Viscous fluid flow in a layer with a free boundary

A partially invariant solution of a three-dimensional problem with a free boundary for the Navier–Stokes equations is studied. The flow domain under consideration is a horizontal layer bounded by a solid plane from below and by a flat free surface from above. The vertical velocity and pressure are independent of the $x$ and $y$ coordinates. Three flow modes can be formed for different initial velocities of the flow: stabilization to the quiescent state with time, solution blow up within a finite time, and self-similar regime in which the layer thickness unboundedly increases with time.