Stationary Shapes Research Articles

On the Existence of Stationary Solutions of a Generalized Mullins Equation

We derive a nonlinear evolution equation which describes the development of a solid surface due to the mechanism of evaporation–condensation under a temperature gradient. The existence of stationary grain boundary grooves is shown and the stationary shapes are evaluated.

Ferrofluid drops in rotating magnetic fields

Drops of a ferrofluid floating in a non-magnetic liquid of thesame density and spun by a rotating magnetic field areinvestigated experimentally and theoretically. The parametersfor the experiment are chosen such that different stationary drop shapes including non-axis-symmetricconfigurations could be observed. Within an approximatetheoretical analysis the character of the occurring shapebifurcations, the different stationary drop forms, as well asthe slow rotational motion of the drop is investigated. Theresults are in qualitative, and often quantitative agreement,with the experimental findings. It is also shown that a smalleccentricity of the rotating field may have a substantial impacton the rotational motion of the drop.

Multiple stationary states for deformable drops in linear Stokes flows

Using a small-deformation expansion and numerical simulations we study stationary shapes of viscous drops in two-dimensional linear Stokes flows with nonzero vorticity. We show that high-viscosity drops in flows with vorticity magnitude β≪1 have two branches of stable stationary states. One branch corresponds to nearly spherical drops stabilized primarily by rotation, and the other to elongated drops stabilized primarily by capillary forces. For drop-to-continuous-phase viscosity ratios beyond a critical value λc, the rotationally stabilized solution exists in the absence of capillary stresses, because the rate of drop deformation (but not rotation) decreases with drop viscosity. We show that λc=10β−2, and the capillary stresses required for drop stability vanish at λc with exponent 1/2, as required by flow-reversal symmetry.

Faceting of a growing crystal surface by surface diffusion.

Consider faceting of a crystal surface caused by strongly anisotropic surface tension, driven by surface diffusion and accompanied by deposition (etching) due to fluxes normal to the surface. Nonlinear evolution equations describing the faceting of 1+1 and 2+1 crystal surfaces are studied analytically, by means of matched asymptotic expansions for small growth rates, and numerically otherwise. Stationary shapes and dynamics of faceted pyramidal structures are found as functions of the growth rate. In the 1+1 case it is shown that a solitary hill as well as periodic hill-and-valley solutions are unique, while solutions in the form of a solitary valley form a one-parameter family. It is found that with the increase of the growth rate, the faceting dynamics exhibits transitions from the power-law coarsening to the formation of pyramidal structures with a fixed average size and finally to spatiotemporally chaotic surfaces resembling the kinetic roughening.

Robustness of spiral waves in two-dimensional reactive–diffusive media

The effects of the reaction terms, diffusion coefficients and size of the computational domain on the dynamics of the two-equation Oregonator model in two-dimensional reactive–diffusive media is studied numerically, and it is shown that spiral waves are robust under truncation or expansion of the computational domain. It is also shown that large diffusion coefficients for the inhibitor may result on either annihilation of the wave or unbounded growth, whereas large diffusion coefficients for the activator concentration may result in unbounded decay of both the activator’s and inhibitor’s concentrations. Small diffusion coefficients of the inhibitor may result in stationary tongue-like shapes, whereas small diffusion coefficients of the activator may result in narrow ring shapes where the activator’s concentration is high. A decrease in the parameter which controls the stiffness of the activator’s reaction rate is found to result in unbounded growth of the activator’s concentration, whereas an increase in the parameter that couples the activator and inhibitor is found to result in either annihilation of the spiral wave or unbounded growth of the concentrations of both the activator and the inhibitor.

Stationary states and scaling shapes of one-dimensional interfaces

Sub-lattice parallel heat bath dynamics is applied to various one dimensional Solid-On-Solid interface models. The existence of invariant product measures in the gradient variables allows to compute exactly the interface speed as function of the slope. This function can have many convex and concave parts, depending on lattice modulation and unboundedness of the state space. This may be associated with the occurrence of corners in the macroscopic scaling shapes.

