= Solution
Scale lateral distance with $L$, transverse distance with $h$, lateral <velocity> with $U$, and the normal <velocity> with $Uh/L$. Balancing lateral <pressure> force with transverse viscous stress gives <pressure> scale $\mu UL/h^2$. The ratio of inertia to that stress is $Uh^2/(\nu L)$, while lateral viscous <derivatives> are smaller by $(h/L)^2$. The two stated smallness conditions therefore reduce the lateral equations to $\mu u_{zz}=p_x$, $\mu v_{zz}=p_y$. The normal component of <balance of linear momentum> gives $p_z=0$ at leading order; continuity remains $u_x+v_y+w_z=0$. The no-slip, impermeable rigid plates require $u=v=w=0$ at both plate surfaces.
Integrating with those boundary values gives
$$
(u,v)=-\frac{z(h-z)}{2\mu}\nabla p,\qquad
\boxed{\bar u=-K\nabla p,\quad K=\frac{h^2}{12\mu}.}
$$
Depth-integrated continuity has no normal boundary flux, so $\nabla\cdot\bar u=0$ and $\boxed{\nabla^2p=0}$ away from the injection point.
Neglecting <surface tension>, the <fluid free boundary> has $p=p_{\rm ext}$, and its normal <velocity> is the fluid <velocity>: $\dot R=\bar u_r(R)$. Any constant <capillary pressure> for a circular interface can instead be included in that spatially uniform boundary <pressure>. Conservation of injected volume gives $2\pi rh\bar u_r=Q$. Hence
$$
\boxed{\bar u_r=\frac{Q}{2\pi hr},\quad
R^2=R_0^2+\frac{Qt}{\pi h},\quad
p(r,t)=p_{\rm ext}+\frac{6\mu Q}{\pi h^3}\log\frac{R(t)}r.}
$$
The logarithmic <pressure> singularity represents the ideal <point source>, which in a physical cell is replaced by a finite inlet. The formula is an exact radial solution of the depth-averaged model for either sign of $Q$, while $R^2>0$.
Its robustness differs sharply between the signs. With viscous fluid inside a much less viscous exterior, injection displaces the less viscous phase and is stabilizing; suction draws the exterior into the viscous region and is prone to fingering. One can see this directly without relying on a slogan. Write the boundary as $R+a_n\cos n\theta$ and its regular <pressure> <perturbation> as $b_nr^n\cos n\theta$. Linearizing the constant-<pressure> condition gives $\delta p(R)=-a_np_0'(R)=a_nQ/(2\pi hKR)$. The <perturbation> radial <velocity> is then $-na_nQ/(2\pi hR^2)$, and evaluation of the base <velocity> at the displaced boundary adds $-a_nQ/(2\pi hR^2)$. Thus
$$
\boxed{\dot a_n=-\frac{(n+1)Q}{2\pi hR^2}a_n.}
$$
Modes decay for injection and grow for suction. Accordingly the suction circle is a mathematically possible symmetric solution but generally an unstable prediction, failing before its formal collapse in a perturbed experiment. This is <Stability of an interior Hele-Shaw blob under injection and suction>.
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