# American Institute of Mathematical Sciences

November  2015, 20(9): 3267-3299. doi: 10.3934/dcdsb.2015.20.3267

## Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon

 1 Grupo de Dinámica No Lineal (DNL), Escuela Técnica Superior de Ingeniería (ICAI), Universidad Pontificia Comillas, Madrid E-28015, Spain, Spain, Spain

Received  June 2013 Revised  May 2014 Published  September 2015

Viscoelastic fluids represent a major challenge both from an engineering and from a mathematical point of view. Recently, we have shown that viscoelasticity induces chaos in closed-loop thermosyphons. This induced behavior might interfere with the engineering choice of using a specific fluid. In this work we show that the addition of a solute to the fluid can, under some conditions, stabilize the system due to thermodiffusion (also known as the Soret effect). Unexpectedly, the role of viscoelasticity is opposite to the case of single-element fluids, where it (generically) induces chaos. Our results are derived by combining analytical results based on the projection of the dynamics on an inertial manifold as well as numerical simulations characterized by the calculation of Lyapunov exponents.
Citation: Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3267-3299. doi: 10.3934/dcdsb.2015.20.3267
##### References:
 [1] H. A. Barnes, A Handbook of Elementary Rheology, Institute of Non-Newtonian Fluid Mechanics, Univesity of Wales, 2000. [2] 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 (1991), 128-135. doi: 10.1007/978-1-4612-0439-8_15. [3] M. E. Bravo-Gutiérrez, C. Mario and A. P. Hernández-Machado, Controlling viscoelastic flow in microchannels with slip, Langmuir, ACS Publications, 27 (2011), 2075-2079. [4] F. Debbasch and J. P. Rivet, The Ludwig-Soret effect and stochastic processes, J. Chem.Thermodynamics, 43 (2011), 300-306. doi: 10.1016/j.jct.2010.09.010. [5] R. Greif, Y. Zvirin and A. Mertol, The transient and stability behavior of a natural convection loop, J. Heat Transfer, 101 (1979), 684-688. doi: 10.1115/1.3451057. [6] W. N. Findley et al., Creep and Relaxation of Nonlinear Viscoelastic Materials, (Dover Publications, New York, 1989). [7] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Equ., 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6. [8] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. [9] J. E. Hart, A model of flow in a closed-loop thermosyphon including the soret effect, J. of Heat Transfer, 107 (1985), 840-849. doi: 10.1115/1.3247512. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics 840, Springer-Verlag, Berlin, New York, 1981. [11] M. A. Herrero and J. J. L. Velazquez, Stability analysis of a closed thermosyphon, European J. Appl. Math., 1 (1990), 1-24. doi: 10.1017/S0956792500000036. [12] St. Hollinger and M. Lucke, Influence of the Soret effect on convection of binary fluids, Phys. Rev. E, 57 (1998), 4238-4249. [13] F. P. Incropera, T. L. Bergman, A. S. Lavine and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, Wiley, 2011. [14] A. Jiménez-Casas, A coupled ODE/PDE system governing a thermosyphon model, Nonlin. Analy., 47 (2001), 687-692. doi: 10.1016/S0362-546X(01)00212-7. [15] 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. doi: 10.1016/S0096-3003(00)00075-8. [16] 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. doi: 10.1002/(SICI)1099-1476(19990125)22:2<117::AID-MMA25>3.0.CO;2-0. [17] A. Jiménez Casas and A. Rodríguez-Bernal, Dinámica no Lineal: Modelos de Campo de Fase y un Termosifón Cerrado, Editorial Académica Española, (Lap Lambert Academic Publishing GmbH and Co. KG, Germany 2012). [18] J. B. Keller, Periodic oscillations in a model of thermal convection, J. Fluid Mech., 26 (1966), 599-606. doi: 10.1017/S0022112066001423. [19] P. A. Lakshminarayana, P. V. S. N. Murthy and R. S. R. Gorla, Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium, J. Fluid Mech., 612 (2008), 1-19. doi: 10.1017/S0022112008002619. [20] A. Liñan, Analytical description of chaotic oscillations in a toroidal thermosyphon, in Fluid Physics, Lecture Notes of Summer Schools, (M.G. Velarde, C.I. Christov, eds.,) World Scientific, River Edge, NJ, (1994), 507-523. [21] F. Morrison, Understanding Rheology, (Oxford University Press, USA, 2001). [22] A. Rodríguez-Bernal, Attractor ansyphon, Journal of Mathematical Analysis and Applications, 193 (1995), 942-965. doi: 10.1006/jmaa.1995.1276. [23] A. Rodríguez-Bernal, Inertial manifolds for dissipative semiflows in Banach spaces, Appl. Anal., 37 (1990), 95-141. doi: 10.1080/00036819008839943. [24] A. Rodríguez-Bernal and E. S. Van Vleck, Complex oscillations in a closed thermosyphon, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 41-56. doi: 10.1142/S0218127498000048. [25] A. Rodríguez-Bernal and E. S. Van Vleck, Diffusion induced chaos in a closed loop thermosyphon, SIAM J. Appl. Math., 58 (1998), 1072-1093. doi: 10.1137/S0036139996304184. [26] A. M. Stuart, Pertubration theory of infinite-dimensiional dyanamical systems, in Theory and Numerics of Ordinary and Partial differential Equations, M.Ainsworth, J. Levesley, W.A.Light and M. Marletta, eds., (Oxford University Press, Oxford, UK, 1994). [27] G. B. Thurston and N. M. Henderson, Effects of flow geometry on blood viscoelasticity, Biorheology, 43 (2006), 729-746. [28] J. J. L. Velázquez, On the dynamics of a closed thermosyphon, SIAM J.Appl. Math., 54 (1994), 1561-1593. doi: 10.1137/S0036139993246787. [29] P. Welander, On the oscillatory instability of a differentially heated fluid loop, J. Fluid Mech., 29 (1967), 17-30. [30] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285-317. doi: 10.1016/0167-2789(85)90011-9. [31] S. Wolfram, The mathematica book, Cambridge University Press, 1999. [32] J. Yasappan, A. Jiménez-Casas and M. Castro, Asymptotic behavior of a viscoelastic fluid in a closed loop thermosyphon: Physical derivation, asymptotic analysis and numerical experiments， Abstract and Applied Analysis, 2013 Art. ID 748683, (2013), 20pp.

