American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 375-384. doi: 10.3934/proc.2013.2013.375

Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid

A. Jiménez-Casas 1, , Mario Castro 2, and  Justine Yassapan 3,

1. 

Grupo Dinámica No Lineal(ICAI), Universidad Pontificia Comillas, C/Alberto Aguilera 23, 28015 Madrid
2. 

Grupo Interdisciplinar de Sistemas Complejos (GISC) and Grupo de Dinmica No Lineal (DNL), Escuela Tcnica Superior de Ingeniera (ICAI), Universidad Pontificia Comillas, E28015, Madrid, Spain
3. 

Grupo de Dinmica No Lineal (DNL), Departamento de Matemtica Aplicada y Computacin, Escuela Tcnica Superior de Ingeniera (ICAI), Universidad Ponti cia Comillas, E28015, Madrid, Spain

Received  July 2012 Published  November 2013

We analyse the motion of a viscoelastic fluid in the interior of a closed loop thermosyphon under the effects of natural convection. We consider a viscoelastic fluid described by the Maxwell constitutive equation. This fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and derive the exact equations of motion in the inertial manifold that characterize the asymptotic behavior. Our work is a generalization of some previous results on standard (Newtonian) fluids.
Keywords: Thermosyphon, viscoelastic fluid, heat flux, asymptotic behaviour..
Mathematics Subject Classification: 35M33, 35B42, 76A1.
Citation: A. Jiménez-Casas, Mario Castro, Justine Yassapan. Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid. Conference Publications, 2013, 2013 (special) : 375-384. doi: 10.3934/proc.2013.2013.375
References:
[1]

A. M. Bloch and E. S. Titi, On the dynamics of rotating elastic beams, in New Trends in System Theory, Progr. Systems Control Theory 7, Birkh$\ddota$user Boston, Boston, MA, 128-135, (1991).  Google Scholar

[2]

C. Foias, G. R. Sell and R. Temam, Inertial Manifolds for Nonlinear Evolution Equations, J. Diff. Equ., 73 (1988), 309-353.  Google Scholar

[3]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, (1988).  Google Scholar

[4]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics, 840 (1982), Springer-Verlag, Berlin, New York.  Google Scholar

[5]

A. Jiménez-Casas and A. M. L. Ovejero, Numerical analysis of a closed-loop thermosyphon including the Soret effect, Appl. Math. Comput., 124 (2001), 289-318.  Google Scholar

[6]

A. Jiménez-Casas and A. Rodríguez-Bernal, Finite-dimensional asymptotic behavior in a thermosyphon including the Soret effect, Math. Meth. in the Appl. Sci., 22, (1999), 117-137.  Google Scholar

[7]

B. Keller, Periodic oscillations in a model of thermal convection, J. Fluid Mech., 26 (1966), 599-606.  Google Scholar

[8]

F. Morrison, Understanding rheology, Oxford University Press, 2001, USA. Google Scholar

[9]

A. Rodríguez-Bernal and E. S. Van Vleck, Diffusion Induced Chaos in a Closed Loop Thermosyphon, SIAM J. Appl. Math., 58 (1998), 1072-1093(electronic).  Google Scholar

[10]

A. M. Stuart, "Perturbation Theory of Infinite-Dimensional Dynamical Systems," in Theory and Numerics of Ordinary and Partial differential Equations, M. Ainsworth, J. Levesley, W.A. Light and M. Marletta, eds. (1994), Oxford University Press, Oxford, UK.  Google Scholar

[11]

J. J. L. Velázquez, On the dynamics of a closed thermosyphon, SIAM J. Appl. Math. 54 (1994), 1561-1593.  Google Scholar

[12]

P. Welander, On the oscillatory instability of a differentially heated fluid loop, J. Fluid Mech., 29 (1967), 17-30. Google Scholar

[13]

J. Yasappan, A. Jiménez-Casas and Mario Castro, "Asymptotic Behavior of a Viscoelastic Fluid in a Closed Loop Thermosyphon: Physical Derivation, Asymptotic Analysis and Numerical Experiments," Abstr. Appl. Anal., (2013).  Google Scholar

[14]

J. Yasappan, A. Jiménez-Casas and Mario Castro, Chaotic behavior of the closed loop thermosyphon model with memory effects, submitted, (2012). Google Scholar

show all references

References:
[1]

A. M. Bloch and E. S. Titi, On the dynamics of rotating elastic beams, in New Trends in System Theory, Progr. Systems Control Theory 7, Birkh$\ddota$user Boston, Boston, MA, 128-135, (1991).  Google Scholar

[2]

C. Foias, G. R. Sell and R. Temam, Inertial Manifolds for Nonlinear Evolution Equations, J. Diff. Equ., 73 (1988), 309-353.  Google Scholar

[3]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, (1988).  Google Scholar

[4]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics, 840 (1982), Springer-Verlag, Berlin, New York.  Google Scholar

[5]

