American Institute of Mathematical Sciences

Advanced
December  2014, 7(6): 1133-1148. doi: 10.3934/dcdss.2014.7.1133

Fluid structure interaction problem with changing thickness beam and slightly compressible fluid

Eugenio Aulisa 1, , Akif Ibragimov 1, and  Emine Yasemen Kaya-Cekin 2,

1. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042
2. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States

Received  January 2013 Revised  September 2013 Published  June 2014

In this work, we consider the dynamical response of a non-linear beam with changing thickness, perturbed in both vertical and axial directions, interacting with a Darcy flow. We explore this fluid-structure interaction problem where the fluid is assumed to be slightly compressible. In an appropriate Sobolev norm, we build an energy functional for the displacement field of the beam and the gradient pressure of the fluid flow. We show that for a class of boundary conditions the energy functional is bounded by the flux of mass through the inlet boundary.
Keywords: Stability, darcy flow., von Karman plate, non-linear plate.
Mathematics Subject Classification: Primary: 74F10, 35Q35; Secondary: 58C3.
Citation: Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1133-1148. doi: 10.3934/dcdss.2014.7.1133
References:
[1]

E. Aulisa, L. Bloshanskaya and A. Ibragimov, Long-term dynamics for well productivity index for nonlinear flows in porous media,, Journal of Mathematical Physics, 52 (2011). doi: 10.1063/1.3536463. Google Scholar

[2]

E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application,, Communications in Computational Physics, 6 (2009), 319. doi: 10.4208/cicp.2009.v6.p319. Google Scholar

[3]

M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 777. doi: 10.1016/j.anihpc.2008.02.004. Google Scholar

[4]

I. D. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Mathematical Methods in the Applied Sciences, 34 (2011), 1801. doi: doi:10.1002/mma.1496. Google Scholar

[5]

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

[6]

Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete and Continuous Dynamical Systems, 9 (2003), 633. doi: 10.3934/dcds.2003.9.633. Google Scholar

[7]

F. Flori and P. Orenga, Fluid-structure interaction: Analysis of a 3-D compressible model,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 17 (2000), 753. doi: 10.1016/S0294-1449(00)00119-0. Google Scholar

[8]

D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction,, Journal of Sound and Vibration, 308 (2007), 231. doi: 10.1016/j.jsv.2007.07.032. Google Scholar

[9]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, SIAM Journal on Mathematical Analysis, 40 (2008), 716. doi: 10.1137/070699196. Google Scholar

[10]

M. Grobbelaar Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, Journal of Mathematical Fluid Mechanics, 10 (2008), 388. doi: 10.1007/s00021-006-0236-4. Google Scholar

[11]

M. Grobbelaar Van Dalsen, Strong stability for a fluid structure interaction model,, Math. Meth. Appl. Sci., 32 (2009), 1452. doi: 10.1002/mma.1104. Google Scholar

[12]

J. Hron and S. Turek, A Monolithic FEM/ Multigrid solver for ALE formulation of fluid structure interaction with applications in biomechanics,, Lecture Notes in Computational Science and Engineering, 53 (2006), 146. doi: 10.1007/3-540-34596-5_7. Google Scholar

[13]

E. Kaya-Cekin, E. Aulisa and A. Ibragimov, Fluid structure interaction problem with changing thickness non-linear beam,, Discrete And Continuous Dynamical Systems, II (2011), 813. Google Scholar

[14]

E. Kaya-Cekin, E. Aulisa and A. Ibragimov, Stability Analysis of non-linear plates coupled with Darcy flows,, Evolution Equation and Control Theory, 2 (2013), 193. doi: 10.3934/eect.2013.2.193. Google Scholar

[15]

E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, A Stability Estimate for Fluid Structure Interaction Problem with Non-Linear Beam,, Discrete And Continuous Dynamical Systems. Supplement, (2009), 424. Google Scholar

[16]

