University of Cambridge > DAMTP > Goldstein Lab > Research

Goldstein Lab

Research

I. Overview

When asked whether I am a theorist or an experimentalist, my reply is that I am a scientist. Our group seeks to understand fundamental principles that govern the behavior of nonequilibrium systems in physics and biology, using a combination of experiment and theory. This research is not easily described by a single, conventional academic label; rather, it involves the domains of condensed matter physics, physical chemistry, biological physics, fluid dynamics, applied mathematics, and geophysics. I subscribe to Poincaré's motivation:

The scientist does not study nature because it is useful;
he studies it because he delights in it, and he delights in it because
it is beautiful. If nature were not beautiful, it would not be worth knowing, and
if nature were not worth knowing, life would not be worth living.

I also believe that some of the best science is close to art, and that Glenn Gould captured this spirit when he said

The purpose of art is not the release of a momentary ejection of adrenaline but rather the gradual,
lifelong construction of a state of wonder and serenity.

Current research in my group falls into two broad categories: Biological Physics and Natural Pattern Formation.

II. Biological Physics

Our group is currently focused on a range of questions centred around the origins of multicellularity. We use extensively the Volvocine green algae as a class of model organisms to understand the driving force behind the emergence of germ-soma differentiation, the nature of flagellar synchronization, the mechanisms of phototaxis, and basic aspects of biological fluid dynamics. A parallel set of investigations on the phenomenon of cytoplasmic streaming aims to answer the basic question of its biological purpose. Here we use the Characean algae as model organisms, and also have a developing collaboration involving streaming in the developing fruit fly oocyte.

Swimming, Stirring, and Scaling in the Volvocales

One of the most fundamental issues in biology is the nature of evolutionary transitions from single cell organisms to multicellular ones. It is a general rule of nature that larger organisms are more complex, at least as measured by the number of distinct types of cells present. This reflects the fitness advantage conferred by a division of labor among specialized cells over homogeneous totipotency. Yet, increasing size has both costs and benefits, and the search for the driving forces behind the evolution of multicellularity is becoming a very active area of research. Not surprisingly for microscopic life in a fluid environment, many of the processes involved are related to transport and locomotion, for efficient exchange of chemical species with the environment is one of the most basic features of life. We study these issues using as model organisms the Volvocales, which comprise algae ranging from Chlamydomonas (A), swimming single cells, to coenobia of undifferentiated cells, such as Gonium (B) and Pandorina (C), to Volvox sp. (E,F), where the surface is covered by thousands of flagellated somatic cells, while the interior contains a far smaller number of gonidia. The existence of these closely related species allows one to study some of the most basic questions in the evolution of multicellularity. What determines the length scale or the cell number at which germ-soma differentiation appears? How do simple multicellular (or colonial) organisms lacking a central nervous system achieve the apparently coordinated motion they exhibit? What developmental program leads to the long-range cellular orientation observed in the larger species? Our first two works in this area laid the groundwork for many of these issues by developing micromanipulation and fluid dynamical methods of studying the flagella-driven flows in these organisms, thereby establishing the high Péclet numbers found in these systems and showing through experiment and theory that such flows can remove what would otherwise be a diffusional bottleneck in nutrient uptake limiting viability of the larger species.

Tracking Protists in Three Dimensions

Many flagellated protists display swimming behavior that is inherently three dimensional, and a number of important questions in biology and physics are associated with how the motion of such organisms is related to their body plan and to external stimuli such as light, dissolved molecular species, gravity, temperature, boundaries, and electromagnetic fields. It is thus desirable to track their position and orientation in 3D with high spatiotemporal resolution and, unless desired, free from systematic bias introduced by external stimuli, background fluid motion, and hydrodynamic surface effects. We have developed such an apparatus, optimized for tracking swimming micro-organisms in the size range of 10-1000 microns, in three dimensions, far from surfaces, and with negligible background convective fluid motion. Charge coupled device cameras attached to two long working distance microscopes synchronously image the sample from two perpendicular directions, with narrow band dark-field or bright-field illumination chosen to avoid triggering a phototactic response. The images from the two cameras can be combined to yield 3D tracks of the organism. Using additional, highly directional broad-spectrum illumination with millisecond timing control the phototactic trajectories in 3D of organisms ranging from Chlamydomonas to Volvox can be studied in detail. Surface-mediated hydrodynamic interactions can also be investigated without convective interference. Minimal modifications to the apparatus allow for studies of chemotaxis and other taxes.

