= Linear pinch-off scaling of a slender viscous bubble
{title2=$a=(\gamma/\mu)\tau A(\zeta),\quad\zeta=\mu\sqrt\lambda(z-z_*)/(\gamma\tau)$}
For $\tau=t_*-t$, balance $\mu a_t/a\sim\gamma/a$ to obtain radius proportional to $\tau$. Axial lubrication conservation then makes the axial length proportional to $\tau$ as well. The displayed variables reduce the <lubrication equation for a bubble with viscous exterior> to $\zeta AA'-A^2=\tfrac1{16}\{A^4[(2\zeta A'+1)/A]'\}'$. The physical interface slope is $\sqrt\lambda A'$, compatible with a slender neck when $\lambda\ll1$ and the dimensionless profile varies on order-one scales.
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