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.

我们考虑在给定外力的作用下，在具有薄边界的薄域中具有不可压缩的Bingham流，并且在该域的整个边界上都具有无滑移边界条件。用数学术语来说，此问题由域上的非线性变化不等式描述，其中小参数\（\ epsilon \）表示域的厚度和边界的粗糙度周期性。通过使用自适应线性展开算子，我们对\（\ epsilon \）时Bingham流的渐近行为进行了详细分析趋于零。我们获得了速度和压力的均化极限问题，该问题保留了流动的非线性特征，并在相应的有效方程中研究了微结构的影响。最后，我们根据非线性达西定律对极限问题进行解释。