Symmetry-breaking Instability Research Articles

How viscous bubbles collapse: Topological and symmetry-breaking instabilities in curvature-driven hydrodynamics

The duality between deformations of elastic bodies and noninertial flows in viscous liquids has been a guiding principle in decades of research. However, this duality is broken when a spheroidal or other doubly curved liquid film is suddenly forced out of mechanical equilibrium, as occurs, e.g., when the pressure inside a liquid bubble drops rapidly due to rupture or controlled evacuation. In such cases, the film may evolve through a noninertial yet geometrically nonlinear surface dynamics, which has remained largely unexplored. We reveal the driver of such dynamics as temporal variations in the curvature of the evolving surface. Focusing on the prototypical example of a floating bubble that undergoes rapid depressurization, we show that the bubble surface evolves via a topological instability and a subsequent front propagation, whereby a small planar zone that includes a singular flow structure, analogous to a disclination in elastic systems, nucleates spontaneously and expands in the spherically shaped film. This flow pattern brings about hoop compression and triggers another, symmetry-breaking instability to the formation of radial wrinkles that invade the flattening film. Our analysis reveals the dynamics as a nonequilibrium branch of "jellium" physics, whereby a rate-of-change of surface curvature in a viscous film is akin to charge in an electrostatic medium that comprises polarizable and conducting domains. We explain key features underlying recent experiments and highlight a qualitative inconsistency between the prediction of linear stability analysis and the observed "wavelength" of surface wrinkles.

Bifurcation tracking on moving meshes and with consideration of azimuthal symmetry breaking instabilities

We present a black-box method to numerically investigate the linear stability of arbitrary multi-physics problems. While the user just has to enter the system's residual in weak formulation, e.g. by a finite element method, all required discretized matrices are automatically assembled based on just-in-time generated and compiled C codes. Based on this method, entire phase diagrams in the parameter space can be obtained by bifurcation tracking and continuation at low computational costs. Particular focus is put on problems with moving domains, e.g. free surface problems in fluid dynamics, since a moving mesh introduces a plethora of complicated nonlinearities to the system. By symbolic differentiation before the code generation, however, these moving mesh problems are made accessible to bifurcation tracking methods. In a second step, our method is generalized to investigate symmetry-breaking instabilities of axisymmetric stationary solutions by effectively utilizing the symmetry of the base state. Each bifurcation type is validated on the basis of results reported in the literature on versatile fluid dynamics problems, for which we subsequently present novel results as well.

Symmetry breaking and high-order instability of telephone cord buckles triggered by in-plane gradients of thin films

Buckle delaminations are ubiquitous in natural and artificial systems including tectonic plates, biological tissues, film structures, and two-dimensional materials. Although the formation and manipulation of various buckle delaminations have been extensively investigated in the past few decades, understanding complex buckle delamination patterns is still a challenge. Here we report on the symmetry breaking and high-order instability of telephone cord (TC) buckles triggered by in-plane gradients of thin films, including strain gradient and thickness gradient. Our experimental observations reveal the unique morphological profile of the asymmetric TC buckle due to the strain gradient. Impact of in-plane strain gradient on morphological changes of TC buckles is elucidated by simulations of buckling and delamination of the thin film. Phase diagrams for five buckle delamination morphologies, i.e., symmetric TC buckle, asymmetric TC buckle, symmetric bilateral branched buckle, asymmetric bilateral branched buckle, and unilateral branched buckle with respect to average strain and strain gradient (or thickness gradient) have been numerically established. This work could promote a better understanding of complex buckle delamination patterns with symmetry breaking and high-order instability and controllable fabrication of ordered buckle delamination patterns by regulating in-plane gradients of thin films.

Instability mechanisms of symmetry breaking in the wakes of two anti-phase pitching foils at zero free-stream velocity

In the present study, we investigate into the symmetry-breaking phenomenon in the wakes produced by two anti-phase pitching foils at zero free-stream velocity. Numerical simulations are performed in a gap ratio range of 0.6≤g≤2.0 and a Reynolds number range of 70≤Re≤200. Six regions are identified in the plane (g, Re), based on distinct symmetry-breaking (or symmetry-preserving) behaviors of the wakes. Floquet stability analysis and dynamic mode decomposition are performed on some selected cases. By means of modal analysis, the behaviors of wakes in different regions are rationalized and the instability mechanisms of symmetry breaking in the unstable regions are unveiled. In addition, the influence of symmetry-breaking instability on propulsive performance is also discussed.

Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Exploring multi-stability in three-dimensional viscoelastic flow around a free stagnation point

Fluid elements passing near a stagnation point experience finite strain rates over long persistence times, and thus accumulate large strains. By the numerical optimization of a microfluidic 6-arm cross-slot geometry, recent works have harnessed this flow type as a tool for performing uniaxial and biaxial extensional rheometry (Haward et al., 2023 [5,6]). Here we use the microfluidic ‘Optimized-shape Uniaxial and Biaxial Extensional Rheometer’ (OUBER) geometry to probe an elastic flow instability which is sensitive to the alignment of the extensional flow. A three-dimensional symmetry-breaking instability occurring for flow of a dilute polymer solution in the OUBER geometry is studied experimentally by leveraging tomographic particle image velocimetry. Above a critical Weissenberg number, flow in uniaxial extension undergoes a supercritical pitchfork bifurcation to a multi-stable state. However, for biaxial extension (which is simply the kinematic inverse of uniaxial extension) the instability is strongly suppressed. In uniaxial extension, the multiple stable states align in an apparently random orientation as flow joining from four neighbouring inlet channels passes to one of the two opposing outlets; thus forming a mirrored asymmetry about the stagnation point. We relate the suppression of the instability in biaxial extension to the kinematic history of flow under the context of breaking the time-reversibility assumption.

Non-linear and non-local plant–plant interactions in arid climate: Allometry, criticality and desertification

We present a model that considers the non-linear and non-local facilitative and competitive interactions in arid and semiarid climates. These two botanical processes are interconnected and operate at different spatial scales. Seed dispersal is modeled as a diffusion process. We consider landscapes populated by a dominant species to be isotropic.and homogeneous environmental conditions. We included the allometric factor in the modeling, considering the age classes. We show that allometry tends to move the critical and tipping points towards a low level of aridity. This alteration of aridity conditions can prevent desertification and thus avoid vegetation collapse. Unlike other interaction redistribution models, we show that even in the absence of allometry, critical and tipping points have a finite biomass. Consequently, a branch of low-biomass density is stabilized. Finally, we show that the allometric factor reduces the range of symmetry-breaking instability and favors the formation of homogeneous cover rather than vegetation patterns.

Self-propulsion of an elliptical phoretic disk emitting solute uniformly

Self-propulsion of chemically active droplets and phoretic disks has been studied widely; however, most research overlooks the influence of disk shape on swimming dynamics. Inspired by experimentally observed prolate composite droplets and elliptical camphor disks, we employ simulations to investigate the phoretic dynamics of an elliptical disk that emits solutes uniformly in the creeping flow regime. By varying the disk's eccentricity $e$ and the Péclet number $Pe$ , we distinguish five disk behaviours: stationary, steady, orbiting, periodic and chaotic. We perform a linear stability analysis (LSA) to predict the onset of instability and the most unstable eigenmode when a stationary disk transitions spontaneously to steady self-propulsion. In addition to the LSA, we use an alternative approach to determine the perturbation growth rate, illustrating the competing roles of advection and diffusion. The steady motion features a transition from a puller-type to a neutral-type swimmer as $Pe$ increases, which occurs as a bimodal concentration profile at the disk surface shifts to a polarized solute distribution, driven by convective solute transport. An elliptical disk achieves an orbiting motion through a chiral symmetry-breaking instability, wherein it repeatedly follows a circular path while simultaneously rotating. The periodic swinging motion, emerging from a steady motion via a supercritical Hopf bifurcation, is characterized by a wave-like trajectory. We uncover a transition from normal diffusion to superdiffusion as eccentricity $e$ increases, corresponding to a random-walking circular disk and a ballistically swimming elliptical counterpart, respectively.