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##### References:
 [1] H. A. Barnes, A Handbook of Elementary Rheology, Institute of Non-Newtonian Fluid Mechanics, Univesity of Wales, 2000. [2] 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 (1991), 128-135. doi: 10.1007/978-1-4612-0439-8_15. [3] M. E. Bravo-Gutiérrez, C. Mario and A. P. Hernández-Machado, Controlling viscoelastic flow in microchannels with slip, Langmuir, ACS Publications, 27 (2011), 2075-2079. [4] F. Debbasch and J. P. Rivet, The Ludwig-Soret effect and stochastic processes, J. Chem.Thermodynamics, 43 (2011), 300-306. doi: 10.1016/j.jct.2010.09.010. [5] R. Greif, Y. Zvirin and A. Mertol, The transient and stability behavior of a natural convection loop, J. Heat Transfer, 101 (1979), 684-688. doi: 10.1115/1.3451057. [6] W. N. Findley et al., Creep and Relaxation of Nonlinear Viscoelastic Materials, (Dover Publications, New York, 1989). [7] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations, J. Diff. Equ., 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6. [8] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. [9] J. E. Hart, A model of flow in a closed-loop thermosyphon including the soret effect, J. of Heat Transfer, 107 (1985), 840-849. doi: 10.1115/1.3247512. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Mathematics 840, Springer-Verlag, Berlin, New York, 1981. [11] M. A. Herrero and J. J. L. Velazquez, Stability analysis of a closed thermosyphon, European J. Appl. Math., 1 (1990), 1-24. doi: 10.1017/S0956792500000036. [12] St. Hollinger and M. Lucke, Influence of the Soret effect on convection of binary fluids, Phys. Rev. E, 57 (1998), 4238-4249. [13] F. P. Incropera, T. L. Bergman, A. S. Lavine and D. P. DeWitt, Fundamentals of Heat and Mass Transfer, Wiley, 2011. [14] A. Jiménez-Casas, A coupled ODE/PDE system governing a thermosyphon model, Nonlin. Analy., 47 (2001), 687-692. doi: 10.1016/S0362-546X(01)00212-7. [15] 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. doi: 10.1016/S0096-3003(00)00075-8. [16] 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. doi: 10.1002/(SICI)1099-1476(19990125)22:2<117::AID-MMA25>3.0.CO;2-0. [17] A. Jiménez Casas and A. Rodríguez-Bernal, Dinámica no Lineal: Modelos de Campo de Fase y un Termosifón Cerrado, Editorial Académica Española, (Lap Lambert Academic Publishing GmbH and Co. KG, Germany 2012). [18] J. B. Keller, Periodic oscillations in a model of thermal convection, J. Fluid Mech., 26 (1966), 599-606. doi: 10.1017/S0022112066001423. [19] P. A. Lakshminarayana, P. V. S. N. Murthy and R. S. R. Gorla, Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium, J. Fluid Mech., 612 (2008), 1-19. doi: 10.1017/S0022112008002619. [20] A. Liñan, Analytical description of chaotic oscillations in a toroidal thermosyphon, in Fluid Physics, Lecture Notes of Summer Schools, (M.G. Velarde, C.I. Christov, eds.,) World Scientific, River Edge, NJ, (1994), 507-523. [21] F. Morrison, Understanding Rheology, (Oxford University Press, USA, 2001). [22] A. Rodríguez-Bernal, Attractor ansyphon, Journal of Mathematical Analysis and Applications, 193 (1995), 942-965. doi: 10.1006/jmaa.1995.1276. [23] A. Rodríguez-Bernal, Inertial manifolds for dissipative semiflows in Banach spaces, Appl. Anal., 37 (1990), 95-141. doi: 10.1080/00036819008839943. [24] A. Rodríguez-Bernal and E. S. Van Vleck, Complex oscillations in a closed thermosyphon, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 41-56. doi: 10.1142/S0218127498000048. [25] A. Rodríguez-Bernal and E. S. Van Vleck, Diffusion induced chaos in a closed loop thermosyphon, SIAM J. Appl. Math., 58 (1998), 1072-1093. doi: 10.1137/S0036139996304184. [26] A. M. Stuart, Pertubration theory of infinite-dimensiional dyanamical systems, in Theory and Numerics of Ordinary and Partial differential Equations, M.Ainsworth, J. Levesley, W.A.Light and M. Marletta, eds., (Oxford University Press, Oxford, UK, 1994). [27] G. B. Thurston and N. M. Henderson, Effects of flow geometry on blood viscoelasticity, Biorheology, 43 (2006), 729-746. [28] J. J. L. Velázquez, On the dynamics of a closed thermosyphon, SIAM J.Appl. Math., 54 (1994), 1561-1593. doi: 10.1137/S0036139993246787. [29] P. Welander, On the oscillatory instability of a differentially heated fluid loop, J. Fluid Mech., 29 (1967), 17-30. [30] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285-317. doi: 10.1016/0167-2789(85)90011-9. [31] S. Wolfram, The mathematica book, Cambridge University Press, 1999. [32] J. Yasappan, A. Jiménez-Casas and M. Castro, Asymptotic behavior of a viscoelastic fluid in a closed loop thermosyphon: Physical derivation, asymptotic analysis and numerical experiments， Abstract and Applied Analysis, 2013 Art. ID 748683, (2013), 20pp.
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