  We consider an incompressible Bingham flow in a thin domain with rough
boundary, under the action of given external forces and with no-slip boundary
condition on the whole boundary of the domain. In mathematical terms, this
problem is described by non linear variational inequalities over domains where
a small parameter $\epsilon$ denotes the thickness of the domain and the
roughness periodicity of the boundary. By using an adapted linear unfolding
operator we perform a detailed analysis of the asymptotic behavior of the
Bingham flow when $\epsilon$ tends to zero. We obtain the homogenized limit
problem for the velocity and the pressure, which preserves the nonlinear
character of the flow, and study the effects of the microstructure in the
corresponding effective equations. Finally, we give the interpretation of the
limit problem in terms of a non linear Darcy law.
