From the previous example in Middleman (3.2.1, page 68-74)  you may recall two points.  First, that P[z] was a linear function and thus P'[z] = const.  We will call this constant, dpdz.  Second, we need a constitutive equation for the stress component, , in terms of the velocity gradient.  Since we cannot easily derive this, we just stated that it is:
        .
        
Now we need to recognize that there are two fluids and thus two values of viscosity for the fluids in different regions.  Thus we will have two versions of eq1, one for fluid I and one for fluid II.  Since we know the constitutive equations and we know that P'[z] is a constant, we might as well substitute and start to solve it.  We also make the restriction that gravity is not important and thus ignore its possible effects by making g=0.

The z momentum equations are: