Spatio-Temporal Dynamics

Unified Morphological Framework – Supplemental Topology and Mathematics Codex extends the Unified Morphological Framework (UMF), a scientific metatheory that unites five primordial morphologies—Spiral, Coil, Wave, Crystal, and Web—into a coherent language for complex systems design. Each morphology represents an archetypal solution to organizational challenges, embodying principles of growth, feedback, communication, order, and connectivity. This codex consolidates the mathematical foundations, symmetry groups, and topological invariants associated with these forms, while introducing higher-dimensional tools such as cobordism theory, fiber bundles, and persistent homology to model transitions between morphologies. By framing morphologies as dynamic phase states, the work outlines how systems can fluidly shift structures through continuous deformations or critical phase changes, enabling adaptive resilience. Serving as a supplemental reference to the SpiralDriveEngine architecture, the codex provides both theoretical constructs and applied design insights for developing intelligent, field-based systems that harmonize continuity, adaptability, and structural integrity.

This paper develops a rigorous mathematical framework for the study of time travel by integrating classical and quantum models of temporality with topological, geometric, and dynamical structures of time. We introduce the theory of Spherical Time, wherein time is modeled not as a linear continuum but as a compact manifold-such as S 1 , S 2 , or higher spheres-thereby naturally allowing for closed timelike curves (CTCs), temporal loops, and novel causal structures. Beginning with an analysis of differential models of time and temporal derivatives on manifolds, we explore how topology, homology, and spectral geometry constrain and permit exotic time-travel phenomena. We examine temporal manifolds endowed with Lorentzian and complex metrics, analyze the behavior of dynamical systems on S 1 , S 2 , and T 2 , and classify the homotopy classes of time-travel paths. Through the use of spectral zeta functions, KMS states, and noncommutative geometry, we investigate thermodynamic and quantum implications for systems traversing cyclic or multi-sheeted time. Visualizations of temporal attractors, Ricci flow on compact time surfaces, and dynamical vector fields illustrate the evolution and deformation of time loops. The culmination of this work is a robust formalism that bridges general relativity, quantum field theory, and topological quantum gravity to provide new insights into the feasibility, mathematics, and physics of time travel.

Surfactants non-monotonically modify the onset of Faraday waves 1 STEPHEN STRICKLAND, Dickinson College, MICHAEL SHEARER, KAREN DANIELS, North Carolina State University -When a water-filled container is vertically vibrated, subharmonic Faraday waves emerge once the driving from the vibrations exceeds viscous dissipation. In the presence of an insoluble surfactant, a viscous boundary layer forms at the contaminated surface to balance the Marangoni and Boussinesq stresses. For linear gravity-capillary waves in an undriven fluid, the surfactant-induced boundary layer increases the amount of viscous dissipation. In our analysis and experiments, we consider whether similar effects occur for nonlinear Faraday (gravity-capillary) waves. Assuming a finite-depth, infinite-breadth, low-viscosity fluid, we derive an analytic expression for the onset acceleration up to second order in ϵ = √

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A large viscosity contrast can be a driver for interfacial instabilities. Striking arrays of viscous fingers can be made to grow at the interface between two stably stratified liquids in an oscillating cylinder provided that the viscosity ratio between the two fluids is large. T.B. Benjamin’s student, Miller (D. Phil Thesis, Oxford University, 1986), analysed the stability of this two-layer flow with inviscid flow in the denser lower layer and Stokes flow in the lighter upper layer. Although a good prediction was made of the critical wavenumber, the predicted critical velocity was twofold larger than in experiments in viscous oil and water. We revisit this comparison with new experiments and identify contact line perturbations as the likely cause of the low stability thresholds measured experimentally by Miller. In fact, we show that the Miller–Benjamin (MB) model makes good predictions of stability thresholds over a wide range of fluid parameters. We find that the critical wave num...

(fluid 1) flows on top of a thick liquid layer (fluid 2), such that their thickness ratio is h 1 = 9h 2. The two liquid layers have the same density but different viscosities η. In particular, we consider the case η 2 < η 1. The problem is described by the shear Reynolds number (Re τ), by the Weber number (W e, which quantifies surface tension effects) and by the viscosity ratio λ between the two fluids. Compared to a single phase flow at the same shear Reynolds number (Re τ = 300), in the two phase flow case we observe an increase of the flow rate of fluid 1 and a strong modification of the turbulence structures near the liquid-liquid interface. Alltogether, these observations support the presence of a significant Drag Reduction (DR), whose efficiency depends strongly on the interface deformability (W e) and on the viscosity ratio between the two fluids (λ).

We perform a linearized local stability analysis for short-wavelength perturbations of a circular Couette flow with the radial temperature gradient. Axisymmetric and nonaxisymmetric perturbations are considered and both the thermal diffusivity and the kinematic viscosity of the fluid are taken into account. The effect of the asymmetry of the heating both on the centrifugally unstable flows and on the onset of the instabilities of the centrifugally stable flows, including the flow with the Keplerian shear profile, is thoroughly investigated. It is found that the inward temperature gradient destabilizes the Rayleigh stable flow either via Hopf bifurcation if the liquid is a very good heat conductor or via steady state bifurcation if viscosity prevails over the thermal conductance.

2014 Nous employons la théorie des milieux continus pour étudier la stabilité d'un échantillon de cristal liquide nématique sous faible épaisseur contenu entre deux longs cylindres concentriques en rotation à vitesse angulaire constante le long de leur axe, et soumis à un gradient de température radial. Nous considérons le cas d'un matériau initialement aligné parallèlement à l'axe des cylindres. Utilisant les méthodes de Galerkin et d'or- thonormalisation, nous déterminons les seuils positif et négatif des gradients thermiques pour lesquels l'apparition d'une instabilité de convection est possible. Les effets de la vitesse angulaire et d'un champ magnétique appliqué sont également examinés. Nous montrons qu'il est possible d'observer l'instabilité à petit nombre d'onde que Velarde et Zuniga [1] ont proposée dans le cas du problème de Rayleigh-Bénard pour les nématiques.

A numerical study of the onset of thermal convection in a rotating circular cylinder of radius-to-depth ratio equal to four is considered in a regime dominated by the Coriolis force where the onset is to so-called wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and descending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dynamics are controlled by an Eckhaus–Benjamin–Feir instability, the most basic features of which had been captured by a ...

equal to 4 is considered in a regime dominated by the Coriolis force where the onset is to wall modes. The wall modes consist of hot and cold pairs of thermal plumes rising and decending in the cylinder wall boundary layer, forming an essentially one-dimensional pattern characterized by the number of hot/cold plume pairs, m. In the limit of zero centrifugal force, this onset of convection at a critical temperature difference across the depth of the cylinder is via a symmetry-breaking supercritical Hopf bifurcation which leads to retrograde precession of the pattern with respect to the rotation of the cylinder. For temperature differences greater than critical, a number of distinct wall modes, distinguished by m, coexist and are stable. Their dyanmics are controlled by an Eckhaus-Benjamin-Feir instability, the most basic features of which are captured by a complex Ginsburg-Landau equation model. This instability in rotating convection has been analyzed for the first time using direct numerical simulations of the Navier-Stokes equations in the Boussinesq approximation. Several properties of the wall modes have been computed, extending the results to far beyond the onset of convection. Extensive favorable comparisons between our numerical results and previous experimental observations and complex Ginsburg-Landau model results of Liu and Ecke (PRL 1997, JFM 1999) are made.

The large-scale laminar/turbulent spiral patterns that appear in the linearly unstable regime of counter-rotating Taylor-Couette flow are investigated from a statistical perspective by means of direct numerical simulation. Unlike the vast majority of previous numerical studies, we analyse the flow in periodic parallelogram-annular domains, following a coordinate change that aligns one of the parallelogram sides with the spiral pattern. The domain size, shape and spatial resolution have been varied and the results compared with those in a sufficiently large computational orthogonal domain with natural axial and azimuthal periodicity. We find that a minimal parallelogram of the right tilt significantly reduces the computational cost without noticeably compromising the statistical properties of the supercritical turbulent spiral. Its mean structure, obtained from extremely long time integrations in a corotating reference frame using the method of slices, bears remarkable similarity with the turbulent stripes observed in plane Couette flow, the centrifugal instability playing only a secondary role.

