Where is the flowrate maximum located?

I can't resist one more calculation.  We can find the maximum enhancement readily.  First take the derivative.

Set it to 0 and find the value of η.  Isn't Mathematica great!!.  

We have observed that it does not change much with M.  Lets see why by looking at the solution for large M.

The value of the maximum is about 0.29.  You now know even more about this problem.