Nonlinear Dynamics of Slipping Flows

Abstract

We develop a new concept for the formation of behavior features of inviscid incompressible fluids on the rigid boundary due to breaking of slipping flows. The breaking possibility is related to the compressibility of such flows due to the boundary. For two- and three-dimensional inviscid Prandtl equations, we analytically obtain the criteria for a gradient catastrophe for slipping flows. For the two-dimensional Prandtl equations, breaking occurs for both the velocity component parallel to the boundary and the vorticity gradient. The explosive growth of the vorticity gradient correlates with the appearance of a jet in the direction perpendicular to the boundary. For the three-dimensional Prandtl flows, breaking (fold formation) leads to an explosive growth for both the symmetric part of the velocity-gradient tensor and its antisymmetric part, i.e., vorticity. The blow-up generation of vorticity is possible due to the fluid suction from the slipping flow with simultaneous formation of a jet perpendicular to the boundary. These factors can be considered as a tornado-formation mechanism. Within the framework of the two-dimensional Euler equations, we numerically study the problem of the formation of increasing velocity gradients for the flows between two parallel plates. It is revealed that on the rigid boundary, the maximum velocity gradient exponentially increases with time simultaneously with an increase in the vorticity gradient according to the double exponential law. This process is also accompanied by a jet formation in the direction perpendicular to the boundary.