Convective flows with multiple spatial periodicities

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.