Slipping flows and their breaking

TL;DR

This paper investigates the formation of singularities in slipping inviscid flows along rigid boundaries, analyzing both analytical criteria and numerical simulations to understand the conditions leading to flow breaking and gradient blow-up.

Contribution

It provides new analytical criteria for gradient catastrophe in 2D and 3D Prandtl flows and links these to numerical findings on velocity gradient growth in Euler flows.

Findings

Gradient catastrophe criteria are established for 2D and 3D flows.

Velocity gradients grow exponentially and double exponentially in time.

Flow folding is identified as the key process in gradient formation.

Abstract

The process of breaking of inviscid incompressible flows along a rigid body with slipping boundary conditions is studied. Such slipping flows are compressible, which is the main reason for the formation of a singularity for the gradient of the velocity component parallel to rigid border. Slipping flows are studied analytically in the framework of two- and three-dimensional inviscid Prandtl equations. Criteria for a gradient catastrophe are found in both cases. For 2D Prandtl equations breaking takes place both for the parallel velocity along the boundary and for the vorticity gradient. For three-dimensional Prandtl flows, breaking, i.e. the formation of a fold in a finite time, occurs for the symmetric part of the velocity gradient tensor, as well as for the antisymmetric part - vorticity. The problem of the formation of velocity gradients for flows between two parallel plates is studied numerically in the framework of two-dimensional Euler equations. It is shown that the maximum velocity gradient grows exponentially with time on a rigid boundary with a simultaneous increase in the vorticity gradient according to a double exponential law. Careful analysis shows that this process is nothing more than the folding, with a power-law relationship between the maximum velocity gradient and its width: $% \max|u_x|\propto \ell^{-2/3}$.