= Solution
For constant-density incompressible inviscid flow with conservative body force, <Euler equations for an inviscid fluid> give $\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u=-\nabla(p/\rho+\Phi)$. Using $(\mathbf u\cdot\nabla)\mathbf u=\nabla(|\mathbf u|^2/2)-\mathbf u\times\boldsymbol\omega$, take the <curl> to obtain $\partial_t\boldsymbol\omega=\nabla\times(\mathbf u\times\boldsymbol\omega)$. Since both divergences vanish, the vector identity expands this to
$$
\boxed{\frac{D\boldsymbol\omega}{Dt}=(\boldsymbol\omega\cdot\nabla)\mathbf u.}
$$
Without incompressibility or the constant-density/barotropic hypothesis, extra compression or baroclinic terms are present; the printed equation uses these flow assumptions.
Let $\kappa(t)=\dot h/h$, with $h>0$. Axisymmetric <mass conservation> is $r^{-1}\partial_r(ru_r)+\partial_zu_z=0$. If $u_r$ is independent of $z$, then $u_z$ is affine in $z$. The impermeable moving-plate conditions $u_z(r,\pm h,t)=\pm\dot h$ force $u_z=\kappa z$, and integrating the radial equation gives $u_r=-\kappa r/2+C(t)/r$. Regularity at the axis excludes $C(t)/r$. The meridional part of the <axisymmetric inviscid flow between moving parallel plates> is therefore
$$
\boxed{u_r=-\frac{\dot h}{2h}r,\qquad u_z=\frac{\dot h}{h}z.}
$$
For no swirl, $u_\theta=0$ completes the field. Continuity alone does not determine azimuthal motion; this must be specified separately. More generally, prescribe a smooth regular initial profile $U(r)$ independent of $z$. The azimuthal part of <Euler equations for an inviscid fluid> is
$$
\partial_tu_\theta-\frac{\kappa r}{2}\partial_ru_\theta-\frac\kappa2u_\theta=0,
$$
so conservation of $r u_\theta$ along radial particle paths gives $u_\theta(r,t)=\lambda U(\lambda r)$, where $\lambda=\sqrt{h/h_0}$. Together with the meridional components above, this gives a family of complete fields, with a suitable <pressure> determined by the radial and vertical equations. The field is therefore not unique without rotational data.
With the stated solid-body rotation, $u_\theta=\Omega(t)r$ and $\boldsymbol\omega=2\Omega\mathbf e_z$. The <vorticity equation> gives $2\dot\Omega=2\Omega\partial_zu_z=2\Omega\kappa$, hence
$$
\boxed{\Omega(t)=\Omega_0\frac{h(t)}{h_0},\qquad
\mathbf u=-\frac{\dot h}{2h}r\,\mathbf e_r+\Omega_0\frac h{h_0}r\,\mathbf e_\theta+\frac{\dot h}{h}z\,\mathbf e_z.}
$$
For a fluid particle, $\dot r=-\kappa r/2$, so $r^2h$ is constant and $r^2\Omega$ is constant. This is <conservation of angular momentum> about the axis: radial spreading when the plates approach decreases the angular velocity. The meridional and rotating fields satisfy the inviscid equations with an appropriate pressure; no tangential no-slip condition is imposed on inviscid plates.
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