Question #42132

A cylindrical tank of constant cross-sectional area $A_1$ contains an ideal, incompressible liquid of density $\rho$. At the bottom of the tank is a circular orifice made of a material with a coefficient of linear thermal expansion $\alpha$. The tank walls have zero thermal expansion. Initially ($t=0$), the liquid is at a height $H$ above the orifice, and the orifice has an area $A_0$ such that $A_1 = 2A_0$. The system is heated such that the temperature of the orifice increases at a constant rate $\frac{dT}{dt} = c$. Assuming the liquid density remains constant, find the initial rate of change of the efflux velocity of the liquid, $\left. \frac{dv_2}{dt} \right|_{t=0}$.