# American Institute of Mathematical Sciences

March  2017, 37(3): 1389-1409. doi: 10.3934/dcds.2017057

## A leading term for the velocity of stationary viscous incompressible flow around a rigid body performing a rotation and a translation

 1 Univ. Littoral Côte d'Opale, Laboratoire de mathématiques pures et appliquées Joseph Liouville, F-62228 Calais, France 2 Department of Technical Mathematics, Czech Technical University, Karlovo nám. 13, 121 35 Prague 2, Czech Republic 3 Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25,115 67 Prague 1, Czech Republic

* Corresponding author

Received  February 2016 Revised  October 2016 Published  December 2016

Fund Project: The Š.N. was supported by Grant Agency of the Czech Republic P201-13-00522S.

We consider the Navier-Stokes system with Oseen and rotational terms describing the stationary flow of a viscous incompressible fluid around a rigid body moving at a constant velocity and rotating at a constant angular velocity. In a previous paper, we proved a representation formula for Leray solutions of this system. Here the representation formula is used as starting point for splitting the velocity into a leading term and a remainder, and for establishing pointwise decay estimates of the remainder and its gradient.

Citation: Paul Deuring, Stanislav Kračmar, Šárka 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, 2017, 37 (3) : 1389-1409. doi: 10.3934/dcds.2017057
##### References:
 [1] N. Abada, T. Z. Boulmezaoud and N. Kerdid, The Stokes flow around a rotating body in the whole space, J. Math. Soc. Japan, 65 (2013), 607-632.  doi: 10.2969/jmsj/06520607. [2] T. K. Aldoss and T. W. Abou-Arab, Experimental study of the flow around a rotating cylinder in crossflow, Experimental Thermal and Fluid Science, 3 (1990), 316-322.  doi: 10.1016/0894-1777(90)90006-S. [3] A. Rosa da Silva, N. Aristeu da Silveira, M. G. de Lima and D. A. Rade, doititle Numerical simulations of flows over a rotating circular cylinder using the immersed boundary method, J. Braz. Soc. Mech. Sci. & Eng. 33 (2011). doi: 10.1590/S1678-58782011000100014. [4] K. I. Babenko, On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Mat. Sb. , 91 (1973), 3-26 (Russian); [English translation: Math. USSR Sbornik, 20 (1973), 1-25. ] [5] P. Deuring and 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 (2004), 86-115.  doi: 10.1002/mana.200310167. [6] P. Deuring, S. Kračmar and Nečasová Š, A representation formula for linearized stationar incompressible viscous flows around rotating and translating bodie, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 237-253.  doi: 10.3934/dcdss.2010.3.237. [7] P. Deuring, S. Kračmar and Nečasová Š, On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies, SIAM J. Math. Anal., 43 (2011), 705-738.  doi: 10.1137/100786198. [8] P. Deuring, S. Kračmar and Nečasová Š, Linearized stationary incompressible flow around rotating and translating bodies: Asymptotic profile of the velocity gradient and decay estimate of the second derivatives of the velocity, J. Differential Equations, 252 (2012), 459-476.  doi: 10.1016/j.jde.2011.08.037. [9] P. Deuring, S. Kračmar and Š Nečasová A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: Improved decay estimates of the velocity and its gradient, Dynamical Systems, Differential Equations and Applications, Ed. by W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, S. Siegmund and J. Voigt. Discrete Contin. Dyn. Syst. , Supplement 2011 (8th AIMS Conference, Dresden, Germany) 1 (2011), 351-361 [10] P. Deuring, S. Kračmar and Nečasová Š, Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity, J. Differential Equations, 255 (2013), 1576-1606.  doi: 10.1016/j.jde.2013.05.016. [11] P. Deuring, S. Kračmar and Nečasová Š, Linearized stationary incompressible flow around rotating and translating bodies -Leray solutions, Discrete Contin. Dyn. Syst., 7 (2014), 967-979.  doi: 10.3934/dcdss.2014.7.967. [12] P. Deuring, S. Kračmar and Š. Nečasová, Asymptotic structure of viscous incompressible flow around a rotating body, with nonvanishing flow field at infinity, arXiv 1511.04378, to appear in ZAMP [13] P. Deuring, S. Kračmar, Nečasová Š and P. Wittwer, Decay estimates for linearized unsteady incompressible viscous flows around rotating and translating bodies, J. Elliptic Parabol. Equ., 1 (2015), 325-333. [14] P. Deuring, S. Kračmar and S. Nečasová, Note to the problem of asymptotic behavior of viscous incompressible flow around a rotating body, Comptes Rendus Mathematique, 354 (2016), 794-798.  doi: 10.1016/j.crma.2016.05.013. [15] R. Farwig, The stationary exterior 3D-problem of Oseen, Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.  doi: 10.1007/BF02571437. [16] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, Banach Center Publications, 70 (2005), 73-84.  doi: 10.4064/bc70-0-5. [17] R. Farwig, G. P. Galdi and M. Kyed, Asymptotic structure of a Leray solution to the Navier Stokes flow around a rotating body, Pacific J. Math., 253 (2011), 367-382.  doi: 10.2140/pjm.2011.253.367. [18] R. Farwig, R. B. Guenther, S. Nečasová and E. A. Thomann, The fundamental solution of the linearized instationary Navier-Stokes equations of motion around a rotating and translating body, Discrete Contin. Dyn. Syst., 34 (2014), 511-529.  doi: 10.3934/dcds.2014.34.511. [19] R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacle, Funkcialaj Ekvacioj, 50 (2007), 371-403.  doi: 10.1619/fesi.50.371. [20] R. Farwig and Hishida T., Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle, Ann. Univ. Ferrara, 55 (2009), 263-277.  doi: 10.1007/s11565-009-0072-6. [21] R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscripta Math., 136 (2011), 315-338.  doi: 10.1007/s00229-011-0479-0. [22] R. Farwig and T. Hishida, Leading term at infinity of steady Navier-Stokes flow around a rotating obstacle, Math. Nachr., 284 (2011), 2065-2077.  doi: 10.1002/mana.200910192. [23] R. Farwig, T. Hishida and D. Müller, Lq-theory of a singular "winding" integral operator arising from fluid dynamics, Pacific J. Math., 215 (2004), 297-312.  doi: 10.2140/pjm.2004.215.297. [24] R. Farwig, M. Krbec and S. Nečasová, A weighted Lq approach to Stokes flow around a rotating body, Ann. Univ. Ferrara, Sez., 54 (2008), 61-84.  doi: 10.1007/s11565-008-0040-6. [25] R. Farwig, M. Krbec and S. Nečasová, A weighted Lq-approach to Oseen flow around a rotating body, Math. Meth. Appl. Sci., 31 (2008), 551-574.  doi: 10.1002/mma.925. [26] R. Farwig and J. Neustupa, On the spectrum of a Stokes-type operator arising from flow around a rotating body, Manuscripta Math., 122 (2007), 419-437.  doi: 10.1007/s00229-007-0078-2. [27] J. Feng, H. H. Hu and D. D. Joseph, Direct simulation of initial value problems for the motion of solid bodies in a Newtonian Fluid, Part 2. Couette and Poiseuille flows, Journal of Fluid Mechanics, 277 (1994), 271-301.  doi: 10.1017/S0022112094002764. [28] R. Finn, Estimates at infinity for stationary solutions of the Navier-Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine, 3 (1959), 387-418. [29] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. [30] E. Feireisl, A. Gallagher and A. Novotný, A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.  doi: 10.1137/100808010. [31] E. Feireisl, I. Gallagher, D. Gerard-Varet and A. Novotný, Multi-scale analysis of compressible viscous and rotating flows, Comm. Math. Phys., 314 (2012), 641-670.  