Pattern formation in photo-controlled bioconvection

The model eukaryotic microalgae Chlamydomonas reinhardtii is well known for its ability to generate bioconvection flows that are associated to intricate concentration patterns. Recently, it was demonstrated that the propensity of these algae to move toward a light source – a phenomenon termed phototaxis – can be exploited to locally concentrate micro-organisms and induce (photo)-bioconvection in algal suspensions by inducing a localised excess of density. In the present study we show experimentally that a cell population in a thin liquid layer self-organises in the presence of a heterogeneous light field and displays remarkable symmetry-breaking instabilities that are ruled by both the width of the light beam and the photo-bioconvection Rayleigh number. Beside circular stable states, fingers, dendrites and wave instabilities are reported, quantified and classified in a general phase diagram. Next, we use lubrication theory to develop an asymptotic model for bioconvection in a thin liquid layer, that includes the influences of both gyrotaxis and phototaxis. We obtain a single nonlinear anisotropic diffusion–drift equation describing the spatiotemporal evolution of the depth-averaged algal population. Analytical and numerical solutions are presented and show a very good agreement with the experimental results. In particular, we show that the dendrite instability arises as a result of a subtle coupling between the nonlinearity of the phototactic response, the gyrotactic effect and the self-induced bioconvective flows. Such complex flow fields might find applications in photo-bioreactors, through the efficient stirring of the harvested biomass.

Uniform-density Bose-Einstein condensates of the Gross-Pitaevskii equationfound by solving the inverse problem for the confining potential.

In this work, we study the existence and stability of constant density (flat-top) solutions to the Gross-Pitaevskii equation(GPE) in confining potentials. These are constructed by using the "inverse problem" approach which corresponds to the identification of confining potentials that make flat-top waveforms exact solutions to the GPE. In the one-dimensional case, the exact solution is the sum of stationary kink and antikink solutions, and in the overlapping region, the density is constant. In higher spatial dimensions, the exact solutions are generalizations of this wave function. In the absence of self-interactions, the confining potential is similar to a smoothed-out finite square well with minima also at the edges. When self-interactions are added, terms proportional to ±gψ^{*}ψ and ±gM with M representing the mass or number of particles in Bose-Einstein condensates get added to the confining potential and total energy, respectively. In the realm of stability analysis, we find (linearly) stable solutions in the case with repulsive self-interactions which also are stable to self-similar deformations. For attractive interactions, however, the minima at the edges of the potential get deeper and a barrier in the center forms as we increase the norm. This leads to instabilities at a critical value of M. Comparing the stability criteria from Derrick's theorem with Bogoliubov-de Gennes (BdG) analysis stability results, we find that both predict stability for repulsive self-interactions and instability at a critical mass M for attractive interactions. However, the numerical analysis gives a much lower critical mass. This is due to the emergence of symmetry-breaking instabilities that were detected by the BdG analysis and violate the symmetry x→-x assumed by Derrick's theorem.

Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. II. History-dependent motion

Isotropic chemically active particles can spontaneously self-propel owing to a symmetry-breaking instability when diffusion of the chemical is sufficiently weak relative to advection. This series revisits the weakly nonlinear theory describing the dynamics of such spontaneous swimmers near the instability threshold. Part II presents an extension to the general framework from part I to include unsteadiness in the form of an integral over the history of the particle motion, representing the interaction of the particle with its own chemical wake. This allows efficient simulation and theoretical analysis of fully three-dimensional unsteady problems in a range of physical scenarios.

Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. I. Adjoint method

Isotropic chemically active particles can spontaneously self-propel owing to a symmetry-breaking instability when diffusion of the chemical is sufficiently weak relative to advection. This series revisits the weakly nonlinear theory describing the dynamics of such spontaneous swimmers near the instability threshold. Part I identifies the adjoint linearized equations, a key element towards a versatile theory that can be used to model general motion in three dimensions including perturbation effects and interactions. The general theoretical framework is demonstrated by deriving and analyzing steady amplitude equations in a range of physical scenarios.

