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

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.
Citation: Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete and 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), 023506, 26 pp. doi: 10.1063/1.3536463.

[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), 319-341. doi: 10.4208/cicp.2009.v6.p319.

[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), 777-813. doi: 10.1016/j.anihpc.2008.02.004.

[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), 1801-1812. doi: doi:10.1002/mma.1496.

[5]

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.

[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), 633-650. doi: 10.3934/dcds.2003.9.633.

[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), 753-777. doi: 10.1016/S0294-1449(00)00119-0.

[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), 231-245. doi: 10.1016/j.jsv.2007.07.032.

[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), 716-737. doi: 10.1137/070699196.

[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), 388-401. doi: 10.1007/s00021-006-0236-4.

[11]

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

[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), 146-170. doi: 10.1007/3-540-34596-5_7.

[13]

E. Kaya-Cekin, E. Aulisa and A. Ibragimov, Fluid structure interaction problem with changing thickness non-linear beam, Discrete And Continuous Dynamical Systems, Supplement, II (2011), 813-823.

[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), 193-232. doi: 10.3934/eect.2013.2.193.

[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), 424-432.

[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), 1447-1462. doi: 10.3934/cpaa.2011.10.1447.

[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), 197-216.

[18]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete And Continuous Dynamical Systems, 32 (2012), 1355-1389. doi: 10.3934/dcds.2012.32.1355.

[19]

M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937. doi: 10.1097/00010694-193808000-00008.

[20]

V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Dover Publication, Inc., 1999.

[21]

N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading, Proceedings: Mathematical, Physical and Engineering Sciences, 462 (2006), 2205-2224. doi: 10.1098/rspa.2006.1673.

[22]

J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam, Journal of Applied Mathematics, (2009), 12 pp. doi: 10.1155/2009/818269.

[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), 310-318. doi: 10.1016/j.jbiomech.2011.10.020.

[24]

M. Sathyamoorthy, Nonlinear Analysis of Structures, CRC, 1998.

[25]

SIAM PDE Conference 2011 San-Diego, Book of Abstractshttp://www.siam.org/meetings/pd11/pd11_abstracts.pdf.

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), 023506, 26 pp. doi: 10.1063/1.3536463.

[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), 319-341. doi: 10.4208/cicp.2009.v6.p319.

[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), 777-813. doi: 10.1016/j.anihpc.2008.02.004.

[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), 1801-1812. doi: doi:10.1002/mma.1496.

[5]

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.

[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), 633-650. doi: 10.3934/dcds.2003.9.633.

[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), 753-777. doi: 10.1016/S0294-1449(00)00119-0.

[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), 231-245. doi: 10.1016/j.jsv.2007.07.032.

[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), 716-737. doi: 10.1137/070699196.

[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), 388-401. doi: 10.1007/s00021-006-0236-4.

[11]

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

[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), 146-170. doi: 10.1007/3-540-34596-5_7.

[13]

E. Kaya-Cekin, E. Aulisa and A. Ibragimov, Fluid structure interaction problem with changing thickness non-linear beam, Discrete And Continuous Dynamical Systems, Supplement, II (2011), 813-823.

[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), 193-232. doi: 10.3934/eect.2013.2.193.

[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), 424-432.

[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), 1447-1462. doi: 10.3934/cpaa.2011.10.1447.

[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), 197-216.

[18]

I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete And Continuous Dynamical Systems, 32 (2012), 1355-1389. doi: 10.3934/dcds.2012.32.1355.

[19]

M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937. doi: 10.1097/00010694-193808000-00008.

[20]

V. V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Dover Publication, Inc., 1999.

[21]

N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading, Proceedings: Mathematical, Physical and Engineering Sciences, 462 (2006), 2205-2224. doi: 10.1098/rspa.2006.1673.

[22]

J. Peradze, A numerical alghorithm for Kirchhoff-Type nonlinear static beam, Journal of Applied Mathematics, (2009), 12 pp. doi: 10.1155/2009/818269.

[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), 310-318. doi: 10.1016/j.jbiomech.2011.10.020.

[24]

M. Sathyamoorthy, Nonlinear Analysis of Structures, CRC, 1998.

[25]

SIAM PDE Conference 2011 San-Diego, Book of Abstractshttp://www.siam.org/meetings/pd11/pd11_abstracts.pdf.

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