PS2 - v , show that these are equivalent to the vector...

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MATH 505: Mathematical Fluid Dynamics Problem Set 2 due Monday, September 24, 2007 1. Find the complex potential for a point source in planar flow, with a velocity field ~u = ( B/r, 0) , (source for B > 0 or sink, for B < 0). Comment on how this could be combined with a point vortex. 2. Consider the complex potential for flow past a circle of radius a F ( z ) = φ + = U 0 ± z + a 2 z ² where U 0 is the flow speed along the x-axis infinitely far away from the circle. Assume additionally a point vortex of circulation strength Γ at the center of the circle. As Γ is increased from zero, the two stagnation points on the circle move towards each other. Find the critical value Γ c at which the two stagnation points merge, and above which the stagnation point detaches into the flow. Find an expression for the location of this stagnation point in the flow. It may help to sketch the flow! 3. Consider a 2D velocity field ~u = ( u, v ). Starting separately from the conservation of x-momentum ρu and y-momentum
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Unformatted text preview: v , show that these are equivalent to the vector conservation law ( ~u ) t + div( ~u ~u ) = ~ f int + ~ f ext where ( ~u ~u ) ij = u i u j is the tensor product. Remember that for each quantity q we assume an advective ux ~ F = q~u . 4. Using Bernoullis equation for steady incompressible irrotational ow, calculate the pressure eld for the ow past a cylinder described in Problem 2, with = 0. 5. The only unphysical aspect of a Point Vortex is that the velocity eld approaches inn-ity at its center (!!). So consider the Rankine Vortex, a ow eld given in cylindrical coordinates by ~u = (0 , v ( r ) , 0), with v ( r ) = r if r a a 2 /r if r > a , (a) Let C b be a circle of radius b centered on the origin, and calculate the circulation integrated around that curve, as a function of b . (b) Calculate the vorticity for the Rankine Vortex....
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This note was uploaded on 07/23/2008 for the course MATH 505 taught by Professor Belmonte,andrewl during the Fall '07 term at Pennsylvania State University, University Park.

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