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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.

The leading lubrication momentum equations and incompressibility condition areAt the solid plane $y=0$, no slip and no penetration give $u=v=0$. At the free surface $y=h(x,t)$, neglecting surface tension and ambient viscous stress givesto leading order. The kinematic boundary condition for a free-surface graph is

Write the free surface as the zero setBecause this is a material surface, the kinematic boundary condition is $DS/Dt=0$. Thus, on $z=\eta$,Thereforeat the free surface. This is the kinematic boundary condition for a free-surface graph.