Viscoplastic plug flow equation
 

Hi! I am having trouble with writing equation for the velocity in the plug flow region of viscoplastic fluid. Any kind of suggestion would be highly appreciated.

Problem: It's a two layer pipe flow with two viscoplastic fluids. The radius of the pipe is (-1<r<1) where there is a plug flow in (-0.5<r<0.5). And the velocity equations are:

v1=Ar^2 - Br + Clnr for (-1<r<-0.5)
v2=A(-0.5)^2 - B(-0.5) + Cln(-0.5) for (-0.5<r<0.5)
v3=A(0.5)^2 + B(0.5) + Cln(0.5) for (-0.5<r<0.5)
v4=Ar^2 + Br + Clnr for (0.5<r<1)

where A,B,C are constants and v2=v3
My confusion: v1 and v4 are correct and we can have the constant value of the plug flow region from these two equations. But there is no term r_i which is the interfacial radius. To be continuous at the interface between two flows- the velocity and the shear stress must be continuous, but according to my calculation, there is no r_i term :confused: :confused: