On Multiple Solutions to the Steady Flow of Incompressible Fluids Subject to Do-nothing or Constant Traction Boundary Conditions on Artificial Boundaries

Lanzendörfer, Martin; Hron, J.

doi:10.1007/s00021-019-0472-z

Cited by 5 publications

(5 citation statements)

References 15 publications

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“…The impossibility of standard energy-type estimates in case of flows with the classical "do nothing" boundary condition (1.8) (which corresponds to 𝜉 = −1 in 1.9) on a part of the boundary is demonstrated on a set of concrete examples in Lanzendorfer and Hron. 15 One can realistically expect that analogous examples can also be constructed for −1 < 𝜉 < 0. The problem lies in the fact that the energy estimates contain the integral ∫ Γ out (u • n) |u| 2 dl, which may have a "bad" sign in the case of "large" reverse flows u on Γ out , that is, flows satisfying u • n < 0.…”

Section: The Navier-stokes Boundary-value Problem On One Spatial Periodmentioning

confidence: 99%

Existence of steady flows of a viscous incompressible fluid through a profile cascade and their L^r‐regularity

Neustupa

2021

Math Methods in App Sciences

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The paper deals with the steady Navier-Stokes problem, describing a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to one spatial period Ω, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves Γ 0 and Γ 1 , the Dirichlet boundary conditions on Γ in and Γ P , and an artificial boundary condition on Γ out (see Figure 1). For "small data," we consider the so-called "do nothing" boundary condition on Γ out and prove the existence and uniqueness of a weak and strong solution in the L r -framework. For "large data," we consider an appropriately modified "do nothing" condition on Γ out and prove the existence of a weak and strong solution.

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Section: The Navier-stokes Boundary-value Problem On One Spatial Periodmentioning

confidence: 99%

Existence of steady flows of a viscous incompressible fluid through a profile cascade and their L^r‐regularity

Neustupa

2021

Math Methods in App Sciences

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“…Without the corrector 1 2 |v| 2 n, relation (1.11) is sometimes called constant traction boundary condition (cf [45]), which is just the do-nothing boundary condition (2.1), but for the symmetric part of the velocity gradient. As we shall see, we have the freedom to replace Dv in the definition of I ε by other types of gradients, leading, for example, to boundary conditions involving ∇v rather than Dv.…”

Section: Interpretation Of (110) and (111)mentioning

confidence: 99%

Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

Bathory

Stefanelli

2022

Nonlinearity

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This paper focuses on the so-called weighted inertia-dissipation-energy variational approach for the approximation of unsteady Leray–Hopf solutions of the incompressible Navier–Stokes system. Initiated in (Ortiz et al 2018 Nonlinearity 31 5664–82), this variational method is here extended to the case of non-Newtonian fluids with power-law index r ⩾ 11/5 in three space dimension and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimisation. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition − 1 2 | v | 2 n + T n = 0 . This provides the first theoretical explanation for the onset of such boundary conditions.

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“…Prescribing the correct boundary conditions, especially at the outflow boundary, is very important in correctly capturing the physiological flow conditions within a subregion of the cardiovascular system that is being modeled [117]. From the analysis point of view, it is now known that some of the numerically convenient outlet boundary conditions, such as the Neumann, or do nothing, outlet boundary conditions, may produce instabilities [132] or ruin well-posedness by producing multiple solutions to the steady flow, as was recently shown in [77]. In this paper we consider dynamic pressure data (2.2) both at the inlet and at the outlet, which is a boundary condition that is consistent with the energy of the coupled problem.…”

Section: Description Of the Main Problemmentioning

confidence: 99%

Moving boundary problems

Čanić

2020

Bull. Amer. Math. Soc.

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Moving boundary problems are ubiquitous in nature, technology, and engineering. Examples include the human heart and heart valves interacting with blood flow, biodegradable microbeads swimming in water to clean up water pollution, a micro camera in the human intestine used for early colon cancer detection, and the design of next-generation vascular stents to prop open clogged arteries and to prevent heart attacks. These are time-dependent, dynamic processes, which involve the interaction between fluids and various structures. Analysis and numerical simulation of fluid-structure interaction (FSI) problems can provide insight into the "invisible" properties of flows and structures, and can be used to advance design of novel technologies and improve the understanding of many physical and biological phenomena. Mathematical analysis of FSI models is at the core of this understanding. In this paper we give a brief survey of recent progress in the area of mathematical well-posedness for moving boundary problems describing fluid-structure interaction between incompressible, viscous fluids and elastic, viscoelastic, and rigid solids.

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On Multiple Solutions to the Steady Flow of Incompressible Fluids Subject to Do-nothing or Constant Traction Boundary Conditions on Artificial Boundaries

Cited by 5 publications

References 15 publications

Existence of steady flows of a viscous incompressible fluid through a profile cascade and their L^r‐regularity

Existence of steady flows of a viscous incompressible fluid through a profile cascade and their L^r‐regularity

Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

Moving boundary problems

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On Multiple Solutions to the Steady Flow of Incompressible Fluids Subject to Do-nothing or Constant Traction Boundary Conditions on Artificial Boundaries

Cited by 5 publications

References 15 publications

Existence of steady flows of a viscous incompressible fluid through a profile cascade and their Lr‐regularity

Existence of steady flows of a viscous incompressible fluid through a profile cascade and their Lr‐regularity

Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

Moving boundary problems

Contact Info

Product

Resources

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Existence of steady flows of a viscous incompressible fluid through a profile cascade and their L^r‐regularity

Existence of steady flows of a viscous incompressible fluid through a profile cascade and their L^r‐regularity