Does it make sense that increasing one of the flow rates will decrease the pressure drop?

There is one more interesting point to make about this calculation.  We can examine how the pressure drop changes as the flowrate of the less viscous fluid is increased.  If we increase the flowrate of a fluid in a single phase flow the pressure drop will increase.  This will be linearly for a laminar flow.  However, for a two-fluid flow will this always be the case?  We will find that it is not the case.  For a range of flow rates, increasing the less viscous fluid flowrate will decrease the pressure drop.  

Calculate the dimensional flowrate of the inside fluid starting with the dimensional velocity profile.  

The flowrate is,

Now do the same for the outside fluid.  

To make the comparison we will have to solve for the pressure drop, dpdz and h.  We will then plot how pressure drop changes with changes in the volumetric flow rate of the outside fluid.  We will use numbers to show this.

Here is what is solved to get the plot.  Note the order of h.    

First find the pressure drop for a single phase flow.  We can do this if h=0.

This looks correct and if we pick some numbers in cgs units for the variables we might find,

So this is the pressure drop with no lubricating flow.  Adding outside fluid will decrease dpdz from this value.  

Now find the pressure drop as a function of qq2.  We need to use a nonlinear solution technique to get the answer since there are at least 6 solutions for different values of h.  You can explore this more if you wish, or just trust me.  Note that h must be positive and less than D.  

(To make this work in Mathematica 4, I needed to pick out root 5 "[[5]]" of the 6 roots.  I also chose the range of the entire calculation "i,1,20" and the scaling of que2, "  3+(i-1)*Exp[i/3.3]" to show the particular region that I wanted.  This type of behavior will exist for almost any "qq1" but you might imagine that it is difficult to show because it involves a six-order equation for h.)

(To make this work in Mathematica 3, you need to pick the first root, [[1]]. )

Here is what the set of two equations look like for one set of numbers: