Two short analytical stories: (1) Thin films, an edge, and a novel similarity solution and (2) A new look at ODEs and resonance
Speaker: Howard Stone, Princeton
Date: Monday, March 15, 2021, 3:45 p.m.
I discuss two recent projects where new analytical results are obtained in problems with classical features. In the first example, we document experimentally the time and (three-dimensional) space variations of the shape of a falling film near the edge of a vertical plate and rationalize the quantitative features using a similarity solution. This example seems unusual since we are able to theoretically show that the shape is described by a nonlinear partial differential equation, involving three independent variables, yet the equation can be reduced by a similarity transformation to a nonlinear ordinary differential equation. The results are in excellent agreement with the experimental measurements. Second, we present a new look at the classical question of obtaining solutions to linear ODEs forced at resonance, or the closely related problem where the linear operator has the equivalent of "repeated roots." We obtain a new analytical structure that is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. The ideas can be introduced at the undergraduate level, but we are not aware of any elementary or advanced text that illustrates these ideas with appropriate generality.