American Institute of Mathematical Sciences

Advanced
October  2011, 4(5): 1213-1225. doi: 10.3934/dcdss.2011.4.1213

Dissipative solitons in binary fluid convection

Isabel Mercader 1, , Oriol Batiste 1, , Arantxa Alonso 1, and  Edgar Knobloch 2,

1. 

Departament de Física Aplicada, Universitat Politècnica de Catalunya, Campus Nord, 08034 Barcelona, Spain, Spain, Spain
2. 

Department of Physics, University of California, California, Berkeley, CA 94720, United States

Received  July 2009 Revised  January 2010 Published  December 2010

A horizontal layer containing a miscible mixture of two fluids can produce dissipative solitons when heated from below. The physics of the system is described, and dissipative solitons are computed using numerical continuation for three distinct sets of experimentally realizable parameter values. The stability of the solutions is investigated using direct numerical integration in time and related to the stability properties of the competing periodic state.
Keywords: Dissipative solitons, binary fluid convection, bifurcation theory..
Mathematics Subject Classification: Primary: 74J35, 76E0634; Secondary: 34K18, 34C3.
Citation: Isabel Mercader, Oriol Batiste, Arantxa Alonso, Edgar Knobloch. Dissipative solitons in binary fluid convection. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1213-1225. doi: 10.3934/dcdss.2011.4.1213
References:
[1]

N. Akhmediev and A. Ankiewicz (eds), "Dissipative Solitons," Lect. Notes in Physics, 661, Springer, Berlin, 2005.  Google Scholar

[2]

P. Assemat, A. Bergeon and E. Knobloch, Spatially localized states in Marangoni convection in binary mixtures, Fluid Dyn. Res., 40 (2008), 852-876. doi: 10.1016/j.fluiddyn.2007.11.002.  Google Scholar

[3]

W. Barten, M. Lücke, M. Kamps and R. Schmitz, Convection in binary fluid mixtures. I. Extended traveling-wave and stationary states, Phys. Rev. E, 51 (1995), 5636-5661. doi: 10.1103/PhysRevE.51.5636.  Google Scholar

[4]

O. Batiste and E. Knobloch, Simulations of localized states of stationary convection in 3He-4He mixtures, Phys. Rev. Lett., 95 (2005), 244501. Google Scholar

[5]

O. Batiste, E. Knobloch, A. Alonso and I. Mercader, Spatially localized binary-fluid convection, J. Fluid Mech., 560 (2006), 149-158. doi: 10.1017/S0022112006000759.  Google Scholar

[6]

M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972. doi: 10.1137/080713306.  Google Scholar

[7]

A. Bergeon and E. Knobloch, Spatially localized states in natural doubly diffusive convection, Phys. Fluids, 20 (2008), 034102. doi: 10.1063/1.2837177.  Google Scholar

[8]

A. Bergeon, J. Burke, E. Knobloch and I. Mercader, Eckhaus instability and homoclinic snaking, Phys. Rev. E, 78 (2008), 046201. doi: 10.1103/PhysRevE.78.046201.  Google Scholar

[9]

S. Blanchflower, Magnetohydrodynamic convectons, Phys. Lett. A, 261 (1999), 74-81. doi: 10.1016/S0375-9601(99)00573-3.  Google Scholar

[10]

S. Blanchflower and N. O. Weiss, Three-dimensional magnetohydrodynamic convectons, Phys. Lett. A, 294 (2002), 297-303. doi: 10.1016/S0375-9601(02)00076-2.  Google Scholar

[11]

C. S. Bretherton and E. A. Spiegel, Intermittency through modulational instability, Phys. Lett. A, 96 (1983), 152-156. doi: 10.1016/0375-9601(83)90491-7.  Google Scholar

[12]

J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation, Phys. Lett. A, 360 (2007), 681-688. doi: 10.1016/j.physleta.2006.08.072.  Google Scholar

[13]

P. Coullet, C. Riera and C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett., 84 (2000), 3069-3072. doi: 10.1103/PhysRevLett.84.3069.  Google Scholar

[14]

J. H. P. Dawes, Localized convection cells in the presence of a vertical magnetic field, J. Fluid Mech., 570 (2007), 385-406. doi: 10.1017/S0022112006002795.  Google Scholar

