= Two-layer plane Poiseuille flow with unequal viscosities
{title2=$u_\pm=U_0+Cy/\mu_\pm-Gy^2/(2\mu_\pm)$}
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 $U_0=Ga^2/(\mu_-+\mu_+)$ 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.
Back to article page
