Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T13:10:14.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Convective flows with multiple spatial periodicities

Published online by Cambridge University Press:  21 April 2006

Mary Lowe ,
B. Steven Albert  and
J. P. Gollub
Mary Lowe
Affiliation:
Physics Department, Haverford College, Haverford, PA19041, USA and Physics Department, the University of Pennsylvania, Philadelphia, PA 19104, USA Current address: Group P-10, MS-K764, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
B. Steven Albert
Affiliation:
Physics Department, Haverford College, Haverford, PA19041, USA and Physics Department, the University of Pennsylvania, Philadelphia, PA 19104, USA
J. P. Gollub
Affiliation:
Physics Department, Haverford College, Haverford, PA19041, USA and Physics Department, the University of Pennsylvania, Philadelphia, PA 19104, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The response of a convective flow to spatially periodic forcing at a period different from the critical wavelength is investigated experimentally. For reasons of experimental convenience, we utilize an electrohydrodynamic instability in a thin layer of nematic liquid. With this system, a sample containing several hundred rolls is easily obtained, and periodic forcing is imposed electrically using a specially designed interdigitated electrode. Several novel flows with multiple periodicities are found. They may be broadly classified as commensurate (phase-locked) or incommensurate (quasi-periodic) flows, depending on whether the dominant periodicity of the perturbed flow and the periodicity of the forcing are in the ratio of small integers. Near the instability threshold and for weak forcing, an approximation of slow spatial variations is satisfied. In that case, the flows can be described quantitatively by an amplitude equation. We note a close connection between these hydrodynamic phenomena and a problem of competing periodicities that occurs in statistical mechanics. This relationship leads us to suggest the use of a discontinuous function for describing certain incommensurate flows in which non-repeating short and long groups of rolls are observed to occur in an irregular sequence. We also note, as predicted by Pal & Kelly, that two-dimensional forcing can lead to a three-dimensional flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 173 , December 1986 , pp. 253 - 272
Copyright
© 1986 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aubrey, S. 1983 The twist map, the extended Frenkel-Kontorova model and the devil's staircase. Physica D 7, 240–258.Google Scholar
Aubrey, S. & André, G. 1979 Analyticity breaking and Anderson localization in incommensurate lattices. Annls Israel Phys. Soc. 3, 133–164.Google Scholar
Behreman, D. W. 1972 Solid surface shape and the alignment of an adjacent nematic liquid crystal. Phys. Rev. Lett. 28, 1683–1686.
Busse, F. H. & Whitehead, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.Google Scholar
Chen, M. M. & Whitehead, J. A. 1968 Evolution of two-dimensional periodic Rayleigh convection cells or arbitrary wave-numbers. J. Fluid Mech. 31, 1–15.Google Scholar
Coullet, P. 1986 Commensurate-incommensurate transitions in non-equilibrium systems. Phys. Rev. Lett. 56, 724–727.Google Scholar
Frank, F. C. & van der Merwe, J. H. 1949 One-dimensional dislocations. Proc. R. Soc. Lond. A198, 205–225.Google Scholar
Goossens, W. J. A. 1978 Electrohydrodynamic instabilities in nematic liquid crystals. In Advances in Liquid Crystals (ed G. H. Brown), pp. 1–39. Academic.
Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions. J. Fluid Mech. 86, 433–456.Google Scholar
Lowe, M. & Gollub, J. P. 1985a Solitons and the commensurate-incommensurate transition in a convecting nematic fluid. Phys. Rev. A 31, 3893–3897.Google Scholar
Lowe, M. & Gollub, J. P. 1985b Pattern selection near the onset of convection: the Eckhaus instability. Phys. Rev. Lett. 55, 2575–2578.Google Scholar
Lowe, M., Gollub, J. P. & Lubensky, T. 1983 Commensurate and incommensurate structures in a nonequilibrium system. Phys. Rev. Lett. 51, 786–789.Google Scholar
Lubensky, T. C. & Ingersent, K. 1986 Patterns in systems with competing incommensurate lengths. In Proc. NATO Advanced Research Workshop on Patterns, Defects, and Microstructures in Non-Equilibrium Systems (Ed L. Reichl). Martinus Nijhoff, (to appear).
Milne-Thomson, L. M. 1972 Jacobian elliptic functions and theta functions; also elliptic integrals. In Handbook of Mathematical Functions (ed. M. Abramowitz & I. A. Stegun), pp. 567–626. Dover.
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid. Mech. 38, 279–303.Google Scholar
Pal, D. & Kelly, R. E. 1978 Thermal convection with spatially periodic nonuniform heating: nonresonant wavelength excitation. In Proc. 6th Intl Heat Transfer Conference, vol. 2, pp. 235–238. National Research Council of Canada.
Pal, D. & Kelly, R. E. 1979 Three-dimensional thermal convection produced by two-dimensional thermal forcing. In Proc. Joint ASME/AIChE 18th National Heat Transfer Conference, paper 79-HT-109. American Society of Mechanical Engineers.
Penz, P. A. 1971 Order parameter distribution of the electrohydrodynamic mode of a nematic liquid crystal. Molec. Crystals and Liquid Crystals 15, 141–160.Google Scholar
Pokrovsky, V. L. & Talapov, A. L. 1984 Theory of Incommensurate Crystals. Harwood.
Segel, L. A. 1969 Distant side-walls cause slow amplitude modulation of cellular convection. J. Fluid. Mech. 38, 203–224.Google Scholar
Uebach, W., Boix, M. & Guyon, E. 1974 Alignment of nematics and smectics on evaporated films. Appl. Phys. Lett. 25, 479–481.Google Scholar