Some solutions of continuum equations for an incompressible viscous fluid

We consider the Navier–Stokes equations for an incompressible fluid that at any specific instant $t\ge 0$ fills an open axially symmetric cylindric layer $D$. We find solutions of these equations in the class of motions described by velocity fields whose lines for $t\ge 0$ coincide with their vortex lines and lie on axially symmetric cylindric surfaces in $D$.