doi: 10.1007/s00220-012-1533-9. [32] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations.Vol. Ⅱ. Nonlinear Steady Problems, Springer, New York, 1994. [33] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations.Vol. Ⅰ. Linearized Steady Problems (rev. ed.), Springer, New York, 1994. [34] G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, Ed. by S. Friedlander, D. Serre, Elsevier, 1 (2002), 653-791. [35] G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity, 71 (2003), 1-31.  doi: 10.1023/B:ELAS.0000005543.00407.5e. [36] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems (2nd ed.), Springer, New York, 2011.  doi: 10.1007/978-0-387-09620-9. [37] G. P. Galdi and M. Kyed, Steady-state Navier-Stokes flows past a rotating body: Leary solutions are physically reasonable, Arch. Rat. Mech. Anal., 200 (2011), 21-58.  doi: 10.1007/s00205-010-0350-6. [38] G. P. Galdi and M. Kyed, Asymptotic behavior of a Leray solution around a rotating obstacle, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 251-266.  doi: 10.1007/978-3-0348-0075-4_13. [39] G. P. Galdi and M. Kyed, A simple proof of Lq-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part Ⅰ: Strong solutions, Proc. Am. Math. Soc., 141 (2013), 573-583.  doi: 10.1090/S0002-9939-2012-11638-7. [40] G. P. Galdi and M. Kyed, A simple proof of Lq-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part Ⅱ: Weak solutions, Proc. Am. Math. Soc., 141 (2013), 1313-1322.  doi: 10.1090/S0002-9939-2012-11640-5. [41] G. P. Galdi and A. L. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body, Arch. Rat. Mech. Anal., 184 (2007), 371-400.  doi: 10.1007/s00205-006-0026-4. [42] M. Geissert, H. Heck and M. 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##### References:
 [1] N. Abada, T. Z. Boulmezaoud and N. Kerdid, The Stokes flow around a rotating body in the whole space, J. Math. Soc. Japan, 65 (2013), 607-632.  doi: 10.2969/jmsj/06520607. [2] T. K. Aldoss and T. W. Abou-Arab, Experimental study of the flow around a rotating cylinder in crossflow, Experimental Thermal and Fluid Science, 3 (1990), 316-322.  doi: 10.1016/0894-1777(90)90006-S. [3] A. Rosa da Silva, N. Aristeu da Silveira, M. G. de Lima and D. A. Rade, doititle Numerical simulations of flows over a rotating circular cylinder using the immersed boundary method, J. Braz. Soc. Mech. Sci. & Eng. 33 (2011). doi: 10.1590/S1678-58782011000100014. [4] K. I. Babenko, On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Mat. Sb. , 91 (1973), 3-26 (Russian); [English translation: Math. USSR Sbornik, 20 (1973), 1-25. ] [5] P. Deuring and 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 (2004), 86-115.  doi: 10.1002/mana.200310167. [6] P. Deuring, S. Kračmar and Nečasová Š, A representation formula for linearized stationar incompressible viscous flows around rotating and translating bodie, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 237-253.  doi: 10.3934/dcdss.2010.3.237. [7] P. Deuring, S. Kračmar and Nečasová Š, On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies, SIAM J. Math. Anal., 43 (2011), 705-738.  doi: 10.1137/100786198. [8] P. Deuring, S. Kračmar and Nečasová Š, Linearized stationary incompressible flow around rotating and translating bodies: Asymptotic profile of the velocity gradient and decay estimate of the second derivatives of the velocity, J. Differential Equations, 252 (2012), 459-476.  doi: 10.1016/j.jde.2011.08.037. [9] P. Deuring, S. Kračmar and Š Nečasová A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: Improved decay estimates of the velocity and its gradient, Dynamical Systems, Differential Equations and Applications, Ed. by W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, S. Siegmund and J. Voigt. Discrete Contin. Dyn. Syst. , Supplement 2011 (8th AIMS Conference, Dresden, Germany) 1 (2011), 351-361 [10] P. Deuring, S. Kračmar and Nečasová Š, Pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity, J. Differential Equations, 255 (2013), 1576-1606.  