Stationary Shapes Research Articles

Attracting dynamical modes of highly elastic fibres settling under gravity in a viscous fluid

The dynamics of a single highly elastic fibre settling under gravity in a very viscous fluid is studied numerically. We employ the bead model and multipole expansion of the Stokes equations, corrected for lubrication that is implemented in the precise Hydromultipole numerical codes. Four attracting regular dynamical modes of highly elastic fibres are found: two stationary shapes (one translating and the other rotating and translating), and two periodic oscillations around such shapes. The phase diagram of these modes is presented. It illustrates that the existence of each mode depends not only on the elasto-gravitation number but also on the fibre aspect ratio. Characteristic time scales, fibre deformation patterns and motion in the different modes are determined.

Validation of the structural response of a 70kW VAWT

This paper describes a project for predicting and measuring the structural response of a 70 kW vertical axis wind turbine (VAWT), while offering insights for optimizing efficiency and ensuring longevity. We employ the traditional frequency response approach and model the turbine behavior using its stationary modal shapes. Results for three key sensor locations – lower central mast bending, upper blade flapwise bending, and central horizontal arm out-of-plane bending – reveal that the codes accurately predict the critical resonance and capture the dominant harmonic amplitudes. In addition, it is shown how a statistical treatment of the steady state solution can lead to predictions of the fatigue cycle distribution enabling accurate life predictions. The program demonstrates the potential to significantly improve the design and operation of VAWT using simple, steady state, simulation codes for capturing VAWT aerodynamics. The model demonstrates sufficient accuracy while offering computational efficiency compared to more complex models.

Effect of Marangoni flow during the solidification of a Fe-0.82wt.%C steel alloy

A two-phase Mixture model is proposed to simulate the liquid-solid phase transition of a Fe-0.82wt%C steel alloy under the effect of Marangoni flow. This model simplifies computations by solving a single momentum and enthalpy equation for the mixture phase using a three-dimensional finite volume method. The simulation involves solidifying a rectangular ingot (100 × 10 × 100 mm3) from the cold bottom surface towards the hot-free surface at the top. To facilitate heat exchange with the surrounding environment, a high heat transfer coefficient of h = 600 W/m2/K was applied on the bottom surface to establish an upward solidification direction. However, a lower heat transfer coefficient of 20 W/m2/K was applied on the top free surface, which was considered flat. This study aims to examine the effect of Marangoni flow generated by surface tension on flow and segregation patterns. The results show that the Marangoni flow emerges at the free surface and penetrates into the liquid depth, leading to the formation of hexagonal patterns along the liquid thickness. Upon full solidification, macro-segregation also exhibits hexagonal structures, mirroring the stationary hexagonal shapes generated by Marangoni flow.

Stationary dimpled drops under linear flow

The axially symmetric deformation of a drop in a viscous fluid, under the influence of an externally imposed flow having simultaneous rotating and compressional or extensional components, is addressed. In the previous studies, two families of stationary drop shapes were constructed by simulating the dynamics of drop deformation: stable singly connected shapes with respect to axisymmetric disturbances, and unstable toroidal shapes. These two branches coexist at the same flow conditions, but were not connected. In this study, we obtain a new family of branches of unstable highly deformed stationary drops connecting with the stable flattened shapes and the toroidal ones. We use a method based on classical control theory. The controller is designed for a two-state dynamic model of the system and is employed on a high-order nonlinear dynamic model of the drop deformation. Through this feedback-control-centred approach, an extended collection of unstable stationary solutions is constructed, which spans the range from the loss of stability to the dimpled shapes almost collapsed at the centre. In the latter region, which was never obtained in previous studies, a multiplicity of solutions is identified.

Stationary shapes of axisymmetric vesicles beyond lowest-energy configurations.

We conduct a systematic exploration of the energy landscape of vesicle morphologies within the framework of the Helfrich model. Vesicle shapes are determined by minimizing the elastic energy subject to constraints of constant area and volume. The results show that pressurized vesicles can adopt higher-energy spindle-like configurations that require the action of point forces at the poles. If the internal pressure is lower than the external one, multilobed shapes are predicted. We utilize our results to rationalize experimentally observed spindle shapes of giant vesicles in a uniform AC electric field.

Deformations of an active liquid droplet

A fluid droplet, in general, deforms if subject to active driving, such as a finite slip velocity or active tractions on its interface. Starting from Stokes equations, we show that these deformations and their dynamics can be computed analytically in a perturbation theory in the inverse of the surface tension γ, by using an approach based on vector spherical harmonics. We consider squirmer models and general active tractions, such as inhomogeneous surface tensions, which may result from the Marangoni effects. In the lowest order, the deformation is of order ε∝1/γ, yet it affects the flow fields inside and outside of the droplet in order to ε0. Hence, a correct description of the flow has to allow for shape fluctuations, —even in the limit of large surface tension. We compute stationary shapes and relaxation times and compare our results to an approach, which discards all effects of deformations on surface tensions. This approach leads to the same propulsion velocity but to significantly different flow fields.

Controlled stabilization of rotating toroidal drops in viscous linear flow

Toroidal drops, embedded in viscous flow, have a large range of stationary shapes that are challenging to compute numerically due to their inherent instability. When both the drop and the outer fluid are Newtonian liquids, the only reported cases of such stable configurations are of highly expanded drops rotating in an axisymmetrical extensional flow. In this study, we propose a method for stabilizing the stationary shapes of inherently unstable rotating toroidal drops, embedded in extensional or biextensional flow, by subjecting the system to feedback control stabilization. The proposed controller is designed using a two-state dynamic model of the system and is tested on a high-order nonlinear dynamic model of the drop deformation. It is demonstrated that, through this simplified feedback-control-centred approach, an extended collection of stabilized stationary solutions is generated, which spans the range from highly expanded drops to almost collapsed ones. In the latter region, that was never obtained in previous studies, multiplicity of solutions is identified. Furthermore, our method is more accurate and more efficient compared with the previously used ones.

Dissociation between Attention-Dependent and Spatially Specific Illusory Shape Responses within the Topographic Areas of the Posterior Parietal Cortex.

The human visual system consists of multiple topographic maps that extend from the early visual cortex (EVC) along the dorsal and ventral processing streams. Responses to illusory shapes within these maps have been demonstrated in the ventral stream areas, in particular the lateral occipital complex (LOC). Recently, the intraparietal sulcus (IPS) of the dorsal stream has been linked to the processing of illusory shapes defined by motion. It remains unclear whether the topographically organized parietal areas also respond to stationary illusory shapes, which would suggest their generic role in representing illusory content. In the current study we measured brain responses using fMRI while 30 human participants (12 male) observed flickering inducers around the fixation task. The inducers either formed an illusory diamond in the center, a triangle in the left or right hemifield, or were inverted such that no illusory figure was formed. We compared responses of parietal regions IPS0-IPS5 and SPL1 to each illusory figure with the nonillusory condition. To determine the role of attentional modulation on illusory shape responses we manipulated the difficulty of the fixation task. Our results show that all IPS areas responded to illusory shapes. The more posterior areas IPS0-IPS3 additionally displayed a preference toward contralateral shapes, while the more anterior areas IPS4 and IPS5 showed response attenuation with increased task difficulty. We suggest that the IPS can represent illusory content generated not only by moving, but also by stationary stimuli, and that there is a functional dissociation between attention-dependent anterior and spatially specific posterior topographic maps.SIGNIFICANCE STATEMENT The traditional view of the ventral visual pathway being solely responsible for representation of objects has recently been challenged by demonstrating illusory shape representation within the dorsal visual pathway with moving bistable stimuli. Our results provide evidence for the dorsal stream contribution to representing not only moving, but also stationary illusory shapes. Our results also show a functional subdivision along the topographic maps, with spatially specific shape responses in the more posterior, and attention-dependent responses in the more anterior areas. These findings have implications for our understanding of the relationship between attention and grouping in healthy individuals and neuropsychological patients. Furthermore, IPS areas should be considered in theoretical accounts and models of how subjective content is generated in the brain.

Field-controlled flow and shape of a magnetorheological fluid annulus.

We investigate the behavior of a magnetorheological (MR) fluid annulus, bounded by a nonmagnetic fluid and confined in a Hele-Shaw cell, under the simultaneous effect of in-plane, external radial and azimuthal magnetic fields. A second-order mode-coupling theory is used to study the early nonlinear stage of the pattern-forming dynamics. We examine changes in the morphology of the MR fluid annular structure as a function of its magnetic-field-tunable rheological properties, as well as the combined magnetic field's intensities, and thickness of the ring. Our weakly nonlinear perturbative results show that, depending on the system control parameters, the MR fluid annulus adopts various stationary shapes. These equilibrium annular structures present slightly bent, asymmetric fingered protrusions which may emerge on the inner, outer, or even on both boundaries of the magnetic fluid ring. On top of these morphological changes, we find that the resulting permanent shape patterns rotate with a well defined angular velocity. We focus on analyzing how the overall shape of the fingered patterns, in particular their sharpness and asymmetric form, as well as the number of resulting fingers are impacted by the magnetic-field-dependent yield stress of the MR fluid annulus. The influence of the magnetically controlled rheological properties of the MR fluid on the angular velocity of the rotating annulus is also scrutinized.

Quasi-static crack front deformations in cohesive materials

When a crack interacts with material heterogeneities, its front distorts and adopts complex tortuous configurations that are reminiscent of the energy barriers encountered during crack propagation. As such, the study of crack front deformations is key to rationalize the effective failure properties of micro-structured solids and interfaces. Yet, the impact of a localized dissipation in a finite region behind the crack front, called the process zone, has often been overlooked. In this work, we derive the equation ruling 3D coplanar crack propagation in heterogeneous cohesive materials where the opening of the crack is resisted by some traction in its wake. We show that the presence of a process zone results in two competing effects on the deformation of crack fronts: (i) it makes the front more compliant to small-wavelength perturbations, and (ii) it smooths out local fluctuations of strength and process zone size, from which emerge heterogeneities of fracture energy. Their respective influence on front deformations is shown to strongly impact the stability of perturbed crack fronts, as well as their stationary shapes when interacting with arrays of tough obstacles. Overall, our theory provides a unified framework to predict the variety of front profiles observed in experiments, even when the small-scale yielding hypothesis of linear elastic fracture mechanics breaks down.

Ferrofluid annulus in crossed magnetic fields.

We study the dynamics and pattern formation of a ferrofluid annulus enveloped by two nonmagnetic fluids in a Hele-Shaw cell, subjected to an in-plane crossed magnetic field configuration involving the combination of radial and azimuthal magnetic fields. A perturbative, second-order mode-coupling analysis is employed to investigate how the ferrofluid annulus responds to variations in the relative strength of the radial and azimuthal magnetic field components, as well as in the thickness of magnetic fluid ring. By tuning the magnetic field components and the annulus' thickness, we have found the development of several stationary annular shapes, presenting polygon-shaped structures typically having skewed, peaked fingers. Such fingered structures may vary their skewness, sharpness, and number and arise on the inner, outer, or even both boundaries of the annulus. In addition to controlling the morphologies of the ferrofluid annuli, the external field can be used to put the annulus into a rotational motion, with an angular velocity having prescribed magnitude, and direction. Our second-order theory is utilized to obtain a correction to the linear stability analysis prediction of such angular velocity, usually resulting in a decreased weakly nonlinear value as compared with the magnitude predicted by purely linear theory. These theoretical results suggest the use of magnetic-field-controlled ferrofluid annuli in Hele-Shaw cells as a potential laboratory for microscale applications related to the manipulation of shape-programmable magnetic fluid objects and tunable fluidic-mixing devices in confined environments.

Bishop and Wölfflin: The Painterly Geometry

AbstractThis article establishes Elizabeth Bishop’s relationship with art historian Heinrich Wölfflin through an in-depth analysis of the former’s landscape poems, “Questions of Travel,” “Cape Breton” and “The End of March.” Bishop’s poetry is shown to exercise Wölfflin’s “painterly” geometry to soften stationary shapes and enhance her language’s ability to paint her poetic environment. This study focuses on Bishop’s geometry of indeterminate lines as incomplete and unclear dimensions of vision, where their preference for diagonal motion references Wölfflin’s description of the “painterly […] depreciation of line” in baroque art. Bishop’s painterly lines of perception help to differentiate her spatial poetics from plastic tendencies. Her poems are not tangible facts, but spaces of dialogue. This article thus proposes that Bishop’s landscape poems show strategies of painterly paintings read as verse.

Action Principle and Ritz Method for Yarn Dynamics in Ring Spinning

AbstractRing spinning is the most relevant method of producing high quality staple yarns. The rotating yarn forms one or multiple balloon shapes governed by inertial, eyelet friction and drag forces. Traditionally, the stationary yarn path has been computed by direct integration of an equation of motion, and stability and natural oscillations have been assessed by application of Galerkin's method to these ([1], [4], and others). Recent technological advances by using frictionless superconducting magnetic bearings allow for increased process speeds and thus motivate a renewed interest in the dynamics of ring spinning.An integral formulation of the yarn dynamics problem is presented using Hamilton's principle for the one‐dimensional continuum and the magnetic bearing. While therefrom the known equations of motions can be deduced elegantly, here Ritz's method is employed to achieve direct discretization of the problem, with cases to be made for both local and global shape functions.While applicable to the classical study of stationary balloon shapes and their stability, the resulting models in particular enable the study of the instationary yarn path and natural and driven yarn oscillations, which both have received limited attention in literature, but are known to affect yarn tension and thus the process viability.

Dynamic and stationary shapes of rotating toroidal drops in viscous linear flows

Abstract

Impact of electric charge and motion of water drops on the inception field strength of partial discharges.

Strong electric fields may deform drops and induce their oscillation or motion on the substrate. Moreover, they can initiate partial discharges (PDs) because of the enhancement of the electric field in the vicinity of the three-phase contact lines. The partial discharges affect the drop spreading which can result in unusual drop shapes. In addition, the partial discharges can also deteriorate the surface properties of the substrate, e.g., of high-voltage composite insulators. In this study the occurrence of partial discharges due to stationary or oscillating sessile drops under the influence of an alternating electric field is investigated using a generic insulator model under well-defined conditions. Drops of a yield stress fluid (a gelatin-water mixture) are used to determine the PD inception field strength for stationary drop shapes. The influence of the volume as well as the distance between the individual drops for two drop configurations on the PD inception threshold is determined. The inception field strength of the partial discharges is measured for various drop volumes, drop charges, as well as for different resonance modes of drop oscillations. Besides the electrical measurement, the location of the partial discharges is optically determined by a UV camera. The detailed knowledge of the influencing factors of the partial discharges improves the understanding of the drop behavior under the impact of strong electric fields.

Planar polymers under cylindrical confinement: geometrical approach

We show that the cylindrical confinement of a planar ribbon gives rise to the formation of stationary helical shapes, previously observed in molecular dynamics simulation studies of some planar polymers inside nanotubes. In the limit of small twisting deformations, we notice that a confined ribbon describes identical equilibrium configurations of an elastic rod on the plane. Also, every locally planar strip adsorbed on a cylinder is forced to follow a helical envelope, as its length increases. We also determine the forces and torques that sustain each ribbon equilibrium configuration through the equilibrium geometry and the confinement surface parameters.

A Leaky Dielectric Drop with Conical Ends in An Electric Field

There has long been interest in the existence of stationary shapes with conical ends for drops freely suspended in an ambient fluid and subjected to a uniform electric field. If the two phases are ...

Closed, Two Dimensional Surface Dynamics

We present dynamic equations for two dimensional closed surfaces and analytically solve it for some simplified cases. We derive final equations for surface normal motions by two different ways. The solution of the equations of motions in normal direction indicates that any closed, two dimensional, homogeneous surface with time invariable surface energy density adopts constant mean curvature shape when it comes in equilibrium with environment. In addition, we show that the shape equation is an approximate solution to our equation of motion in the normal direction and is valid for stationary or near to stationary shapes. As an example, we apply the formalism to analyze equilibrium shapes of micelles and explain why they adopt spherical, lamellar and cylindrical shapes. Theoretical calculation for micellar optimal radius is in good agreement with all atom simulations and experiments.

Fingering patterns in magnetic fluids: Perturbative solutions and the stability of exact stationary shapes

Pattern formation processes in magnetic fluids confined in a Hele-Shaw cell and subjected to a radial magnetic field are studied. It is shown that the weakly nonlinear flow drives an initially circular interface towards nontrivial exact stationary shapes, which are dynamically stable.

Efficient steady-state solver for hierarchical quantum master equations.

Steady states play pivotal roles in many equilibrium and non-equilibrium open system studies. Their accurate evaluations call for exact theories with rigorous treatment of system-bath interactions. Therein, the hierarchical equations-of-motion (HEOM) formalism is a nonperturbative and non-Markovian quantum dissipation theory, which can faithfully describe the dissipative dynamics and nonlinear response of open systems. Nevertheless, solving the steady states of open quantum systems via HEOM is often a challenging task, due to the vast number of dynamical quantities involved. In this work, we propose a self-consistent iteration approach that quickly solves the HEOM steady states. We demonstrate its high efficiency with accurate and fast evaluations of low-temperature thermal equilibrium of a model Fenna-Matthews-Olson pigment-protein complex. Numerically exact evaluation of thermal equilibrium Rényi entropies and stationary emission line shapes is presented with detailed discussion.