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Spontaneous formation of travelling localized structures and their asymptotic behaviour in binary fluid convection

Published online by Cambridge University Press:  28 September 2012

Takeshi Watanabe ,
Makoto Iima  and
Yasumasa Nishiura
Takeshi Watanabe*
Affiliation:
Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8521, Japan
Makoto Iima
Affiliation:
Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8521, Japan
Yasumasa Nishiura
Affiliation:
WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
*
†Email address for correspondence: [email protected]
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Abstract

We study spontaneous pattern formation and its asymptotic behaviour in binary fluid flow driven by a temperature gradient. When the conductive state is unstable and the size of the domain is large enough, finitely many spatially localized time-periodic travelling pulses (PTPs), each containing a certain number of convection cells, are generated spontaneously in the conductive state and are finally arranged at non-uniform intervals while moving in the same direction. We found that the role of PTP solutions and their strong interactions (collision) are important in characterizing the asymptotic state. Detailed investigations of pulse–pulse interactions showed the differences in asymptotic behaviour between that in a finite but large domain and in an infinite domain.

JFM classification

Instability: Absolute/convective instability Nonlinear Dynamical Systems: Bifurcation Convection: Double diffusive convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 219 - 243
Copyright
©2012 Cambridge University Press

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References

Barten, W., Lücke, M. & Kamps, M. 1991 Localized traveling-wave convection in binary-fluid mixtures. Phys. Rev. Lett. 66, 2621–2624.Google Scholar
Barten, W., Lücke, M., Kamps, M. & Schmitz, R. 1995 Convection in binary fluid mixtures. Part 2. Localized traveling waves. Phys. Rev. E 51, 5662–5680.Google Scholar
Batiste, O., Knobloch, E., Alonso, A. & Mercader, I. 2006 Spatially localized binary-fluid convection. J. Fluid Mech. 560, 149–158.CrossRefGoogle Scholar
Beaume, C., Bergeon, A. & Knobloch, E. 2011 Homoclinic snaking of localized states in doubly diffusive convection. Phys. Fluids 23, 093102.Google Scholar
Beck, M., Knobloch, J., Lloyd, D. J. B., Sandstede, B. & Wagenknecht, T. 2009 Snakes, ladders, and isolas of localized patterns. SIAM J. Math. Anal 41, 936–972.CrossRefGoogle Scholar
Burke, J. & Dawes, J. H. P. 2012 Localized states in an extended Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 11, 261–284.CrossRefGoogle Scholar
Burke, J. & Knobloch, E. 2006 Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73, 056211.Google Scholar
Burke, J. & Knobloch, E. 2007 Homoclinic snaking: structure and stability. Chaos 17, 037102.Google Scholar
Busse, F. H. 1964 The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Chapman, S. J. & Kozyreff, G. 2009 Exponential asymptotics of localised patterns and snaking bifurcation diagrams. Physica D 238, 319–354.Google Scholar
Cross, M. C. 1986 Traveling and standing waves in binary-fluid convection in finite geometries. Phys. Rev. Lett. 23, 2935–2938.Google Scholar
Heinrichs, R., Ahlers, G. & Cannell, D. S. 1987 Traveling waves and spatial variation in the convection of a binary mixture. Phys. Rev. A 35, 2761–2764.CrossRefGoogle ScholarPubMed
Houghton, S. M. & Knobloch, E. 2011 Swift–Hohenberg equation with broken cubic–quintic nonlinearity. Phys. Rev. E 84, 016204.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Iima, M. & Nishiura, Y. 2009 Unstable periodic solution controlling collision of localized convection cells in binary fluid mixture. Physica D 238, 449–460.Google Scholar
Jung, D., Matura, P. & Lücke, M. 2004 Oscillatory convection in binary mixtures: thermodiffusion, solutal buoyancy, and advection. Eur. Phys. J. E 15, 293–304.CrossRefGoogle ScholarPubMed
Katsuki, A., Nishimori, H., Endo, N. & Taniguchi, K. 2005 Collision dynamics of two barchan dunes simulated using a simple model. J. Phys. Soc. Japan 74, 538–541.Google Scholar
Kolodner, P. 1991 Collisions between pulses of traveling-wave convection. Phys. Rev. A 44, 6466–6479.CrossRefGoogle ScholarPubMed
Kolodner, P., Bensimon, D. & Surko, C. M. 1988 Traveling-wave convection in an annulus. Phys. Rev. Lett. 60, 1723–1726.Google Scholar
Mercader, I., Batiste, O., Alonso, A. & Knobloch, E. 2011 Dissipative solutions in binary fluid convection. Disc. Cont. Dyn. Sys. S 4, 1213–1225.Google Scholar
Niemela, J. J., Ahlers, G. & Cannell, D. S. 1990 Localized traveling-wave states in binary-fluid convection. Phys. Rev. Lett. 64, 1365–1368.CrossRefGoogle ScholarPubMed
Ning, L., Harada, Y. & Yahata, H. 1996 Localized traveling waves in binary fluid convection. Prog. Theor. Phys. 96, 669.Google Scholar
Nishiura, Y., Teramoto, T. & Ueda, K. 2003a Scattering and separators in dissipative systems. Phys. Rev. E 67, 056210.CrossRefGoogle ScholarPubMed
Nishiura, Y., Teramoto, T. & Uera, K. 2003b Dynamic transitions through scattors in dissipative systems. Chaos 13, 962–972.CrossRefGoogle ScholarPubMed
Nishiura, Y., Teramoto, T. & Ueda, K. 2005 Scattering of traveling spots in dissipative system. Chaos 15, 047509.Google Scholar
Platten, J. K. 2006 The Soret effect: a review of recent experimental results. J. Appl. Mech. 73, 5–15.Google Scholar
Platten, J. K. & Legros, J. C. 1984 Convection in Liquids. Springer.Google Scholar
Pomeau, Y. 1986 Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23, 3–11.Google Scholar
van Saarloos, W. & Hohenberg, P. C. 1992 Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations. Physica D 56, 303–367.Google Scholar
Schneider, T. M., Gibson, J. F. & Burke, J. 2010 Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.Google Scholar
Teramoto, T., Ueda, K. & Nishiura, Y. 2004 Phase-dependent output of scattering process for traveling breathers. Phys. Rev. E 69, 056224.Google Scholar
Thual, O. & Fauve, S. 1988 Localized structures generated by subcritical instabilities. J. Phys. France 49, 1829–1833.Google Scholar
Toyabe, K. 2009 Collision dynamics of localized convection cells in binary fluid mixtures: network structure of collision orbit (in Japanese). Master’s thesis, Graduate School of Science, Hokkaido University.Google Scholar
Watanabe, T., Toyabe, K., Iima, M. & Nishiura, Y. 2010 Time-periodic traveling solutions of localized convection cells in binary fluid mixture. Theor. Appl. Mech. Japan 59, 211–219.Google Scholar
Woods, P. D. & Champneys, A. R. 1999 Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian–Hopf bifurcation. Physica D 129, 147–170.Google Scholar