Mechanics of Fluids

466 Flow with a free surface
In particular, at the free surface,
0 = cη + A sinh{m(h + η)}sin mx (10.47)
Hence, if, as is usual for ocean waves (except when close to the shore), η is
small compared with both h and λ, the free surface has a sinusoidal form
given by
η ≏
�
− A
�
sinh mh sin mx A (10.48)
c
(In what follows equations marked A are subject to the restrictions η ≪
h, η ≪ λ.)
Since we are considering steady flow, the elevation η of the free surface
can be related to the velocity there by Bernoulli’s equation. We therefore
require expressions for velocity and pressure.
Using values for components of velocity from eqn 9.3 (with z in place of y)
and making use of eqn 10.46, we have
(Velocity) 2 =
� ∂ψ
∂x
� 2
�
+ − ∂ψ
�2 ∂z
= A 2 m 2 sinh 2 {m(h + z)}cos 2 mx
+ � −c − Am cosh{m(h + z)}sin mx � 2
(10.49)
Putting z = η gives the velocity at the free surface. Then substituting for
A from eqn 10.47 and neglecting terms in m 2 η 2 (on the assumption that
η ≪ h) we obtain
(Surface velocity ) 2 = c 2� 1 − 2mη coth{m(h + η)} �
≏ c 2� 1 − 2mη coth mh �
A (10.50)
The pressure above the surface is atmospheric but, because the surface
is not plane, the pressure in the liquid is, in general, modified by surface
tension (γ ). The surface tension force divided by the distance perpendicular
to the plane of Fig. 10.44 is γ and at the position P its vertical component
is γ sin θ .AtQ the vertical component is
γ sin θ + d
(γ sin θ)δx
dx
Hence the net upwards surface tension force on PQ is
d
(γ sin θ)δx
dx
If a mean gauge pressure p acts over PR, the total upwards force on the fluid
in the control volume PQR
= d
(γ sin θ)δx + p δx
dx
= Rate of increase of vertical momentum of the fluid through PQR