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worksheet4

# worksheet4 - Fluid Dynamics 3 2011/12 Sheet 4 Questions...

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Fluid Dynamics 3 2011/12 Sheet 4 Questions 1,2,3 to be handed in on 11th November 1. Consider a vortex with centre along the ˆ z -axis, so the flow field is of the form u = f ( r ) ˆ θ in cylindrical polars. (i) Show that ∇ · u = 0. (ii) Using Euler’s equation in cylindrical coordinates, show that the pressure satisfies p ( r ) = ρ f 2 ( r ) r . (iii) Compute the vorticity ω = ∇ × u and show that it vanishes if f = A/r . Compute the pressure for this case. 2. Consider the impingement and subsequent spreading of a horizontal, two-dimensional jet of water of speed U and width d onto a plane surface, inclined at an angle α to the horizontal. Sufficiently far from the point of impingement the jet flow becomes smooth, uniform and parallel to the inclined plane and the pressure returns to atmospheric pressure. U U 2 U 1 b 2 b 1 d α (i) Use an expression of the conservation of mass to show that U 1 b 1 + U 2 b 2 = Ud, where U 1 and b 1 are the velocity and breath of the flow along the plane on one side of

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worksheet4 - Fluid Dynamics 3 2011/12 Sheet 4 Questions...

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