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

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

Justine Yasappan 1, , Ángela Jiménez-Casas 1, and  Mario Castro 1,

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.
Keywords: viscoelastic fluid, Thermosyphon, thermodiffusion, non-newtonian fluids, heat flux, soret effect, asymptotic behavior., numerical analysis.
Mathematics Subject Classification: Primary: 34D45, 35B40, 35K, 35Q, 58F; Secondary: 74D0.
Citation: Justine Yasappan, Ángela Jiménez-Casas, Mario Castro. Stabilizing interplay between thermodiffusion and viscoelasticity in a closed-loop thermosyphon. Discrete & 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, (2000).   Google Scholar

[2]

A. M. Bloch and E. S. Titi, On the dynamics of rotating elastic beams,, in New Trends in System Theory, 7 (1991), 128.  doi: 10.1007/978-1-4612-0439-8_15.  Google Scholar

[3]

M. E. Bravo-Gutiérrez, C. Mario and A. P. Hernández-Machado, Controlling viscoelastic flow in microchannels with slip,, Langmuir, 27 (2011), 2075.   Google Scholar

[4]

F. Debbasch and J. P. Rivet, The Ludwig-Soret effect and stochastic processes,, J. Chem.Thermodynamics, 43 (2011), 300.  doi: 10.1016/j.jct.2010.09.010.  Google Scholar

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R. Greif, Y. Zvirin and A. Mertol, The transient and stability behavior of a natural convection loop,, J. Heat Transfer, 101 (1979), 684.  doi: 10.1115/1.3451057.  Google Scholar

[6]

W. N. Findley et al., Creep and Relaxation of Nonlinear Viscoelastic Materials,, (Dover Publications, (1989).   Google Scholar

[7]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations,, J. Diff. Equ., 73 (1988), 309.  doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[8]

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

[9]

J. E. Hart, A model of flow in a closed-loop thermosyphon including the soret effect,, J. of Heat Transfer, 107 (1985), 840.  doi: 10.1115/1.3247512.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lectures Notes in Mathematics 840, (1981).   Google Scholar

[11]

M. A. Herrero and J. J. L. Velazquez, Stability analysis of a closed thermosyphon,, European J. Appl. Math., 1 (1990), 1.  doi: 10.1017/S0956792500000036.  Google Scholar

[12]

St. Hollinger and M. Lucke, Influence of the Soret effect on convection of binary fluids,, Phys. Rev. E, 57 (1998), 4238.   Google Scholar

[13]

F. P. Incropera, T. L. Bergman, A. S. Lavine and D. P. DeWitt, Fundamentals of Heat and Mass Transfer,, Wiley, (2011).   Google Scholar

[14]

A. Jiménez-Casas, A coupled ODE/PDE system governing a thermosyphon model,, Nonlin. Analy., 47 (2001), 687.  doi: 10.1016/S0362-546X(01)00212-7.  Google Scholar

[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.  doi: 10.1016/S0096-3003(00)00075-8.  Google Scholar

[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.  doi: 10.1002/(SICI)1099-1476(19990125)22:2<117::AID-MMA25>3.0.CO;2-0.  Google Scholar

[17]

A. Jiménez Casas and A. Rodríguez-Bernal, Dinámica no Lineal: Modelos de Campo de Fase y un Termosifón Cerrado,, Editorial Académica Española, (2012).   Google Scholar

[18]

J. B. Keller, Periodic oscillations in a model of thermal convection,, J. Fluid Mech., 26 (1966), 599.  doi: 10.1017/S0022112066001423.  Google Scholar

[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.  doi: 10.1017/S0022112008002619.  Google Scholar

[20]

A. Liñan, Analytical description of chaotic oscillations in a toroidal thermosyphon,, in Fluid Physics, (1994), 507.   Google Scholar

[21]

F. Morrison, Understanding Rheology,, (Oxford University Press, (2001).   Google Scholar

[22]

A. Rodríguez-Bernal, Attractor ansyphon,, Journal of Mathematical Analysis and Applications, 193 (1995), 942.  doi: 10.1006/jmaa.1995.1276.  Google Scholar

[23]

A. Rodríguez-Bernal, Inertial manifolds for dissipative semiflows in Banach spaces,, Appl. Anal., 37 (1990), 95.  doi: 10.1080/00036819008839943.  Google Scholar

[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.  doi: 10.1142/S0218127498000048.  Google Scholar

[25]

A. Rodríguez-Bernal and E. S. Van Vleck, Diffusion induced chaos in a closed loop thermosyphon,, SIAM J. Appl. Math., 58 (1998), 1072.  doi: 10.1137/S0036139996304184.  Google Scholar

[26]

A. M. Stuart, Pertubration theory of infinite-dimensiional dyanamical systems,, in Theory and Numerics of Ordinary and Partial differential Equations, (1994).   Google Scholar

[27]

G. B. Thurston and N. M. Henderson, Effects of flow geometry on blood viscoelasticity,, Biorheology, 43 (2006), 729.   Google Scholar

[28]

J. J. L. Velázquez, On the dynamics of a closed thermosyphon,, SIAM J.Appl. Math., 54 (1994), 1561.  doi: 10.1137/S0036139993246787.  Google Scholar

[29]

