Figure 5.9 depicts the two-dimensional flow beneath a sluice gate inserted in a wide channel of flowing water. Far upstream and downstream, the flow velocities and , respectively, are horizontal and uniform from the bottom to the air/water interface at the top, and the depth of the fluid layers, and , are also constant. Close to the gate, the velocity and layer depth vary, as the incoming fluid accelerates to pass under the opening in the gate. We therefore select a control volume enclosing the gate which extends far enough upstream and downstream that the flow conditions are known.
In applying the linear momentum theorem to the control volume shown by a dashed line in figure 5.9, we will need to evaluate the integral of the pressure on the control surface. However, we need only to consider the portion of the flow where the pressure differs from atmospheric by the amount , which occurs in the upstream and downstream water layers since a uniform pressure of integrates to zero. In these water layers, the streamlines are horizontal and uncurved and the pressure distribution is hydrostatic:
so that the net pressure integral on the upstream flow, for example, becomes:
Now we may utilize the linear momentum theorem 5.11 in the horizontal direction to find the horizontal force per unit width of gate, , needed to restrain the sluice gate:
where we have considered the viscous force acting on the stream bed as negligible. We may now eliminate from this expression by applying mass conservation to the control volume:
Substituting in equation 5.33 and simplifying,
The first term of this expression is simply the force that would exist if the sluice gate were closed so as to separate two static layers of depths and . The second term reduces this amount in proportion to the flow through the sluice gate.
To simplify this expression further, we assume that the flow through the gate is an inviscid flow so that Bernoulli's equation for steady flow may be applied to a streamline along the air/water interface:
where we have used mass conservation equation 5.34 in the second line to eliminate . If we now substitute this expression for into equation 5.35 for the force on the sluice gate and simplify the ensuing expression, we will find: