 Two-layer plane Poiseuille flow with unequal viscosities

Two viscous layers occupy $-a<y<0$ and $0<y<a$, with viscosities ${\mu }_{-}$ and ${\mu }_{+}$. A uniform driving pressure gradient $G=-p_x>0$, a stationary flat interface and no-slip outer walls give the displayed plane Poiseuille flow, where $U_0=G{a}^2/({\mu }_{-}+{\mu }_{+})$ and $C=Ga({\mu }_{-}-{\mu }_{+})/[2({\mu }_{-}+{\mu }_{+})]$. These constants follow from continuity of velocity and shear stress at the interface and zero wall velocities. The upper-to-lower volume-flux ratio is $(1+7r)/[r(7+r)]$, with $r={\mu }_{+}/{\mu }_{-}$. As $r\to \infty$, the interface becomes effectively stationary, the upper flux vanishes, and the lower layer approaches flow between two stationary plates a distance $a$ apart. The upper wall stress remains finite despite its vanishing velocity gradient.