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June  2013, 2(2): 193-232. doi: 10.3934/eect.2013.2.193

Stability analysis of non-linear plates coupled with Darcy flows

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  October 2012 Revised  January 2013 Published  March 2013

In this paper we study the dynamical response of a non-linear plate with viscous damping perturbed in both vertical and axial directions and interacting with Darcy flow. We first consider the problem for non-linear elastic body with damping coefficient; existence and uniqueness of the solution for the steady state problem is proven. The stability of the dynamical non-linear plate problem under certain conditions on the applied loads is investigated. Second, we explore the fluid structure interaction problem with Darcy flow in porous media. Energy functional for the displacement field of the plate and the gradient pressure of the fluid flow is built in an appropriate Sobolev type norm. We show that for a class of boundary conditions the energy functional is limited by the flux of mass through the inlet boundary.
Keywords: non-linear plate, Stability, von Karman plate, Darcy flow..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations and Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193
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]

Annalisa Quaini, Suncica Canic, Roland Glowinski, Stephen Igo, Craig J Hartley, William Zoghbi and Stephen Little, Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture, Journal of Biomechanics, 45 (2012), 310-318.

[4]

I. D. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, Journal of Differential Equations, 233 (2007), 42-86. doi: 10.1016/j.jde.2006.09.019.

[5]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of the AMS, (2008).

[6]

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: 10.1002/mma.1496.

[7]

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.

[8]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, AMS, Providence, RI, 1998.

[9]

F. Flori and P. Orenga, Fluid-structure interaction: Analysis of a 3-D compressible model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 753-777. doi: 10.1016/S0294-1449(00)00119-0.

[10]

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.

[11]

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.

[12]

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.

[13]

M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813. doi: 10.1016/j.anihpc.2008.02.004.

[14]

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.

[15]

L. Hoang and A. Ibragimov, Structural stability of generalized Forchheimer equations for compressible fluids in porous media, Nonlinearity, 24 (2011), 1-41. doi: 10.1088/0951-7715/24/1/001.

[16]

J. Hron and S. Turek, A monolithic FEM/multigrid solver for ALE formulation of fluid-structure interaction with application in biomechanics, in "Fluid-structure interaction," Lecture Notes in Computational Science and Engineering, 53, Springer, Berlin, (2006), 146-170. doi: 10.1007/3-540-34596-5_7.

[17]

E. Kaya, 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.

[18]

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.

[19]

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

[20]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, Journal of Mathematical Analysis and Applications, 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.

[21]

H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, in "Evolution Equations, Semigroups and Functional Analysis" (Milano, 2000), Progress in Nonlinear Differential Equations and their Applications, 50, Birkhäuser, Basel, (2002), 197-216.

[22]

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.

[23]

J. E. Lagnese, Modelling and stabilization of nonlinear plates, in "Estimation and Control of Distributed Parameter Systems" (Vorau, 1990), International Series of Numerical Mathematics, 100, Birkhäuser, Basel, (1991), 247-264.

[24]

J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[25]

I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422. doi: 10.1137/S0363012996301907.

[26]

V. G. Maz'ya, "Sobolev Spaces," $2^{nd}$ augmented edition, Grundlehren der mathematischen Wissenschaften, Vol. 342, Springer, 2011.

[27]

M. Muskat, "The Flow of Homogeneous Fluids Through Porous Media," McGraw-Hill, New York, 1937.

[28]

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

[29]

J. Y. Park and J. R. Kang, Global existence and stability for a von Karman equations with memory in noncylindrical domains, Journal of Mathematical Physics, 50 (2009), 112701, 13 pp. doi: 10.1063/1.3253977.

[30]

J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Kármán equations, SIAM J. Control Optim., 33 (1995), 255-273. doi: 10.1137/S0363012992228350.

[31]

N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2205-2224. doi: 10.1098/rspa.2006.1673.

[32]

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

[33]

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

[34]

D. Tataru and M. Tucsnak, On the Cauchy problem for the full von Kármán system, Nonlinear Differential Equations Appl., 4 (1997), 325-340. doi: 10.1007/s000300050018.

[35]

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.

