NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID AROUND A ROTATING AND TRANSLATING RIGID BODY

Authors

  • Paul Deuring Université du Littoral Côte d’Opale, Centre Universitaire de la Mi-Voix 50, rue F.Buisson CS 80699, 62228 Calais Cedex, France
  • Stanislav Kračmar Czech Technical University in Prague, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic; Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 11567 Praha 1, Czech Republic
  • Šárka Nečasová Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 11567 Praha 1, Czech Republic

DOI:

https://doi.org/10.14311/AP.2021.61.0005

Keywords:

Incompressible fluid , rigid body, exterior domain, estimates of pressure, leading terms, artificial boundary conditions

Abstract

We consider the linearized and nonlinear systems describing the motion of incompressible flow around a rotating and translating rigid body Ɗ in the exterior domain Ω = ℝ3 \ D , where Ɗ ⊂ ℝ3 is open and bounded, with Lipschitz boundary. We derive the L∞-estimates for the pressure and investigate the leading term for the velocity and its gradient. Moreover, we show that the velocity essentially behaves near the infinity as a constant times the first column of the fundamental solution of the Oseen system. Finally, we consider the Oseen problem in a bounded domain ΩR := BR ∩ Ω under certain artificial boundary conditions on the truncating boundary ∂BR, and then we compare this solution with the solution in the exterior domain Ω to get the truncation error estimate.

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References

C. Conca, J. San Martín H., M. Tucsnak. Motion of a rigid body in a viscous fluid. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics 328(6):473 – 478, 1999. doi:10.1016/S0764-4442(99)80193-1.

B. Desjardins, M. J. Esteban. Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Archive for Rational Mechanics and Analysis 146:59–71, 1999. doi:10.1007/s002050050136.

M. Gunzburger, H.-C. Lee, G. Seregin. Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. Journal of Mathematical Fluid Mechanics 2:219–266, 2000. doi:10.1007/PL00000954.

K. H. Hoffmann, V. N. Starovoitov. On a motion of a solid body in a viscous fluid. Two dimensional case. Advances in Mathematical Sciences and Applications 9:633–648, 1999.

G. P. Galdi. Handbook of Mathematical Fluid Dynamics, vol. 1, chap. On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, pp. 653–791. North-Holland, Amsterdam, 2002.

R. Farwig, M. Krbec, Š. Necasová. A weighted Lq-approach to Oseen flow around a rotating body. Mathematical Methods in the Applied Sciences 31(5):551–574, 2008. doi:10.1002/mma.925.

R. Farwig. An lq-analysis of viscous fluid flow past a rotating obstacle. Tôhoku Math J 58:129–147, 2006. doi:10.2748/tmj/1145390210.

R. Farwig. Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Regularity and other aspects of the Navier-Stokes equations 70:73–84, 2005.

R. Farwig, T. Hishida, D. Muller. Lq-theory of a singular “winding” integral operator arising from fluid dynamics. Pacific Journal of Mathematics 215:297–313, 2004. doi:10.2140/pjm.2004.215.297.

T. Hishida. An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle. Archive for Rational Mechanics and Analysis 150:307–348, 1999. doi:10.1007/s002050050190.

T. Hishida. The Stokes operator with rotation effect in exterior domains. Analysis 19(1):51 – 68, 1999. doi:10.1524/anly.1999.19.1.51.

T. Hishida. lq estimates of weak solutions to the stationary Stokes equations around a rotating body. Journal of the Mathematical Society of Japan 58:743–767, 2006.

T. Takahashi. Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Advances in Differential Equations 8(12):1499 – 1532, 2003.

T. Takahashi, T. Marius. Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. Journal of Mathematical Fluid Mechanics 6:53–77, 2004. doi:10.1007/s00021-003-0083-4.

P. Cumsille, T. Takahashi. Well posedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid. Czechoslovak Mathematical Journal 58:961–992, 2008. doi:10.1007/s10587-008-0063-2.

R. Farwig, R. Guenther, E. Thomann, Š. Necasová. The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body. Discrete and Continuous Dynamical Systems 34:511–529, 2014. doi:10.3934/dcds.2014.34.511.

M. Geissert, T. Hansel. A non-autonomous model problem for the Oseen-Navier-Stokes flow with rotating effects. Journal of The Mathematical Society of Japan 63, 2010. doi:10.2969/jmsj/06331027.

G. P. Galdi, M. Kyed. Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable. Archive for Rational Mechanics and Analysis 200:21–58, 2011. doi:10.1007/s00205-010-0350-6.

R. Farwig, G. Galdi, M. Kyed. Asymptotic structure of a leray solution to the navier-stokes flow around a rotating body. Pacific Journal of Mathematics 253(2):367–382, 2011. doi:10.2140/pjm.2011.253.367.

P. Deuring, S. Kračmar, Š. Nečasová. Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity. Journal of Differential Equations 255(7):1576 – 1606, 2013. doi:10.1016/j.jde.2013.05.016.

M. Kyed. On the asymptotic structure of a Navier-Stokes flow past a rotating body. Journal of the Mathematical Society of Japan 66(1):1–16, 2014. doi:10.2969/jmsj/06610001.

G. P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I. Linearised Steady Problems. Springer, New York, 1998.

P. Deuring, S. Kračmar, Š. Nečasová. Asymptotic structure of viscous incompressible flow around a rotating body, with nonvanishing flow field at infinity. Zeitschrift für angewandte Mathematik und Physik 68, 2016. doi:10.1007/s00033-016-0760-x.

P. Deuring, S. Kračmar, Š. Nečasová. A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation. Discrete and Continuous Dynamical Systems 37:1389–1409, 2016. doi:10.3934/dcds.2017057.

P. Deuring, S. Kračmar, Š. Nečasová. On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies. SIAM J Math Analysis 43:705–738, 2011. doi:10.1137/100786198.

P. Deuring, S. Kračmar, Š. Nečasová. Linearized stationary incompressible flow around rotating and translating bodies – leray solutions. Discrete and Continuous Dynamical Systems - Series S 7:967–979, 2014. doi:10.3934/dcdss.2014.7.967.

P. Deuring. Finite element methods for the stokes system in three-dimensional exterior domains. Mathematical Methods in the Applied Sciences 20(3):245–269, 1997.

doi:10.1002/(SICI)1099-1476(199702)20:3<245::AID-MMA856>3.0.CO;2-F.

P. Deuring, S. Kračmar. Artificial boundary conditions for the oseen system in 3D exterior domains. Analysis 20(1):65 – 90, 2000. doi:10.1524/anly.2000.20.1.65.

G. P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations. Springer, New York, 2nd edn., 2011.

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Published

2021-02-10

Issue

Vol. 61 No. SI (2021): Special Issue

Section

Non-Refereed Articles

License

Copyright (c) 2021 Sarka Necasova, Stanislav Kracmar, Paul Deuring

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Deuring, P., Kračmar, S., & Nečasová, Šárka. (2021). NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID AROUND A ROTATING AND TRANSLATING RIGID BODY. Acta Polytechnica, 61(SI), 5-13. https://doi.org/10.14311/AP.2021.61.0005
Received 2020-02-19
Accepted 2020-07-30
Published 2021-02-10