On the Weak Solution of the Fluid-Structure Interaction Problem for Shear-Dependent Fluids

Hundertmark, Anna; Lukáčová-Medviďová, Mária; Nečasová, Šárka

doi:10.1007/978-3-0348-0939-9_16

Cited by 4 publications

(10 citation statements)

References 8 publications

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“…In [7], [8] we have studied the existence of weak solution to the similar fluidstructure interaction problem for shear-dependent fluids with viscosity obeying the power-law µ(|e(v)|) = µ(1 + |e(v)| 2 ) p−2 2…”

Section: Discussionmentioning

confidence: 99%

“…In the case of a twodimensional fluid flow and one-dimensional structure the authors obtained in [5,6] the existence of a weak solution for the same regularity of the domain deformation as in [3] using the Schauder fixed point argument, without giving any details on the construction of the domain deformation and without providing the uniqueness of the solution. Using a similar fixed point procedure for the geometry the existence of weak solution for a fluid-structure interaction problem for a generalized power-law shear-dependent fluid including the Newtonian case has been shown in our recent joint works [7,8], see also [9].…”

Section: Introductionmentioning

confidence: 93%

“…Theorem 3.1 (Continuous dependence on data). Let (u 1 , η 1 ), (u 2 , η 2 ) be two weak solutions of the initial boundary value problem (1)- (8) transformed to the rectangular fixed rectangular domain D satisfying (13). Let the corresponding domain deformation be given by some functions h 1 , h 2 satisfying (9).…”

Section: Final Estimatementioning

confidence: 99%

“…In the following lines we present a corrected proof of those integral equicontinuity estimate for u κ , which is independent on ε. To this end we however shall simplify our artificial compressibility condition and consider only elliptic equation for pressure, −ε∆p κ + divv κ = 0 in Ω(h), ∇p κ · n = 0 on ∂Ω(h), Ω(h) p κ = 0 in our (κ, ε)-approximation of (1)- (8). Note, that the proof of existence of weak solution for fixed ε, κ with only the elliptic (instead of parabolic) compressibility approximation can be obtained analogously as the proof in [7,Section 4], compare also [3], using the same techniques.…”

Section: Acknowledgmentmentioning

confidence: 99%

See 3 more Smart Citations

On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

Hundertmark-Zaušková

2019

Journal of Differential Equations

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 93%

Section: Final Estimatementioning

confidence: 99%

Section: Acknowledgmentmentioning

confidence: 99%

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On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

Hundertmark-Zaušková

2019

Journal of Differential Equations

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“…When the shell is replaced by a plate (a fourth order operator added to (1.1) above), existence has been established [15,33,35,38]. The fourth order term seems essential to the existence of solutions.…”

mentioning

confidence: 99%

Analysis of a Coupled Fluid-Structure Model with Applications to Hemodynamics

Rebollo¹,

Girault²,

Murat³

et al. 2016

SIAM J. Numer. Anal.

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Abstract. We propose and analyze a simplified fluid-structure coupled model for flows with compliant walls. As in [F. Nobile and C. Vergara, SIAM J. Sci. Comput., 30 (2008), pp. 731-763], the wall reaction to the fluid is modeled by a small displacement viscoelastic shell where the tangential stress components and displacements are neglected. We show that within this small displacement approximation a transpiration condition can be used which does not require an update of the geometry at each time step, for pipe flow at least. Such simplifications lead to a model which is well posed and for which a semi-implicit time discretization can be shown to converge. We present some numerical results and a comparison with a standard test case taken from hemodynamics. The model is more stable and less computer demanding than full models with moving mesh. We apply the model to a three-dimensional arterial flow with a stent. 1. Introduction. Fluid-structure interaction (FSI) is computationally challenging because it involves moving geometries and the coupling of Lagrangian and Eulerian models [31,41]; most popular applications are for biofluid dynamics, hemodynamics, and aerospace. This paper is a contribution to FSI algorithms, not to hemodynamics as such; but since we need to compare solutions we chose this field because it is well documented. Other applications like aircraft design and tires for instance have additional intrinsic difficulties which complicate the comparison with a simplified model.Computational hemodynamics has important applications (see [62,27] or [61,46], and the references therein). Modeling flow in a large blood vessel can be done with incompressible Navier-Stokes equations. The blood vessel is more difficult to model as it is a complex material for which the rheology is unclear because it is different in vitro from in vivo [62]. No doubt future computers will be able to handle this complexity and one will use large displacement nonlinear models for the structure [63]. However in the meantime there is a need for fast, well understood, and appropriate though less accurate models.To handle the complexity of moving walls, the method of immersed boundaries has been used-if not invented-by Peskin, the pioneer of computational hemodynamics [49,48,50,64]. The mathematical analysis of this method is difficult [5,42] and it is also hard to incorporate an elaborate viscoelastic model for the vessels, how-

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Existence of Global Strong Solutions to a Beam–Fluid Interaction System

Grandmont

Hillairet

2016

Arch Rational Mech Anal

102

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Abstract. We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure.

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On the Weak Solution of the Fluid-Structure Interaction Problem for Shear-Dependent Fluids

Cited by 4 publications

References 8 publications

On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

Analysis of a Coupled Fluid-Structure Model with Applications to Hemodynamics

Existence of Global Strong Solutions to a Beam–Fluid Interaction System

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