Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

Avitabile, Daniele; Desroches, Mathieu; Knobloch, Edgar; Krupa, Martin

doi:10.1098/rspa.2017.0018

Cited by 13 publications

(6 citation statements)

References 38 publications

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“…The relevant fluid mechanical reasons for this sensitivity to the inclination are discussed further in Section IV with the relevant mathematical considerations likely resembling those in Ref. [31]. In any case, we see that for these very small values of the inclination α the localized states are not connected to the LSF branch which continues monotonically to large values of the Rayleigh number Ra (Fig.…”

Section: Resultssupporting

confidence: 55%

Effect of small inclination on binary convection in elongated rectangular cells

et al. 2019

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We analyze the effect of a small inclination on the well-studied problem of two-dimensional binary fluid convection in a horizontally extended closed rectangular box with a negative separation ratio, heated from below. The horizontal component of gravity generates a shear flow that replaces the motionless conduction state when inclination is not present. This large scale flow interacts with the convective currents resulting from the vertical component of gravity. For very small inclinations the primary bifurcation of this flow is a Hopf bifurcation that gives rise to chevrons and blinking states similar to those obtained with no inclination. For larger but still small inclinations this bifurcation disappears and is superseded by a fold bifurcation of the large scale flow. The convecton branches, i.e., branches of spatially localized states consisting of counterrotating rolls, are strongly affected, with the snaking bifurcation diagram present in the non-inclined system destroyed already at small inclinations. For slightly larger but still small inclinations we obtain small amplitude localized states consisting of corotating rolls that evolve continuously when the primary large scale flow is continued in the Rayleigh number. These localized states lie on a solution branch with very complex behavior strongly dependent on the values of the system parameters. In addition, several disconnected branches connecting solutions in the form of corotating rolls, counterrotating rolls, and mixed corotating and counterrotating states are also obtained.

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Section: Resultssupporting

confidence: 55%

Effect of small inclination on binary convection in elongated rectangular cells

et al. 2019

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“…These curves were computed using the region of (x, t) space for which |h(x, t) − | was above a given tolerance. The boundary of this region is wavy, and so the curves were computed by fitting through the farthest upstream points (similar to figure 8 in [65]). There is an initial transient phase in which the wavepacket is convected downstream in all cases -we found this is difficult to avoid by choosing better initial data due to DNS constraints.…”

Section: Direct Numerical Simulations: Pulse Initializationmentioning

confidence: 99%

Instability and dripping of electrified liquid films flowing down inverted substrates

2020

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We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flat substrate inclined at a non-zero angle to the horizontal. The dynamics of the thin film is influenced by an electric field which is set up parallel to the substrate surface -this nonlocal physical mechanism has a linearly stabilizing effect on the interfacial dynamics. Our particular interest is in fluid films that are hanging from the underside of the substrate; these films may drip depending on physical parameters, and we investigate whether a sufficiently strong electric field can suppress such nonlinear phenomena. For a non-electrified flow, it was observed by Brun et al. (Phys. Fluids 27, 084107, 2015) that the thresholds of linear absolute instability and dripping are reasonably close. In the present study, we incorporate an electric field and analyse the absolute/convective instabilities of a hierarchy of reduced-order models to predict the dripping limit in parameter space. The spatial stability results for the reduced-order models are verified by performing an impulse-response analysis with direct numerical simulations (DNS) of the Navier-Stokes equations coupled to the appropriate electrical equations. Guided by the results of the linear theory, we perform DNS on extended domains with inflow/outflow conditions (mimicking an experimental set-up) to investigate the dripping limit for both non-electrified and electrified liquid films. For the latter, we find that the absolute instability threshold provides an order-of-magnitude estimate for the electric field strength required to suppress dripping; the linear theory may thus be used to determine the feasibility of dripping suppression given a set of geometrical, fluid and electrical parameters. * ruben.tomlin11@imperial.ac.uk arXiv:1906.08757v2 [physics.flu-dyn]

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“…In [5], the authors use a framework called spatial dynamics to obtain a global picture of bifurcations in a wide class of systems, and suggest how various aspects of these bifurcation diagrams are in some sense universal. Spatial dynamics and related approaches study solutions of time-independent reaction–diffusion systems on R by thinking of the spatial variable as ‘time’, and considering the two-component system as a flow in double-struckR4, where one can then make use of all of the machinery for such systems, such as numerical continuation [6–8].…”

Section: Introductionmentioning

confidence: 99%

Modern perspectives on near-equilibrium analysis of Turing systems

Krause

Gaffney

Maini

et al. 2021

Phil. Trans. R. Soc. A.

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In the nearly seven decades since the publication of Alan Turing’s work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction–diffusion theory. Some of these developments were nascent in Turing’s paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction–diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of ‘trivial’ base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

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Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system

Cited by 13 publications

References 38 publications

Effect of small inclination on binary convection in elongated rectangular cells

Effect of small inclination on binary convection in elongated rectangular cells

Instability and dripping of electrified liquid films flowing down inverted substrates

Modern perspectives on near-equilibrium analysis of Turing systems

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