# American Institute of Mathematical Sciences

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

## Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid

 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.
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). [2] C. Foias, G. R. Sell and R. Temam, Inertial Manifolds for Nonlinear Evolution Equations, J. Diff. Equ., 73 (1988), 309-353. [3] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, (1988). [4] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics, 840 (1982), Springer-Verlag, Berlin, New York. [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. [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. [7] B. Keller, Periodic oscillations in a model of thermal convection, J. Fluid Mech., 26 (1966), 599-606. [8] F. Morrison, Understanding rheology, Oxford University Press, 2001, USA. [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). [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. [11] J. J. L. Velázquez, On the dynamics of a closed thermosyphon, SIAM J. Appl. Math. 54 (1994), 1561-1593. [12] P. Welander, On the oscillatory instability of a differentially heated fluid loop, J. Fluid Mech., 29 (1967), 17-30. [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). [14] J. Yasappan, A. Jiménez-Casas and Mario Castro, Chaotic behavior of the closed loop thermosyphon model with memory effects, submitted, (2012).

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). [2] C. Foias, G. R. Sell and R. Temam, Inertial Manifolds for Nonlinear Evolution Equations, J. Diff. Equ., 73 (1988), 309-353. [3] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, (1988). [4] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics, 840 (1982), Springer-Verlag, Berlin, New York. [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. [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. [7] B. Keller, Periodic oscillations in a model of thermal convection, J. Fluid Mech., 26 (1966), 599-606. [8] F. Morrison, Understanding rheology, Oxford University Press, 2001, USA. [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). [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. [11] J. J. L. Velázquez, On the dynamics of a closed thermosyphon, SIAM J. Appl. Math. 54 (1994), 1561-1593. [12] P. Welander, On the oscillatory instability of a differentially heated fluid loop, J. Fluid Mech., 29 (1967), 17-30. [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). [14] J. Yasappan, A. Jiménez-Casas and Mario Castro, Chaotic behavior of the closed loop thermosyphon model with memory effects, submitted, (2012).
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