Past exam of the theoretical mechanics course of the University of Oxford / 2007 / A / 3 / 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 $U{h}^2/(\nu L)$, 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 givesDepth-integrated continuity has no normal boundary flux, so $\nabla \cdot \bar{u}=0$ and $\boxed{{\nabla }^{2}p=0}$ away from the injection point.

Neglecting surface tension, the fluid free boundary has $p=p_{\mathrm{ext}}$, 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 $2\pi rh{\bar{u}}_r=Q$. HenceThe 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 $b_n{r}^n\cos n\theta$. Linearizing the constant-pressure condition gives $\delta p(R)=-a_n{p}_0^{\prime }(R)=a_{n}Q/(2\pi hKR)$. The perturbation radial velocity is then $-n{a}_{n}Q/(2\pi h{R}^2)$, and evaluation of the base velocity at the displaced boundary adds $-a_{n}Q/(2\pi h{R}^2)$. ThusModes 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.