= Two-layer falling film of equal-density viscous fluids
Let the lower fluid have thickness $h$ and <viscosity> $\mu$, and the upper fluid thickness $\alpha h$ and <viscosity> $\beta\mu$, with common density $\rho$. For steady flow down a plane inclined at angle $\theta$, set $H=(1+\alpha)h$, $A=\rho g\sin\theta/\mu$. The common <hydrostatic pressure> is $p_{\rm atm}+\rho g\cos\theta(H-y)$. No slip at the wall, continuous <velocity> and <shear stress> at $y=h$, and zero upper surface shear give $u_1=A(Hy-y^2/2)$ and $u_2(h+z)=Ah^2(\alpha+1/2)+(A/\beta)(\alpha hz-z^2/2)$. These follow by integrating $\mu_i u_i''=-\rho g\sin\theta$ in each layer. The stress applied to the lower fluid equals the downslope weight of the upper layer, independent of its <viscosity>; the upper <viscosity> affects its shear rate but not that transmitted stress.
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