Fluid Dynamics | Part IB, 2001

A long straight canal has rectangular cross-section with a horizontal bottom and width w(x)w(x) that varies slowly with distance xx downstream. Far upstream, ww has a constant value WW, the water depth has a constant value HH, and the velocity has a constant value UU. Assuming that the water velocity is steady and uniform across the channel, use mass conservation and Bernoulli's theorem, which should be stated carefully, to show that the water depth h(x)h(x) satisfies

(Ww)2=(1+2F)(hH)22F(hH)3 where F=U2gH\left(\frac{W}{w}\right)^{2}=\left(1+\frac{2}{F}\right)\left(\frac{h}{H}\right)^{2}-\frac{2}{F}\left(\frac{h}{H}\right)^{3} \text { where } F=\frac{U^{2}}{g H}

Deduce that for a given value of FF, a flow of this kind can exist only if w(x)w(x) is everywhere greater than or equal to a critical value wcw_{c}, which is to be determined in terms of FF.

Suppose that w(x)>wcw(x)>w_{c} everywhere. At locations where the channel width exceeds WW, determine graphically, or otherwise, under what circumstances the water depth exceeds H.H .

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