1. Consider a trapped, cylindrical liquid bubble (viscosity mu2) in a flow field of another liquid (viscosity mu1), as shown below. Assume that the bubble is an infinitely long cylinder. The pressure is not radially deformable. However, as liquid flows around it, the interface moves giving rise to circulation within the bubble, V, at bubble interface r = R is given by V theta = f(R)f(theta). You are required to find this interface velocity and the velocity components around the bubble using a stream function approach. This is a creeping flow problem.
a) Write down all the appropriate velocity boundary conditions for the flow around the bubble (one of the BCs may not be known explicitly). Determine an appropriate form for the stream function psi(r, theta) for use in part b).
b) Obtain a general expression for psi(r, theta) . You do not have to evaluate the constants of your expression for this part.
c) Use boundary conditions to determine psi(r, theta) and the velocity components explicitly for the flow around the bubble. Determine what f(R)f(theta) are for at r = R.
d) If you were to solve the flow field of the liquid within the bubble, what boundary conditions would you have used to obtain Vr and V theta. Mention only three BCs.
