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This work studies two-dimensional fixed-flux Rayleigh–Bénard convection with periodic boundary conditions in both horizontal and vertical directions and analyses its dynamics using numerical continuation, secondary instability analysis and direct numerical simulation. The fixed-flux constraint leads to time-independent elevator modes with a well-defined amplitude. Secondary instability of these modes leads to tilted elevator modes accompanied by horizontal shear flow. For$$Pr=1$$, where$$Pr$$is the Prandtl number, a subsequent subcritical Hopf bifurcation leads to hysteresis behaviour between this state and a time-dependent direction-reversing state, followed by a global bifurcation leading to modulated travelling waves without flow reversal. Single-mode equations reproduce this moderate Rayleigh number behaviour well. At high Rayleigh numbers, chaotic behaviour dominated by modulated travelling waves appears. These transitions are characteristic of high wavenumber elevator modes since the vertical wavenumber of the secondary instability is linearly proportional to the horizontal wavenumber of the elevator mode. At a low$$Pr$$, relaxation oscillations between the conduction state and the elevator mode appear, followed by quasi-periodic and chaotic behaviour as the Rayleigh number increases. In the high$$Pr$$regime, the large-scale shear weakens, and the flow shows bursting behaviour that can lead to significantly increased heat transport or even intermittent stable stratification.
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Dynamical behaviour of a logarithmically sensitive chemotaxis model under time-dependent boundary conditions
Abstract This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the$$H^2$$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.
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- Award ID(s):
- 2316699
- PAR ID:
- 10616395
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- European Journal of Applied Mathematics
- Volume:
- 36
- Issue:
- 3
- ISSN:
- 0956-7925
- Page Range / eLocation ID:
- 638 to 664
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0956792524000329
