Question

4. Stokes Flow: In class we solved the problem of Stokes flow around a sphere with the no-slip boundary condition at the surface of the sphere. In this problem we

relax this assumption. Inplace of the no-slip condition, we allow the tangential velocity at the surface of the sphere to be proportional to the tangential stress there. Let 1/B be the coefficient of proportionality. a. Write out this and the other boundary conditions, together with the governing equations interms of the stream function in the appropriate coordinate system. b. Please read carefully so as to avoid doing unnecessary algebra. Solve these equations aboundary conditions for the stream function only. c. Recall from class that we derived an expression for the hydrodynamic force on the sphere interms of the coefficients in the expression for the stream function. Do not re-derive this. Just use it to write the resulting hydrodynamic force and the velocity in terms of the force balance.Discuss the limiting cases of zero and infinite B and their physical meanings.

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