Gabriel Faivre: his legacy on eutectic solidification

In the honor of Gabriel Faivre (1944-2020), I will present a review of major scientific contributions to the understanding of the dynamics of eutectic growth patterns. From the end of the 1980s, Gabriel Faivre undertook a systematic research in solidification guided by the new concepts of the nonlinear physics of out-of-equilibrium pattern formation. Drawing on his outstanding capabilities as an experimentalist, he refined the method of in situ directional solidification of model alloys. With constant reference to physics and metallurgy, he succeeded in carrying out a high-level research, keen to reach strong qualitative impact and quantitative accuracy. Gabriel Faivre made key discoveries, together with coworkers and young researchers in Paris, and in collaboration with materials scientists and physicists in France and abroad. From symmetry breaking instabilities to eutectic cells and dendrites, over rod-like and labyrinth patterns, full light has been shed onto new phenomena, fascinating to the eye and the mind. During the last decade, Gabriel Faivre mentored an in-depth analysis of interfacial-anisotropy effects on coupled-growth patterns, thus reconciliating the theories of regular eutectics and crystal-orientation dependent eutectic-grain growth. Being both a rigorous scientist and a generous colleague, he left us a vast legacy of prospective research topics in solidification and crystal-growth science. Sharing his knowledge of fine arts and humanities, Gabriel Faivre also instilled the best of intellectual thinking in those who were fortunate enough to work with him.

Stabilization of Axisymmetric Airy Beams by Means of Diffraction and Nonlinearity Management in Two-Dimensional Fractional Nonlinear Schrödinger Equations

The propagation dynamics of two-dimensional (2D) ring-Airy beams is studied in the framework of the fractional Schrödinger equation, which includes saturable or cubic self-focusing or defocusing nonlinearity and Lévy index ((LI) alias for the fractionality) taking values 1≤α≤2. The model applies to light propagation in a chain of optical cavities emulating fractional diffraction. Management is included by making the diffraction and/or nonlinearity coefficients periodic functions of the propagation distance, ζ. The management format with the nonlinearity coefficient decaying as 1/ζ is considered too. These management schemes maintain stable propagation of the ring-Airy beams, which maintain their axial symmetry, in contrast to the symmetry-breaking splitting instability of ring-shaped patterns in 2D Kerr media. The instability driven by supercritical collapse at all values α<2 in the presence of the self-focusing cubic term is eliminated, too, by the means of management.

Homogeneous nucleation of dislocations as a pattern formation phenomenon

Dislocation nucleation in homogeneous crystals begins as a symmetry-breaking elastic instability. In the absence of explicit nucleation centers, such instability develops simultaneously all over the crystal and due to the dominance of long range elastic interactions it advances into the nonlinear stage as a collective phenomenon through pattern formation. In this paper we use a novel mesoscopic tensorial model (MTM) of crystal plasticity to study the delicate role of crystallographic symmetry in the development of the dislocation nucleation patterns in defect free crystals. The model is formulated in 2D and we systematically compare the unfolding of the pattern formation phenomenon in lattices with square and triangular symmetry. To avoid the dominance of the conventional plastic mechanisms, we consider the loading paths represented by pure shears applied on the boundary of the otherwise unloaded body. These peculiar loading protocols can be qualified as exploiting the ‘softest’ and the ‘hardest’ directions and we show that the associated dislocation patterns are strikingly different.

Vegetation covers phase separation in inhomogeneous environments

Vegetation patterns in arid and semi-arid ecosystems as a self-organized response to resource scarcity is a well-documented issue. Their formation is often attributed to the symmetry-breaking type of instability. In this contribution, we focus on a regime far from any symmetry-breaking instability and consider a bistable regime involving uniformly vegetated covers and a bare state. We show that vegetation populations exhibit non-random two-phase structures where high biomass density regions are separated by sparsely covered areas or even bare soil. These structures are referred to as phase separation vegetation covers. We provide observations of this phenomenon in Gabon, Angola, Argentina, and Mexico. The inhomogeneities in environmental conditions are crucial to explain the origin of phase separation vegetation covers. We derive a simple equation from ecologically relevant models to explain various field observations. The bifurcation diagrams obtained from this model allow us to prove that inhomogeneity in the aridity parameter is a source of resilience for vegetation covers, avoiding collapsing towards a bare state. We characterize the natural observations and the equilibria from the model by using Fourier transform technique, spatial autocorrelation analysis, and size distribution of patches analysis.