A derivation is given of the amplitude equations governing pattern formation in surface tension gradient-driven Bénard–Marangoni convection. The amplitude equations are obtained from the continuity, the Navier–Stokes and the Fourier equations in the Boussinesq approximation neglecting surface deformation and buoyancy. The system is a shallow liquid layer heated from below, confined below by a rigid plane and above with a free surface whose surface tension linearly depends on temperature. The amplitude equations of the convective modes are equations of the Ginzburg–Landau type with resonant advective non-variational terms. Generally, and in agreement with experiment, above threshold solutions of the equations correspond to an hexagonal convective structure in which the fluid rises in the centre of the cells. We also analytically study the dynamics of pattern formation leading not only to hexagons but also to squares or rolls depending on the various dimensionless parameters like Pran...

Measurements of fluctuations and convection patterns in horizontal layers of fluid heated from below and near the onset of Rayleigh–Bénard convection (RBC) are reported under conditions where the fluid properties vary strongly over the temperature range ΔT = Tb − Tt (Tb and Tt are the temperatures at the bottom and top of the sample, respectively). To facilitate a comparison with the data, the theory of Busse (J. Fluid Mech., vol. 30, 1967, p. 625) of these so called non-Oberbeck–Boussinesq (NOB) effects, which applies to the case of relatively weak (and linear) temperature dependences, was extended to arbitrary variations with temperature. It is conceptually useful to divide the variations with temperature of the fluid properties into two disjunct parts. One part is chosen so that it preserves the reflection symmetry of the system about the horizontal midplane, while the remainder breaks that symmetry. The latter, exclusively considered by Busse, leads (in contrast to the formation...

The stability of the circular Couette flow with two differentially heated cylinders is studied in the special cases of hydrodynamically stable rotation regimes. A one-dimensional model is developed to derive the condition of flow stability. This condition combines the curvature of the cylinders, the applied temperature difference between the two cylinders, and the diffusion properties of the fluid. The three-dimensional analysis is performed for two different rotation regimes: the Keplerian regime and the regime where the inner cylinder is stationary. The main results of this analysis is that, for a given radius ratio of the two cylinders, a single parameter combining the Prandtl number and the thermal expansion parameter can describe the critical state of the system. The description is in good agreement with the result of the one-dimensional model. An energy analysis shows a subtle role played by the shear stress in these two rotation regimes.

6294-We are interested in the study of the transition to turbulence in the Taylor-Couette flow, the flow between two independently rotating coaxial cylinders. Once the geometry is fixed, the flow is controlled by the inner and outer Reynolds numbers and present a large variety of flow regimes. In counter-rotation, the transition is characterized by a succession of more or less turbulent flow regimes: intermittency with turbulent spots, spiral turbulence, featureless turbulence. For larger values of the inner Reynolds number, turbulent Taylor roll re-emerge from the featureless turbulence and remain for very large values of the Reynolds numbers. Bifurcations between different turbulent rolls states are even observed in the ultimate turbulence regime. Nevertheless the transition from the featureless turbulence to the turbulent rolls still requires a detailed study and the mechanism which causes and sustains turbulent spots or turbulent spirals remains unknown. In this study we present new experimental information on the organization of the flow for the different regimes with turbulence. The experiments are conducted in a Taylor-Couette flow with η = 0.8. Stereo-Particle Image Velocimetry measurements and visualizations of the different flow regimes are realized and discussed.

A rich variety of flow regimes in a Newtonian fluid inside a vertical large-aspect ratio and a wide-gap Taylor-Couette system with a radial temperature gradient has been determined in experiments and in direct numerical simulations (DNS). Compared to previous experiments and numerical studies, a wider range of temperature differences (i.e. of the Grashof number Gr) and of the rotation rate (the Taylor number Ta) has been covered. The combined effect of rotation and of the radial temperature gradient is the occurrence of helicoidal vortices or modulated waves at the onset. Stationary axisymmetric vortices are found for very weak temperature differences. A good agreement was found for critical states between results from experiments, linear stability analysis and DNS. Higher instability modes have been determined for a wide range of parameters and a state diagram of observable flow regimes has been established in the plane spanned by Gr and Ta. Some higher states observed in experiments were retrieved in DNS.

The transition between hexagons and rolls in convective patterns has been studied in pure water under non-Boussinesq conditions. The transition is characterized qualitatively by shadowgraph images and quantitatively by heat-flow measurements and a local technique based on the deflections of a laser beam. We also show that with this local technique one can recover global quantities such as the convective heat flow.

Dependence of magnetic field generation on the rotation rate is explored by direct numerical simulation of magnetohydrodynamic convective attractors in a plane layer of conducting fluid with square periodicity cells for the Taylor number varied from zero to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We observe 5 types of hydrodynamic (amagnetic) attractors: two families of two-dimensional (i.e. depending on two spatial variables) rolls parallel to sides of periodicity boxes of different widths and parallel to the diagonal, travelling waves and three-dimensional "wavy" rolls. All types of attractors, except for one family of rolls, are capable of kinematic magnetic field generation. We have found 21 distinct nonlinear convective MHD attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge. In addition, we have observed a family of periodic, twofrequency quasiperiodic and chaotic regimes, as well as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The system is highly symmetric. We have found two novel global bifurcations reminiscent of the SNIC bifurcation, which are only possible in the presence of symmetries. The universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid on continuing the increase) remains unaltered, but we demonstrate that this "large-scale" picture lacks many significant details.

Fully developed velocity profiles of longitudinal convection rolls in mixed convection between horizontal plates were measured in nitrogen by laser Doppler anemometry for a range 2472 < Ra < 8300 and 15 < Re < 150. It is shown analytically and experimentally that the transverse velocities of the longitudinal convection rolls are independent of the forced flow. The experimentally and numerically obtained w-profiles (Pr = 0.71) are in good agreement with theoretical predictions (Pr + co) and other experimental results (Pr = 11.1 and 930) for Rayleigh-Benard convection. A detailed study of the longitudinal velocity modulation Au[w,,,(Ra), Re] is presented. Also, asymmetric roll patterns were found in spite of the small temperature differences used between the horizontal plates.

We report numerical investigations of three-dimensional pattern formation of binary mixtures in a vertical cylindrical container heated from below. Negative separation ratio mixtures, for which the onset of convection occurs via a subcritical Hopf bifurcation, are considered. We focus on the dynamics in the neighbourhood of the initial oscillatory instability and analyze the spatiotemporal properties of the patterns for different values of the aspect ratio of the cell, Γ ≲ ≲ 0.25 11 (Γ ≡ R d, where R is the radius of the cell and d its height). Despite the oscillatory nature of the primary instability, for highly constrained geometries, Γ ≲ 2.5, only pure thermal stationary modes are selected after long transients. As the aspect ratio of the cell increases, for intermediate aspect ratio cells such as Γ = 3, multistability and coexistence of stationary and time-dependent patterns is observed. In highly extended cylinders, Γ ≈ 11, the dynamics near the onset is completely different from the pure fluid case, and a startling diversity of confined patterns is observed. Many of these patterns are consistent with experimental observations. Remarkably, though, we have obtained persistent large amplitude highly localized states not reported previously.

First and the foremost I wish to acknowledge is Alastair Rucklidge who gave the opportunity to me for doctoral studies. Apart from being my principal supervisor he was a strength, an inspiration, a critique and a dearly friend to me. My co-supervisor Steve Tobias brought unique perspectives to my research, enriching it greatly. I would thankfully remember his encouragement and enthusiasm. My appreciation also goes to Ian Melbourne for his valuable suggestions. I am also grateful for some helpful conversations with Edgar Knobloch, Stephen Morris and Paul Matthews. There is nothing like the parents' love. While I was tirelessly tied up with the thesis, they held me close to their hearts, watching my wellbeing from their far away home. Words cant express how much I owe to Sanjeeva Witharana, who stayed by my side in the most needed times. I am forever indebted to my parents and Sanjeeva for their understanding, endless patience and encouragement when it was most required-it is to them that this thesis is dedicated. Many thanks also to my colleagues and friends, especially Beverley and Rowan for being wonderful. Lastly I would like to express my gratitude for financial support provided by the School of Mathematics at University of Leeds. Famous French mathematician Joseph Fourier said "the profound study of nature is the most fertile source of mathematical discoveries". I studied naturally occurring patterns and their formations. I hope this Doctoral thesis did the justice to all who stood by me during this courageous endeavour.

We discuss the irreversibility, nonlocality, and fluctuations, as well as the Lyapunov and hydrodynamic instabilities characterizing atomistic, smooth-particle, and finite-difference solutions of the two-dimensional Rayleigh-Bénard problem. To speed up the numerical analysis we control the time-dependence of the Rayleigh number, R(t) , so as to include many distinct flow morphologies in a single simulation. The relatively simple nature of these computational algorithms and the richness of the results they can yield make such studies and their interpretation particularly well suited to graduate-level research.

wake behind a sphere, rotating about an axis aligned with the streamwise direction, has been experimentally investigated in a water tunnel using LIF visualizations and PIV measurements. The measurements focused on the evolution of the flow regimes that appears depending of two control parameters, namely the Reynolds number Re and the dimensionless rotation or swirl rate Ω which is the ratio of the maximum azimuthal velocity of the body to the free stream velocity. In the present investigation, we covers the range of Re smaller than 400 and Ω from 0 and 1.5. Different wakes regimes such as an axisymmetric base flow, a low frequency frozen state, and an single and double helicoidal mode are represented in the (Re, Ω) parameter plane.

Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest in physics. The following fields are emph asized: elementar y par ticle physics, solid-state physics. complex systems, and fundamental astrophysics.

Recently, Banerjee et al. [Phys. Rev. E 102, 013107 (2020)] investigated overstable rotating convection in the presence of an external horizontal magnetic field and reported a rich bifurcation structure near the onset. However, the bifurcation structure near the onset of overstable rotating convection in the presence of a vertical magnetic field has not been explored yet. We address the issue here by performing three dimensional direct numerical simulations and low-dimensional modeling of the system using a Rayleigh–Bénard convection model. The control parameters, namely, the Taylor number (Ta), the Chandrasekhar number (Q), and the Prandtl number (Pr) are varied in the ranges 750≤Ta≤106, 0&lt;Q≤103, and 0&lt;Pr≤0.5. Our investigation reveals two qualitatively different onset scenarios including bistability (coexistence of subcritical and supercritical convections). Analysis of the low-dimensional model shows that a supercritical Hopf bifurcation is responsible for the supercritical...

We present the results of our investigation on nonlinear overstable rotating magnetoconvection (RMC) in presence of vertical external magnetic field. We focus on the dynamics appearing near the onset of convection by varying the system control parameters, namely, the Taylor number (Ta), the Chandrasekhar number (Q) and the Prandtl number (Pr) in the ranges 750 Ta 10 6 , 0 < Q 10 3 and 0 < Pr 0.5. Three dimensional (3D) direct numerical simulations (DNS) of the governing equations and low-dimensional modeling of the system are performed for this purpose. Extensive DNS in the specified parameter space shows two qualitatively different onsets depending on Ta, Q and Pr. In the first one, bistability appears at the onset, where both subcritical and supercritical convection coexist, while only supercritical convection is observed in the second one. Analysis of the low-dimensional model reveals that a supercritical Hopf bifurcation is responsible for the supercritical onset and a subcritical pitchfork bifurcation is responsible for the subcritical onset. It is also observed that appearance of subcritical convection at the onset has strong dependence on all three control parameters Ta, Q and Pr. The scenario of subcritical convection is found to disappear as Pr is increased for fixed Ta and Q. However, most striking findings of the investigation is that the increment in Ta for fixed Q and Pr opposes the subcritical convection, whereas the increment in Q for fixed Ta and Pr favors it. This is in sharp contrast with the earlier results reported in RMC.

Time-periodic Taylor vortex flow between two concentric cylinders is studied. With the outer cylinder at rest and the inner one rotating with angular velocity A(t) =O, (1+6 cosset), the stability boundary of circular Couette flow and the fully nonlinear Taylor vortices are investigated. Results are obtained by a finite-difference numerical simulation of the full Navier-Stokes equations for a radius ratio g=0. 65 and by an analytical Galerkin approximation with four modes for arbitrary gap width. Data obtained by both methods agree very well. Modulation is found to destabilize the basic flow in agreement with earlier theoretical results. The time dependence of Taylor vortices is determined in detail and compared with recent experiments. Their response to the modulated driving is elucidated and explained by investigating various limiting behaviors and by comparison with the amplitude equation. Subharmonic response is found for large modulation amplitudes when the driving Q(t) becomes supercritical in both rotation directions during one period. Differences and similarities of linear and nonlinear flow properties with modulated convection in Rayleigh-Benard systems are discussed in detail. That low-frequency small-amplitude modulation stabilizes the basic conductive state in the latter system while it destabilizes circular Couette flow is shown to be caused by the different coupling of the driving to the secondary flow field.

We give a brief review of several prominent fluid instabilities representing transitions driven by gravity, surface tension, thermal energy, and applied motion/acceleration. Strategies for controlling these instabilities, including their pattern formation properties, are discussed. The importance of gravity for many common fluid instabilities is emphasized and used to understand the sometimes dramatically different behavior of fluids in microgravity environments. This is illustrated in greater detail, using recent results, for the case of the frozen wave instability, which leads to large columnar structures in the absence of gravity. The development of these highly nonlinear states is often complex, but can be manipulated through an appropriate choice of forcing amplitude, container length and height, initial inclination of the surface, and other parameters affecting the nonlinear and inhomogeneous growth process. The increased opportunity for controlling fluids and their instabilit...

The wakes behind open rings of two different aspect ratios are computed using a spectral-element method, and the stability of the solutions are analysed by the application of a linear stability analysis. The predicted instabilities are compared with the computed three-dimensional flows and with experimentally obtained dye visualisations. The flows computed at the chosen aspect ratios show that the bifurcations from steady axisymmetric flow produce remarkably different wakes. This phenomenon is related to the existence of a transition aspect ratio that delineates annular and cylinder-like wakes at small and large aspect ratios, respectively. Background: The Stability of Bluff-Body Flows The subject of instability and the bifurcation of fluid flows is as rich as it is diverse. The motion of a fluid around even the simplest of geometries is governed by the mathematically formidable Navier-Stokes equations; a set of non-linear partial differential equations for which analytical solutions to practical flows are rare.

The onset of thermal convection in a double layer of two superimposed immiscible fluids heated from below is investigated. The linearized equations of the problem are analysed in a much wider region of the parameter space than has been studied before. It is shown that the onset of steady convection in the two layers may occur in the form of either viscously or thermally coupled motions. In addition to the oscillatory interfacial instability, which depends on a non-vanishing distortion of the interface, there exists another oscillatory instability which corresponds to a cyclic variation between viscous and thermal coupling. Conditions for the onset of this instability are outlined and its connections with the other modes of the system are demonstrated in bifurcation diagrams. In the experiments the shadowgraph method is used for the visualization of the onset of convection and for the measurement of its wavelength. Changeovers between viscous and thermal coupling can be identified, but the experimental realization of an oscillatory onset has been elusive so far.

Heat-transport measurements in a bulk normal-fluid 'He-He mixture heated from below and over the range-0.02 & P & 0.02 of the separation ratio P reveal a forward bifurcation with an initial slope S =-0 of the Nusselt number for large Q, and a backward bifurcation for P & Pt, =0.006. At pt"S =-1. The critical line 5 T, (p) has two branches which meet at p =-0.003, and which we attribute to the expected stationary and Hopf bifurcation lines. However, stable oscillations bifurcating from the conduction state exist only for P &-0.015. PACS numbers: 47.20.k, 47.25.c In a horizontal layer of fluid heated from below, a transition, or bifurcation, occurs from conduction to convection when the temperature difference 5 T reaches b, T,. Depending on the values of relevant externally controlled parameters, the fluid velocity may grow continuously as 5 T increases beyond 6 T"

Heat-transport measurements in a normal-fluid He-4He mixture contained in a porous medium and heated from below show a bifurcation to steady or oscillatory flow, depending on the mean temperature. With increasing Rayleigh number 8 the oscillatory state is entered via a forward Hopf bifurcation with frequency w, & 0. As the stationary bifurcation is approached by change of the mean temperature, cu, vanishes. With increasing 8, the oscillatory state terminates with vanishing frequency and finite amplitude in a hysteretic bifurcation to a steady state. Those observations agree with recent theoretical predictions.

A report about experimental studies of the dynamics and interaction of topological defects in the roll structure of electrohydrodynamic convection in nematic liquid crystals is given. It is found that the motion of defects of opposite topological charges towards annihilation has two stages. At large distances they move with a constant velocity that depends mainly linearly on the wave-number mismatch. At a later stage, when the defects come closer, they are accelerated due to attraction. In order to compare these results quantitatively with available theories that are based on the Ginzburg-Landau equation, all coeScients of this equation are measured.

A new experimental system showing a transition to spatiotemporal intermittency is presented. It consists of a ring of hundred oscillating ferrofluidic spikes. Four of five of the measured critical exponents of the system agree with those obtained from a theoretical model of directed percolation.

The stability of convection rolls in a fluid heated from below is limited by secondary instabilities, including the skew-varicose and crossroll instabilities. We observe a stability boundary defined by the same instabilities in stripe patterns in a vertically oscillated granular layer. Molecular dynamics simulations show that the mechanism of the skew-varicose instability in granular patterns is similar to that in convection. These results suggest that pattern formation in granular media can be described by continuum models analogous to those used in fluid systems.

2014 Par un développement formel en gradient de la variable de phase nous obtenons les équations qui gouvernent la relaxation des zigzags. L'application a été développée dans le cas de la convection près du seuil au moyen d'un développement en 03B5 de la solution avec condition aux limites de non-glissement pour la vitesse. Les valeurs numériques qui interviennent dans D~ sont données dans le texte. Les points de comparaison avec d'autres théories et les résultats antérieurs concernant l'instabilité zigzag sont discutés. Abstract. 2014 By a formal expansion in gradient of the phase variable we obtain the equations governing the relaxation of zigzags. The application has been developed for convection close to the threshold. We use an 03B5-expansion of the solution for the realistic case of no slip (rigid) velocity boundary conditions. We obtain : D~ = 0 3 B E 2 0 / 0 3 C 4 0 (03B4k/kc + F(Pr) 03B4Ra/Rac) where the numerical values of the different quantities are given in the text. Points of comparison with other theories and earlier results concerning zigzag instability are discussed.

Marangoni patterns are created by instabilities caused by thermocapillary and solutocapillary stresses on the deformable free surface of a thin liquid layer. In the present paper, we consider the influence of the insoluble surfactant on the selection and modulational instability of stationary Marangoni patterns near their onset threshold. The basic governing parameters of the problem are the Biot number characterizing the heat-transfer resistances of and at the surface, the Galileo number indicating the role of gravity via viscous forces, and the elasticity number specifying the influence of insoluble surfactant on the interfacial dynamics of the liquid. The paper includes a review of the previous results obtained in that problem as well as new ones.

The stability of a non-isothermal circular Couette flow is analyzed when subjected to a dielectrophoretic force field. Outward and inward heating configurations are considered when the inner cylinder is rotating and the outer cylinder is at rest. In addition, an alternating voltage is applied between the two cylinders to induce a radial electric buoyancy that acts on the dielectric fluid. The linear stability analysis provides the threshold for the first transition to instability, as well as the corresponding wavenumber and frequency of the modes. This article is part of the theme issue ‘Taylor–Couette and related flows on the centennial of Taylor’s seminal Philosophical Transactions paper (part 1)’.

The flow stability of a dielectric fluid confined in the cylindrical annulus is investigated by linear stability analysis and direct numerical simulations. The annulus is in solid-body rotation about its axis and it is subject to a radial temperature gradient and a high-frequency electric tension, which produce a radial dielectrophoretic force. We analyze the effects of the rotation of the annulus on the thermoelectric convection and of the alternating electric tension on the onset of the rotation-induced thermal convection. Critical states are predicted by linear stability analysis for different values of the rotation rate of the cylindrical annulus and of the electric tension. Direct numerical simulations allow for the validation of the critical states, for the determination of their bifurcation nature, and for the visualization of instantaneous velocity and temperature fields.

The scalings near the wall for v' and & in the non-linear Navier-Stokes equations are the same as for the linear Orr-Sommerfeld equation as nonlinear terms are not dominant in the viscous sublayer. This follows from x-momentum balance in a timeasymptotic stationary state when periodic boundary conditions are applied downstream.

The behavior of an extra roll extending into an otherwise regular convection pattern is studied as a function of Rayleigh number, Prandtl number, P, and wavelength, by means of a fully resolved numerical simulation of the Boussinesq equations with free-slip boundary conditions. For .reduced Rayleigh numbers of order one or less and P &40, numerical simulations of the lowest-order amplitude equations reproduce the Boussinesq results semiquantitatively. In particular, we find that when this class of defects is stable, they move with constant velocity v, parallel to the roll axis and give rise to a slow modulation of the roll pattern of the form f(x,yvt). Both f and v have been calculated analytically within a linearized theory. The envelope function f depends in an essential way on v such that the limit v~0 cannot be sensibly taken.

We present an experimental study of highly localized, solitonlike structures that propagate on the two-dimensional surface of highly dissipative fluids. Like the well-known Faraday instability, these highly dissipative structures are driven by means of the spatially uniform, vertical acceleration of a thin fluid layer. These structures, harmonically coupled to the external driving frequency, are observed above a critical ratio of the viscous boundary layer height to the depth of the fluid layer for a wide range of fluid viscosities and system parameters. [S0031-9007(96)00186-X]

While non-Boussinesq hexagonal convection patterns are well known to be stable close to threshold (i.e. for Rayleigh numbers R ≈ R c), it has often been assumed that they are always unstable to rolls already for slightly higher Rayleigh numbers. Using the incompressible Navier-Stokes equations for parameters corresponding to water as a working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime (ǫ ≡ R − R c /R c = O(1)). We find 'reentrant' behavior of the hexagons, i.e. as ǫ is increased they can lose and regain stability. This can occur for values of ǫ as low as ǫ = 0.2. We identify two factors contributing to the reentrance: i) the hexagons can make contact with a hexagon attractor that has been identified recently in the nonlinear regime even in Boussinesq convection (Assenheimer & Steinberg (1996); Clever & Busse (1996)) and ii) the non-Boussinesq effects increase with ǫ. Using direct simulations for circular containers we show that the reentrant hexagons can prevail even for side-wall conditions that favor convection in the form of the competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons become stable even over the whole ǫ-range considered, 0 ≤ ǫ ≤ 1.5. Contents 1 Introduction 1 2 Basic Equations 3 3 Linear Stability of Hexagons 8 3.1 Amplitude Instabilities 8 3.2 Side-Band Instabilities 10 4 Origin of Reentrant Hexagons 12 5 Numerical Simulations 16 6 Conclusions 18

We employ numerical computations of the full Navier-Stokes equations to investigate non-Boussinesq convection in a rotating system using water as the working fluid. We identify two regimes. For weak non-Boussinesq effects the Hopf bifurcation from steady to oscillating (whirling) hexagons is supercritical and typical states exhibit defect chaos that is systematically described by the cubic complex Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and the oscillations exhibit localized chaotic bursting, which is modeled by a quintic complex Ginzburg-Landau equation.