Question #92471

A large cylindrical tank of cross-sectional area $A$ contains a transparent liquid of density $\rho$ and refractive index $\mu$. At the bottom of the tank, there is a small drainage hole of area $a$ ($a \ll A$) through which the liquid drains out freely under gravity. Also at the bottom of the tank, there is a circular window of radius $r$ covered by a thin, perfectly transparent, elastic membrane with a constant surface tension $\gamma$. The liquid bulges out through this window, forming a plano-convex liquid lens. The pressure outside the tank is atmospheric pressure $P_0$. Assume the membrane's shape is always a spherical cap and its tension $\gamma$ remains constant. Find the magnitude of the rate of change of the focal length of this liquid lens, $\left| \frac{df}{dt} \right|$, as a function of its instantaneous focal length $f$.