Vibrational flows of viscous incompressible fluids for high Reinolds numbers

The article presents the high-frequency asymptotics of the Navier–Stokes system, which describes the motion of a viscous incompressible fluid in the region bounded by a vibrating surface. The boundary conditions require the coincidence of the velocity vectors of the material particle of the fluid and the point of the boundary in which the particle is located. Consequently, the fluid is not allowed either to slip along the boundary (the no-slip condition) or to penetrate through it. It is assumed that the motion of the boundary surface is given and periodic in time, and the domain confined within it stays at rest on average but, generally speaking, can be changing its shape. The frequency of oscillations of the boundary tends to infinity, and the amplitude tends to zero, but the ratio of the amplitude to the Stokes's layer thickness remains of the order of unity. The main result is the explicit form of the equations and boundary conditions that determine the mean flow in the most general case, without special assumptions about the problem data. On this basis, a number of specific flows have been investigated, in particular, a flow in a circular pipe, caused by the normal vibration of its walls.