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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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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