General solutions to these differential equations

Let's solve what we can.

Start with the middle of the fluid.  We can first get a general solution to this equation.  

Back to conclusions

It is convenient to examine this solution to see if an obvious simplification arises.  It does.  At r=0, we don't anticipate any reason for the velocity to be infinite.  Since there is no other term to cancel log(r), we must make =0.  We can't do anything with the other constant at this moment.  Recall however that in the single phase example, was adjusted to make the velocity = 0 at the fixed wall.   

This real ugly command sets =0, and changes the name of to f1 (to sort of resemble Middleman's notation.).  It also picks out just the equation without the braces.

Equation 2 solution

We can solve the second equation the same way.  This time the domain of r does not go to 0 and there is no need (or reason) to exclude the log(r) term -- so it stays.

Change the name of the constants to e2 and f2 and extract the equation as "eq2a"