We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.
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Benjamin Obando, Takéo Takahashi, Existence of weak solutions for a Bingham fluid-rigid body system. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1281–1309DOI 10.1016/J.ANIHPC.2018.12.001