Citation: |
[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. |