[36]

L. Yang and C. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 3802-3810. doi: 10.1016/j.na.2007.10.016.

[37]

SIAM PDE Conference 2011 San-Diego, Book of Abstracts,, Available from: , (). 

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]

Annalisa Quaini, Suncica Canic, Roland Glowinski, Stephen Igo, Craig J Hartley, William Zoghbi and Stephen Little, Validation of a 3D computational fluid-structure interaction model simulating flow through an elastic aperture, Journal of Biomechanics, 45 (2012), 310-318.

[4]

I. D. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping, Journal of Differential Equations, 233 (2007), 42-86. doi: 10.1016/j.jde.2006.09.019.

[5]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of the AMS, (2008).

[6]

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: 10.1002/mma.1496.

[7]

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.

[8]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, AMS, Providence, RI, 1998.

[9]

F. Flori and P. Orenga, Fluid-structure interaction: Analysis of a 3-D compressible model, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 753-777. doi: 10.1016/S0294-1449(00)00119-0.

[10]

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.

[11]

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.

[12]

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.

[13]

M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 777-813. doi: 10.1016/j.anihpc.2008.02.004.

[14]

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.

[15]

L. Hoang and A. Ibragimov, Structural stability of generalized Forchheimer equations for compressible fluids in porous media, Nonlinearity, 24 (2011), 1-41. doi: 10.1088/0951-7715/24/1/001.

[16]

J. Hron and S. Turek, A monolithic FEM/multigrid solver for ALE formulation of fluid-structure interaction with application in biomechanics, in "Fluid-structure interaction," Lecture Notes in Computational Science and Engineering, 53, Springer, Berlin, (2006), 146-170. doi: 10.1007/3-540-34596-5_7.

[17]

E. Kaya, 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.

[18]

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.

[19]

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

[20]

A. Kh. Khanmamedov, Global attractors for von Karman equations with nonlinear interior dissipation, Journal of Mathematical Analysis and Applications, 318 (2006), 92-101. doi: 10.1016/j.jmaa.2005.05.031.

[21]

H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, in "Evolution Equations, Semigroups and Functional Analysis" (Milano, 2000), Progress in Nonlinear Differential Equations and their Applications, 50, Birkhäuser, Basel, (2002), 197-216.

[22]

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.

[23]

J. E. Lagnese, Modelling and stabilization of nonlinear plates, in "Estimation and Control of Distributed Parameter Systems" (Vorau, 1990), International Series of Numerical Mathematics, 100, Birkhäuser, Basel, (1991), 247-264.

[24]

J. E. Lagnese, "Boundary Stabilization of Thin Plates," SIAM Studies in Applied Mathematics, 10, SIAM, Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[25]

I. Lasiecka, Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998), 1376-1422. doi: 10.1137/S0363012996301907.

[26]

V. G. Maz'ya, "Sobolev Spaces," $2^{nd}$ augmented edition, Grundlehren der mathematischen Wissenschaften, Vol. 342, Springer, 2011.

[27]

M. Muskat, "The Flow of Homogeneous Fluids Through Porous Media," McGraw-Hill, New York, 1937.

[28]

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

[29]

J. Y. Park and J. R. Kang, Global existence and stability for a von Karman equations with memory in noncylindrical domains, Journal of Mathematical Physics, 50 (2009), 112701, 13 pp. doi: 10.1063/1.3253977.

[30]

J.-P. Puel and M. Tucsnak, Boundary stabilization for the von Kármán equations, SIAM J. Control Optim., 33 (1995), 255-273. doi: 10.1137/S0363012992228350.

[31]

N. Peake and S. V. Sorokin, A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2205-2224. doi: 10.1098/rspa.2006.1673.

[32]

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

[33]

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

[34]

D. Tataru and M. Tucsnak, On the Cauchy problem for the full von Kármán system, Nonlinear Differential Equations Appl., 4 (1997), 325-340. doi: 10.1007/s000300050018.

[35]

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.

[36]

L. Yang and C. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 3802-3810. doi: 10.1016/j.na.2007.10.016.

[37]

SIAM PDE Conference 2011 San-Diego, Book of Abstracts,, Available from: , (). 

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