Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality

Kračmar, Stanislav; Neustupa, Jiřı́

doi:10.1002/mana.201700228

Cited by 24 publications

(24 citation statements)

References 29 publications

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“…A different approach has been suggested in papers [13][14][15]. There the authors have considered the stationary and non-stationary problems and impose an additional condition on Γ 2 , that enables one to estimate the kinetic energy of a possible reverse flow and obtain an a priori estimate of the energy type.…”

Section: On Some Previous Related Existential Resultsmentioning

confidence: 99%

“…at first on the level of approximations and then considering an appropriate limit transition) in papers [14] and [15]. However, it must be noted that while the convex set, corresponding to our K c t , is defined in a rather artificial way in [14] and [15], our K c t has a good physical sense. Naturally, the change of set K c t requires a new technique in the derivation of approximations.…”

Section: Resultsmentioning

confidence: 99%

“…By analogy with [15], one can show that if v is a solution of a problem (P) then there exists an associated pressure p as a distribution in Ω × (0, T ). The pair (v, p) satisfies the equations (1), (2) in the sense of distributions in Ω × (0, T ).…”

Section: Remarkmentioning

confidence: 99%

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Modeling of Flows Through a Channel by the Navier–stokes Variational Inequalities

Kračmar

Neustupa

2021

Acta Polytech

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We deal with a mathematical model of a flow of an incompressible Newtonian fluid through a channel with an artificial boundary condition on the outflow. We explain how several artificial boundary conditions formally follow from appropriate variational formulations and the wayone expresses the dynamic stress tensor. As the boundary condition of the “do nothing”–type, that is predominantly considered to be the most appropriate from the physical point of view, does not enable one to derive an energy inequality, we explain how this problem can be overcome by using variational inequalities. We derive a priori estimates, which are the core of the proofs, and present theorems on the existence of solutions in the unsteady and steady cases.

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Section: On Some Previous Related Existential Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

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Modeling of Flows Through a Channel by the Navier–stokes Variational Inequalities

Kračmar

Neustupa

2021

Acta Polytech

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“…At present, (DN) is a well-established boundary condition to represent natural outflows (see, e.g., [7] and the references therein). For a recent discussion on the derivation and physical meaning of the do-nothing condition (DN), we refer to [31].…”

Section: Re ∂V ∂Nmentioning

confidence: 99%

“…This approach was introduced by Bruneau and Fabrie in [13] and then followed by several authors (see, e.g., [2,12,19,33]). The other is due to Kračmar and Neustupa and consists in supplementing the do-nothing condition by a bound for the backflows (see [29][30][31]).…”

Section: Re ∂V ∂Nmentioning

confidence: 99%

The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow

Ceretani

Rautenberg

2018

Z. Angew. Math. Phys.

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A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. The problem is motivated by modeling energy systems in rooms that possess an outlet where the fluid can freely flow, known as an "open boundary". Numerical and experimental investigations available from the literature on heated cavities with open boundaries suggest that the heat transfer at the outlet depends on the temperature at the boundary, the velocity of the fluid, and the outside temperature. Aiming to include this feature in the model, we propose a novel non-smooth boundary condition which is deduced from physical assumptions. We show that this condition is compatible with two approaches at dealing with the "do-nothing" boundary condition for the fluid: (1) the "directional do-nothing" condition and (2) the "do-nothing" condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem.

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The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

Neustupa

2022

Mathematische Nachrichten

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We study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the Lr$L^r$‐setup. The mathematical model used here is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated using three types of boundary conditions: the conditions of periodicity on the “lower” and “upper” parts of the boundary, the Dirichlet boundary conditions on the “inflow” and on the profile and an artificial “do nothing”‐type boundary condition on the “outflow.” Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in W1,r(Ω)$\mathbf {W}^{1,r}(\Omega )$ and its continuous dependence on the data. We explain the sense in which the “do nothing” boundary condition on the “outflow” is satisfied.

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Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality

Cited by 24 publications

References 29 publications

Modeling of Flows Through a Channel by the Navier–stokes Variational Inequalities

Modeling of Flows Through a Channel by the Navier–stokes Variational Inequalities

The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow

The weak Stokes problem associated with a flow through a profile cascade in Lr$L^r$‐framework

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