E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, Stability Analysis of Inhomogeneous Equilibrium for Axially and Transversely Excited Nonlinear Beam,, Communications on Pure and Applied Analysis, 10 (2011), 1447. doi: 10.3934/cpaa.2011.10.1447. Google Scholar

[17]

H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems,, Progress in Nonlinear Differential Equations and Their Applications, 50 (2002), 197. Google Scholar

[18]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem,, Discrete And Continuous Dynamical Systems, 32 (2012), 1355. doi: 10.3934/dcds.2012.32.1355. Google Scholar

[19]

M. Muskat, The Flow of Homogeneous Fluids Through Porous Media,, McGraw-Hill, (1937). doi: 10.1097/00010694-193808000-00008. Google Scholar

[20]

V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity,, Dover Publication, (1999). Google Scholar

[21]

N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading,, Proceedings: Mathematical, 462 (2006), 2205. doi: 10.1098/rspa.2006.1673. Google Scholar

[22]

J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam,, Journal of Applied Mathematics, (2009). doi: 10.1155/2009/818269. Google Scholar

[23]

A. Quaini, S. Canic, R. Glowinski, S. Igo, C. Hartley, W. Zoghbi and S. Little, Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture,, Journal of Biomechanic, 45 (2012), 310. doi: 10.1016/j.jbiomech.2011.10.020. Google Scholar

[24]

M. Sathyamoorthy, Nonlinear Analysis of Structures,, CRC, (1998). Google Scholar

[25]

SIAM PDE Conference 2011 San-Diego, Book of Abstracts,, , (). Google Scholar

show all references

References:
[1]

E. Aulisa, L. Bloshanskaya and A. Ibragimov, Long-term dynamics for well productivity index for nonlinear flows in porous media,, Journal of Mathematical Physics, 52 (2011). doi: 10.1063/1.3536463. Google Scholar

[2]

E. Aulisa, A. Cervone, S. Manservisi and P. Seshaiyer, A multilevel domain decomposition approach for studying coupled flow application,, Communications in Computational Physics, 6 (2009), 319. doi: 10.4208/cicp.2009.v6.p319. Google Scholar

[3]

M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations,, Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 777. doi: 10.1016/j.anihpc.2008.02.004. Google Scholar

[4]

I. D. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Mathematical Methods in the Applied Sciences, 34 (2011), 1801. doi: doi:10.1002/mma.1496. Google Scholar

[5]

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

[6]

Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete and Continuous Dynamical Systems, 9 (2003), 633. doi: 10.3934/dcds.2003.9.633. Google Scholar

[7]

F. Flori and P. Orenga, Fluid-structure interaction: Analysis of a 3-D compressible model,, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 17 (2000), 753. doi: 10.1016/S0294-1449(00)00119-0. Google Scholar

[8]

D. G. Gorman, I. Trendafilova, A. J. Mulholland and J. Horacek, Analytical modeling and extraction of the modal behavior of a cantilever beam in fluid interaction,, Journal of Sound and Vibration, 308 (2007), 231. doi: 10.1016/j.jsv.2007.07.032. Google Scholar

[9]

C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, SIAM Journal on Mathematical Analysis, 40 (2008), 716. doi: 10.1137/070699196. Google Scholar

[10]

M. Grobbelaar Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, Journal of Mathematical Fluid Mechanics, 10 (2008), 388. doi: 10.1007/s00021-006-0236-4. Google Scholar

[11]

M. Grobbelaar Van Dalsen, Strong stability for a fluid structure interaction model,, Math. Meth. Appl. Sci., 32 (2009), 1452. doi: 10.1002/mma.1104. Google Scholar

[12]

J. Hron and S. Turek, A Monolithic FEM/ Multigrid solver for ALE formulation of fluid structure interaction with applications in biomechanics,, Lecture Notes in Computational Science and Engineering, 53 (2006), 146. doi: 10.1007/3-540-34596-5_7. Google Scholar

