47 research outputs found
Bifurcations in annular electroconvection with an imposed shear
We report an experimental study of the primary bifurcation in
electrically-driven convection in a freely suspended film. A weakly conducting,
submicron thick smectic liquid crystal film was supported by concentric
circular electrodes. It electroconvected when a sufficiently large voltage
was applied between its inner and outer edges. The film could sustain rapid
flows and yet remain strictly two-dimensional. By rotation of the inner
electrode, a circular Couette shear could be independently imposed. The control
parameters were a dimensionless number , analogous to the Rayleigh
number, which is and the Reynolds number of the
azimuthal shear flow. The geometrical and material properties of the film were
characterized by the radius ratio , and a Prandtl-like number . Using measurements of current-voltage characteristics of a large number of
films, we examined the onset of electroconvection over a broad range of
, and . We compared this data quantitatively to
the results of linear stability theory. This could be done with essentially no
adjustable parameters. The current-voltage data above onset were then used to
infer the amplitude of electroconvection in the weakly nonlinear regime by
fitting them to a steady-state amplitude equation of the Landau form. We show
how the primary bifurcation can be tuned between supercritical and subcritical
by changing and .Comment: 17 pages, 12 figures. Submitted to Phys. Rev. E. Minor changes after
refereeing. See also http://mobydick.physics.utoronto.c
Instability of small-amplitude convective flows in a rotating layer with stress-free boundaries
We consider stability of steady convective flows in a horizontal layer with
stress-free boundaries, heated below and rotating about the vertical axis, in
the Boussinesq approximation (the Rayleigh-Benard convection). The flows under
consideration are convective rolls or square cells, the latter being
asymptotically equal to the sum of two orthogonal rolls of the same wave number
k. We assume, that the Rayleigh number R is close to the critical one, R_c(k),
for the onset of convective flows of this wave number: R=R_c(k)+epsilon^2; the
amplitude of the flows is of the order of epsilon. We show that the flows are
always unstable to perturbations, which are a sum of a large-scale mode not
involving small scales, and two large-scale modes, modulated by the original
rolls rotated by equal small angles in the opposite directions. The maximal
growth rate of the instability is of the order of max(epsilon^{8/5},(k-k_c)^2),
where k_c is the critical wave number for the onset of convection.Comment: Latex, 12 pp., 15 refs. An improved version of the manuscript
submitted to "Mechanics of fluid and gas", 2006 (in Russian; English
translation "Fluid Dynamics"