Lack of Collision Between Solid Bodies in a 2D Incompressible Viscous Flow

Hillairet, Matthieu

doi:10.1080/03605300601088740

Cited by 116 publications

(147 citation statements)

References 12 publications

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“…This situation heavily contrasts to the one encountered for viscous fluids; see Ref. [19], [20], [33] and [38]. Considering again deformable bodies and assuming additional regularity for the fluid's boundary and for the deformations, we can make the ODE (2.10) explicit.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

On the Self-Displacement of Deformable Bodies in a Potential Fluid Flow

Munnier

2008

Math. Models Methods Appl. Sci.

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Abstract. Understanding fish-like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanism. In this article, we study a coupled system of partial differential equations and ordinary differential equations which models the motion of self-propelled deformable bodies (called swimmers) in an potential fluid flow. The deformations being prescribed, we apply the least action principle of Lagrangian mechanics to determine the equations of the inferred motion. We prove that the swimmers degrees of freedom solve a second order system of nonlinear ordinary differential equations. Under suitable smoothness assumptions on the fluid's domain boundary and on the given deformations, we prove the existence and regularity of the bodies rigid motions, up to a collision between two swimmers or between a swimmer with the boundary of the fluid. Then we compute explicitly the Euler-Lagrange equations in terms of the geometric data of the bodies and of the value of the fluid's harmonic potential on the boundary of the fluid.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

On the Self-Displacement of Deformable Bodies in a Potential Fluid Flow

Munnier

2008

Math. Models Methods Appl. Sci.

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“…Si le modèle ci-dessus est utilisé pour décrire la dynamique, on obtient le résultat paradoxal selon lequel la sphère ne peut pas toucher le plan, aussi dense soit-elle. Nous renvoyons aux travaux de Brenner [9] pour une justification formelle, et au papier [10] pour une réelle preuve mathéma-tique (voir aussi [15] pour un résultat préliminaire). Cette absence de collision est aberrante, et a interpellé de nombreux physiciens, désireux d'expliquer ce paradoxe.…”

Section: Collision De Solides Immergésunclassified

“…L'unicité jusqu'à collision a été montrée ensuite par Takahashi [17]. Quand il n'y a pas collision, par exemple pour II-14 les configurations étudiées par Hillairet [10], ces résultats fournissent une unique solution régulière.…”

Section: Ii): Les Fonctions Scalaires ρ ρ I Définies En (15) Et Le Cunclassified

Quelques problèmes d’irrégularité dans l’interaction fluide-solide

Gérard-Varet¹

2010

Journées Équations Aux Dérivées Partielles

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“…Assuming, as we have in this model, that particle surfaces are smooth and that Stokes model is valid at any scale, it is known that contacts are not supposed to happen (see [9,15]). In fact, two passive or active particles can get infinitely close to each other, but they will never collide.…”

Section: Fully Microscopic Modelmentioning

confidence: 99%

Microscopic Modelling of Active Bacterial Suspensions

Decoene

Maury

2011

Math. Model. Nat. Phenom.

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Abstract. We present two-dimensional simulations of chemotactic self-propelled bacteria swimming in a viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on an associated region in the fluid representing the flagellar bundle. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation written on the whole domain, strongly coupling the fluid and the rigid particle problems: rigid motion is enforced by penalizing the strain rate tensor on the rigid domain, while incompressibility is treated by duality. This model allows to achieve an accurate description of fluid motion and hydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopic model is also used, in which the size of the bacteria is supposed to be much smaller than the elements of fluid: the perturbation of the fluid due to propulsion and motion of the swimmers is neglected, and the fluid is only subjected to the buoyant forcing induced by the presence of the bacteria, which are denser than the fluid. Although this model does not accurately take into account hydrodynamic interactions, it is able to reproduce complex collective dynamics observed in concentrated bacterial suspensions, such as bioconvection. From a mathematical point of view, both models lead to a minimization problem which is solved with a standard Finite Element Method. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. We also reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation on the oxygen concentration is solved in the fluid domain, with a source term accounting for oxygen consumption by the bacteria. The orientations of the individual bacteria are subjected to random changes, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient.

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Lack of Collision Between Solid Bodies in a 2D Incompressible Viscous Flow

Cited by 116 publications

References 12 publications

On the Self-Displacement of Deformable Bodies in a Potential Fluid Flow

On the Self-Displacement of Deformable Bodies in a Potential Fluid Flow

Quelques problèmes d’irrégularité dans l’interaction fluide-solide

Microscopic Modelling of Active Bacterial Suspensions

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