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On global regular solutions to the Navier-Stokes equations with heat convection

Piotr Kacprzyk (2013)

Annales Polonici Mathematici

Global existence of regular solutions to the Navier-Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in a cylindrical pipe is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First we prove long time existence of regular solutions in [kT,(k+1)T]. Having T sufficiently large and imposing some decay estimates on | | f ( t ) | | L ₂ ( Ω ) , | | f , x ₃ ( t ) | | L ₂ ( Ω ) we continue the local solutions step by step up to a global one.

On Large Eddy Simulation and Variational Multiscale Methods in the numerical simulation of turbulent incompressible flows

Volker John (2006)

Applications of Mathematics

Numerical simulation of turbulent flows is one of the great challenges in Computational Fluid Dynamics (CFD). In general, Direct Numerical Simulation (DNS) is not feasible due to limited computer resources (performance and memory), and the use of a turbulence model becomes necessary. The paper will discuss several aspects of two approaches of turbulent modeling—Large Eddy Simulation (LES) and Variational Multiscale (VMS) models. Topics which will be addressed are the detailed derivation of these...

On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion

Piotr Mucha, Wojciech Zajączkowski (2000)

Applicationes Mathematicae

The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved. We show the existence of solutions with lowest possible regularity for this problem such that u ∈ W r 2 , 1 ( Ω ˜ T ) with r>3. The existence is proved by the method of successive approximations where the solvability of the Cauchy-Neumann problem for the Stokes system is applied. We have to underline that in the L p -approach the Lagrangian coordinates must be used. We are looking...

On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations

Xuejun Xu, C. O. Chow, S. H. Lui (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, a Dirichlet-Neumann substructuring domain decomposition method is presented for a finite element approximation to the nonlinear Navier-Stokes equations. It is shown that the Dirichlet-Neumann domain decomposition sequence converges geometrically to the true solution provided the Reynolds number is sufficiently small. In this method, subdomain problems are linear. Other version where the subdomain problems are linear Stokes problems is also presented.

On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations

Xuejun Xu, C. O. Chow, S. H. Lui (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a Dirichlet-Neumann substructuring domain decomposition method is presented for a finite element approximation to the nonlinear Navier-Stokes equations. It is shown that the Dirichlet-Neumann domain decomposition sequence converges geometrically to the true solution provided the Reynolds number is sufficiently small. In this method, subdomain problems are linear. Other version where the subdomain problems are linear Stokes problems is also presented.

On some free boundary problems for Navier-Stokes equations

Ewa Zadrzyńska (2005)

Banach Center Publications

In this survey we report on existence results for some free boundary problems for equations describing motions of both incompressible and compressible viscous fluids. We also present ways of controlling free boundaries in two cases: a) when the free boundary is governed by surface tension, b) when surface tension does not occur.

On stability of the P n mod / P n element for incompressible flow problems

Petr Knobloch (2006)

Applications of Mathematics

It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of n th order accuracy in the energy norm called P n elements. For n ≤ 3 we show that the stability condition holds if the velocity...

On suitable inlet boundary conditions for fluid-structure interaction problems in a channel

Jan Valášek, Petr Sváček, Jaromír Horáček (2019)

Applications of Mathematics

We are interested in the numerical solution of a two-dimensional fluid-structure interaction problem. A special attention is paid to the choice of physically relevant inlet boundary conditions for the case of channel closing. Three types of the inlet boundary conditions are considered. Beside the classical Dirichlet and the do-nothing boundary conditions also a generalized boundary condition motivated by the penalization prescription of the Dirichlet boundary condition is applied. The fluid flow...

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