Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid

Boulakia, Muriel

doi:10.1016/j.crma.2004.11.003

Cited by 10 publications

(7 citation statements)

References 16 publications

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“…The result is proved in any time interval (0, T ), where T > 0 and for a small perturbation of a stable constant state provided there is no collision between the rigid body and the boundary ∂Ω of the fluid domain. In [5] the existence of weak solution is obtained in three dimension for an elastic structure immersed in a compressible fluid. The structure equation considered in [5] is strongly regularized in order to obtain suitable estimates on the S. Mitra JMFM elastic deformations.…”

Section: Bibliographical Commentsmentioning

confidence: 99%

Local Existence of Strong Solutions of a Fluid–Structure Interaction Model

Mitra

2020

J. Math. Fluid Mech.

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We are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.

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Section: Bibliographical Commentsmentioning

confidence: 99%

Local Existence of Strong Solutions of a Fluid–Structure Interaction Model

Mitra

2020

J. Math. Fluid Mech.

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“…similar to the classical form of the mass conservation in the Biot system. Note finally that, even in large displacement cases, when existence results exist they usually make use of energy estimate like (66), see for instance [10] for comparable systems.…”

Section: Energy Balancementioning

confidence: 99%

General coupling of porous flows and hyperelastic formulations—From thermodynamics principles to energy balance and compatible time schemes

Chapelle

Moireau

2014

European Journal of Mechanics - B/Fluids

120

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We formulate a general poromechanics model -within the framework of a two-phase mixture theory -compatible with large strains and without any simplification in the momentum expressions, in particular concerning the fluid flows. The only specific assumptions made are fluid incompressibility and isothermal conditions. Our formulation is based on fundamental physical principles -namely, essential conservation and thermodynamics laws -and we thus obtain a Clausius-Duhem inequality which is crucial for devising compatible constitutive laws. We then propose to model the solid behavior based on a generalized hyperelastic free energy potential -with additional viscous effects -which allows to represent a wide range of mechanical behaviors. The resulting formulation takes the form of a coupled system similar to a fluid-structure interaction problem written in an Arbitrary Lagrangian-Eulerian formalism, with additional volume-distributed interaction forces. We achieve another important objective by identifying the essential energy balance prevailing in the model, and this paves the way for further works on mathematical analyses, and time and space discretizations of the formulation.

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“…Let 6/5 < r < p. By standard embedding theorems we have W 1,r (Ω) ֒→ Hs(Ω) for somes > 0. In order to prove the claim we can proceed exactly like in the proof of [29, Proposition 2.28] once we showed that extension by 0 defines a bounded, linear operator from Hs(Ω) to H s (R 3 ) for some 0 < s <s. 4 To this end, it suffices to estimate the integral…”

Section: Variable Domainsmentioning

confidence: 99%

“…Fluid-solid interaction problems involving moving interfaces have been studied intensively during the last two decades. The interaction with elastic solids has proven to be particularly difficult, due to apparent regularity incompatibilities between the parabolic fluid phase and the hyperbolic or dispersive solid phase, see, e.g., [3,4,13,14,8,9,7,20,30,29] and the references therein. In [7,20] the global-in-time existence of weak solutions for the interaction of an incompressible, Newtonian fluid with a Kirchhoff-Love plate is shown.…”

Section: Introductionmentioning

confidence: 99%

Weak Solutions for An Incompressible, Generalized Newtonian Fluid Interacting with a Linearly Elastic Koiter Type Shell

Lengeler¹

2014

SIAM J. Math. Anal.

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In this paper we analyze the interaction of an incompressible, generalized Newtonian fluid with a linearly elastic Koiter shell whose motion is restricted to transverse displacements. The middle surface of the shell constitutes the mathematical boundary of the three-dimensional fluid domain. We show that weak solutions exist as long as the magnitude of the displacement stays below some (possibly large) bound which is determined by the geometry of the undeformed shell.

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Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid

Cited by 10 publications

References 16 publications

Local Existence of Strong Solutions of a Fluid–Structure Interaction Model

Local Existence of Strong Solutions of a Fluid–Structure Interaction Model

General coupling of porous flows and hyperelastic formulations—From thermodynamics principles to energy balance and compatible time schemes

Weak Solutions for An Incompressible, Generalized Newtonian Fluid Interacting with a Linearly Elastic Koiter Type Shell

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