Critical behavior of drops in linear flows. I. Phenomenological theory for drop dynamics near critical stationary states

The dynamics of viscous drops in linear creeping flows are investigated near the critical flow strength at which stationary drop shapes cease to exist. According to our theory the near-critical behavior of drops is dominated by a single slow mode evolving on a time scale that diverges at the critical point with exponent 1/2. The theory is based on the assumption that the system undergoes a saddle-node bifurcation. The predictions have been verified by numerical simulations for drops in axisymmetric straining flow and in two-dimensional flows with less vorticity than in shear flow. Application of our theory to the accurate determination of critical parameters is discussed.

Shapes of nearly cylindrical, axisymmetric bilayer membranes

Shapes of nearly cylindrical sections of axisymmetric phospholipid membranes are studied theoretically. Describing the shape of such sections by their deviation from a reference cylinder, the well-established shape equation for axisymmetric bilayer membranes is expanded in terms of this deviation, and it is then solved analytically. The phase diagram shows the resulting stationary shapes as functions of system parameters and external conditions, i.e., the pressure difference across the membrane, the membrane tension, the difference between the tensions of the two monolayers, and the axial force acting on the vesicle. The accuracy of the approximate analytical solution is demonstrated by comparison with numerical results. The obtained analytical solution allows to extend the analysis to include shapes where numerical methods have failed.

Expansion of a compressible gas bubble in Stokes flow

The flow-induced deformation of an inviscid bubble occupied by a compressible gas and suspended in an ambient viscous liquid is considered at low Reynolds numbers with particular reference to the pressure developing inside the bubble. Ambient fluid motion alters the bubble pressure with respect to that established in the quiescent state, and requires the bubble to expand or contract according to an assumed equation of state. When changes in the bubble volume are prohibited by a global constraint on the total volume of the flow, the ambient pressure is modified while the bubble pressure remains constant during the deformation. A numerical method is developed for evaluating the pressure inside a two-dimensional bubble in an ambient Stokes flow on the basis of the normal component of the interfacial force balance involving the capillary pressure, the normal viscous stress, and the pressure at the free surface on the side of the liquid; the last is computed by evaluating a strongly singular integral. Dynamical simulations of bubble deformation are performed using the boundary integral method properly implemented to remove the multiplicity of solutions due to the a priori unknown rate of expansion, and three particular problems are discussed in detail: the shrinkage of a bubble at a specified rate, the deformation of a bubble subject to simple shear flow, and the deformation of a bubble subject to a purely elongational flow. In the case of shrinkage, it is found that the surface tension plays a critical role in determining the behaviour of the bubble pressure near the critical time when the bubble disappears. In the case of shear or elongational flow, it is found that the bubble contracts during an initial period of deformation from the circular shape, and then it expands to obtain a stationary shape whose area is higher than that assumed in the quiescent state. Expansion may destabilize the bubble by raising the capillary number above the critical threshold under which stationary shapes can be found.

Hydrodynamical Modeling Of Oceanic Vortices

Mesoscale coherent vortices are numerous in the ocean.Though they possess various structures in temperature and salinity,they are all long-lived, fairly intense and mostly circular. Thephysical variable which best describes the rotation and the density anomaly associated with coherent vortices is potential vorticity. It is diagnostically related to velocity and pressure, when the vortex is stationary. Stationary vortices can be monopolar (circular or elliptical) or multipolar; their stability analysis shows thattransitions between the various stationary shapes are possible when they become unstable. But stable vortices can also undergo unsteady evolutions when perturbed by environmental effects, likelarge-scale shear or strain fields, β-effect or topography. Changes in vortex shapes can also result from vortex interactions. such as the pairing, merger or vertical alignment of two vortices, which depend on their relative polarities and depths. Such interactions transfer energy and enstrophy between scales, and are essential in two-dimensional and in geostrophic turbulence. Finally, in relation with the observations, we describe a few mechanisms of vortex generation.

