= 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 $\Omega=\Omega_0h/h_0$. Along a particle path, $r^2h$ and $r^2\Omega$ are constant, illustrating <conservation of angular momentum>.
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