Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 1 / a / ii

The interface is material: fluid particles on $F=z-\eta (x,t)=0$ remain there. Its material derivative therefore vanishes, giving $w_j={\eta }_t+u_j{\eta }_x=D_j\eta /Dt$ on the interface. Since $w_j={\partial }_z{\varphi }_j$, this is the kinematic boundary condition for a free-surface graph on each side. The material derivatives use the respective tangential fluid velocities; they enforce the same moving normal boundary.

The Unsteady Bernoulli equation gives $p_j+{\rho }_j[{\partial }_t{\varphi }_j+|\nabla {\varphi }_j{|}^2/2+gz]=C_j(t)$. Absorb the spatially constant functions $C_j$ into the gauges of the velocity potentials. Evaluating at $z=\eta$ and subtracting the equations yields the stated dynamic boundary condition for an inviscid interface. Surface tension supplies the pressure jump, rather than pressure continuity. Thus the first condition enforces material motion, and the second enforces the Bernoulli pressure balance.