Control of purely-elastic instabilities in cross-slot geometries

<h2>Abstract</h2> The cross-slot stagnation point flow is one of the benchmark problems in fluid mechanics as it allows large strains to develop and can therefore be used in extensional rheometry. In such a flow, for purely-elastic cases in creeping flow regimes, elasticity can break symmetry which is perhaps an unwanted phenomenon if used as a rheometer and will limit the maximum deformation rate in which these tests can be performed, or beneficial once used as mixing device. Here, this instability will be investigated in more detail using a combination of numerical, experimental and analytical analysis and a series of methods will be proposed that can potentially be used to delay and control the start point of the instability. The first part of this presentation focuses on the effect of elongational dominated flow on the onset criteria of symmetry-breaking purely-elastic instability in the cross-slot geometry by applying a fundamental change on the kinematics of the flowfield in this region. Here, the standard geometry is modified by adding a cylinder at the geometric centre and replacing the free stagnation point by pinned stagnation points at the cylinder walls. Next, two-phase flows of Newtonian and/or viscoelastic fluids in a "cross-slot" geometry will be investigated in the creeping-flow limit. In this part, the effect of injecting two fluids with different elastic properties from each inlet arm, and effects of interfacial tension and the viscosity ratio of these flow streams will be investigated.

Temperature-linear resistivity in twisted double bilayer graphene

We report an experimental study of carrier density ($n$), displacement field, and twist angle (\ensuremath{\theta}) dependence of temperature ($T$)-linear resistivity in twisted double-bilayer graphene (TDBG). For a large twist angle ($\ensuremath{\theta}&gt;1.{5}^{\ensuremath{\circ}}$) where correlated insulating states are absent, we observe a $T$-linear resistivity (with a slope on the order of $\ensuremath{\sim}10\phantom{\rule{0.28em}{0ex}}\mathrm{\ensuremath{\Omega}}/\mathrm{K}$) over a wide range of carrier densities, and its slope decreases with increasing $n$, which is in agreement with the acoustic phonon scattering model semiquantitatively. The slope of $T$-linear resistivity is nonmonotonically dependent on the displacement field with a single peak structure. For a device with $\ensuremath{\theta}\ensuremath{\sim}1.{23}^{\ensuremath{\circ}}$ at which correlated states emerge, the slope of $T$-linear resistivity is found to be a maximum ($\ensuremath{\sim}100\phantom{\rule{0.28em}{0ex}}\mathrm{\ensuremath{\Omega}}/\mathrm{K}$) at the boundary of the halo structure where phase transition occurs, with signatures of continuous phase transition, Planckian dissipation, and diverging effective mass. These observations are in line with quantum critical behaviors, which might be due to symmetry-breaking instability at the critical points. Our results shed new light on correlated physics in TDBG and other twisted moir\'e systems.

Interaction of multiple spiral rotors in a reaction-diffusion system.

Rotors of reaction and diffusion are phase singularities that give rise to spiral waves of chemical activity, which are very similar to spatiotemporal patterns observed across several excitable media. Here we carry out experiments with the Belousov-Zhabotinsky reaction system and numerical simulations based on a reaction-diffusion model to show the possible interactions of multiple spiral rotors. When the cores of two spirals come very close to each other, they could either repel, attract, or remain stationary, depending on their relative chirality, phase, and distance separating them. Multiple pairs of spiral waves, in proximity to each other, could alter the paths of the individual rotors. A spiral core will be influenced most by the rotor that is closest to it, depending on the nature of the corresponding spiral wave arm. We observed rotors lying within a limiting distance of each other attract and annihilate. Otherwise, they push each other away until they reach a critical distance, beyond which all interactions seem to cease. We have established a relationship of this critical distance to the properties of the spiral wave. We also observed spontaneous symmetry-breaking instability for a system of up to eight rotors. Our experiments with the Belousov-Zhabotinsky reaction have successfully demonstrated the validity of the numerical results. A thorough understanding of the dynamics of several spiral rotors within a small area could help us perceive the nature of such excitation waves in cardiac tissue and cell membranes.

Symmetry-breaking instability in a charge-controlled dielectric film: Large electro-actuation and high stored energy

In this paper, we study the electromechanical behaviors of a charge-controlled dielectric film that may encounter symmetry-breaking instability. With electric charges and the radial dead load, a circular film would gradually expand its area and become a large circular film; however, it will deform into an elliptical film when electromechanical loads increase to the threshold. This symmetry-breaking instability brings the change of shape and makes the dielectric film achieve large electro-actuation and high stored energy. Since the functionality of the dielectric devices is also limited by the electric breakdown, we give the phase diagram to show the competition between the symmetry-breaking instability and the electric breakdown in detail. This paper is desirable to further harness the symmetry-breaking instability for improving functionalities of actuators and harvesters.