Solutions of the Navier–Stokes equations with various types of boundary conditions

Beneš, Michal; Kučera, Petr

doi:10.1007/s00013-012-0387-x

Cited by 17 publications

(38 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There appear various artificial boundary conditions in the literature, see e.g. [1][2][3][4][5][6][7][8]. Automatically following from an appropriate weak formulation of the considered problem if one a priori assumes a sufficient regularity of a solution, boundary conditions are usually called the "do nothing" conditions.…”

Section: The Considered Initial-boundary Value Problemmentioning

confidence: 99%

See 1 more Smart Citation

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

Kračmar

Neustupa

2021

Acta Polytech

View full text Add to dashboard Cite

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.

show abstract

Section: The Considered Initial-boundary Value Problemmentioning

confidence: 99%

“…(See e.g. [1,6,9] for more details.) An example, and probably the most often used artificial boundary condition is − pn + ν ∂v ∂n = g on Γ 2 × (0, T ),…”

Section: The Considered Initial-boundary Value Problemmentioning

confidence: 99%

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

Kračmar

Neustupa

2021

Acta Polytech

View full text Add to dashboard Cite

show abstract

“…This is the reason why the known existential results for the problem (1.1)-(1.5) assume that the given data of the problem are in some sense "small", or the time interval (0, ) is "sufficiently short". (See [1,17,18].) The global in time existence of a weak solution of the problem (1.1)-(1.5) for "large" data, which is well known for the Navier-Stokes equations with other boundary conditions than (1.4), is an open problem.…”

Section: The Question Of Solvability Of the Problem (11)-(15) And Rmentioning

confidence: 99%

“…(See e.g. , , for more details.) In this paper, we use the inhomogeneous “do nothing” boundary condition

- p n + ν \nabla u \cdot n = g on {normalΓ}_{2} \times false(0, T false),

where n denotes the outer normal vector field and g is a given function.…”

Section: Introduction and Notationmentioning

confidence: 99%

See 1 more Smart Citation

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

Kračmar

Neustupa

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

We prove the global in time existence of a weak solution to the variational inequality of the Navier–Stokes type, simulating the unsteady flow of a viscous fluid through the channel, with the so‐called “do nothing” boundary condition on the outflow. The condition that the solution lies in a certain given, however arbitrarily large, convex set and the use of the variational inequality enables us to derive an energy‐type estimate of the solution. We also discuss the use of a series of other possible outflow “do nothing” boundary conditions.

show abstract

Analysis of viscous incompressible flows of micropolar fluidswith thermal convection and mixed boundary conditions

Beneš

2023

Proc Appl Math and Mech

Self Cite

View full text Add to dashboard Cite

The Navier‐Stokes equations do not take into account the microstructure of the fluid in the sense that they do not consider the angular momentum of small particles of the fluid due to their rotation. The model of micropolar fluid represents a generalization of the well‐established Navier‐Stokes equations, in such a way that it introduces a new kinematic vector field called microrotation (the angular velocity field of rotation of particles) and adds a new vectorial equation, expressing the conservation of the angular momentum. We will be concerned with the initial boundary value problem for the flow of micropolar heat conducting fluids in a two‐dimensional channel with mixed boundary conditions. The considered boundary conditions are of three types: the Dirichlet boundary conditions on the inflow, the Navier type conditions on solid surfaces and Neumann‐type boundary conditions on the outflow of the channel. The homogeneous Dirichlet boundary conditions on solid surfaces for the microrotation is commonly used in practice. However, imposing such condition is doubtful from the physical point of view. For that reason, more general boundary conditions for the microrotation were proposed throughout the engineering literature to take into account the rotation of the microelements on the solid boundary, linking the velocity and microrotation through the so‐called boundary viscosity. The well‐posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present contribution is devoted to the analysis of the existence and uniqueness of the solution.

show abstract

Solutions of the Navier–Stokes equations with various types of boundary conditions

Cited by 17 publications

References 12 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

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

Analysis of viscous incompressible flows of micropolar fluidswith thermal convection and mixed boundary conditions

Contact Info

Product

Resources

About