[15]

Q. Feng, J. V. Moloney and A. C. Newell, Transverse patterns in lasers, Phys. Rev. A, 50 (1994), 3601-3604. doi: 10.1103/PhysRevA.50.R3601.  Google Scholar

[16]

K. Ghorayeb and A. Mojtabi, Double diffusive convection in a vertical rectangular cavity, Phys. Fluids, 9 (1997), 2339-2348. doi: 10.1063/1.869354.  Google Scholar

[17]

D. Jung and M. Lücke, Bistability of moving and self-pinned fronts of supercritical localized convection structures, Europhys. Lett., 80 (2007), 14002, 1-6. Google Scholar

[18]

E. Knobloch, A. E. Deane, J. Toomre and D. R. Moore, Doubly diffusive waves, in "Multiparameter Bifurcation Theory" (eds. M. Golubitsky and J. Guckenheimer), Contemp. Math., 56 (1986), American Mathematical Society, Providence, R.I., 203-216.  Google Scholar

[19]

P. Kolodner, Observations of the Eckhaus instability in one-dimensional traveling-wave convection, Phys. Rev. A, 46 (1992), 1739-1742. doi: 10.1103/PhysRevA.46.R1739.  Google Scholar

[20]

P. Kolodner, Coexisting traveling waves and steady rolls in binary-fluid convection, Phys. Rev. E, 48 (1993), R665-668. doi: 10.1103/PhysRevE.48.R665.  Google Scholar

[21]

P. Kolodner, J. A. Glazier and H. L. Williams, Dispersive chaos in one-dimensional traveling-wave convection, Phys. Rev. Lett., 65 (1990), 1579-1582. doi: 10.1103/PhysRevLett.65.1579.  Google Scholar

[22]

I. Mercader, A. Alonso and O. Batiste, Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures, Eur. Phys. J. E, 15 (2004), 311-318. doi: 10.1140/epje/i2004-10071-7.  Google Scholar

[23]

I. Mercader, A. Alonso and O. Batiste, Spatiotemporal dynamics near the onset of convection for binary mixtures in cylindrical containers, Phys. Rev. E, 77 (2008), 036313. doi: 10.1103/PhysRevE.77.036313.  Google Scholar

[24]

I. Mercader, O. Batiste and A. Alonso, Continuation of travelling-wave solutions of the Navier-Stokes equations, Int. J. Num. Methods in Fluids, 52 (2006), 707-721. doi: 10.1002/fld.1196.  Google Scholar

[25]

I. Mercader, O. Batiste, A. Alonso and E. Knobloch, Localized pinning states in closed containers: Homoclinic snaking without bistability, Phys. Rev. E, 80 (2009), 025201(R). doi: 10.1103/PhysRevE.80.025201.  Google Scholar

[26]

I. Mercader, O. Batiste, A. Alonso and E. Knobloch, Convectons in periodic and bounded domains, Fluid Dyn. Res., 42 (2010), 025505. doi: 10.1088/0169-5983/42/2/025505.  Google Scholar

[27]

D. R. Ohlsen, S. Y. Yamamoto, C. M. Surko and P. Kolodner, Transition from traveling-wave to stationary convection in fluid mixtures, Phys. Rev. Lett., 65 (1990), 1431-1434. doi: 10.1103/PhysRevLett.65.1431.  Google Scholar

[28]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11. doi: 10.1016/0167-2789(86)90104-1.  Google Scholar

[29]

P. D. Woods and A. R. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation, Physica D, 129 (1999), 147-170. doi: 10.1016/S0167-2789(98)00309-1.  Google Scholar

show all references

References:
[1]

N. Akhmediev and A. Ankiewicz (eds), "Dissipative Solitons," Lect. Notes in Physics, 661, Springer, Berlin, 2005.  Google Scholar

[2]

P. Assemat, A. Bergeon and E. Knobloch, Spatially localized states in Marangoni convection in binary mixtures, Fluid Dyn. Res., 40 (2008), 852-876. doi: 10.1016/j.fluiddyn.2007.11.002.  Google Scholar

[3]

W. Barten, M. Lücke, M. Kamps and R. Schmitz, Convection in binary fluid mixtures. I. Extended traveling-wave and stationary states, Phys. Rev. E, 51 (1995), 5636-5661. doi: 10.1103/PhysRevE.51.5636.  Google Scholar

