Manuel Villanueva Pesqueira

arXiv2020

Asymptotic behavior of a Bingham Flow in thin domains with rough boundary

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 $ε$ 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 $ε$ 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.

math.AP↗

arXiv2016

Straight rod with different order of thickness

In this paper, we consider rods whose thickness vary linearly between $\eps$ and $\eps^2$. Our aim is to study the asymptotic behavior of these rods in the framework of the linear elasticity. We use a decomposition method of the displacement fields of the form $u=U\_e + \bar{u}$, where $U\_e$ stands for the translation-rotations of the cross-sections and $\bar{u}$ is related to their deformations. We establish a priori estimates. Passing to the limit in a fixed domain gives the problems satisfied by the bending, the stretching and the torsion limit fields which are ordinary differential equations depending on weights.

math.AP↗