Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 65 / 4 / a / Solution

Let $h_0=h(0)$ and $S=\pi R^2$. The closed tank has constant mass, soTherefore $({\rho }_l-{\rho }_u)h=({\rho }_l-{\rho }_u(0))h_0$. Under the Boussinesq approximation, define $g^{\prime }=g({\rho }_l-{\rho }_u)/{\rho }_l$ and obtainThe gravitational potential energy is especially easy to evaluate with the stipulated downward depth coordinate: the gravitational potential per unit mass is $-gz$. HenceThis proves the potential energy of two-layer entrainment relationThe positive sign expresses the lifting and homogenization of dense entrained fluid within the upper layer. The fixed thin interface is neglected in this leading two-layer energy integral.