Hydrodynamic Bound States of Swimming Algae

Because of its bottom-heaviness, in the absence of phototactic cues Volvox swims upward against gravity. Using glass chambers we discovered that when nearby colonies reach the chamber ceiling they are attracted together and can form a stable bound state in which they "waltz" around each other. The attractive interaction was shown to be a surface-mediated effect associated with the density offset of the colonies relative to water (so that in the far-field each is described by a downward-pointing Stokeslet), in the presence of a no-slip wall. Quantitative agreement with experimental observations on infalling trajectories was achieved with no free parameters. Lubrication theory for the dynamics of nearby spinning, bottom-heavy colonies can be used to explain the orbiting dynamics of the bound states. A second "minueting" dynamics occurs with older colonies that hover near the lower chamber wall. These phenomena are suggested to underlie observed clustering of Volvox at surfaces.

Synchronization of Eukaryotic Flagella

The coordination of eukaryotic flagella is essential for many of the most basic processes of life (motility, sensing, and development), yet its emergence and regulation and its connection to locomotion are poorly understood. Previous studies show that the unicellular alga and Chlamydomonas reinhardtii, widely regarded as an ideal system in which to study flagellar biology, swims forward by the synchronous action of its two flagella. Using high-speed imaging over long intervals, we found a richer behavior: A cell swimming in the dark stochastically switches between synchronous and asynchronous flagellar beating. The synchronous state is interrupted stochastically by phase slips. The dynamics of slips and the statistics of phase-locked intervals are consistent with a low-dimensional stochastic model of hydrodynamically coupled oscillators, with a noise amplitude set by the intrinsic fluctuations of single flagellar beats. Three-dimensional tracking shows that the alternation between synchronous and asynchronous beating regimes leads, respectively, to nearly straight swimming and to abrupt large reorientations, which yield a eukaryotic version of the "run-and-tumble" motion of peritrichously flagellated bacteria.

Fidelity of Adaptive Phototaxis

Along the evolutionary path from single cells to multicellular organisms with a central nervous system are species of intermediate complexity that move in ways suggesting high-level coordination, yet have none. Instead, organisms of this type possess many autonomous cells endowed with programs that have evolved to achieve concerted responses to environmental stimuli. In our recent paper experiment and theory are used to develop a quantitative understanding of how cells of such organisms coordinate to achieve phototaxis, by using the colonial alga Volvox carteri as a model. It is shown that the surface somatic cells act as individuals but are orchestrated by their relative position in the spherical extracellular matrix and their common photoresponse function to achieve colony-level coordination. Analysis of models that range from the minimal to the biologically faithful shows that, because the flagellar beating displays an adaptive down-regulation in response to light, the colony needs to spin around its swimming direction and that the response kinetics and natural spinning frequency of the colony appear to be mutually tuned to give the maximum photoresponse. These models further predict that the phototactic ability decreases dramatically when the colony does not spin at its natural frequency, a result confirmed by phototaxis assays in which colony rotation was slowed by increasing the fluid viscosity.

Emergence of Synchronized Beating During the Regrowth of Eukaryotic Flagella

A fundamental issue in the biology of eukaryotic flagella is the origin of synchronized beating observed in tissues and organisms containing multiple flagella. Our recent studies of the biflagellate unicellular alga Chlamydomonas reinhardtii provided the first evidence that the interflagellar coupling responsible for synchronization is of hydrodynamic origin. To investigate this mechanism in detail we have studied synchronization in Chlamydomonas as its flagella slowly regrow after mechanically-induced self-scission. The duration of synchronized intervals is found to be strongly dependent on flagellar length. Analysis within a stochastic model of coupled phase oscillators is used to extract the length dependence of the interflagellar coupling and the intrinsic beat frequencies of the two flagella. Physical and biological considerations that may explain these results are proposed.

