A Weak Solvability of the Navier-Stokes Equation with Navier’s Boundary Condition Around a Ball Striking theWall

Neustupa, Jiřı́; Penel, Patrick

doi:10.1007/978-3-642-04068-9_24

Cited by 28 publications

(32 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The non-slip condition prescribes the adherence of fluid particles to the solid boundaries and, as a consequence of a regularity of the fluid velocity, permits the creation of fine boundary layer that does not allow the contact of the solids.Another method for coupling of the fluid and of the bodies admits the slippage of fluid particles at the boundaries, which is described by Navier's boundary condition. The first step in this direction of the study of Navier's condition was done by Neustupa, Penel [22], [23], who demonstrate that the collision with a wall can occur for a prescribed movement of a solid ball, when the slippage was allowed on both boundaries. We refer for a discussion of Navier's boundary condition to Introduction of [20].…”

mentioning

confidence: 99%

Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition

Chemetov

Nečasová

Muha

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

We consider a coupled PDE-ODE system describing the motion of the rigid body in a container filled with the incompressible, viscous fluid. The fluid and the rigid body are coupled via Navier's slip boundary condition. We prove that the local in time strong solution is unique in the larger class of weak solutions on the interval of its existence. This is the first weak-strong uniqueness result in the area of fluid-structure interaction with a moving boundary.1 kinematic coupling condition in the literature is the standard no-slip boundary condition: the continuity of velocities of the fluid and the solids at the fluid-body interfaces. Such approach has been investigated by many authors, e.g. [3]-[8], [17,24] and references within. Despite lot of progress there are only few uniqueness results for weak solutions. The uniqueness of weak solution for the fluid-rigid body system in 2D was proven in [14]. The uniqueness of weak solution to 3D Navier-Stokes equation is a famous open problem (see e.g. [10]). However, it is well known that strong solution (i.e. solution with some extra regularity or integrability) is unique in a larger class of weak solution on interval of its existence [25]. These type of results are called weak-strong uniqueness results. The main result of this paper is to prove a weak-strong uniqueness result for the fluid-rigid body system. We also mention that in [6] the authors studied the motion of rigid body containing a cavity filled with the fluid and proved a weak-strong uniqueness result for that problem.However, it has been shown in [15], [16], [26] that the non-slip condition exhibits an unrealistic phenomenon: two smooth solids can not touch each other. The non-slip condition prescribes the adherence of fluid particles to the solid boundaries and, as a consequence of a regularity of the fluid velocity, permits the creation of fine boundary layer that does not allow the contact of the solids.Another method for coupling of the fluid and of the bodies admits the slippage of fluid particles at the boundaries, which is described by Navier's boundary condition. The first step in this direction of the study of Navier's condition was done by Neustupa, Penel [22], [23], who demonstrate that the collision with a wall can occur for a prescribed movement of a solid ball, when the slippage was allowed on both boundaries. We refer for a discussion of Navier's boundary condition to Introduction of [20]. In this last work a local in time existence result was demonstrated for the motion of the fluid and an elastic structure with prescribed Navier's condition on the boundaries. Motivated by these works, in this paper we study coupling via Navier's slip boundary condition. Also we mention the article [12] where a local existence up to collisions of a weak solution for a fluid-solid structure was proved. The existence of a strong solution in 2D case was proven by Wang [29].The global in time existence of the weak solution was proven in [2] for a mixed case, when Navier's condition was given on the solid boundar...

show abstract

mentioning

confidence: 99%

Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition

Chemetov

Nečasová

Muha

2019

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…This definition of a guarantees that a is divergence-free. The validity of (3) as well as conditions (a1)-(a5) crucially depends on the form and properties of the function w. The details can be found in our paper [6]. It is important to mention that the validity of (a4), namely inequality (7), leads to the restriction, that |δ t | is "sufficiently small" in comparison with coefficients ν and γ for t in a certain neighbourhood of the instant of collision t c .…”

Section: Example: the Flow Around Two Striking Bodies With Ball-shapementioning

confidence: 97%

“…In addition to the considered situation, when bodies B t 1 and B t 2 strike with ball-like surfaces, we also discuss the case of more general front surfaces of B t 1 and B t 2 in paper [6]. We give a hint how to construct the auxiliary function a.…”

Section: Example: the Flow Around Two Striking Bodies With Ball-shapementioning

confidence: 98%

See 1 more Smart Citation

The Navier–Stokes equations with Navier's boundary condition around moving bodies in presence of collisions

Neustupa

Penel

2009

Comptes Rendus. Mathématique

Self Cite

View full text Add to dashboard Cite

“…The investigation of incompressible fluids in time dependent domains started with a seminal paper of Ladyzhenskaya [20], Fujita et al [15], see also [24,25,26,29] for more recent results in this direction.…”

Section: Introductionmentioning

confidence: 99%

Flow of heat conducting fluid in a time-dependent domain

Kreml

Mácha

Nečasová

et al. 2018

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

We consider a flow of heat conducting fluid inside a moving domain whose shape in time is prescribed. The flow in this case is governed by the Navier-Stokes-Fourier system consisting of equation of continuity, momentum balance, entropy balance and energy equality. The velocity is supposed to fulfill the full-slip boundary condition and we assume that the fluid is thermally isolated. In the presented article we show the existence of a variational solution.

show abstract

A Weak Solvability of the Navier-Stokes Equation with Navier’s Boundary Condition Around a Ball Striking theWall

Cited by 28 publications

References 10 publications

Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition

Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition

The Navier–Stokes equations with Navier's boundary condition around moving bodies in presence of collisions

Flow of heat conducting fluid in a time-dependent domain

Contact Info

Product

Resources

About