[4]

O. Batiste and E. Knobloch, Simulations of localized states of stationary convection in 3He-4He mixtures, Phys. Rev. Lett., 95 (2005), 244501. Google Scholar

[5]

O. Batiste, E. Knobloch, A. Alonso and I. Mercader, Spatially localized binary-fluid convection, J. Fluid Mech., 560 (2006), 149-158. doi: 10.1017/S0022112006000759.  Google Scholar

[6]

M. Beck, J. Knobloch, D. J. B. Lloyd, B. Sandstede and T. Wagenknecht, Snakes, ladders, and isolas of localized patterns, SIAM J. Math. Anal., 41 (2009), 936-972. doi: 10.1137/080713306.  Google Scholar

[7]

A. Bergeon and E. Knobloch, Spatially localized states in natural doubly diffusive convection, Phys. Fluids, 20 (2008), 034102. doi: 10.1063/1.2837177.  Google Scholar

[8]

A. Bergeon, J. Burke, E. Knobloch and I. Mercader, Eckhaus instability and homoclinic snaking, Phys. Rev. E, 78 (2008), 046201. doi: 10.1103/PhysRevE.78.046201.  Google Scholar

[9]

S. Blanchflower, Magnetohydrodynamic convectons, Phys. Lett. A, 261 (1999), 74-81. doi: 10.1016/S0375-9601(99)00573-3.  Google Scholar

[10]

S. Blanchflower and N. O. Weiss, Three-dimensional magnetohydrodynamic convectons, Phys. Lett. A, 294 (2002), 297-303. doi: 10.1016/S0375-9601(02)00076-2.  Google Scholar

[11]

C. S. Bretherton and E. A. Spiegel, Intermittency through modulational instability, Phys. Lett. A, 96 (1983), 152-156. doi: 10.1016/0375-9601(83)90491-7.  Google Scholar

[12]

J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift-Hohenberg equation, Phys. Lett. A, 360 (2007), 681-688. doi: 10.1016/j.physleta.2006.08.072.  Google Scholar

[13]

P. Coullet, C. Riera and C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett., 84 (2000), 3069-3072. doi: 10.1103/PhysRevLett.84.3069.  Google Scholar

[14]

J. H. P. Dawes, Localized convection cells in the presence of a vertical magnetic field, J. Fluid Mech., 570 (2007), 385-406. doi: 10.1017/S0022112006002795.  Google Scholar

[15]

Q. Feng, J. V. Moloney and A. C. Newell, Transverse patterns in lasers, Phys. Rev. A, 50 (1994), 3601-3604. doi: 10.1103/PhysRevA.50.R3601.  Google Scholar

[16]

K. Ghorayeb and A. Mojtabi, Double diffusive convection in a vertical rectangular cavity, Phys. Fluids, 9 (1997), 2339-2348. doi: 10.1063/1.869354.  Google Scholar

[17]

D. Jung and M. Lücke, Bistability of moving and self-pinned fronts of supercritical localized convection structures, Europhys. Lett., 80 (2007), 14002, 1-6. Google Scholar

[18]

E. Knobloch, A. E. Deane, J. Toomre and D. R. Moore, Doubly diffusive waves, in "Multiparameter Bifurcation Theory" (eds. M. Golubitsky and J. Guckenheimer), Contemp. Math., 56 (1986), American Mathematical Society, Providence, R.I., 203-216.  Google Scholar

[19]

P. Kolodner, Observations of the Eckhaus instability in one-dimensional traveling-wave convection, Phys. Rev. A, 46 (1992), 1739-1742. doi: 10.1103/PhysRevA.46.R1739.  Google Scholar

[20]

P. Kolodner, Coexisting traveling waves and steady rolls in binary-fluid convection, Phys. Rev. E, 48 (1993), R665-668. doi: 10.1103/PhysRevE.48.R665.  Google Scholar

[21]

P. Kolodner, J. A. Glazier and H. L. Williams, Dispersive chaos in one-dimensional traveling-wave convection, Phys. Rev. Lett., 65 (1990), 1579-1582. doi: 10.1103/PhysRevLett.65.1579.  Google Scholar

[22]