Impairment in the perception of 2D shape by adults with mild intellectual disability: An exploratory investigation

Following a necessarily brief review of evidence for perceptual impairment associated with mild intellectual disability, two experiments are reported. Adults with mild intellectual disability and age- and gender-matched adults with average or above-average intelligence identified 16 two-dimensional random shapes after their progressive exposure moving past an aperture and after their brief full exposure while stationary. Identifications were made by pointing at the previously presented target shape randomly positioned among five similar shapes. In the first experiment, the aperture was narrow so that the target shape was never fully visible, and in the second it was wider so that it was briefly fully visible while moving. In both experiments, participants with intellectual disability made significantly more identification errors than those of the age-and sex-matched controls. The data are interpreted as showing that individuals with intellectual disability are impaired in their perception of both moving and stationary two-dimensional shapes. It is suggested that the deficit shown by individuals with intellectual disability may be due to the “impoverishment” of the moving stimulus shapes by progressive exposure in an aperture.

Numerical studies of singularity formation at free surfaces and fluid interfaces in two-dimensional Stokes flow

We consider the analytic structure of interfaces in several families of steady and unsteady two-dimensional Stokes flows, focusing on the formation of corners and cusps. Previous experimental and theoretical studies have suggested that, without surface tension, the interfaces spontaneously develop such singular points. We investigate whether and how corners and cusps actually develop in a time-dependent flow, and assess the stability of stationary cusped shapes predicted by previous authors. The motion of the interfaces is computed with high resolution using a boundary integral method for three families of flows. In the case of a bubble that is subjected to the family of straining flows devised by Antanovskii, we find that a stationary cusped shape is not likely to occur as the asymptotic limit of a transient deformation. Instead, the pointed ends of the bubble disintegrate in a process that is reminiscent of tip streaming. In the case of the flow due to an array of point-source dipoles immersed beneath a free surface, which is the periodic version of a flow proposed by Jeong & Moffatt, we find evidence that a cusped shape indeed arises as the result of a transient deformation. In the third part of the numerical study, we show that, under certain conditions, the free surface of a liquid film that is levelling under the action of gravity on a horizontal or slightly inclined surface develops an evolving corner or cusp. In certain cases, the film engulfs a small air bubble of ambient fluid to obtain a composite shape. The structure of a corner or a cusp in an unsteady flow does not have a unique shape, as it does at steady state. In all cases, a small amount of surface tension is able to prevent the formation of a singularity, but replacing the inviscid gas with a viscous liquid does not have a smoothing effect. The ability of the thin-film lubrication equation to produce mathematical singularities at the free surface of a levelling film is also discussed.

Mössbauer investigation of intermixing during ball milling of Fe0.3Cr0.7 and Fe0.5W0.5 powder mixtures.

The intermixing of Fe and T (T=Cr,W) during ball milling of elemental powder mixtures ${\mathrm{Fe}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{T}}_{\mathit{x}}$, with x=0.70 for T=Cr and x=0.50 for T=W, has been followed by $^{57}\mathrm{Fe}$ M\"ossbauer spectroscopy at room temperature (RT) and by magnetization measurements for T=W. The chemical compositions have been chosen to yield final alloys or compounds which are nonmagnetic at RT to better follow the evolution of magnetic phases with milling times. For a long period of milling time ${\mathit{t}}_{\mathit{m}}$ before reaching the final stationary state, the hyperfine magnetic field distributions remain stationary in shape for both T=Cr and T=W. Only the relative weight of the magnetic contribution decreases with ${\mathit{t}}_{\mathit{m}}$. For T=W, the average moment of magnetic Fe atoms is further shown to remain constant with ${\mathit{t}}_{\mathit{m}}$. Stationary hyperfine field distribution shapes are found to be similar not only for T=Cr and W but also for T=Si (x=0.50) while published spectra suggest to add T=Al, Ti, V, Ta, Re to the latter nonexhaustive list. The stationary shape is characterized by a narrow peak located at a field close to the field of alpha iron at RT (330 kG) and by a broad, almost featureless, band from \ensuremath{\sim}50--100 kG to \ensuremath{\sim}300--320 kG. The broad band represents about 2/3 of the normalized field distribution. We deduce that the interpretation which consists in attributing the x-ray diffraction peaks of Fe-based bcc solid solutions to a single Fe-rich homogeneous solid solution must be done with care for intermediate milling times. We cannot infer from such hyperfine measurements a detailed description of the regions of the powders which are responsible for such magnetic features. We argue however that irregular interfaces between nanometer-sized Fe-rich zones and T-rich zones may play a role to explain the observed shape of the hyperfine field distributions. The general conditions (process and materials) in which such phenomena may occur remain to be clarified. \textcopyright{} 1996 The American Physical Society.

