= Solution
Write $d_+=\sigma+h$ and $d_-=h-\sigma$ for the lower and upper layer thicknesses. A steady unidirectional velocity $\mathbf u=u(y)\mathbf e_x$ with no pressure gradient satisfies the <Navier-Stokes equations> through $u''=0$ in each layer. Continuity of velocity and tangential stress at the interface gives one constant <shear stress>
$$
\tau=\mu_+u_+'=\mu_-u_-'
=\frac{U_--U_+}{d_+/\mu_++d_-/\mu_-}.
$$
The <two-layer Couette flow> is therefore
$$
\boxed{
u_+(y)=U_++\frac\tau{\mu_+}(y+h),
\quad -h\leq y\leq\sigma,}
$$
$$
\boxed{
u_-(y)=U_--\frac\tau{\mu_-}(h-y),
\quad \sigma\leq y\leq h.}
$$
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