Anomalous Tracer Statistics in Active Suspensions

In contexts such as suspension feeding in marine ecologies there is an interplay between Brownian motion of nonmotile particles and their advection by flows from swimming microorganisms. One appealing point of view is that the sea of swimming organisms constitutes an effective "thermal bath" analogous to the multitudes of molecules responsible for Brownian motion, where each encounter of a tracer particle with a swimmer provides a random kick. In conventional Brownian motion, e.g., with micron-size particles in water, there is an enormous separation of time scales between the duration of molecular collisions (ps) and the observed particle motion (ms). In contrast, in a suspension of microorganisms it is possible to resolve the encounters with tracer particles, and the dynamical problem involves correlated advective trajectories in the presence of true Brownian noise. As a laboratory realization of this, we have studied passive tracers in suspensions of eukaryotic swimmers, the alga Chlamydomonas reinhardtii. While the cells behave ballistically over short intervals, the tracers behave diffusively, with a time-dependent but self-similar probability distribution function of displacements consisting of a Gaussian core and robust exponential tails. We emphasize the role of flagellar beating in creating oscillatory flows that exceed Brownian motion far from each swimmer.

Direct Measurements of the Flow Fields Around Swimming Microorganisms

Swimming microorganisms create flows that influence their mutual interactions and modify the rheology of their suspensions. While extensively studied theoretically, these flows have not been measured in detail around any freely-swimming microorganism. Using a synthesis of tracking microscopy, particle imaging velocimetry, and particle tracking velocimetry, we achieved such measurements for the microphytes Volvox carteri, and Chlamydomonas reinhardtii. The minute (~0.3%) density excess of V. carteri over water leads to a strongly dominant Stokeslet contribution, with the widely-assumed stresslet flow only a correction to the subleading source dipole term. This implies that suspensions of V. carteri have features similar to suspensions of sedimenting particles (see the next section for an explanation of the consequences of this). The flow in the region around C. reinhardtii where significant hydrodynamic interaction is likely to occur differs qualitatively from a puller stresslet, and can be described by a simple three-Stokeslet model.

Fluid Dynamics and Noise in Bacterial Cell-Cell and Cell-Surface Interactions

Bacterial processes ranging from gene expression to motility and biofilm formation are constantly challenged by internal and external noise. While the importance of stochastic fluctuations has been appreciated for chemotaxis, it is currently believed that deterministic long-range fluid dynamical effects govern cell-cell and cell-surface scattering, the elementary events that lead to swarming and collective swimming in active suspensions and to the formation of biofilms. We have succeeded in making the first direct measurements of the bacterial flow field generated by individual swimming Escherichia coli both far from and near to a solid surface. These experiments allowed us to examine the relative importance of fluid dynamics and rotational diffusion for bacteria. For cell-cell interactions it is shown that thermal and intrinsic stochasticity drown the effects of long-range fluid dynamics, implying that physical interactions between bacteria are determined by steric collisions and near-field lubrication forces. This dominance of short-range forces closely links collective motion in bacterial suspensions to self-organization in driven granular systems, assemblages of biofilaments, and animal flocks. For the scattering of bacteria with surfaces, long-range fluid dynamical interactions are also shown to be negligible before collisions; however, once the bacterium swims along the surface within a few microns after an aligning collision, hydrodynamic effects can contribute to the experimentally observed, long residence times. Because these results are based on purely mechanical properties, they apply to a wide range of microorganisms.

Fluid Velocity Fluctuations in a Suspension of Swimming Protists

In dilute suspensions of swimming microorganisms the local fluid velocity is a random superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions decaying as inverse powers of distance from the organism. We have studied the relationship between the decay exponent of the dominant multipole contribution and the statistics of velocity fluctuations, and derived the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocities are satisied. This holds when the leading force singularity is a Stokeslet, but not when it is any higher multipole. These results are confirmed by numerical studies and by experiments on suspensions of the alga Volvox carteri, which show that deviations from Gaussianity arise from near-field effects. Such observations are complementary to the observation (see above) of non-Gaussian finite-time particle displacement distribution functions in suspensions of C. reinhardtii.