[13]

E. Kaya-Cekin, E. Aulisa and A. Ibragimov, Fluid structure interaction problem with changing thickness non-linear beam,, Discrete And Continuous Dynamical Systems, II (2011), 813. Google Scholar

[14]

E. Kaya-Cekin, E. Aulisa and A. Ibragimov, Stability Analysis of non-linear plates coupled with Darcy flows,, Evolution Equation and Control Theory, 2 (2013), 193. doi: 10.3934/eect.2013.2.193. Google Scholar

[15]

E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, A Stability Estimate for Fluid Structure Interaction Problem with Non-Linear Beam,, Discrete And Continuous Dynamical Systems. Supplement, (2009), 424. Google Scholar

[16]

E. Kaya-Cekin, E. Aulisa, A. Ibragimov and P. Seshaiyer, Stability Analysis of Inhomogeneous Equilibrium for Axially and Transversely Excited Nonlinear Beam,, Communications on Pure and Applied Analysis, 10 (2011), 1447. doi: 10.3934/cpaa.2011.10.1447. Google Scholar

[17]

H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems,, Progress in Nonlinear Differential Equations and Their Applications, 50 (2002), 197. Google Scholar

[18]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem,, Discrete And Continuous Dynamical Systems, 32 (2012), 1355. doi: 10.3934/dcds.2012.32.1355. Google Scholar

[19]

M. Muskat, The Flow of Homogeneous Fluids Through Porous Media,, McGraw-Hill, (1937). doi: 10.1097/00010694-193808000-00008. Google Scholar

[20]

V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity,, Dover Publication, (1999). Google Scholar

[21]

N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading,, Proceedings: Mathematical, 462 (2006), 2205. doi: 10.1098/rspa.2006.1673. Google Scholar

[22]

J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam,, Journal of Applied Mathematics, (2009). doi: 10.1155/2009/818269. Google Scholar

[23]

A. Quaini, S. Canic, R. Glowinski, S. Igo, C. Hartley, W. Zoghbi and S. Little, Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture,, Journal of Biomechanic, 45 (2012), 310. doi: 10.1016/j.jbiomech.2011.10.020. Google Scholar

[24]

M. Sathyamoorthy, Nonlinear Analysis of Structures,, CRC, (1998). Google Scholar

[25]

SIAM PDE Conference 2011 San-Diego, Book of Abstracts,, , (). Google Scholar

[1]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations & Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193
[2]

Pascal Cherrier, Albert Milani. Hyperbolic equations of Von Karman type. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 125-137. doi: 10.3934/dcdss.2016.9.125
[3]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[4]

Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184
[5]

Franca Franchi, Barbara Lazzari, Roberta Nibbi. Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2111-2132. doi: 10.3934/dcdsb.2014.19.2111
[6]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178
[7]

Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations & Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241
[8]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214
[9]

Irena Lasiecka, Justin Webster. Eliminating flutter for clamped von Karman plates immersed in subsonic flows. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1935-1969. doi: 10.3934/cpaa.2014.13.1935
[10]

Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : ⅰ-ⅳ. doi: 10.3934/dcdss.201702i
[11]

Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165
[12]

Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.

[13]

Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675
[14]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[15]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
[16]

Shun Li, Peng-Fei Yao. Modeling of a nonlinear plate. Evolution Equations & Control Theory, 2012, 1 (1) : 155-169. doi: 10.3934/eect.2012.1.155
[17]

Michela Eleuteri, Jana Kopfov, Pavel Krej?. Fatigue accumulation in an oscillating plate. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 909-923. doi: 10.3934/dcdss.2013.6.909
[18]

Orazio Arena. A problem of boundary controllability for a plate. Evolution Equations & Control Theory, 2013, 2 (4) : 557-562. doi: 10.3934/eect.2013.2.557
[19]

Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure & Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815
[20]

Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences