Branching Instability Research Articles

Kozai–Lidov disc instability

Recent results by Martin et al. (2014) showed in 3D SPH simulations that tilted discs in binary systems can be unstable to the development of global, damped Kozai-Lidov (KL) oscillations in which the discs exchange tilt for eccentricity. We investigate the linear stability of KL modes for tilted inviscid discs under the approximations that the disc eccentricity is small and the disc remains flat. By using 1D equations, we are able to probe regimes of large ratios of outer to inner disc edge radii that are realistic for binary systems of hundreds of AU separations and are not easily probed by multi-dimensional simulations. For order unity binary mass ratios, KL instability is possible for a window of disc aspect ratios H/r in the outer parts of a disc that roughly scale as (n_b/n)^2 < H/r < n_b/n, for binary orbital frequency n_b and orbital frequency n at the disc outer edge. We present a framework for understanding the zones of instability based on the determination of branches of marginally unstable modes. In general, multiple growing eccentric KL modes can be present in a disc. Coplanar apsidal-nodal precession resonances delineate instability branches. We determine the range of tilt angles for unstable modes as a function of disc aspect ratio. Unlike the KL instability for free particles that involves a critical (minimum) tilt angle, disc instability is possible for any nonzero tilt angle depending on the disc aspect ratio.

Antiplane two-scale model for dynamic failure

A new dynamic damage law is proposed for the anti-plane loading case. The model is deduced from the energy criterion describing the dynamic propagation of microcracks by using the mathematical homogenization method based on asymptotic expansions. A study of the local macroscopic response predicted by the new model is conducted to highlight the influence of parameters like the size of the microstructure and the rate of loading on the evolution of damage. Results of macroscopic simulations of dynamic failure and the associated branching instabilities are presented and compared with those reported by experimental observations and theoretical studies on dynamic fracture in brittle materials.

Superfluid flow past an obstacle in annular Bose–Einstein condensates

We investigate the flow of a one-dimensional nonlinear Schrödinger model with periodic boundary conditions past an obstacle, motivated by recent experiments with Bose–Einstein condensates in ring traps. Above certain rotation velocities, localized solutions with a nontrivial phase profile appear. In striking difference from the infinite domain, in this case there are many critical velocities. At each critical velocity, the steady flow solutions disappear in a saddle-center bifurcation. These interconnected branches of the bifurcation diagram lead to additions of circulation quanta to the phase of the associated solution. This, in turn, relates to the manifestation of persistent current in numerous recent experimental and theoretical works, the connections to which we touch upon. The complex dynamics of the identified waveforms and the instability of unstable solution branches are demonstrated.

Instabilities of interacting vortex rings generated by an oscillating disk.

We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c, undergoing oscillations of frequency f_{0} and amplitude A. We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly on the azimuthal (integer) wave number m. We vary two key control parameters, the Keulegan-Carpenter number K_{C}=2πA/c and the Stokes number β=f_{0}c^{2}/ν, where ν is the kinematic viscosity of the fluid. We observe two distinct flow regimes. First, for sufficiently small β, and hence low frequency of oscillation corresponding to relatively weak interaction between sequentially shedding vortex rings, symmetry breaking occurs directly to a single unstable mode with m=1. Second, for sufficiently large yet fixed values of β, corresponding to a higher oscillation frequency and hence stronger ring-ring interaction, the onset of asymmetry is predicted to occur due to two branches of high m instabilities as the amplitude is increased, with m=1 structures being dominant only for sufficiently large values of K_{C}. These two branches can be distinguished by the phase properties of the vortical structures above and below the disk. The region in (K_{C},β) parameter space where these two high m instability branches arise can be described accurately in terms of naturally defined Reynolds numbers, using appropriately chosen characteristic length scales. We subsequently carry out direct numerical simulations of the fully three-dimensional flow to verify the principal characteristics of the Floquet analysis, in particular demonstrating that high wave-number symmetry-breaking generically occurs when vortex rings sequentially interact sufficiently strongly.

An integrative toy model of cell flattening, spreading, and ruffling.

The processes of cell spreading and crawling are frequently associated with mysterious waves and ruffling cycles of the leading edge. To develop a physical model that can account for these phenomena based on a few simple and plausible rules governing adhesion, contractility, polymerization of cytoskeleton, and membrane tension. Extension of a continuum mechanical model of phagocytosis [J Cell Sci. (2006);119(Pt 9):1903-13] adding a simple coupling between membrane curvature and cytoskeletal polymerization. We show that our generalized model has just the right nonlinearity needed for triggering of stochastic/chaotic cycles of ruffling similar to those that are observed in real cells. The cycles are caused by a branching instability at the leading edge that leads to bifurcations of protrusion into forward moving lamellipodium and upward and rearward folding ruffles. The amplitude of the instability is modulated by the surface tension, with higher tension stabilizing against ruffling (but inhibiting protrusion) and lower tension promoting ruffling and protrusion.

Emerging morphologies in round bacterial colonies: comparing volumetric versus chemotactic expansion.

Biological experiments performed on living bacterial colonies have demonstrated the microbial capability to develop finger-like shapes and highly irregular contours, even starting from an homogeneous inoculum. In this work, we study from the continuum mechanics viewpoint the emergence of such branched morphologies in an initially circular colony expanding on the top of a Petri dish coated with agar. The bacterial colony expansion, based on either a source term, representing volumetric mitotic processes, or a nonconvective mass flux, describing chemotactic expansion, is modeled at the continuum scale. We demonstrate that the front of the colony is always linearly unstable, having similar dispersion curves to the ones characterizing branching instabilities. We also perform finite element simulations, which not only prove the emergence of branching, but also highlight dramatic differences between the two mechanisms of colony expansion in the nonlinear regime. Furthermore, the proposed combination of analytical and numerical analysis allowed studying the influence of different model parameters on the selection of specific patterns. A very good agreement has been found between the resulting simulations and the typical structures observed in biological assays. Finally, this work provides a new interpretation of the emergence of branched patterns in living aggregates, depicted as the results of a complex interplay among chemical, mechanical and size effects.

On the instability of a buoyancy-driven downflow

Gravity currents flowing downslope, namely downflows, were observed to have a larger scale instability on high slope angles and such violent instability was absent for downflows on low slope angles. By linear theory, it is found that two branches of instability occur for slope angle in the range of 0°<θ<90°. The ensuing instability is on the upper branch for low slope angles and on the lower branch for high slope angles. There also exists a transitional slope angle, θE≈0.04°, at which the onset instability switches from one branch to the other. The scale of instability is found to increase and tend to skew towards the upper edge of the downflow as the ensuing instability switches from the upper branch to the lower one. Our findings surprisingly resonate with previously reported observations. Critical Reynolds number, below which the flow is stable to infinitesimal disturbances, is found to increase as the slope angle decreases. The role played by the bottom slope is essentially twofold. On one hand, the downslope component of gravity acts as the driving force for downflows. On the other hand, the wall-normal component of gravity acts for the stratification effect. Therefore, as the slope angle decreases, the driving force diminishes and the stratification intensifies, which can explain that the critical Reynolds number increases as the slope angle decreases. When a downflow propagates onto a sufficiently low slope angle, the low driving force and intensified stratification effect would make the downflow less prone to sustain a turbulent state of flow, which ultimately leads to the final stage of a gravity current event.

Branching instability in expanding bacterial colonies.

Self-organization in developing living organisms relies on the capability of cells to duplicate and perform a collective motion inside the surrounding environment. Chemical and mechanical interactions coordinate such a cooperative behaviour, driving the dynamical evolution of the macroscopic system. In this work, we perform an analytical and computational analysis to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish. The continuous mathematical model addresses the growth and the chemotactic migration of the living monolayer, together with the diffusion and consumption of nutrients in the agar. The governing equations contain four dimensionless parameters, accounting for the interplay among the chemotactic response, the bacteria-substrate interaction and the experimental geometry. The spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations. The typical length scales of such fingers, which align in the radial direction and later undergo further branching, are controlled by the size parameters of the problem, whereas the emergence of branching is favoured if the diffusion is dominant on the chemotaxis. The model is able to predict the experimental morphologies, confirming that compact (resp. branched) patterns arise for fast (resp. slow) expanding colonies. Such results, while providing new insights into pattern selection in bacterial colonies, may finally have important applications for designing controlled patterns.

Stability range of tilted dendritic arrays during directional solidification

The stability range of primary spacing of the tilted dendritic arrays in directional solidification has been studied by quantitative phase-field simulations. Results show that both the real growth direction and morphological shapes of dendritic arrays change with the primary spacing for different misorientation angles (θ0). It has been found that the lower limit of primary spacing is independent of θ0, but the upper limit of primary spacing is strongly influenced by that. The two kinds of tertiary branching instabilities result in different behaviors of the variation of the upper limit with misorientation angle for different pulling velocities.

Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

AbstractLinear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz &amp; Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich &amp; Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

Developmental instability as an index of adaptation to drought stress in a Mediterranean oak

An increase in temperature and water deficits caused by the ongoing climate change might lead to a decline growth rates and threaten the persistence of tree species in drought-prone areas within the Mediterranean Basin. Developmental instability (the error in development caused by stress) may provide an index of the adaptability of woody plants to withstand climatic stressors such as water shortage. This study evaluated the effects of drought stress on growth variables in three stands of a Mediterranean oak (Quercus faginea) exposed to differing climatic conditions (xeric, mesic and cooler) along an altitudinal gradient in northeastern Spain, in two climatically contrasting years (wet and dry years). Two indices of developmental instability, fluctuating and translational asymmetries, which reflect environmental stress, were measured in leaves and current-year shoots, respectively. We also measured branch biomass and fractal complexity of branches as indicators of the species’ performance. After a period of drought the individuals’ at the most xeric site presented lower developmental instability and less branch biomass than did the individuals from the mesic and cooler sites. We interpret that difference as an adaptive response to drought which reflects a trade-off between maintenance of homeostasis and growth when water is scarce. The study demonstrated that developmental instability constitutes a useful index to assess the degree of adaptation to stressful environmental conditions. The assessment of developmental instability in sites and years with contrasting climatic conditions provides a means of quantifying the capacity of plants to develop plastic adaptive responses to climatic stress.

Fractographic aspects of crack branching instability using a phase-field model

A phase-field model of a crack front propagating in a three-dimensional brittle material is used to study the fractographic patterns induced by the branching instability. The numerical results of this model give rise to crack surfaces that are similar to those obtained in various experimental situations. Depending on applied loading configurations and initial conditions, we show that the branching instability is either restricted to a portion of the crack front or revealed through quasi-two-dimensional branches. For the former, the crack front leaves on the main broken surface either aligned or disordered parabolic marks. For the latter, fractography reveals the so called échelon cracks showing that branching instability can also induce crack front fragmentation.

Branching Mechanisms in Surfactant Micellar Growth

We present a phase-field model to study the morphological transitions of surfactant micelles in supersaturated dilute solution. Simulations reveal that multiply connected micellar structure can be produced by interface branching instability of a growing micelle at relatively large supersaturation and intermediate spontaneous curvatures. Two branching mechanisms, i.e., a disk-to-cylinder shape transition and a tip bifurcation process, are identified for disklike and cylindrical micelles, respectively. We propose that dynamic branching at the micelle growth front provides an important kinetic pathway for the formation of branched wormlike micelles that are observed in many surfactant systems.

Crucial flow stabilization and multiple instability branches of gravity-driven films over topography

In this paper, we present experimental results on the linear instability of gravity-driven viscous films flowing down a strongly undulated incline. To systematically investigate the relation between the eddies which form in the troughs of sufficiently steep undulations and the corresponding stability maps, we vary the liquid's viscosity experimentally. We report on a rich variety of phenomena, which is provoked by the corrugation of the underlying substrate, including: (a) transitions from long-wave to short-wave type instability, (b) a disjoining of the instability branches leading to formation of isles in the stability map, (c) flow destabilization, but also, (d) very strong stabilization of the flow up to a factor of two for arbitrary linear disturbances and even up to a factor of four for linear short-wave disturbances. To our knowledge, this is the first experimental work, which reports on any of these phenomena for gravity-driven liquids flowing down an undulated incline. Since structured substrates turn out to hold an enormous potential for crucial film flow stabilization, we present a first approach for a topography shape, based on the shape of the eddy, which is optimized in terms of flow stability.

Propagation mechanism of brittle cracks

We employ a recently developed model that allows the study of two-dimensional brittle crack propagation under fixed grip boundary conditions. The crack development highlights the importance of voids which appear ahead of the crack as observed in experiments on the nanoscale. The appearance of these voids is responsible for roughening the crack path on small scales, in agreement with theoretical expectations. With increasing speed of propagation one observes the branching instabilities that were reported in experiments. The simulations allow understanding the phenomena by analyzing the elastic stress field that accompanies the crack dynamics.

Collective chemotactic dynamics in the presence of self-generated fluid flows

In microswimmer suspensions locomotion necessarily generates fluid motion, and it is known that such flows can lead to collective behavior from unbiased swimming. We examine the complementary problem of how chemotaxis is affected by self-generated flows. A kinetic theory coupling run-and-tumble chemotaxis to the flows of collective swimming shows separate branches of chemotactic and hydrodynamic instabilities for isotropic suspensions, the first driving aggregation, the second producing increased orientational order in suspensions of "pushers" and maximal disorder in suspensions of "pullers." Nonlinear simulations show that hydrodynamic interactions can limit and modify chemotactically driven aggregation dynamics. In puller suspensions the dynamics form aggregates that are mutually repelling due to the nontrivial flows. In pusher suspensions chemotactic aggregation can lead to destabilizing flows that fragment the regions of aggregation.

Regions for Brillouin seed pulse growth in relativistic laser-plasma interaction

Parametric plasma processes received renewed interest in the context of generating ultraintense and ultrashort laser pulses up to the exawatt-zetawatt regime. For Brillouin scattering and seed pulse amplification at high intensities, the strong coupling regime is of special interest. The intensity-driven low-frequency modes depend on the amplitudes of the laser fields. It is investigated here how these modes develop in the relativistic regime. Then, a unified treatment of Raman and Brillouin processes becomes necessary. Assuming circular polarization, it is shown that with increasing intensity an overlap of the originally different Raman, Brillouin, and modulational instability branches occurs. Numerical simulations with a linearized Maxwell-fluid code confirm the analytically predicted behavior.

Stability of the poiseuille flow in a longitudinal magnetic field

The stability to small perturbations of a 2D flow of a conducting viscous fluid with large Reynolds numbers in a longitudinal magnetic field is investigated. A complete linearized system of magnetohydrodynamics equations is considered using the method of collocations and the differential sweep method. The dependences of the critical Reynolds numbers on the electrical conductivity are analyzed in detail. A new instability branch for large Reynolds numbers and a jumpwise variation of the critical Reynolds numbers are discovered.

Experimental study of toroidal Alfvén modes in the Globus-M spherical tokamak

In the experiments carried out on the Globus-M tokamak in regimes with injection of 26-keV neutral beams with a power of 0.75–0.85 MW, two branches of instabilities excited by fast ions were observed in the early stage of a discharge: a low-frequency energetic particle mode (EPM) in the frequency range of 5–30 kHz and a high-frequency mode in the range of 50–200 kHz, identified as a toroidal Alfven eigenmode (TAE). The TAE developed in the initial phase of the discharge at q(0) > 1 and terminated when sawtooth oscillations were excited at q(0) < 1. The spectrum and spatial localization of the mode agree with predictions of the linear theory. The modes observed in the Globus-M tokamak possess both properties common to other tokamaks and their own specific features.

On the stability of scalar-vacuum space-times

We study the stability of static, spherically symmetric solutions to the Einstein equations with a scalar field as the source. We describe a general methodology of studying small radial perturbations of scalar-vacuum configurations with arbitrary potentials V(\phi), and in particular space-times with throats (including wormholes), which are possible if the scalar is phantom. At such a throat, the effective potential for perturbations V_eff has a positive pole (a potential wall) that prevents a complete perturbation analysis. We show that, generically, (i) V_eff has precisely the form required for regularization by the known S-deformation method, and (ii) a solution with the regularized potential leads to regular scalar field and metric perturbations of the initial configuration. The well-known conformal mappings make these results also applicable to scalar-tensor and f(R) theories of gravity. As a particular example, we prove the instability of all static solutions with both normal and phantom scalars and V(\phi) = 0 under spherical perturbations. We thus confirm the previous results on the unstable nature of anti-Fisher wormholes and Fisher's singular solution and prove the instability of other branches of these solutions including the anti-Fisher "cold black holes".