8 For an ideal, incompressible fluid of density p, subject to a gravitational field g= -Vo' (here ' is the gravitational potential) the Euler equation is: v = Vp + g D Dt- Dv=v+ (v.V)v (a) (b) D Dt- Use vector identity v × (Vxv) = −(v. V)v+Vv²) to derive Bernoulli's equation: path) in steady flow. v² - 1½p - ' = 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 12 2g + h E where E is a constant. (ii) Show that h satisfies the cubic equation h³ – Eh² +22 = 0 (iii) (c) A possible value of h is 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 (iv) 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.