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

Dissipative solitons in binary fluid convection

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.
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 (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. 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. 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). Google Scholar

[5]

O. Batiste, E. Knobloch, A. Alonso and I. Mercader, Spatially localized binary-fluid convection,, J. Fluid Mech., 560 (2006), 149. 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. doi: 10.1137/080713306. Google Scholar

[7]

A. Bergeon and E. Knobloch, Spatially localized states in natural doubly diffusive convection,, Phys. Fluids, 20 (2008). 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). doi: 10.1103/PhysRevE.78.046201. Google Scholar

[9]

S. Blanchflower, Magnetohydrodynamic convectons,, Phys. Lett. A, 261 (1999), 74. 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. 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. 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. 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. 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. 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. 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. 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), 1. Google Scholar

[18]

E. Knobloch, A. E. Deane, J. Toomre and D. R. Moore, Doubly diffusive waves,, in, 56 (1986), 203. Google Scholar

[19]

P. Kolodner, Observations of the Eckhaus instability in one-dimensional traveling-wave convection,, Phys. Rev. A, 46 (1992), 1739. 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), 665. 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. 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. 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). 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. 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). 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). 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. doi: 10.1103/PhysRevLett.65.1431. Google Scholar

[28]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics,, Physica D, 23 (1986), 3. 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. 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 (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. 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. 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). Google Scholar

[5]

O. Batiste, E. Knobloch, A. Alonso and I. Mercader, Spatially localized binary-fluid convection,, J. Fluid Mech., 560 (2006), 149. 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. doi: 10.1137/080713306. Google Scholar

[7]

A. Bergeon and E. Knobloch, Spatially localized states in natural doubly diffusive convection,, Phys. Fluids, 20 (2008). 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). doi: 10.1103/PhysRevE.78.046201. Google Scholar

[9]

S. Blanchflower, Magnetohydrodynamic convectons,, Phys. Lett. A, 261 (1999), 74. 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. 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. 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. 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. 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. 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. 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. 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), 1. Google Scholar

[18]

E. Knobloch, A. E. Deane, J. Toomre and D. R. Moore, Doubly diffusive waves,, in, 56 (1986), 203. Google Scholar

[19]

P. Kolodner, Observations of the Eckhaus instability in one-dimensional traveling-wave convection,, Phys. Rev. A, 46 (1992), 1739. 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), 665. 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. 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. 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). 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. 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). 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). 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. doi: 10.1103/PhysRevLett.65.1431. Google Scholar

[28]

Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics,, Physica D, 23 (1986), 3. 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. doi: 10.1016/S0167-2789(98)00309-1. Google Scholar

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