Mechanics of Fluids

434 Flow with a free surface
Fig. 10.20
Differentiation with respect to h gives
�
∂E Q2
= 1 + −
∂h 2g
2
A3 �
∂A
∂h
Now, from Fig. 10.20, the small increase of area δA, which corresponds to
a small increase δh in the depth, is given by Bδh where B is the breadth of
the surface. Hence, as δh → 0, δA/δh → ∂A/∂h = B.
∴ ∂E
∂h
Q2
= 1 − B
gA3 which is zero when Q 2 = gA 3 /B, that is, when gA/B = Q 2 /A 2 = u 2 .IfA/B
is regarded as the mean depth of the section and is represented by h then
uc = (gA/B) 1/2 =
� �1/2 gh
(10.20)
(This mean depth h must not be confused with the hydraulic mean
depth A/P.)
The great importance of the critical conditions is that they separate two
distinct types of flow: that in which the velocity is less than the critical value
and that in which the velocity exceeds the critical value. As we have seen in
Section 10.8, the critical velocity uc = (gh) 1/2 corresponds to the velocity
of propagation (relative to the undisturbed liquid) of a small surface wave
in shallow liquid. Thus, when the velocity of flow is less than the critical
velocity, it is possible for a small surface wave to be propagated upstream
as well as downstream. Any small disturbance to the flow can cause a small
surface wave to be formed, and this wave may be regarded as carrying, to
the liquid further away, information about the disturbance. If the wave,
as messenger, can be propagated against the flow then the liquid upstream
will be informed of the disturbance and its behaviour will be influenced
accordingly. When the flow velocity is less than (gh) 1/2 , then the behaviour
of the liquid upstream can be influenced by events downstream.
If, on the other hand, the flow velocity is greater than the critical, the liquid
travels downstream faster than a small wave can be propagated upstream.
Information about events downstream cannot therefore be transmitted to
the liquid upstream, and so the behaviour of the liquid is not controlled