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Take Home #3 - Mag Flow past sphere

Take Home #3 - Mag Flow past sphere - u is as found in the...

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Take Home Final 3. Magnetized Flow past Sphere (25 points) (extension of Problem#2 in TH) Suppose we now allow a B field in Problem #2, flow past sphere. Let the field be weak, in the sense u ~ V A << c s , where V A is the Alfven speed. For this part, use the full, magnetized ideal MHD eqns. Find a steady state solution as follows. 0. Write down the full ideal MHD equations in steady state. Box it. 1. Show that p = p 0 to lowest order, as before. Show that this still implies .u =0 and n=n 0 . Of course, .B = 0. 2. Annihilate p 2 as before and so find an eqn that involves x u and x B . Box it. A useful vector identity makes this easier (as in Problem 2). 3. Identify one more equation that involves u and B . Note the symmetry between u and B in the set of eqns that include the divergence eqns from 1, the annihilated eqn from 2, and the eqn identified from 3. Box it. 4. Show that a solution exists such that
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Unformatted text preview: u is as found in the unmagnetized case and B = C u , where C = const. Do you think this solution is consistent with frozen-in? Explain. 5. Suppose the sphere is an excellent conductor and it was introduced into the flow field quickly enough that B = 0 inside it for quite some time. From the Maxwell eqn ∇ .B = 0, what is the jump condition on B at the plasma-conductor interface? Box it. If so, is your solution to B found in 4 consistent with this? Thus, do we have a true steady state? What are the currents that must flow inside the sphere? Box it. 6. (optional) Suppose the sphere is an insulator. Then there can be no currents inside, but there may be a non zero vacuum field inside. Is your outside solution for B as found now consistent with possible vacuum fields inside? (just give your best guess) Thus, do we have a true steady state in the insulator case?...
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