For an ideal, incompressible fluid of density p, subject to a gravitational field g = −√ø' (here p' is the gravitational potential) the Euler equation is: (a) (b) Ꭰ Dv = − 1 Vp +g, Ꭰ Dt at Dv=v + (v. v)v Use vector identity v × (V × v) = −(v. V)v + V(±²²) to derive Bernoulli's equation: -½v² — — —½— p − 0' = path) in steady flow. - = const, along any streamline (dye This question concerns free surface flow and hydraulic jumps. Incompressible inviscid fluid (water) is in steady, free-surface flow in a very long "open" channel which is of rectangular cross-section, unit width and horizontal. Let h denote the water's level relative the channel bed, u its velocity and Q the channel discharge. In this question, take g = 10ms². (i) Show that for this flow u2 2g +hE where E is a constant. (ii) Show that h satisfies the cubic equation h³ - Eh² + 02 = 0 2g (iii) (c) (iv) A possible value of his 1m when u = 2ms 1. Find the other possible values of h. (Hint. When you know one solution of the cubic in (ii) you can factorize it, to obtain a quadratic). u² Compute a Froude number, F = for appropriate values of h in gh' Imagine running water from a tap onto the back of a dinner plate. Relate you observation of the flow on the back of the plate your results in part (b) (iii) above.