Axisymmetric inviscid flow between moving parallel plates

For plates $z=\pm h(t)$, incompressibility, impermeability and regularity on the axis determine a z-independent radial velocity $u_r=-r\dot{h}/(2h)$ and vertical velocity $u_z=z\dot{h}/h$. A possible azimuthal velocity requires additional rotational data; continuity alone does not determine it. For a smooth regular initial profile $U(r)$ independent of $z$, its azimuthal Euler equation transports it as $u_{\theta }(r,t)=\lambda U(\lambda r)$, where $\lambda =\sqrt{h/h_0}$. If the rotation is solid-body with angular velocity $\Omega (t)$, the vorticity equation gives $\Omega ={\Omega }_{0}h/h_0$. Along a particle path, $r^{2}h$ and $r^2\Omega$ are constant, illustrating conservation of angular momentum.