I. Mercader, A. Alonso and O. Batiste, Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures, Eur. Phys. J. E, 15 (2004), 311-318. doi: 10.1140/epje/i2004-10071-7.  Google Scholar

[23]

I. Mercader, A. Alonso and O. Batiste, Spatiotemporal dynamics near the onset of convection for binary mixtures in cylindrical containers, Phys. Rev. E, 77 (2008), 036313. doi: 10.1103/PhysRevE.77.036313.  Google Scholar

[24]

I. Mercader, O. Batiste and A. Alonso, Continuation of travelling-wave solutions of the Navier-Stokes equations, Int. J. Num. Methods in Fluids, 52 (2006), 707-721. doi: 10.1002/fld.1196.  Google Scholar

[25]

I. Mercader, O. Batiste, A. Alonso and E. Knobloch, Localized pinning states in closed containers: Homoclinic snaking without bistability, Phys. Rev. E, 80 (2009), 025201(R). doi: 10.1103/PhysRevE.80.025201.  Google Scholar

[26]

I. Mercader, O. Batiste, A. Alonso and E. Knobloch, Convectons in periodic and bounded domains, Fluid Dyn. Res., 42 (2010), 025505. doi: 10.1088/0169-5983/42/2/025505.  Google Scholar

[27]

D. R. Ohlsen, S. Y. Yamamoto, C. M. Surko and P. Kolodner, Transition from traveling-wave to stationary convection in fluid mixtures, Phys. Rev. Lett., 65 (1990), 1431-1434. doi: 10.1103/PhysRevLett.65.1431.  Google Scholar

[28]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D, 23 (1986), 3-11. doi: 10.1016/0167-2789(86)90104-1.  Google Scholar

[29]

P. D. Woods and A. R. Champneys, Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian Hopf bifurcation, Physica D, 129 (1999), 147-170. doi: 10.1016/S0167-2789(98)00309-1.  Google Scholar

[1]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591
[2]

Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591
[3]

Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11
[4]

Chun-Hao Teng, I-Liang Chern, Ming-Chih Lai. Simulating binary fluid-surfactant dynamics by a phase field model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1289-1307. doi: 10.3934/dcdsb.2012.17.1289
[5]

Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Bifurcation and stability of two-dimensional double-diffusive convection. Communications on Pure & Applied Analysis, 2008, 7 (1) : 23-48. doi: 10.3934/cpaa.2008.7.23
[6]

Eduard Feireisl, Šárka Nečasová, Reimund Rautmann, Werner Varnhorn. New developments in mathematical theory of fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : i-ii. doi: 10.3934/dcdss.2014.7.5i
[7]

Jingli Ren, Gail S. K. Wolkowicz. Preface: Recent advances in bifurcation theory and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : i-ii. doi: 10.3934/dcdss.2020417
[8]

Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387
[9]

Gunog Seo, Gail S. K. Wolkowicz. Pest control by generalist parasitoids: A bifurcation theory approach. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3157-3187. doi: 10.3934/dcdss.2020163
[10]

J. F. Toland. Path-connectedness in global bifurcation theory. Electronic Research Archive, 2021, 29 (6) : 4199-4213. doi: 10.3934/era.2021079
[11]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141
[12]

Nurul Hafizah Zainal Abidin, Nor Fadzillah Mohd Mokhtar, Zanariah Abdul Majid. Onset of Benard-Marangoni instabilities in a double diffusive binary fluid layer with temperature-dependent viscosity. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 413-421. doi: 10.3934/naco.2019040
[13]

Toshiyuki Ogawa. Bifurcation analysis to Rayleigh-Bénard convection in finite box with up-down symmetry. Communications on Pure & Applied Analysis, 2006, 5 (2) : 383-393. doi: 10.3934/cpaa.2006.5.383
[14]

Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic & Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003
[15]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
[16]

Klaus Reiner Schenk-Hoppé. Random attractors--general properties, existence and applications to stochastic bifurcation theory. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 99-130. doi: 10.3934/dcds.1998.4.99
[17]

Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529
[18]

Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133
[19]

Jaume Llibre, Amar Makhlouf, Sabrina Badi. $3$ - dimensional Hopf bifurcation via averaging theory of second order. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1287-1295. doi: 10.3934/dcds.2009.25.1287
[20]

Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy