I'm trying to figure out if it is correct or incorrect to assume that the pressure gradient in a core-annular flow system is the same in the core and the annulus (steady state, vertical flow, annulus has larger density and viscosity):
$\begin{array}{ll}\mu_1 \frac{1}{r} \frac{d}{d r}\left(r \frac{d u_1}{d r}\right)=\frac{d P}{d z}+\rho_1 g, & r \in[0, \zeta] \\ \mu_2 \frac{1}{r} \frac{d}{d r}\left(r \frac{d u_2}{d r}\right)=\frac{d P}{d z}+\rho_2 g, & r \in[\zeta, R]\end{array}$
These are Stokes flow equations, where the appropriate boundary conditions are no-slip at the outside walls ($R$), and continuity of velocity and stress at the interface ($\zeta$). The fluids are immiscible.
Can I assume that $\frac{d P}{d z}$ is the same in both fluids? Is there a continuity of pressure?
