In this paper, we study the time evolution of a finite number of homogeneous rigid disks within a viscous homogeneous incompressible fluid in the whole domain R 2 . The motion of the fluid is governed by Navier-Stokes equations, whereas the movement of each rigid body is described by the standard conservation laws of linear and angular momentum. The motion of the rigid bodies inside the fluid makes the fluid domain time dependent and unknown a priori. At first, we prove the existence and uniqueness of strong solutions of the considered problem up to collision. Then, we prove that contact between rigid bodies cannot occur for almost arbitrary configurations.

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Cell migration is a complex phenomenon that plays an important role in many biological processes. Our aim here is to build and study models of reduced complexity to describe some aspects of cell motility in tissues. Precisely, we study the impact of some biochemical and mechanical cues on the cell dynamics in a 2D framework. For that purpose, we model the cell as an active particle with a velocity solution to a particular Stochastic Differential Equation that describes the intracellular dynamics as well as the presence of some biochemical cues. In the 1D case, an asymptotic analysis puts to light a transition between migration dominated by the cell’s internal activity and migration dominated by an external signal. In a second step, we use the contact algorithm introduced in [15,18] to describe the cell dynamics in an environment with obstacles. In the 2D case, we study how a cell submitted to a constant directional force that mimics the action of chemoattractant, behaves in the presence of obstacles. We numerically observe the existence of a velocity value that the cell can not exceed even if the directional force intensity increases. We find that this threshold value depends on the number of obstacles. Our result confirms a result that was already observed in a discrete framework in [3,4].