At an impermeable boundary, the flow of fluid relative to the boundary must be tangential to it. If the boundary is fixed in space, this means that the component of fluid velocity normal to the boundary must be zero. An example using this special condition was just investigated in Art. 1.111. However, if the boundary is moving, then the normal component of the fluid must be equal to the velocity of the boundary normal to itself. This applies whether there is a solid on one side of the boundary or not. If not, then we have the case of a free surface of the fluid, such as the sea surface
Let 
represent the velocity of the fluid boundary
and 
be the fluid velocity of the boundary. Then
the kinematic boundary condition is simply
where 
is the unit normal to the boundary.
Eq. (223) represents the kinematic boundary condition for an impermeable boundary only, i.e. a boundary across which there is no flux of matter of the macroscopic scale. In those cases where a change of state of the fluid occurs at the boundary or where the boundary surface displays a selective permeability phenomenon, the condition (228) is no longer rigourously applicable. A specific case in point is the effect of a flux of evaporation at the sea surface. This is certainly important in thermodynamic problems although in nearly all kinematic or dynamical problems the effect of evaporation can be ignored so far as the kinematic boundary condition is concerned. This matter is discussed further in Chapter IV.
The boundary condition can be written in a more convenient form for a free surface of the fluid. In any physically real situation a free surface simply represents the boundary between two different fluids, e.g. water and air, water and steam, oil and water, etc. That is, the free surface exists by virtue of a difference in the neighboring fluids. The difference can be a chemical difference or merely a physical difference (difference in phase of the same substance, e.g. water/water vapor). The important thing in connection with our present discussion is that there exist certain physical or thermal properties of the two fluids which are continuous across the free surface. An example is temperature. Pressure is another example in those cases where the effect of surface tension can be ignored.
Suppose f(x,y,z,t) represents the field of some property in the fluid concerned and f'(x,y,z,t) the field of the same property in the adjacent fluid. If the property f is one which is continuous across the free surface (which delineates the boundary between these two fluids) then
it at the boundary. If the functions f and f' are known, then Eq. (229) actually defines the boundary, i.e.
is the implicit equation of the free surface. Usually this is written in the explicit form
where 
denotes the function representing the elevation
of the free surface above some reference plane at
position x, y and time t.
The time rate of change of the property f following the
motion of the fluid at the boundary is 
.
The condition that the fluid at the boundary flows along
the boundary and never leaves the boundary is equivalent
to the condition that the individual rate of change of
f at the boundary is equivalent to the rate of change
of f' on the other side of the boundary as one follows
with the particle motion of the first fluid. Thus
represents the kinematic boundary condition. In view of Eq. (231), it will be noted that
so that Eq. (232) can be written as
or
where 
, 
, 
apply at the boundary.
If the free surface is stationary (i.e.
) and slopes only in
the x-direction, then
This expression can be verified by reference to Fig. 1.12-2. If a particle starting at position A is to remain on the surface as itmoves over the differential distance ds, then we must have
which is equivalent to (233a).
If the surface has no slope but is ascending at the
rate of 
then it is obvious
that in order that the fluid remain at the surface, then
must satisfy the relation
Eq. (233) is applicable when both effects are present, including the effect of surface slope in the y-direction as well as the x-direction.
In many problems the nonlinear terms in Eq. (233) are of second
order compared with and Eq. (233b)
represents a good approximation. However, one should always
investigate the order of magnitude of these terms before
making such an approximation.