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Uniform axial strain rate $E$ in an annular Newtonian fluid gives radial velocity $u_r=-Er/2+C/r$. For radii $R_i,R_e$, area parameter $A=R_e^2-R_i^2$, and constant gas pressures,The opposite signs of inner and outer curvature in the Young–Laplace equation give these expressions. The kinematic boundary condition implies $\dot{A}=-EA$ and

A moving boundary excites an internal gravity wave through its kinematic boundary condition. For wavenumber $k$ and angular frequency $\omega$, the vertical velocity field satisfies $W^{\prime \prime }+k^2(N^2/{\omega }^2-1)W=0$. The radiation condition selects outward group velocity; when $\omega >N$, the bounded response is an evanescent wave.

Let fluid 1 lie above fluid 2, with mass densities ${\rho }_1<{\rho }_2$ and dynamic viscosities ${\mu }_1<{\mu }_2$. Define $\Delta \rho ={\rho }_2-{\rho }_1>0$, $\Delta \mu ={\mu }_2-{\mu }_1>0$, porosity $\varphi$, permeability of a porous medium $k$, and downward interface speed $W$. With Darcy flux $\varphi W$, no capillarity, and two unbounded layers, Darcy's law givesFor a Fourier mode of interface displacement, the two perturbation pressures decay exponentially into the respective layers. The kinematic boundary condition makes their amplitudes proportional to $\varphi {\mu }_1\sigma /(k|\alpha |)$ and $-\varphi {\mu }_2\sigma /(k|\alpha |)$. Linearized pressure continuity adds the base-gradient difference $\varphi W\Delta \mu /k-g\Delta \rho$. Eliminating the amplitudes proves the formula. The critical interface speed is $kg\Delta \rho /(\varphi \Delta \mu )$; the critical Darcy velocity is $kg\Delta \rho /\Delta \mu$. Without surface tension this model has unbounded short-wave growth rate on its unstable side.

For steady equal-pressure flow, mass conservation gives $Aw=A_0{w}_0$ and the kinematic boundary condition gives $(D\sqrt{w}{)}_z=\gamma /(2\mu \sqrt{w})$, where $D=R_e-R_i$. Therefore finite-height closure occurs exactly whenFor positive velocity the integral is increasing. If it diverges, closure is finite; if its limiting value lies below the right-hand threshold, there is no finite closure. Quadratic freely falling profiles have logarithmically divergent integrals, while sufficiently strongly drawn exponentially accelerating profiles can have finite integrals. Thus nozzle speed alone is insufficient to guarantee closure: the axial tension or downstream condition matters.

A flexural-gravity wave bends a floating elastic plate while displacing the underlying water. For a uniform plate with areal mass $m$ and bending stiffness $D$ above inviscid deep water of mass density ${\rho }_w$, put $\alpha =m/{\rho }_w$ and $\beta =D/{\rho }_w$. Combining the plate equation with the water's kinematic boundary condition gives the displayed dispersion relation. Both gravitational acceleration and plate bending restore displacement; water and plate inertia resist acceleration. It assumes small amplitude, no prestress and negligible shear deformation of the plate.

A free surface is a fluid boundary whose position is part of the solution and whose traction is prescribed by the adjoining medium. Its motion obeys a kinematic boundary condition, while its pressure or stress supplies a dynamic boundary condition.

At a material interface without surface tension, continuity of displacement and of the pressure evaluated on the displaced interface givesThe first follows from the kinematic boundary condition $ik(U-c)\hat{\eta }=\hat{w}$. The second uses $\hat{p}={\rho }_0[(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$ and hydrostatic displacement $[\hat{p}-g\rho \hat{\eta }]=0$. These conditions apply to density, velocity and vorticity jumps within the Boussinesq approximation.

Two deep layers of incompressible flow with upper mass density ${\rho }_1$, lower mass density ${\rho }_2$, and tangential velocities $U_1,U_2$ have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For $k>0$ the squared departure of the phase velocity $c$ from the density-weighted mean velocity is $g({\rho }_2-{\rho }_1)/[k({\rho }_1+{\rho }_2)]-{\rho }_1{\rho }_2(U_1-U_2{)}^2/({\rho }_1+{\rho }_2{)}^2$. A negative value gives Kelvin-Helmholtz instability despite stable density stratification.