doi: 10.1016/j.jde.2013.05.016. [11] P. Deuring, S. Kračmar and Nečasová Š, Linearized stationary incompressible flow around rotating and translating bodies -Leray solutions, Discrete Contin. Dyn. Syst., 7 (2014), 967-979.  doi: 10.3934/dcdss.2014.7.967. [12] P. Deuring, S. Kračmar and Š. Nečasová, Asymptotic structure of viscous incompressible flow around a rotating body, with nonvanishing flow field at infinity, arXiv 1511.04378, to appear in ZAMP [13] P. Deuring, S. Kračmar, Nečasová Š and P. Wittwer, Decay estimates for linearized unsteady incompressible viscous flows around rotating and translating bodies, J. Elliptic Parabol. Equ., 1 (2015), 325-333. [14] P. Deuring, S. Kračmar and S. Nečasová, Note to the problem of asymptotic behavior of viscous incompressible flow around a rotating body, Comptes Rendus Mathematique, 354 (2016), 794-798.  doi: 10.1016/j.crma.2016.05.013. [15] R. Farwig, The stationary exterior 3D-problem of Oseen, Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.  doi: 10.1007/BF02571437. [16] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, Banach Center Publications, 70 (2005), 73-84.  doi: 10.4064/bc70-0-5. [17] R. Farwig, G. P. Galdi and M. Kyed, Asymptotic structure of a Leray solution to the Navier Stokes flow around a rotating body, Pacific J. Math., 253 (2011), 367-382.  doi: 10.2140/pjm.2011.253.367. [18] R. Farwig, R. B. Guenther, S. Nečasová and E. A. Thomann, The fundamental solution of the linearized instationary Navier-Stokes equations of motion around a rotating and translating body, Discrete Contin. Dyn. Syst., 34 (2014), 511-529.  doi: 10.3934/dcds.2014.34.511. [19] R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacle, Funkcialaj Ekvacioj, 50 (2007), 371-403.  doi: 10.1619/fesi.50.371. [20] R. Farwig and Hishida T., Asymptotic profiles of steady Stokes and Navier-Stokes flows around a rotating obstacle, Ann. Univ. Ferrara, 55 (2009), 263-277.  doi: 10.1007/s11565-009-0072-6. [21] R. Farwig and T. Hishida, Asymptotic profile of steady Stokes flow around a rotating obstacle, Manuscripta Math., 136 (2011), 315-338.  doi: 10.1007/s00229-011-0479-0. [22] R. Farwig and T. Hishida, Leading term at infinity of steady Navier-Stokes flow around a rotating obstacle, Math. Nachr., 284 (2011), 2065-2077.  doi: 10.1002/mana.200910192. [23] R. Farwig, T. Hishida and D. Müller, Lq-theory of a singular "winding" integral operator arising from fluid dynamics, Pacific J. Math., 215 (2004), 297-312.  doi: 10.2140/pjm.2004.215.297. [24] R. Farwig, M. Krbec and S. Nečasová, A weighted Lq approach to Stokes flow around a rotating body, Ann. Univ. Ferrara, Sez., 54 (2008), 61-84.  doi: 10.1007/s11565-008-0040-6. [25] R. Farwig, M. Krbec and S. Nečasová, A weighted Lq-approach to Oseen flow around a rotating body, Math. Meth. Appl. Sci., 31 (2008), 551-574.  doi: 10.1002/mma.925. [26] R. Farwig and J. Neustupa, On the spectrum of a Stokes-type operator arising from flow around a rotating body, Manuscripta Math., 122 (2007), 419-437.  doi: 10.1007/s00229-007-0078-2. [27] J. Feng, H. H. Hu and D. D. Joseph, Direct simulation of initial value problems for the motion of solid bodies in a Newtonian Fluid, Part 2. Couette and Poiseuille flows, Journal of Fluid Mechanics, 277 (1994), 271-301.  doi: 10.1017/S0022112094002764. [28] R. Finn, Estimates at infinity for stationary solutions of the Navier-Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine, 3 (1959), 387-418. [29] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. [30] E. Feireisl, A. Gallagher and A. Novotný, A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.  doi: 10.1137/100808010. [31] E. Feireisl, I. Gallagher, D. Gerard-Varet and A. Novotný, Multi-scale analysis of compressible viscous and rotating flows, Comm. Math. Phys., 314 (2012), 641-670.  doi: 10.1007/s00220-012-1533-9. [32] G. P. 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