A. Jiménez-Casas and A. M. L. Ovejero, Numerical analysis of a closed-loop thermosyphon including the Soret effect, Appl. Math. Comput., 124 (2001), 289-318.  Google Scholar

[6]

A. Jiménez-Casas and A. Rodríguez-Bernal, Finite-dimensional asymptotic behavior in a thermosyphon including the Soret effect, Math. Meth. in the Appl. Sci., 22, (1999), 117-137.  Google Scholar

[7]

B. Keller, Periodic oscillations in a model of thermal convection, J. Fluid Mech., 26 (1966), 599-606.  Google Scholar

[8]

F. Morrison, Understanding rheology, Oxford University Press, 2001, USA. Google Scholar

[9]

A. Rodríguez-Bernal and E. S. Van Vleck, Diffusion Induced Chaos in a Closed Loop Thermosyphon, SIAM J. Appl. Math., 58 (1998), 1072-1093(electronic).  Google Scholar

[10]

A. M. Stuart, "Perturbation Theory of Infinite-Dimensional Dynamical Systems," in Theory and Numerics of Ordinary and Partial differential Equations, M. Ainsworth, J. Levesley, W.A. Light and M. Marletta, eds. (1994), Oxford University Press, Oxford, UK.  Google Scholar

[11]

J. J. L. Velázquez, On the dynamics of a closed thermosyphon, SIAM J. Appl. Math. 54 (1994), 1561-1593.  Google Scholar

[12]

P. Welander, On the oscillatory instability of a differentially heated fluid loop, J. Fluid Mech., 29 (1967), 17-30. Google Scholar

[13]

J. Yasappan, A. Jiménez-Casas and Mario Castro, "Asymptotic Behavior of a Viscoelastic Fluid in a Closed Loop Thermosyphon: Physical Derivation, Asymptotic Analysis and Numerical Experiments," Abstr. Appl. Anal., (2013).  Google Scholar

[14]

J. Yasappan, A. Jiménez-Casas and Mario Castro, Chaotic behavior of the closed loop thermosyphon model with memory effects, submitted, (2012). Google Scholar

[1]

Kun Wang, Yangping Lin, Yinnian He. Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 657-677. doi: 10.3934/dcds.2012.32.657
[2]

Giovambattista Amendola, Sandra Carillo, John Murrough Golden, Adele Manes. Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815
[3]

Tomás Caraballo, I. D. Chueshov, Pedro Marín-Rubio, José Real. Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 253-270. doi: 10.3934/dcds.2007.18.253
[4]

Tomás Caraballo, Francisco Morillas, José Valero. Asymptotic behaviour of a logistic lattice system. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4019-4037. doi: 10.3934/dcds.2014.34.4019
[5]

Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197
[6]

Yinnian He, Yi Li. Asymptotic behavior of linearized viscoelastic flow problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 843-856. doi: 10.3934/dcdsb.2008.10.843
[7]

Youjun Deng, Hongyu Liu, Xianchao Wang, Dong Wei, Liyan Zhu. Simultaneous recovery of surface heat flux and thickness of a solid structure by ultrasonic measurements. Electronic Research Archive, , () : -. doi: 10.3934/era.2021027
[8]

Dorin Ieşan. Strain gradient theory of porous solids with initial stresses and initial heat flux. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2169-2187. doi: 10.3934/dcdsb.2014.19.2169
[9]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden, Adele Manes. Energy stability for thermo-viscous fluids with a fading memory heat flux. Evolution Equations & Control Theory, 2015, 4 (3) : 265-279. doi: 10.3934/eect.2015.4.265
[10]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5305-5319. doi: 10.3934/dcdsb.2020344
[11]

Roberto Triggiani, Jing Zhang. Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay. Evolution Equations & Control Theory, 2018, 7 (1) : 153-182. doi: 10.3934/eect.2018008
[12]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
[13]

Xiaoli Wang, Peter Kloeden, Meihua Yang. Asymptotic behaviour of a neural field lattice model with delays. Electronic Research Archive, 2020, 28 (2) : 1037-1048. doi: 10.3934/era.2020056
[14]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[15]

Maria Rosaria Lancia, Paola Vernole. The Stokes problem in fractal domains: Asymptotic behaviour of the solutions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1553-1565. doi: 10.3934/dcdss.2020088
[16]

Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206
[17]

María Anguiano, P.E. Kloeden. Asymptotic behaviour of the nonautonomous SIR equations with diffusion. Communications on Pure & Applied Analysis, 2014, 13 (1) : 157-173. doi: 10.3934/cpaa.2014.13.157
[18]

Jorge Ferreira, Mauro De Lima Santos. Asymptotic behaviour for wave equations with memory in a noncylindrical domains. Communications on Pure & Applied Analysis, 2003, 2 (4) : 511-520. doi: 10.3934/cpaa.2003.2.511
[19]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665
[20]

Victor Zvyagin, Vladimir Orlov. On one problem of viscoelastic fluid dynamics with memory on an infinite time interval. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3855-3877. doi: 10.3934/dcdsb.2018114

 Impact Factor: 

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences