# MODELING OF FLOWS THROUGH A CHANNEL BY THE NAVIER–STOKES VARIATIONAL INEQUALITIES

## DOI:

https://doi.org/10.14311/AP.2021.61.0089## Keywords:

Variational inequality, Navier--Stokes equation, “do nothing” outflow boundary condition## Abstract

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 way

one 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.

## References

M. Beneš, P. Kučera. Solutions of the navier–stokes equations with various types of boundary conditions. Archiv der Mathematik 98:487–497, 2012. doi:10.1007/s00013-012-0387-x.

C.-H. Bruneau, P. Fabrie. New efficient boundary conditions for incompressible Navier-Stokes equations: A well-posedness result. Mathematical Modelling and Numerical Analysis 30(7):815–840, 1996. doi:10.1051/m2an/1996300708151.

M. Feistauer, T. Neustupa. On some aspects of analysis of incompressible flow through cascades of profiles. Operator Theory, Advances and Applications 147:257–276, 2004.

M. Feistauer, T. Neustupa. On non-stationary viscous incompressible flow through a cascade of profiles. Mathematical Methods in the Applied Sciences 29(16):1907–1941, 2006. doi:10.1002/mma.755.

M. Feistauer, T. Neustupa. On the existence of a weak solution of viscous incompressible flow past a cascade of profiles with an arbitrarily large inflow. Journal of Mathematical Fluid Mechanics 15(15):701–715, 2013. doi:10.1007/s00021-013-0135-4.

J. G. Heywood, R. Rannacher, S. Turek. Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations. International Journal for Numerical Methods in Fluids 22(5):325–352, 1996. doi:10.1002/(SICI)1097- 0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y.

T. Neustupa. A steady flow through a plane cascade of profiles with an arbitrarily large inflow – the mathematical model, existence of a weak solution. Applied Mathematics and Computation 272:687–691, 2016. doi:10.1016/j.amc.2015.05.066.

T. Neustupa. The weak solvability of the steady problem modelling the flow of a viscous incompressible heat–conductive fluid through the profile cascade. International Journal of Numerical Methods for Heat & Fluid Flow 27(7):1451–1466, 2017. doi:10.1108/HFF-03-2016-0104.

P. Kučera, Z. Skalak. Local solutions to the Navier–Stokes equations with mixed boundary conditions. Acta Applicandae Mathematicae 54:275–288, 1998. doi:10.1023/A:1006185601807.

P. Kučera. Basic properties of the non-steady Navier– Stokes equations with mixed boundary conditions in a bounded domain. Ann Univ Ferrara 55:289–308, 2009.

M. Braack, P. B. Mucha. Directional do-nothing condition for the navier-stokes equations. Journal of Computational Mathematics 32(5):507–521, 2014. doi:10.4208/jcm.1405-m4347.

M. Lanzendörfer, J. Stebel. On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities. Applications of Mathematics 56(3):265–285, 2011. doi:10.1007/s10492-011-0016-1.

S. Kračmar, J. Neustupa. Modelling of flows of a viscous incompressible fluid through a channel by means of variational inequalities. ZAMM 74(6):637–639, 1994.

S. Kračmar, J. Neustupa. A weak solvability of a steady variational inequality of the Navier–Stokes type with mixed boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 47(6):4169–4180, 2001. Proceedings of the Third World Congress of Nonlinear Analysts, doi:10.1016/S0362-546X(01)00534-X.

S. Kračmar, J. Neustupa. Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality. Mathematische Nachrichten 291(11–12):1801–1814, 2018. doi:10.1002/mana.201700228.

P. Deuring, S. Kračmar. Artificial boundary conditions for the Oseen system in 3D exterior domains. Analysis 20:65–90, 2012.

P. Deuring, S. Kračmar. Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: Approximation by flows in bounded domains. Mathematische Nachrichten 269–270:86–115, 2004. doi:10.1002/mana.200310167.

R. Temam. Navier-Stokes Equations. North-Holland, Amsterdam, 1977.

J. L. Lions, E. Magenes. Nonhomogeneous Boundary Value Problems and Applications I. Springer–Verlag, New York, 1972. doi:10.1007/978-3-642-65161-8.

I. Ekeland, R. Temam. Convex Analysis and Variational Problems. North Holland Publishing Company, Amsterdam–Oxford–New York, 1976. doi:10.1137/1.9781611971088.BM.

G. P. Galdi. An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady–State Problems. Springer–Verlag, 2nd edn., 2011.

J. L. Lions. Quelques méthodes de résolution des problèmes âux limites non linéaire. Dunod, Gauthier–Villars, Paris, 1969.

J. Marschall. The trace of sobolev–slobodeckij spaces on lipschitz domains. Manuscipta Mathematica 58:47–65, 1987. doi:10.1007/BF01169082.

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