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

[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.  doi: 10.1016/0167-2789(85)90011-9.  Google Scholar

[31]

S. Wolfram, The mathematica book,, Cambridge University Press, (1999).   Google Scholar

[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 (2013).   Google Scholar

show all references

References:
[1]

H. A. Barnes, A Handbook of Elementary Rheology,, Institute of Non-Newtonian Fluid Mechanics, (2000).   Google Scholar

[2]

A. M. Bloch and E. S. Titi, On the dynamics of rotating elastic beams,, in New Trends in System Theory, 7 (1991), 128.  doi: 10.1007/978-1-4612-0439-8_15.  Google Scholar

[3]

M. E. Bravo-Gutiérrez, C. Mario and A. P. Hernández-Machado, Controlling viscoelastic flow in microchannels with slip,, Langmuir, 27 (2011), 2075.   Google Scholar

[4]

F. Debbasch and J. P. Rivet, The Ludwig-Soret effect and stochastic processes,, J. Chem.Thermodynamics, 43 (2011), 300.  doi: 10.1016/j.jct.2010.09.010.  Google Scholar

[5]

R. Greif, Y. Zvirin and A. Mertol, The transient and stability behavior of a natural convection loop,, J. Heat Transfer, 101 (1979), 684.  doi: 10.1115/1.3451057.  Google Scholar

[6]

W. N. Findley et al., Creep and Relaxation of Nonlinear Viscoelastic Materials,, (Dover Publications, (1989).   Google Scholar

[7]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations,, J. Diff. Equ., 73 (1988), 309.  doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[8]

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

[9]

J. E. Hart, A model of flow in a closed-loop thermosyphon including the soret effect,, J. of Heat Transfer, 107 (1985), 840.  doi: 10.1115/1.3247512.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lectures Notes in Mathematics 840, (1981).   Google Scholar

[11]

M. A. Herrero and J. J. L. Velazquez, Stability analysis of a closed thermosyphon,, European J. Appl. Math., 1 (1990), 1.  doi: 10.1017/S0956792500000036.  Google Scholar

[12]

St. Hollinger and M. Lucke, Influence of the Soret effect on convection of binary fluids,, Phys. Rev. E, 57 (1998), 4238.   Google Scholar

[13]

F. P. Incropera, T. L. Bergman, A. S. Lavine and D. P. DeWitt, Fundamentals of Heat and Mass Transfer,, Wiley, (2011).   Google Scholar

[14]

A. Jiménez-Casas, A coupled ODE/PDE system governing a thermosyphon model,, Nonlin. Analy., 47 (2001), 687.  doi: 10.1016/S0362-546X(01)00212-7.  Google Scholar

[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.  doi: 10.1016/S0096-3003(00)00075-8.  Google Scholar

[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.  doi: 10.1002/(SICI)1099-1476(19990125)22:2<117::AID-MMA25>3.0.CO;2-0.  Google Scholar

[17]

A. Jiménez Casas and A. Rodríguez-Bernal, Dinámica no Lineal: Modelos de Campo de Fase y un Termosifón Cerrado,, Editorial Académica Española, (2012).   Google Scholar

[18]

J. B. Keller, Periodic oscillations in a model of thermal convection,, J. Fluid Mech., 26 (1966), 599.  doi: 10.1017/S0022112066001423.  Google Scholar

[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.  doi: 10.1017/S0022112008002619.  Google Scholar

[20]

A. Liñan, Analytical description of chaotic oscillations in a toroidal thermosyphon,, in Fluid Physics, (1994), 507.   Google Scholar

[21]

F. Morrison, Understanding Rheology,, (Oxford University Press, (2001).   Google Scholar

[22]

A. Rodríguez-Bernal, Attractor ansyphon,, Journal of Mathematical Analysis and Applications, 193 (1995), 942.  doi: 10.1006/jmaa.1995.1276.  Google Scholar

[23]

A. Rodríguez-Bernal, Inertial manifolds for dissipative semiflows in Banach spaces,, Appl. Anal., 37 (1990), 95.  doi: 10.1080/00036819008839943.  Google Scholar

[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.  doi: 10.1142/S0218127498000048.  Google Scholar

[25]

A. Rodríguez-Bernal and E. S. Van Vleck, Diffusion induced chaos in a closed loop thermosyphon,, SIAM J. Appl. Math., 58 (1998), 1072.  doi: 10.1137/S0036139996304184.  Google Scholar

[26]

A. M. Stuart, Pertubration theory of infinite-dimensiional dyanamical systems,, in Theory and Numerics of Ordinary and Partial differential Equations, (1994).   Google Scholar

[27]

G. B. Thurston and N. M. Henderson, Effects of flow geometry on blood viscoelasticity,, Biorheology, 43 (2006), 729.   Google Scholar

[28]

J. J. L. Velázquez, On the dynamics of a closed thermosyphon,, SIAM J.Appl. Math., 54 (1994), 1561.  doi: 10.1137/S0036139993246787.  Google Scholar

[29]

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

[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.  doi: 10.1016/0167-2789(85)90011-9.  Google Scholar

[31]

S. Wolfram, The mathematica book,, Cambridge University Press, (1999).   Google Scholar

[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 (2013).   Google Scholar

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