American Institute of Mathematical Sciences

Advanced
December  2016, 5(4): 605-629. doi: 10.3934/eect.2016021

On interaction of circular cylindrical shells with a Poiseuille type flow

Igor Chueshov 1, and  Tamara Fastovska 2,

1. 

Kharkov National Universit, Department of Mathematics and Mechanics, 4 Svobody sq, 61077 Kharkov
2. 

Department of Mathematics and Mechanics, Kharkov Karazin National University, 4, Svobody sq., Kharkov 61077

Received  July 2016 Revised  September 2016 Published  October 2016

We study dynamics of a coupled system consisting of the 3D Navier--Stokes equations which is linearized near a certain Poiseuille type flow between two unbounded circular cylinders and nonlinear elasticity equations for the transversal displacements of the bounding cylindrical shells. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.
Keywords: Fluid--structure interaction, nonlinear circular shell, attractor., Poiseulle flow, linearized 3D Navier--Stokes equations.
Mathematics Subject Classification: Primary: 74F10, 35B41; Secondary: 35Q30, 74K2.
Citation: Igor Chueshov, Tamara Fastovska. On interaction of circular cylindrical shells with a Poiseuille type flow. Evolution Equations and Control Theory, 2016, 5 (4) : 605-629. doi: 10.3934/eect.2016021
References:
[1]

M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-14648-5.

[2]

G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z.

[3]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, New prospects in direct, inverse and control problems for evolution equations, Springer INdAM Series, 10 (2014), 49-78. doi: 10.1007/978-3-319-11406-4_3.

[4]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic- hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Ser.S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417.

[5]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[6]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and Waves, Contemp. Math., vol. 440, AMS, Providence, RI, (2007), 55-82. doi: 10.1090/conm/440/08476.

[7]

A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-22903-4.

[9]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 2002. http://www.emis.de/monographs/Chueshov/.

[10]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Meth. Appl. Sci., 34 (2011), 1801-1812. doi: 10.1002/mma.1496.

[11]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.

[12]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[13]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635.

[14]

I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Diff. Eqs., 254 (2013), 1833-1862. doi: 10.1016/j.jde.2012.11.006.

[15]

I. Chueshov and I. Ryzhkova, On the interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Mathematical Journal, 65 (2013), 158-177. doi: 10.1007/s11253-013-0771-0.

[16]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7.

[17]

E. H. Donnell, A new theory for the buckling of thin cylinders under axial compression and bending, Trans. ASME, 56 (1934), 795-806.

[18]

D. A. Evensen, Nonlinear Fexural Vibrations of Thin-Walled Circular Cylinders, NASA TN D-4090., 1967.

[19]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.

[20]

G. Galdi, C. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$, Math. Annalen, 331 (2005), 41-74. doi: 10.1007/s00208-004-0573-7.

[21]

N. Große and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr., 286 (2013), 1586-1613. doi: 10.1002/mana.201300007.

[22]

Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633.

[23]

D. Fujiwara, Concrete characterizations of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 ( 1967), 82-86. doi: 10.3792/pja/1195521686.

[24]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737. doi: 10.1137/070699196.

[25]

M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model, Applicable Analysis 88 (2009), 1053-1065. doi: 10.1080/00036810903114841.

[26]

M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. doi: 10.1002/mma.1104.

[27]

M. Guidorzi, M. Padula and P. I. Plotnikov, Hopf solutions to a fluid-elastic interaction model, MMAS, 18 (2008), 215-269. doi: 10.1142/S0218202508002668.

[28]

O. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[29]

J.-L. Lions and E. Magenes, Problémes Aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.

[30]

A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 497-513. doi: 10.1051/cocv:1999119.

[31]

M. E. Taylor, Partial Differential Equations 1. Basic theory, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.

[32]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[33]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

[34]

H. Triebel, Theory of Function Spaces 2, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[35]

H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.

show all references

References:
[1]

M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-14648-5.

[2]

G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z.

[3]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, New prospects in direct, inverse and control problems for evolution equations, Springer INdAM Series, 10 (2014), 49-78. doi: 10.1007/978-3-319-11406-4_3.

[4]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic- hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Ser.S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417.

[5]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[6]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and Waves, Contemp. Math., vol. 440, AMS, Providence, RI, (2007), 55-82. doi: 10.1090/conm/440/08476.

[7]

A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech., 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-22903-4.

[9]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 2002. http://www.emis.de/monographs/Chueshov/.

[10]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Meth. Appl. Sci., 34 (2011), 1801-1812. doi: 10.1002/mma.1496.

[11]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.

[12]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[13]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635.

[14]

I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Diff. Eqs., 254 (2013), 1833-1862. doi: 10.1016/j.jde.2012.11.006.

[15]

I. Chueshov and I. Ryzhkova, On the interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Mathematical Journal, 65 (2013), 158-177. doi: 10.1007/s11253-013-0771-0.

[16]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7.

[17]

E. H. Donnell, A new theory for the buckling of thin cylinders under axial compression and bending, Trans. ASME, 56 (1934), 795-806.

[18]

D. A. Evensen, Nonlinear Fexural Vibrations of Thin-Walled Circular Cylinders, NASA TN D-4090., 1967.

[19]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.

[20]

G. Galdi, C. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$, Math. Annalen, 331 (2005), 41-74. doi: 10.1007/s00208-004-0573-7.

[21]

N. Große and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr., 286 (2013), 1586-1613. doi: 10.1002/mana.201300007.

[22]

Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633.

[23]

D. Fujiwara, Concrete characterizations of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 ( 1967), 82-86. doi: 10.3792/pja/1195521686.

[24]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737. doi: 10.1137/070699196.

[25]

M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model, Applicable Analysis 88 (2009), 1053-1065. doi: 10.1080/00036810903114841.

[26]

M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. doi: 10.1002/mma.1104.

[27]

M. Guidorzi, M. Padula and P. I. Plotnikov, Hopf solutions to a fluid-elastic interaction model, MMAS, 18 (2008), 215-269. doi: 10.1142/S0218202508002668.

[28]

O. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.

[29]

J.-L. Lions and E. Magenes, Problémes Aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.

[30]

A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 497-513. doi: 10.1051/cocv:1999119.

[31]

M. E. Taylor, Partial Differential Equations 1. Basic theory, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.

[32]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.

[33]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.

[34]

H. Triebel, Theory of Function Spaces 2, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[35]

H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.

[1]

Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151
[2]

Luca Bisconti, Davide Catania. Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 59-75. doi: 10.3934/dcdsb.2015.20.59
[3]

Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012
[4]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503
[5]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101
[6]

Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189
[7]

Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4489-4522. doi: 10.3934/dcds.2022062
[8]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[9]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[10]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[11]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
[12]

Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081
[13]

Yutaka Tsuzuki. Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains. Conference Publications, 2015, 2015 (special) : 1079-1088. doi: 10.3934/proc.2015.1079
[14]

Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481
[15]

C. Foias, M. S Jolly, O. P. Manley. Recurrence in the 2-$D$ Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 253-268. doi: 10.3934/dcds.2004.10.253
[16]

Xiuqing Wang, Yuming Qin, Alain Miranville. Approximation of the trajectory attractor of the 3D smectic-A liquid crystal flow equations. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3805-3827. doi: 10.3934/cpaa.2020168
[17]

Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151
[18]

T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure and Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
[19]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[20]

Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193

2021 Impact Factor: 1.169

Tools

Metrics

  • PDF downloads (173)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy