# Force to hold a sluice gate in place

1. Mar 24, 2013

### George32

1. The problem statement, all variables and given/known data
An incompressible, inviscid liquid of density ρ flows, under the influence of gravity, beneath the sluice gate shown. The height and velocity upstream of the gate are h and V, respectively, while downstream of the gate the height is l. Show that the force per unit width necessary to hold the gate in place is given by
$$\frac {\rho g(h-l)^3} {2(h+l)}$$
Hint : Requires consideration of hydrostatic pressure, a momentum balance over a control
volume incorporating three forces acting on three vertical surfaces, and use of the Bernoulli equation.

2. Relevant equations
Using a conservation of mass, $$Vh=Ul$$ where U is the velocity downstream.
Using Bernoulli, $$V^2(1 - \frac {h^2} {l^2}) = g(l-h)$$

3. The attempt at a solution
Taking $$\dot {m} = \rho v h$$ and $\dot {m} v = \rho h v^2$
Using the Steady Flow Momentum Equation:

$$\Sigma \dot{m_i} V_in - \Sigma \dot{m_o} V_out = F - \Sigma pA$$
So filling into this I get: $$\rho h v^2(1-\frac {h}{l}) = F - \frac {1}{2} \rho g (h^2 -l^2)$$
Filling in $V^2 = \frac{g(l-h)}{1-\frac{h^2}{l^2}}$ and rearranging:
$$F = \rho g( (h-\frac{h^2}{l})(\frac{l^2 (l-h)}{l^2 - h^2}) + \frac{1}{2}(h-l)(h+l) )$$
Working through the rearrangement:
$$F = \rho g( (h-\frac{h^2}{l})(\frac{l^2}{l+h}) + \frac{1}{2}(h-l)(h+l))$$
$$F = \rho g( \frac{hl^2 - lh^2}{l+h} + \frac{1}{2}(h-l)(h+l) )$$
$$F = \rho g( \frac{2(hl^2 - lh^2) + h^3 +h^2 l - l^2 h - l^3}{l+h} )$$
$$F= \frac{\rho g}{l+h} ( h^3 - h^2 l + h l^2 - l^3)$$
Which comes down to:
$$F= \frac{\rho g}{l+h} (h-l)(h^2 + l^2)$$

which is obviously not the right answer. In order to get the $(h-l)^3$ that they have in their answer, I would need: $2(h(l^2) - l(h^2)) + (h^2 + l^2)(h-l)$ On the top line in the top row (after adding fractions)

I remember having a similar issue in a tutorial, which was also a show that question, which came out to the same issue (the tutor couldn't make it work).

I just want to know if anyone else can make it work, and if so, if they could nudge me, otherwise I think it's possible that the question is wrong...

Thanks