## Asymptotic behavior of a viscoelastic fluid in a closed loop thermosyphon: physical derivation, asymptotic analysis, and numerical experiments.(English)Zbl 1345.76013

Summary: Fluids subject to thermal gradients produce complex behaviors that arise from the competition with gravitational effects. Although such sort of systems have been widely studied in the literature for simple (Newtonian) fluids, the behavior of viscoelastic fluids has not been explored thus far. We present a theoretical study of the dynamics of a Maxwell viscoelastic fluid in a closed-loop thermosyphon. This sort of fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and the exact equations of motion in the inertial manifold that characterizes the asymptotic behavior. We derive, for the first time, the mathematical derivations of the motion of a viscoelastic fluid in the interior of a closed-loop thermosyphon under the effects of natural convection and a given external temperature gradient.

### MSC:

 76A10 Viscoelastic fluids

Mathematica
Full Text:

### References:

 [1] Cross, M. C.; Greenside, H., Pattern Formation and Dynamics in Nonequilibrium Systems (2009), New York, NY, USA: Cambridge University Press, New York, NY, USA · Zbl 1177.82002 [2] Keller, J. B., Periodic oscillations in a model of thermal convection, Journal of Fluid Mechanics, 26, 3, 599-606 (1966) [3] Welander, P., On the oscillatory instability of a differentially heated fluid loop, Journal of Fluid Mechanics, 29, 1, 17-30 (1967) · Zbl 0163.20702 [4] Morrison, F., Understanding Rheology (2001), New York, NY, USA: Oxford University Press, New York, NY, USA · Zbl 1012.76500 [5] Greif, R.; Zvirin, Y.; Mertol, A., The transient and stability behavior of a natural convection loop, Journal of Heat Transfer, 107, 684-688 (1987) [6] Jiménez Casas, A.; Rodríguez-Bernal, A., Dinámica no Lineal: Modelos de Campo de Fase y un Termosifon Cerrado (2012), Editorial Acadmica Espaola, Lap Lambert Academic Publishing [7] Jiménez-Casas, A.; Rodríguez-Bernal, A., Finite-dimensional asymptotic behavior in a thermosyphon including the Soret effect, Mathematical Methods in the Applied Sciences, 22, 2, 117-137 (1999) · Zbl 0926.35020 [8] Jiménez-Casas, A., A coupled ODE/PDE system governing a thermosyphon model, Nonlinear Analysis, 47, 687-692 (2001) · Zbl 1042.76562 [9] Velázquez, J. J. L., On the dynamics of a closed thermosyphon, SIAM Journal on Applied Mathematics, 54, 6, 1561-1593 (1994) · Zbl 0823.35151 · doi:10.1137/S0036139993246787 [10] Jiménez Casas, A.; Ovejero, A. M. L., Numerical analysis of a closed-loop thermosyphon including the Soret effect, Applied Mathematics and Computation, 124, 3, 289-318 (2001) · Zbl 1022.80003 · doi:10.1016/S0096-3003(00)00075-8 [11] Rodríguez-Bernal, A.; van Vleck, E. S., Complex oscillations in a closed thermosyphon, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 8, 1, 41-56 (1998) · Zbl 1004.34035 · doi:10.1142/S0218127498000048 [12] Herrero, M. A.; Velázquez, J. J. L., Stability analysis of a closed thermosyphon, European Journal of Applied Mathematics, 1, 1, 1-23 (1990) · Zbl 0701.76086 · doi:10.1017/S0956792500000036 [13] Liňan, A.; Velarde, M. G.; Christov, C. I., Analytical description of chaotic oscillations in a toroidal thermosyphon, Fluid Physics. Fluid Physics, Lecture Notes of Summer Schools, 507-523 (1994), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA [14] Rodríguez-Bernal, A.; Van Vleck, E. S., Diffusion induced chaos in a closed loop thermosyphon, SIAM Journal on Applied Mathematics, 58, 4, 1072-1093 (1998) · Zbl 0913.35080 · doi:10.1137/S0036139996304184 [15] Henry, D., Geometric Theory of Semilinear Parabolic Equations. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840 (1981), Berlin, Germany: Springer, Berlin, Germany · Zbl 0456.35001 [16] Rodríguez-Bernal, A., Attractors and inertial manifolds for the dynamics of a closed thermosyphon, Journal of Mathematical Analysis and Applications, 193, 3, 942-965 (1995) · Zbl 0854.76088 · doi:10.1006/jmaa.1995.1276 [17] Hale, J. K., Asymptotic Behavior of Dissipative Systems (1988), Providence, RI, USA: AMS, Providence, RI, USA · Zbl 0642.58013 [18] Incropera, F. P.; Bergman, T. L.; Lavine, A. S.; DeWitt, D. P., Fundamentals of Heat and Mass Transfer (2011), John Wiley & Sons [19] Bloch, A. M.; Titi, E. S., On the dynamics of rotating elastic beams, New Trends in Systems Theory. New Trends in Systems Theory, Progress in Systems and Control Theory, 7, 128-135 (1991), Boston, Mass, USA: Birkhäauser, Boston, Mass, USA · Zbl 0759.73036 [20] Stuart, A. M.; Ainsworth, M.; Levesley, J.; Light, W. A.; Marletta, M., Pertubration theory of infinite-dimensional dynamical systems, Theory and Numerics of OrdInary and Partial Differential Equations (1994), Oxford, UK: Oxford University Press, Oxford, UK [21] Foias, C.; Sell, G. R.; Temam, R., Inertial manifolds for nonlinear evolutionary equations, Journal of Differential Equations, 73, 2, 309-353 (1988) · Zbl 0643.58004 · doi:10.1016/0022-0396(88)90110-6 [22] Rodríguez-Bernal, A., Inertial manifolds for dissipative semiflows in Banach spaces, Applicable Analysis, 37, 1-4, 95-141 (1990) · Zbl 0678.35006 · doi:10.1080/00036819008839943 [23] Fink, A. M., Almost Periodic Differential Equations. Almost Periodic Differential Equations, Lecture Notes in Mathematics, 377 (1974), Berlin, Germany: Springer, Berlin, Germany · Zbl 0325.34039 [24] Wolfram, S., The Mathematica book (1999), Cambridge University Press · Zbl 0924.65002 [25] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press · Zbl 0777.34002 [26] Verhulst, F., Methods and Applications of Singular Perturbations (2005), New York, NY, USA: Springer, New York, NY, USA · Zbl 1148.35006 · doi:10.1007/0-387-28313-7
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