Question

For an ideal, incompressible fluid of density p, subject to a gravitational field g= -Vº' (here Ø' is the gravitational potential) the Euler equation is: Div=-- Vp+g, Div = .v + (v.7)v Use vector identity v * (I xv) = -(v.7)v + rv2) to derive Bernoulli's equation: --v2--p - d' = const, along any streamline (dye path) in steady flow. (a) Use y (b) 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 4- + h = E where E is a constant. 29 (ii) Show that h satisfies the cubic equation h3 – Eh2 + 2 = 0 (iii) 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). (iv) Compute a Froude number, F = 1, for appropriate values of h in 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. (c)

Answer 1

Given,

For an ideal, incompressible fluid of density p ,subject to a gravitational field.

The Euler equation is,

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1 show that zor this flow of th=e where, È is the constant Heae, we are consider point A and which are separated by a long the height h and both of the point are open to atmosphere, Therefore, Applying Bernoullis therom at point A and 6, e non aver match that she the Ii Pn=re] the ver constant = E [Vg= 125h = consi] 50, Bath = elConstant 29 Hence, the therom is 11 show that h satisfies proved. the cubic equation that h Satisfies the h-en tarino Now, G is the rate of How of fleeid = 0.62 As the retangular cross-section has unit-width Therefore 1 uz ne Put u = e in the above equation we have athi nazo

1) A possible va ne the cubic equation proved A possible value of his sm when U= 2ms! To find: The other possible value of h. Given - 2 24 th= € we put, u= 2 m/s and hasm we obtain Agata € or Badran Put Eatt in the cibove equation Hence Possible value of his tahun