Physics and Biology of Cytoplasmic Streaming

Since Bonaventura Corti's discovery in 1774 of the persistent circulation of the cytoplasm of plant cells, the phenomenon now known as cytoplasmic streaming or cyclosis has been conjectured to play an important role in metabolism. It occurs in organisms as diverse as amoebae, algae and terrestrial plants, and fungi. In plants it is driven by multitudes of the motor protein myosin moving along bundled actin at the boundary of the cytoplasm, carrying microscopic particles or organelles, and entraining fluid. The motion of protoplasmic granules entrained in the flow includes unidirectional streaming, "fountain streaming" (in which the motion near the central axis of the cell is opposite to that near the periphery), and spiral "rotational streaming." The fact that transport by fluid motion becomes necessary to outrun the slow pace of diffusion in larger organisms, as emphasized in the celebrated essay by Haldane on size in biology, has been a theme in discussions of cytoplasmic streaming for many years. Yet, there has been little theoretical work and fewer experiments that have quantified the full implications of cytoplasmic streaming for transport and mixing. We have embarked on a research program aimed at answering some of the most basic open questions in this field: What purpose does cytoplasmic streaming have in cells? How does it impact on homeostasis and development? What gives rise to the often complex flow geometries found in streaming?

Microfluidics of Cytoplasmic Streaming

In the more than two centuries since its discovery, streaming has frequently been conjectured to aid in transport and mixing of molecular species in the cytoplasm and, by implication, in cellular homeostasis, yet no theoretical analysis has been presented to quantify these processes. We show by solution of the coupled dynamics of fluid flow and diffusion appropriate to the archetypal "rotational streaming" of algal species such as Chara and Nitella that internal mixing and the transient dynamical response to changing external conditions can indeed be enhanced by streaming, but to an extent that depends strongly on the pitch of the helical flow. The possibility that this may have a developmental consequence is illustrated by the coincidence of the exponential growth phase of Nitella and the point of maximum enhancement of those processes. Key to these results is the discovery and analysis of a circulatory flow transverse to the cylinder's long axis, akin to Dean vortices at finite Reynolds numbers, which arises from the chiral geometry. Strongly enhanced lateral transport and longitudinal homogenization occur if the transverse Péclet number is sufficiently large, with scaling laws arising from boundary layers.

Measurement of Cytoplasmic Streaming by Magnetic Resonance Velocimetry

In the giant cylindrical cells found in Characean algae, multitudes of the molecular motor myosin transport the cytoplasm along opposing spiralling bands covering the inside of the cell wall, generating a helical shear flow in the large central vacuole. It has been suggested that such flows enhance mixing within the vacuole and thereby play a role in regulating metabolism. For this to occur the membrane that encloses the vacuole, namely the tonoplast, must transmit efficiently the hydrodynamic shear generated in the cytoplasm. Existing measurements of streaming flows are of insufficient spatial resolution and extent to provide tests of fluid mechanical theories of such flows and information on the shear transmission. We have used magnetic resonance velocimetry (MRV) to obtain the first measurements of cytoplasmic streaming velocities in single living cells. The spatial variation of the longitudinal velocity field in cross-sections of internodal cells of Chara corallina was obtained and shown to be in quantitative agreement with our theoretical analysis of rotational cytoplasmic streaming driven by bidirectional helical forcing in the cytoplasm, with direct shear transmission by the tonoplast.

Shear-Driven Circulation Patterns in Lipid Membrane Vesicles

Recent experiments have shown that when a hemispherical lipid vesicle attached to a solid surface is subjected to a simple shear flow it exhibits a pattern of membrane circulation much like a dipole vortex. This is in marked contrast to the toroidal circulation that would occur in the related problem of a drop of immiscible fluid attached to a surface and subjected to shear. This profound difference in floow patterns arises from the lateral incompressibility of the membrane, which restricts the observable flows to those in which the velocity field in the membrane is two-dimensionally divergence free, so there is no return flow to the bulk. We have studied these circulation patterns within the simplest model of membrane fluid dynamics. A systematic expansion of the flow field based on Papkovich-Neuber potentials is developed for general viscosity ratios between the membrane and the surrounding fluids. Comparison with experimental results [C. Vezy, G. Massiera, and A. Viallat, Soft Matter 3, 844 (2007)] is made, and it is shown how such studies can allow measurements of the membrane viscosity. Issues of symmetry-breaking and pattern selection are discussed.

Fluctuations, Dynamics, and the Stretch-Coil Transition of Single Actin Filaments in Extensional Flows

Recent work on the motion of elastic filaments subject to hydrodynamic forces has revealed complex nonlinear dynamics in the neighborhood of hyperbolic stagnation points in the flow. Unlike the simpler orbits of rigid elongated objects in the presence of shear and vorticity, these dynamics arise from the tension induced in the filament by an extensional flow, which beyond a critical value can induce an instability analogous to Euler buckling of a filament with thrust at its two ends. This predicted `stretch-coil' transition, which is complementary to the `coil-stretch' transition of exible polymers, has recently been observed with macroscopic fibers in cellular flows generated by electrodynamic forcing. Motivated by the role that semiflexible polymers subject to hydrodynamic forcing play in cytoskeletal motions in the cell, particularly when filaments guide molecular motors whose motions create flows, we have used a microfluidic cross-flow geometry to provide the first comprehensive study of the interplay between tension, fluctuations, and buckling of biopolymers, including a fluctuation-rounded stretch-coil transition of actin filaments.

Coupling of Active Motion and Advection Shapes Intracellular Cargo Transport

Intracellular cargo transport can arise from passive diffusion, active motor-driven transport along cytoskeletal filament networks, and passive advection by fluid flows entrained by such motor/cargo motion. Active and advective transport are thus intrinsically coupled as related, yet different representations of the same underlying network structure. We have used a reaction-advection-diffusion system to show that this coupling affects the transport and localization of a passive tracer in a confined geometry. For sufficiently low diffusion, cargo localization to a target zone is optimized either by low reaction kinetics and decoupling of bound and unbound states, or by a mostly disordered cytoskeletal network with only weak directional bias. These generic results may help to rationalize subtle features of cytoskeletal networks, for example as observed for microtubules in fly oocytes.

Cytoplasmic Streaming in Drosophila Oocytes Varies with Kinesin
Activity and Correlates with the Microtubule Cytoskeleton Architecture

Cells can localize molecules asymmetrically through the combined action of cytoplasmic streaming, which circulates their fluid contents, and specific anchoring mechanisms. Streaming also contributes to the distribution of nutrients and organelles such as chloroplasts in plants, the asymmetric position of the meiotic spindle in mammalian embryos, and the developmental potential of the zygote, yet little has been known quantitatively about the relationship between streaming and the motor activity which drives it. In a collaboration with Dr. Isabel Palacios and her student Dr. Lucy S. Williams (Zoology, Cambridge) we have used Particle Image Velocimetry (PIV) to quantify the statistical properties of Kinesin-dependent streaming during mid-oogenesis in Drosophila. We find that streaming can be used to detect subtle changes in Kinesin activity and that the flows reflect the architecture of the microtubule cytoskeleton. Furthermore, based on characterization of the rheology of the cytoplasm in vivo, we establish estimates of the number of Kinesins required to drive the observed streaming. Using this in vivo data as the basis of a model for transport, we suggest that the disordered character of transport at mid-oogenesis, as revealed by streaming, is an important component of the localization dynamics of the body plan determinant oskar mRNA.

Spontaneous Circulation in Confined Active Suspensions

Many active fluid systems encountered in biology are set in total geometric confinement. Cytoplasmic streaming in plant cells is a prominent and ubiquitous example, in which cargo-carrying molecular motors move along polymer filaments and generate cell-scale flow. When filaments are not fixed to the cell periphery, a situation found both in vivo and in vitro, we observe that the basic dynamics of streaming are closely related to those of a nonmotile stresslet suspension. Under this model, we have demonstrated that confinement makes possible a stable circulating state; a linear stability analysis reveals an activity threshold for spontaneous autocirculation. Numerical analysis of the longtime behavior reveals a phenomenon akin to defect separation in nematic liquid crystals and a high-activity bifurcation to an oscillatory regime. These observations provide a possible interpretation to the experiments of Yotsuyanagi from 1953, in which drops of cytoplasm extracted from the aquatic plant Chara were observed over time to undergo a transition from disordered random internal motion to organized persistent circulation.

