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Unformatted text preview: Fluid Dynamics 3  Solutions to Sheet 4 1. (i) For the given flow field, u = f ( r ) in cylindrical polars. Thus, applying the formula from chapter 0, u = 1 r u = 0 . (ii) We have to solve the steady Euler equation ( u ) u = p/. According to chapter 0, the convective term on the left is ( u ) u = u 2 r r = f 2 r r . Thus p can only depend on r , so that p = p ( r ) r , and the required equation follows. (iii) According to chapter 0, u = z r rf r , which vanishes if f = A/r . From (ii), we have p ( r ) = A 2 r 3 , and so p ( r ) = A 2 2 r 2 + p atm . 2. (i) Conservation of mass is volume flux in equals volume flux out, so Ud = U 1 b 1 + U 2 b 2 (ii) Use the momentum integral theorem. Need to choose a control surface  this should be a boundary where we know stuff about the solu tion. So follow example in lectures and take it to be around the flow cutting the inflow and outflows far away from the impingement. Let S be the contour ABCDEF in the figure: Deal with pressure term first: A B C D E F integraldisplay S p n d S = integraldisplay S ( p p atm ) n d S + integraldisplay S p atm n d S The last term is zero by the divergence theo rem (see lectures). The first term is zero along all parts of S apart from the section DE . The other term is integraldisplay...
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 Fall '11
 Eggers

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