= Stability of an interior Hele-Shaw blob under injection and suction
{c}
{title2=$\dot a_n=-\frac{(n+1)Q}{2\pi hR^2}a_n$}
For viscous fluid inside a circular <fluid free boundary> between parallel plates of separation $h$, the depth-averaged <Hele-Shaw flow> satisfies $\bar u=-K\nabla p$, $K=h^2/(12\mu)$. With volume injection rate $Q$, $\bar u_r=Q/(2\pi hr)$ and $R^2=R_0^2+Qt/(\pi h)$. Neglecting <surface tension> and exterior viscosity, pressure is constant at the interface. Perturb its radius by $a_n\cos n\theta$ and solve the interior <Laplace equation> with a regular pressure term proportional to $r^n\cos n\theta$. The linearized pressure boundary condition gives $\delta p(R)=a_nQ/(2\pi hKR)$, hence velocity perturbation $-na_nQ/(2\pi hR^2)$. The displaced base velocity adds another $-a_nQ/(2\pi hR^2)$, proving the displayed growth law. Injection stabilizes the circular blob; suction destabilizes it. An exact radial solution can therefore exist while giving a fragile experimental prediction. These signs concern viscous fluid inside the blob, rather than the opposite geometry of a low-viscosity injected bubble.
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