3. (a) [6 marks] An inviscid incompressible fluid moves in a steady irrotational manner. Define the complex potential w(

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3 A 6 Marks An Inviscid Incompressible Fluid Moves In A Steady Irrotational Manner Define The Complex Potential W 1 (102.88 KiB) Viewed 19 times

3. (a) [6 marks] An inviscid incompressible fluid moves in a steady irrotational manner. Define the complex potential w(2), z = x + iy, for the flow. Explain how the complex velocity and the stagnation points are related to w(2). (b) (24 marks] (i) Verify that the complex potential w(2) defined by W(z) = aUt coth (6) corresponds to the steady flow of an inviscid incompressible fluid in the upper half plane, defined by y > 0, with a circular cylinder of radius a lying on the boundary and touching the boundary at z = 0. In the absence of the cylinder the flow is uniform, of magnitude U and parallel to the boundary. (ii) Show that the resulting velocity field u satisfies 4 lel? ? (7) |ul2 = u? + v2 = U2 V" (97)* (colha 2 2пах 2πα! COS where r2 = x2 + y2. (ii) Show that the difference in the pressure between the top and bottom points of the cylinder is PA+U2 32