Motion of an Elastic Solid inside an Incompressible Viscous Fluid

Coutand, Daniel; Shkoller, Steve

doi:10.1007/s00205-004-0340-7

Cited by 175 publications

(212 citation statements)

References 16 publications

(22 reference statements)

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“…We denote by N (u 0 ) a generic constant depending on u 0 3 . With these requirements, we will only get the H 7 2 regularity of the moving domain η(Ω) and not of the mapping η.…”

Section: Optimal Regularitymentioning

confidence: 99%

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Well-posedness of the free-surface incompressible Euler equations with or without surface tension

Coutand

Shkoller

2007

J. Amer. Math. Soc.

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359

498

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We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.

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“…We denote by N (u 0 ) a generic constant depending on u 0 3 . With these requirements, we will only get the H 7 2 regularity of the moving domain η(Ω) and not of the mapping η.…”

Section: Optimal Regularitymentioning

confidence: 99%

“…We now come to H 1 , which will require more care, and will provide us with the regularity ofη lκ (Ω) in H 7 2 independent of κ. We have…”

Section: Well-posedness Of the Free-surface Euler Equations 897mentioning

confidence: 99%

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

Coutand

Shkoller

2007

J. Amer. Math. Soc.

Self Cite

359

498

View full text Add to dashboard Cite

show abstract

“…This system couples parabolic (fluid) and hyperbolic (structure) equations, and different behavior of these equations induces a lack of regularity on the interface between the fluid and the structure, which is also mentioned in Section 3.2.3. The existence and uniqueness of such problem is proved for a special kind of structural material [10] . In addition, the free boundary problem is also a challenging aspect in the FSI problem, while the analysis of a free boundary problem for fluid dynamics and species motion is obtained [77] .…”

Section: Review Of Theoretical Resultsmentioning

confidence: 98%

Mathematical modeling and simulation of the evolution of plaques in blood vessels

et al. 2015

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The formation of plaques is one of the main causes for the blockage of arteries. This can lead to ischaemic brain or myocardial infarctions as well as other cardiovascular diseases. Possible biochemical and biomechanical processes contribute to the development of plaque growth and rupture. The main biochemical processes are the penetration of monocytes and the accumulation of foam cells in the vessel wall, leading to the formation and growth of plaques. The biomechanical forces can be measured by observing stresses in the blood flow and the vessel wall, which may lead to the rupture of plaques.In this thesis, we formulate an appropriate model to describe the evolution of plaques. The model consists of both the interaction between the blood flow and the vessel wall, and the growth of plaques due to the penetration of monocytes from the blood flow into the vessel wall. The Navier-Stokes equations and the elastic structure equations are used to describe the dynamics of fluid (blood flow) and the mechanics of structure (vessel wall). The motion of monocytes is described by the convection-diffusion-reaction equation, coupled with an equation for the accumulation of foam cells. Finally the metric of growth is introduced to accurately determine the stress tensor, and its evolution equation is derived. The variational formulation of the model is transformed into the ALE (Arbitrary Lagrangian-Eulerian) formulation, and all the equations are rewritten in the fixed domain. Temporal discretization is achieved with finite differences and spatial discretization is based on the Galerkin finite element method. The nonlinear systems are linearized and solved by the Newton method.Based on the model and the numerical methods above, numerical simulations are performed by using the software Gascoigne. The obtained numerical results make an agreement with the observation, and support the assumption that the penetration of monocytes and the accumulation of foam cells lead to the formation and growth of plaques, and that the evolution of plaques induces the increase of stresses in the vessel wall, which is an indicator of plaque rupture. Zusammenfassung

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“…A variety of marine animals are understood to propel themselves through the shedding of coherent vortex structures; dynamic models and control strategies for such mechanisms promise to impact the development of agile and efficient biomimetic robotic vehicles. There has been quite a bit of theoretical research in the past decade related to the existence and uniqueness of solutions and functional analytic aspects of such coupled systems (for cases including deformable and rigid bodies in inviscid and viscous frameworks, see, for example, [12] and [11] and references therein). However, the dynamics and control of such coupled systems, especially with a focus on vortical structures, remains a fairly open area for theoretical research.…”

Section: Discussionmentioning

confidence: 99%

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

Shashikanth

Sheshmani

Kelly

et al. 2007

Theor. Comput. Fluid Dyn.

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We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot-Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie-Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.

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Motion of an Elastic Solid inside an Incompressible Viscous Fluid

Cited by 175 publications

References 16 publications

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

Mathematical modeling and simulation of the evolution of plaques in blood vessels

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

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