Analytical Solution of the Problem on the Collapse of an Attached Cavity after Cavitation Impact of a Circular Disk

We consider the axisymmetric problem of the vertical separation impact of a circular disk that hermetically closes the bottom of a pool in the form of a layer. After the impact, the disk moves along the gravity vector (outside the layer) at a constant speed. It is assumed that the disk slides along the solid cylindrical walls like a piston. A feature of this problem is that after the impact, an attached cavity is formed and a new internal free boundary of the fluid appears. It is required to study the process of collapse of the cavity at low velocities of the disk that correspond to small Froude numbers. In the leading asymptotic approximation, a problem with one-sided constraints is formulated on the basis of which the dynamics of the separation line is determined and the process of collapse of the cavity is described taking into account the rise of the internal free boundary of the liquid. By separation of variables in cylindrical coordinates and the technique of paired integral equations, this problem is reduced to a coupled nonlinear problem that includes a transcendental equation for the radius of the circular separation line and a Fredholm integral equation of the second kind with a smooth kernel. For a large layer thickness, a good agreement of analytical results with direct numerical calculations is shown.