solution domain into a boundary layer and a far field, he was able to simplify the equations that describe the global domain enough to solve them for the case he needed. Even today, many difficult problems can be usefully approached this way, such as problems of buoyant convection adjacent to vertical walls. For a good review of these and other successes of boundary layer theory, see Bejan [2004].