On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications

Galdi, Giovanni P.

doi:10.1016/s1874-5792(02)80014-3

Cited by 183 publications

(170 citation statements)

References 58 publications

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“…It should be noted that the classical mathematical issue of the well-posedness of a swimming model as a system of PDEs was addressed for the first time by Galdi (1999) (see also Galdi, 2002) for a model of swimming micromotions in R 3 in the fluid governed by the stationary Stokes equation with the swimmer serving as the reference frame.…”

Section: Introduction and 3-d Model Settingmentioning

confidence: 99%

The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

Khapalov

2013

International Journal of Applied Mathematics and Computer Science

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We introduce and investigate the well-posedness of a model describing the self-propelled motion of a small abstract swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, typically associated with low Reynolds numbers. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by rotational and elastic Hooke forces. Models like this are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids.

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Section: Introduction and 3-d Model Settingmentioning

confidence: 99%

The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

Khapalov

2013

International Journal of Applied Mathematics and Computer Science

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“…Let us mention also the pioneering contribution to this research area of G. P. Galdi. He investigates, in the case of a fluid-rigid system filling the whole space, the existence of self-propelled motion in Ref. [14], [13].…”

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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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“…This system is a simplified model corresponding to the motion of a rigid body into a viscous incompressible fluid (see [24,10,11,9,19,25,15,17] for some references). In our case, we replace the Navier-Stokes system by the viscous Burgers equation and the rigid body is reduced to a point particle.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Feedback stabilization of a simplified 1d fluid–particle system

Badra

Takahashi

2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider the feedback stabilization of a simplified 1d model for a fluid-structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ < σ0. An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.Mathematics Subject Classification (2010): 74F10, 35Q35, 76D55, 93C20, 93D15.

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On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications

Cited by 183 publications

References 58 publications

The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

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

Feedback stabilization of a simplified 1d fluid–particle system

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