A moving fluid interface is not specified only by its shape at one instant. Its motion must also be compatible with the velocity of the fluid at the interface.

Represent the interface implicitly by

$$F(\mathbf r,t)=0.$$

If the interface is a material surface—fluid particles on the interface remain on it—then following any such particle must keep $F$ equal to zero. Therefore

$$\boxed{\frac{DF}{Dt}=0}.$$

Using the material derivative,

$$\boxed{\frac{\partial F}{\partial t}+\mathbf v\cdot\nabla F=0}$$

on the interface.

This is the kinematic boundary condition.

Graph form of an interface

In two dimensions, suppose the interface is written as

$$y=h(x,t).$$

Define

$$F(x,y,t)=y-h(x,t).$$

Then

$$\frac{\partial F}{\partial t}=-\frac{\partial h}{\partial t},$$

$$\nabla F= -\frac{\partial h}{\partial x}\hat{\mathbf x} +\hat{\mathbf y}.$$

For fluid velocity

$$\mathbf v=u\hat{\mathbf x}+v\hat{\mathbf y},$$

the material-surface condition gives

$$-\frac{\partial h}{\partial t} -u\frac{\partial h}{\partial x} +v=0.$$

Hence

$$\boxed{ \frac{\partial h}{\partial t} +u\frac{\partial h}{\partial x} =v }$$

at $y=h(x,t)$.

The vertical velocity of the interface is therefore not generally equal to $\partial h/\partial t$ alone. If the surface is sloped, horizontal motion also carries fluid particles up or down the graph.

Simple limits

Flat interface moving vertically

If $h$ is independent of $x$,

$$\frac{\partial h}{\partial x}=0,$$

so

$$\boxed{\frac{dh}{dt}=v}.$$

The interface simply moves with the normal fluid velocity.

Steady sloping interface

If the shape is steady,

$$\frac{\partial h}{\partial t}=0,$$

then

$$\boxed{v=u\frac{dh}{dx}}.$$

A particle following a sloping stationary interface must have the vertical velocity required to remain tangent to that surface.

Relation to impermeable solid boundaries

For a moving impermeable solid boundary, the same geometric principle states that fluid cannot cross the boundary. The fluid and wall must have the same normal velocity.

For a free surface or an interface between fluids, the kinematic condition likewise prevents material from crossing an interface that is assumed to move with the fluid.

This is separate from a dynamic boundary condition, which balances stresses such as pressure, viscous traction, or surface tension across the interface.

Worked check

Suppose a steady interface is

$$h(x)=0.10x$$

and fluid moves along it with horizontal component

$$u=2.0,\mathrm{m/s}.$$

The kinematic condition requires

$$v=u\frac{dh}{dx} =(2.0)(0.10) =\boxed{0.20,\mathrm{m/s}}.$$

A particle moving $1,\mathrm m$ horizontally rises $0.10,\mathrm m$, exactly matching the interface slope.

The kinematic boundary condition is therefore a geometric conservation law: it ensures that the assumed moving boundary and the fluid velocity describe the same material motion.