Fluid Vesicles in Shear Flow.

The shape dynamics of fluid vesicles is governed by the coupling of the flow within the two-dimensional membrane to the hydrodynamics of the surrounding bulk fluid. We present a numerical scheme which is capable of solving this flow problem for arbitrarily shaped vesicles using the Oseen tensor formalism. For the particular problem of simple shear flow, stationary shapes are found for a large range of parameters. The dependence of the orientation of the vesicle and the membrane velocity on shear rate and vesicle volume can be understood from a simplified model.

Shear flow over a liquid drop adhering to a solid surface

The hydrostatic shape, transient deformation, and asymptotic shape of a small liquid drop with uniform surface tension adhering to a planar wall subject to an overpassing simple shear flow are studied under conditions of Stokes flow. The effects of gravity are considered to be negligible, and the contact line is assumed to have a stationary circular or elliptical shape. In the absence of shear flow, the drop assumes a hydrostatic shape with constant mean curvature. Families of hydrostatic shapes, parameterized by the drop volume and aspect ratio of the contact line, are computed using an iterative finite-difference method. The results illustrate the effect of the shape of the contact line on the distribution of the contact angle around the base, and are discussed with reference to contact-angle hysteresis and stability of stationary shapes. The transient deformation of a drop whose viscosity is equal to that of the ambient fluid, subject to a suddenly applied simple shear flow, is computed for a range of capillary numbers using a boundary-integral method that incorporates global parameterization of the interface and interfacial regriding at large deformations. Critical capillary numbers above which the drop exhibits continued deformation, or the contact angle increases beyond or decreases below the limits tolerated by contact angle hysteresis are established. It is shown that the geometry of the contact line plays an important role in the transient and asymptotic behaviour at long times, quantified in terms of the critical capillary numbers for continued elongation. Drops with elliptical contact lines are likely to dislodge or break off before drops with circular contact lines. The numerical results validate the assumptions of lubrication theory for flat drops, even in cases where the height of the drop is equal to one fifth the radius of the contact line.

Vesicular instabilities: The prolate-to-oblate transition and other shape instabilities of fluid bilayer membranes.

The equilibrium shapes of fluid-phase phospholipid vesicles in an aqueous solution are controlled by bending elasticity. The regime of nonvesiculated shapes at reduced volume v\ensuremath{\ge}1/ \ensuremath{\surd}2 involves the interplay of five branches of distinct stationary shapes: pears, prolates, oblates, stomatocytes, plus a branch of nonaxisymmetric shapes with the symmetry ${\mathit{D}}_{2\mathit{h}}$. We exploit a method for calculating explicitly the stability of arbitrary axisymmetric shapes to map out in a numerically exact way both the stable phases and the metastability of the low-lying shape branches. To obtain additional required information about nonaxisymmetric shapes, we calculate these by numerical minimization of the curvature energy on a triangulated surface. Combining these two methods allows us to construct the full (shape) phase diagram and the full stability diagram in this region. We provide explicit results for values of the bending constants appropriate to stearoyl-oleoyl-phosphatidylcholine; generalization to other values is straightforward. (c) 1995 The American Physical Society

Stationary natural frequencies and mode shapes of composite laminates

The natural vibration of composite laminates is studied in this article. It is found that for a composite laminate of constant thickness the natural frequency approaches monotonically to a stationary value, and the natural vibration mode a stationary shape, as the number of its constituent plies increases. The stationary value and the stationary shape can easily be determined by utilizing the effective modulus and the effective density of the composite laminate, for which exact expressions are derived. The effective modulus in natural vibrations equals that in static problems in all cases, whereas the effective density, depending on frequency and other parameters in general, coincides with the mean density of the constituent plies only in simple vibration modes.

Supercoiling transitions of closed DNA.

Conformations of closed DNA molecules are considered within a simple elastic model. The interplay of elastic energies of bending and twisting can lead to supercoiling induced by variations of the linking difference. These shape transformations are studied in the limit of a large length to thickness ratio of the molecule. Stationary shapes and energy diagrams are obtained by solving shape equations for closed rings. Four different families of stationary shapes can be distinguished: (i) planar circles, (ii) nonplanar rings, (iii) self-interacting rings, and (iv) interwound configurations. They all occur as shapes of minimal energy in the phase diagram of supercoiling. The transitions between the different regions in this phase diagram can be either continuous or discontinuous. The sequence of shape transitions turns out to be sensitive to the precise values of the elastic parameters. The buckling instability of the circle changes within the range of physically accessible values from continuous to discontinuous behavior.

Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow

The steady and transient deformation of a drop that is immersed in an ambient simple shear flow is studied in the limit of Stokes flow. The flow is examined as a function of the viscosity ratio λ between the drop and the suspending fluid, and the capillary number Ca. The motion of the drop is considered in an unbounded fluid and in proximity to a bounding plane wall. The problem is formulated in terms of the boundary integral method and is solved using a boundary element numerical procedure. The results provide information on the shape of stationary and transient deformed drops for a wide range of λ and Ca. The streamlines inside and outside deformed drops are analyzed and are shown to exhibit different patterns depending on λ and to a lesser degree on Ca. The kinematics of the flow on the surface of the drop is studied including the travel time of interfacial markers and the straining motion of interfacial patches. The stationary drop shapes are used to compute the effective stress tensor of a dilute emulsion of drops in shearing motion. It is found that a dilute emulsion behaves like an elastic, shear-thinning medium, for all values of λ. The effect of a plane wall is examined with reference to the time-scales of drop deforamation and migration away from the wall. The computed migration velocities are compared with those predicted by available asymptotic theories and experimental data and some differences are identified and discussed.

Phase diagrams and shape transformations of toroidal vesicles

Shapes of vesicles with toroidal topology are studied in the context of curvature models for the membrane. For two simplified curvature models, the spontaneous-curvature (SC) model and the bilayer-couple (BC) model, the structure of energy diagrams, sheets of stationary shapes and phase diagrams are obtained by solving shape equations for axisymmetric shapes. Three different sheets of axisymmetric shapes are investigated systematically: I) discoid tori; it) sickle-shaped tori and iii) toroidal stomatocytes. A stability analysis of axisymmetric shapes with respect to symmetry breaking conformal transformations reveals that large regions of the phase diagrams of toroidal vesicles are non-axisymmetric. Non-axisymmetric shapes are determined approxim~tely using conformal transformations. To compare the theory with experiments, a generalization of the SC and BC model, the area-difference-elasticity-model (ADlGmodel), which is a more realistic curvature model for lipid bilayers, is discussed. Shapes of toroidal vesicles which have been observed recently can be located in the phase diagram of the ADlGmodel. We predict the effect oftemperature changes on the observed shapes-The new class of shapes, the toroidal stomatocytes, have not yet been observed.