The Painlevé paradox in three dimensions: resolution with regularization

The classical Painlevé paradox consists of a slender rigid rod slipping on a rigid rough surface. If the coefficient of friction μ is high enough, the governing equations predict that the rod would be driven into the surface. The paradox is well studied in two dimensions, in which the paradox is resolved via regularization, where the rod tip meets the surface. In this paper, we consider the three-dimensional problem. There are two significant differences in three dimensions. Firstly, sticking now occurs on a co-dimension 2 surface. This results in a non-smooth problem, even when the three-dimensional problem is regularized. Secondly, unlike the highly singular two-dimensional problem, trajectories can now enter the inconsistent region from slipping, requiring a completely new analysis. We use blowup to investigate the problem and show that a key part of the dynamics of the regularized three-dimensional Painlevé problem is governed by a type I Painlevé equation.