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October  2014, 7(5): 925-965. doi: 10.3934/dcdss.2014.7.925

Flow-plate interactions: Well-posedness and long-time behavior

1. 

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

2. 

University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152, United States

3. 

Oregon State University, Department of Mathematics, 368 Kidder Hall, Corvallis, OR 97330, United States

Received  March 2013 Revised  June 2013 Published  May 2014

We consider flow-structure interactions modeled by a modified wave equation coupled at an interface with equations of nonlinear elasticity. Both subsonic and supersonic flow velocities are treated with Neumann type flow conditions, and a novel treatment of the so called Kutta-Joukowsky flow conditions are given in the subsonic case. The goal of the paper is threefold: (i) to provide an accurate review of recent results on existence, uniqueness, and stability of weak solutions, (ii) to present a construction of finite dimensional, attracting sets corresponding to the structural dynamics and discuss convergence of trajectories, and (iii) to state several open questions associated with the topic. This second task is based on a decoupling technique which reduces the analysis of the full flow-structure system to a PDE system with delay.
Citation: Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925
References:
[1]

A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

A. V. Balakrishnan, Aeroelasticity. Continuum Theory, Springer, New York, 2012. doi: 10.1007/978-1-4614-3609-6.

[3]

A. V. Balakrishnan, Nonlinear aeroelastic theory: Continuum models, in Control Methods in PDE-Dynamical Systems, Contemp. Math., 426, Amer. Math. Soc., Providence, RI, 2007, 79-101. doi: 10.1090/conm/426/08185.

[4]

A. V. Balakrishnan and M. A. Shubov, Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1057-1091. doi: 10.1098/rspa.2003.1217.

[5]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.

[6]

R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, John Wiley and Sons, Inc., New York-London 1962; also Dover, New York, 1975.

[7]

L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping, J. Diff. Eqs., 253 (2012), 3568-3609. doi: 10.1016/j.jde.2012.08.004.

[8]

V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963.

[9]

A. Boutet de Monvel and I. Chueshov, The problem of interaction of von Karman plate with subsonic flow gas, Math. Methods in Appl. Sc., 22 (1999), 801-810. doi: 10.1002/(SICI)1099-1476(19990710)22:10<801::AID-MMA61>3.0.CO;2-T.

[10]

L. Boutet de Monvel and I. Chueshov, Non-linear oscillations of a plate in a flow of gas, C. R. Acad. Sci. Paris, Ser.I, 322 (1996), 1001-1006.

[11]

L. Boutet de Monvel and I. Chueshov, Oscillation of von Karman's plate in a potential flow of gas, Izvestiya RAN: Ser. Mat., 63 (1999), 219-244. doi: 10.1070/im1999v063n02ABEH000237.

[12]

L. Boutet de Monvel, I. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Comm. PDEs, 22 (1997), 1453-1474. doi: 10.1080/03605309708821307.

[13]

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.

[14]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer, New York, 1993.

[15]

I. Chueshov, On a system of equations with delay that arises in aero-elasticity, (in Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen, No. 54, (1990), 123-130; translation in J. Soviet Math., 58 (1992), 385-390. doi: 10.1007/BF01097291.

[16]

I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, (in Russian) Acta, Kharkov, 1999; English translation in Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/

[17]

I. Chueshov, Dynamics of von Karman plate in a potential flow of gas: Rigorous results and unsolved problems, in Proceedings of the 16th IMACS World Congress, Lausanne, Switzerland, 2000, 1-6.

[18]

I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 650-665. doi: 10.1016/j.na.2013.10.018.

[19]

I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid, Commun. Pure Appl. Anal., to appear, (2014). doi: 10.1016/j.na.2013.10.018.

[20]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.

[21]

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

[22]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[23]

I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions, Jour. Abstr. Differ. Equ. Appl., 3 (2012), 1-27.

[24]

I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, submitted, arXiv:1208.5245, 2012.

[25]

I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups for supersonic flow-plate interactions, J. of Diff. Eqs., 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009.

[26]

I. Chueshov and A. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. 1, 321 (1995), 607-612.

[27]

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.

[28]

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.

[29]

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

[30]

I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in System Modeling and Optimization. 25th IFIP TC7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers (eds. D. Hömberg and F. Tröltzsch), IFIP Advances in Information and Communication Technology, 391, Springer, Berlin, 2013, 328-337. doi: 10.1007/978-3-642-36062-6_33.

[31]

P. Ciarlet and P. Rabier, Les Équations de Von Kármán, Lecture Notes in Mathematics, 826, Springer, Berlin, 1980.

[32]

C. Mei, K. Abdel-Motagaly and R. Chen, Review of nonlinear panel flutter at supersonic and hypersonic speeds, Appl. Mech. Rev., 52 (1999), 321-332.

[33]

K. F. Clancey, On finite Hilbert transforms, Transactions AMS, 212 (1975), 347-354. doi: 10.1090/S0002-9947-1975-0377598-5.

[34]

D. G. Crighton, The Kutta condition in unsteady flow, Ann. Rev. Fluid Mech., 17 (1985), 411-445. doi: 10.1146/annurev.fluid.17.1.411.

[35]

O. Diekmann, S. van Gils, S. Lunel and H. O. Walther, Delay Equations, Springer, 1995.

[36]

E. Dowell, Nonlinear oscillations of a fluttering plate, I and II, AIAA J., 4 (1966), 1267-1275; and 5 (1967), 1857-1862.

[37]

E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells, AIAA Journal, 8 (1970), 385-399.

[38]

E. Dowell, O. Bendiksen, J. Edwards and T. Strganac, Transonic Nonlinear Aeroelasticity, Encyclopedia of Aerospace Engineering, 2003. doi: 10.1002/9780470686652.eae151.

[39]

, E. Dowell,, Private Communication., (). 

[40]

E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004.

[41]

C. Eloy, C. Souilliez and L. Schouveiler, Flutter of a rectangular plate, J. Fluids and Structures, 23 (2007), 904-919. doi: 10.1016/j.jfluidstructs.2007.02.002.

[42]

A. Favini, M. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation, Diff. Int. Eqs, 9 (1966), 267-294.; Addendum, Diff. Int. Eqs., 10 (1997), 197-220.

[43]

W. Frederiks, H. C. J. Hilbering and J. A. Sparenberg, On the Kutta condition for the flow along a semi-infinite elastic plate, J. Engin. Math., 20 (1986), 27-50. doi: 10.1007/BF00039321.

[44]

P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear damping localized in a boundary layer, J. of Diff. Eqs., 254 (2013), 1193-1229. doi: 10.1016/j.jde.2012.10.016.

[45]

D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge Univ. Press, 2002. doi: 10.1115/1.1566393.

[46]

P. Holmes and J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica, 14 (1978), 367-384. doi: 10.1016/0005-1098(78)90036-5.

[47]

M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a free boundary fluid-structure model, J. of Math. Phys., 53 (2012), 115624-115624. doi: 10.1063/1.4766724.

[48]

T. Von Kármán, Festigkeitsprobleme in Maschinenbau, Encyklopedie der Mathematischen Wissenschaften, 4, Leipzig, 1910, 348-352.

[49]

A. K. Khanmmamedov, Global attractors for von Karman equations with non-linear dissipation, J. Math. Anal. Appl., 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.

[50]

A. Kornecki, E. H. Dowell and J. O'Brien, On the aeroelastic instability of two-dimensional panes in uniform incompressible flow, 47 (1976), 163-178.

[51]

E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow, (in Russian) {Nauka}, Moscow, 1978.

[52]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.

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J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821.

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show all references

References:
[1]

A. Babin and M. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

A. V. Balakrishnan, Aeroelasticity. Continuum Theory, Springer, New York, 2012. doi: 10.1007/978-1-4614-3609-6.

[3]

A. V. Balakrishnan, Nonlinear aeroelastic theory: Continuum models, in Control Methods in PDE-Dynamical Systems, Contemp. Math., 426, Amer. Math. Soc., Providence, RI, 2007, 79-101. doi: 10.1090/conm/426/08185.

[4]

A. V. Balakrishnan and M. A. Shubov, Asymptotic behaviour of the aeroelastic modes for an aircraft wing model in a subsonic air flow, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1057-1091. doi: 10.1098/rspa.2003.1217.

[5]

H. M. Berger, A new approach to the analysis of large deflections of plates, J. Appl. Mech., 22 (1955), 465-472.

[6]

R. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, John Wiley and Sons, Inc., New York-London 1962; also Dover, New York, 1975.

[7]

L. Bociu and D. Toundykov, Attractors for non-dissipative irrotational von Karman plates with boundary damping, J. Diff. Eqs., 253 (2012), 3568-3609. doi: 10.1016/j.jde.2012.08.004.

[8]

V. V. Bolotin, Nonconservative Problems of Elastic Stability, Pergamon Press, Oxford, 1963.

[9]

A. Boutet de Monvel and I. Chueshov, The problem of interaction of von Karman plate with subsonic flow gas, Math. Methods in Appl. Sc., 22 (1999), 801-810. doi: 10.1002/(SICI)1099-1476(19990710)22:10<801::AID-MMA61>3.0.CO;2-T.

[10]

L. Boutet de Monvel and I. Chueshov, Non-linear oscillations of a plate in a flow of gas, C. R. Acad. Sci. Paris, Ser.I, 322 (1996), 1001-1006.

[11]

L. Boutet de Monvel and I. Chueshov, Oscillation of von Karman's plate in a potential flow of gas, Izvestiya RAN: Ser. Mat., 63 (1999), 219-244. doi: 10.1070/im1999v063n02ABEH000237.

[12]

L. Boutet de Monvel, I. Chueshov and A. Rezounenko, Long-time behaviour of strong solutions of retarded nonlinear PDEs, Comm. PDEs, 22 (1997), 1453-1474. doi: 10.1080/03605309708821307.

[13]

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.

[14]

A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, 3rd edition, Springer, New York, 1993.

[15]

I. Chueshov, On a system of equations with delay that arises in aero-elasticity, (in Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen, No. 54, (1990), 123-130; translation in J. Soviet Math., 58 (1992), 385-390. doi: 10.1007/BF01097291.

[16]

I. Chueshov, Introduction to the Theory of Infinite Dimensional Dissipative Systems, (in Russian) Acta, Kharkov, 1999; English translation in Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/

[17]

I. Chueshov, Dynamics of von Karman plate in a potential flow of gas: Rigorous results and unsolved problems, in Proceedings of the 16th IMACS World Congress, Lausanne, Switzerland, 2000, 1-6.

[18]

I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 650-665. doi: 10.1016/j.na.2013.10.018.

[19]

I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid, Commun. Pure Appl. Anal., to appear, (2014). doi: 10.1016/j.na.2013.10.018.

[20]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.

[21]

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

[22]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[23]

I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions, Jour. Abstr. Differ. Equ. Appl., 3 (2012), 1-27.

[24]

I. Chueshov, I. Lasiecka and J. T. Webster, Attractors for delayed, non-rotational von Karman plates with applications to flow-structure interactions without any damping, submitted, arXiv:1208.5245, 2012.

[25]

I. Chueshov, I. Lasiecka and J. T. Webster, Evolution semigroups for supersonic flow-plate interactions, J. of Diff. Eqs., 254 (2013), 1741-1773. doi: 10.1016/j.jde.2012.11.009.

[26]

I. Chueshov and A. Rezounenko, Global attractors for a class of retarded quasilinear partial differential equations, C. R. Acad. Sci. Paris, Ser. 1, 321 (1995), 607-612.

[27]

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.

[28]

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.

[29]

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

[30]

I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in System Modeling and Optimization. 25th IFIP TC7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers (eds. D. Hömberg and F. Tröltzsch), IFIP Advances in Information and Communication Technology, 391, Springer, Berlin, 2013, 328-337. doi: 10.1007/978-3-642-36062-6_33.

[31]

P. Ciarlet and P. Rabier, Les Équations de Von Kármán, Lecture Notes in Mathematics, 826, Springer, Berlin, 1980.

[32]

C. Mei, K. Abdel-Motagaly and R. Chen, Review of nonlinear panel flutter at supersonic and hypersonic speeds, Appl. Mech. Rev., 52 (1999), 321-332.

[33]

K. F. Clancey, On finite Hilbert transforms, Transactions AMS, 212 (1975), 347-354. doi: 10.1090/S0002-9947-1975-0377598-5.

[34]

D. G. Crighton, The Kutta condition in unsteady flow, Ann. Rev. Fluid Mech., 17 (1985), 411-445. doi: 10.1146/annurev.fluid.17.1.411.

[35]

O. Diekmann, S. van Gils, S. Lunel and H. O. Walther, Delay Equations, Springer, 1995.

[36]

E. Dowell, Nonlinear oscillations of a fluttering plate, I and II, AIAA J., 4 (1966), 1267-1275; and 5 (1967), 1857-1862.

[37]

E. Dowell, Panel flutter-A review of the aeroelastic stability of plates and shells, AIAA Journal, 8 (1970), 385-399.

[38]

E. Dowell, O. Bendiksen, J. Edwards and T. Strganac, Transonic Nonlinear Aeroelasticity, Encyclopedia of Aerospace Engineering, 2003. doi: 10.1002/9780470686652.eae151.

[39]

, E. Dowell,, Private Communication., (). 

[40]

E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004.

[41]

C. Eloy, C. Souilliez and L. Schouveiler, Flutter of a rectangular plate, J. Fluids and Structures, 23 (2007), 904-919. doi: 10.1016/j.jfluidstructs.2007.02.002.

[42]

A. Favini, M. Horn, I. Lasiecka and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation, Diff. Int. Eqs, 9 (1966), 267-294.; Addendum, Diff. Int. Eqs., 10 (1997), 197-220.

[43]

W. Frederiks, H. C. J. Hilbering and J. A. Sparenberg, On the Kutta condition for the flow along a semi-infinite elastic plate, J. Engin. Math., 20 (1986), 27-50. doi: 10.1007/BF00039321.

[44]

P. G. Geredeli, I. Lasiecka and J. T. Webster, Smooth attractors of finite dimension for von Karman evolutions with nonlinear damping localized in a boundary layer, J. of Diff. Eqs., 254 (2013), 1193-1229. doi: 10.1016/j.jde.2012.10.016.

[45]

D. H. Hodges and G. A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge Univ. Press, 2002. doi: 10.1115/1.1566393.

[46]

P. Holmes and J. Marsden, Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica, 14 (1978), 367-384. doi: 10.1016/0005-1098(78)90036-5.

[47]

M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness for a free boundary fluid-structure model, J. of Math. Phys., 53 (2012), 115624-115624. doi: 10.1063/1.4766724.

[48]

T. Von Kármán, Festigkeitsprobleme in Maschinenbau, Encyklopedie der Mathematischen Wissenschaften, 4, Leipzig, 1910, 348-352.

[49]

A. K. Khanmmamedov, Global attractors for von Karman equations with non-linear dissipation, J. Math. Anal. Appl., 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.

[50]

A. Kornecki, E. H. Dowell and J. O'Brien, On the aeroelastic instability of two-dimensional panes in uniform incompressible flow, 47 (1976), 163-178.

[51]

E. A. Krasil'shchikova, The Thin Wing in a Compressible Flow, (in Russian) {Nauka}, Moscow, 1978.

[52]

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.

[53]

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