= Solution
Let $z\in[0,h]$ measure distance through the thin <Bingham plastic> layer, with a stationary wall at $z=0$ and large-scale velocity $u_b$ at $z=h$. To leading lubrication order the shear stress $\tau_b$ is uniform across the layer. If $|\tau_b|\leq\tau_c$, the material is unyielded and the no-slip wall makes the entire layer stationary. After yield,
$$
\frac{du}{dz}=\frac1\mu
\left(\tau_b-\tau_c\operatorname{sgn}\tau_b\right).
$$
Integrating across the depth gives the <Bingham sliding law>
$$
\boxed{u_b=\frac h\mu
\left(|\tau_b|-\tau_c\right)_+
\operatorname{sgn}\tau_b}.
$$
Equivalently, for nonzero sliding,
$$
\boxed{\tau_b=\tau_c\operatorname{sgn}u_b+\frac\mu h u_b}.
$$
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