Help solve nozzle flow with Bubble dynamics

cfdoy

Hello: I have the following set up of equations for a quasi-flow in a 1D nozzle:

alpha = vol fraction A = area of nozzle U = Velocity

Continuity:

d/dt(alpha)+1/A*d/dx(alpha*U*A)=0

Momentum Eq:

d/dt(alpha*U)+1/A*d/dx(alpha*U*U*A)=-d/dx(P)+mu*laplacian(U)

Additional eqn for Alpha based on few more differential eqns.

I was wondering, if someone could help me set up a MAC like scheme for solving the momentum equation ??

Thanks,

CFDtoy