Metachronal Waves on Volvox carteri

From unicellular ciliates to the respiratory epithelium, carpets of cilia display metachronal waves, long-wavelength phase modulations of the beating cycles. A large body of theoretical work over the past few decades has suggested that these waves may arise from hydrodynamic coupling between the beating flagella. Experimental study of this phenomenon has been limited by a lack of organisms for which the flagella and the flows they create can be visualized with ease. Using time-resolved particle image velocimetry and micropipette manipulation, we report the discovery of metachronal waves on the surface of the colonial alga Volvox carteri, whose large size and ease of growth and visualization make it an ideal model organism for these studies. The flagella of Volvox are relatively far apart compared to the celebrated ciliate Paramecium, and thus more nearly in the weak-coupling limit amenable to theory. V. carteri robustly displays symplectic metachronal waves, those for which the wave propagation direction is that of the power stroke of each flagellum. An elastohydrodynamic model of weakly coupled compliant oscillators, recast as interacting phase oscillators, reveals that orbit compliance can produce fast, robust synchronization in a manner essentially independent of boundary conditions, and offers an intuitive understanding of a possible mechanism leading to the emergence of metachronal waves.

III. Natural Pattern Formation

The Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles

A bundle of hair, whether a paintbrush or a ponytail, adopts a shape determined by the interplay of the stiffness and weight of the individual fibers and their intrinsic waviness or curliness. Since a typical bundle may have ten thousand individual hairs, each with a distinct profile of intrinsic curvatures, the determination of its shape is a problem in the statistical physics of disordered systems. In work with Patrick B. Warren and Robin C. Ball we have developed a variational theory for the properties of hair bundles. The theory is based on the local density and mean orientation of hairs, with the effects of random curvatures subsumed into a local energy functional. This decomposition reveals how the concept of an `equation of state' of hair appears naturally in the force balance that governs the equilibrium shape of the bundle. Specializing to long, narrow, axisymmetric bundles, an ansatz of self-similarity allows the many-body problem to be reduced to an equivalent single-fiber one for the envelope of the bundle. The interplay between filament stiffness and weight, and the pressure arising from random curvatures defines two main regimes of ponytail shapes. This reduction in turn allows the equation of state of hair to be determined from measurements of ponytail shapes. A robust image-processing method for determining the three-dimensional shapes of individual hairs and a procedure for extracting the envelope of a bundle are described in detail. Measurements of those properties for commercially-available hair switches are presented, and it is shown that the equation of state thus determined is essentially Hookean, with a spring constant determined by the bending elasticity and the spectrum of random curvatures of individual fibers.

A Ratchet Trap for Leidenfrost Droplets

When a Leidenfrost drop is levitated above a surface with parallel asymmetric sawtooth-shaped ridges it is known to be propelled in a unique direction by the interaction of the vapour layer with the surface. We have exploited this effect to construct a `ratchet trap' for Leidenfrost drops: a surface with concentric circular ridges, each asymmetric in cross section. A combination of experiment and theory is used to study the dynamics of drops in these traps, whose centre is a stable fixed point. Numerical analysis of the vapour flow over a ratchet surface suggests new insights into the mechanism of motion rectification that are incorporated into the simplest equations of motion for ratchet-driven motion of a Leidenfrost body; these resemble a central force problem in celestial mechanics with mass loss and drag. A phase plane analysis of experimental trajectories is used to extract more detailed information about the ratcheting phenomenon. Orbiting drops are found to exhibit substantial deformations; those with large internal angular momentum can even undergo binary fission. Such ratchet traps may thus prove useful in the controlled study of many properties of Leidenfrost drops.

Topological Transitions of Minimal Surfaces

It is well known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. We have studied this problem by experiment and theory and found that the process consists of a collapse of the film toward the boundary that produces a previously unrecognized finite-time twist singularity that changes the linking number of the film's Plateau border and the centerline of the wire. A movie of this fascinating phenomenon is here. We conjecture that it is a general feature of this type of transition that the singularity always occurs at the surface boundary. The change in linking number is shown to be a consequence of a viscous reconnection of the Plateau border at the moment of the singularity. High-speed imaging of the collapse dynamics of the film's throat, similar to that of the central opening of a catenoid, reveals a crossover between two power laws. Far from the singularity, it is suggested that the collapse is controlled by dissipation within the fluid film surrounding the wire, whereas closer to the transition the power law has the classical form arising from a balance between air inertia and surface tension. Analytical and numerical studies of minimal surfaces and ruled surfaces are used to gain insight into the energetics underlying the transition and the twisted geometry in the neighborhood of the singularity.

Coiling, Entrainment, and Synchronization of Viscous Fluid Jets

From algal suspensions to magma upwellings and galactic dynamics one finds fluid jets which exhibit complex symmetry-breaking instabilities as they are decelerated by their surroundings. We have studied the simplest system that captures this complexity yet allows direct experimental control; a saline jet descending through a salinity gradient. The descending jet coils like a corkscrew within a conduit of viscously entrained fluid whose upward recirculation braids the jet. The underlying jet structure and certain scaling relations can be understood through similarity solutions to the fluid equations and the physics of Kelvin-Helmholtz instabilities. The image shows vorticity maps (color) obtained from PIV with suspended microspheres that illustrate the recirculating flow within the conduit in a time-averaged manner (left) and instantaneously (right), and a streak photograph (b) offering another view of the swirling flows that braid the coiling jet. Panels (d,e) and (f,g) show from and side views of two interacting jets that are, respectively, sharing a single conduit and just further apart than a conduit width. In the latter case, the jets synchronize as mirror-images. Panel h reveals that when jets are even further apart they nest together. More recently we have done extensive numerical and analytical work on the underlying mechanism of the instability in a two-dimensional version of this problem.

Speleothem Morphology

The chemical mechanisms underlying the growth of cave formations such as stalactites are well-known, yet no theory had been proposed which successfully accounts for the dynamic evolution of their shapes. We have developed such a theory (see also a longer paper) that considers the interplay of thin-film fluid dynamics, calcium carbonate chemistry, and CO₂ transport in the cave to show that stalactites evolve according to a novel local geometric growth law which exhibits extreme amplification at the tip as a consequence of the locally-varying fluid layer thickness. Studies of this model show that a broad class of initial conditions is attracted to an ideal shape which is strikingly close to a statistical average of natural stalactites. This ideal shape has the remarkable property of being completely parameter-free, save for an overall scaling, as for the Platonic ideals like the circle and the square. The figure shows (left) three examples of natural stalactites in Kartchner Caverns (Benson, AZ) and the ideal shape which most closely matches them. At the right is a composite average shape (in blue, with statistical uncertainties in red) compared with our calculated Platonic ideal of stalactites (black). The agreement is striking. We are currently extending these ideas to understand non-axisymmetric shapes found in caves and the surface ripple instability commonly seen.

Icicle Morphology

In light of the results described above on stalactite growth it was natural to wonder how it is that icicles obtain their characteristic long and pointy shapes, and we have begun to investigate the growth and melting of icicles as free-boundary problems. Our first work describes perhaps the simplest synthesis of atmospheric heat transfer, geometrical considerations, and thin-film fluid dynamics. Crucial to this analysis is the power-law form of the thermal boundary layer rising upwards adjacent to the icicle surface. It acts as an insulating thermal blanket whose thickness self-consistently determines the icicle growth rate. This analysis led to the discovery of a nonlinear ordinary differential equation for the shape of a uniformly advancing icicle, the solution to which defines a parameter-free shape which compares very favorably with that of natural icicles. Away from the tip, the solution has a power-law form identical to that found for the growth of stalactites by precipitation of calcium carbonate. This analysis thereby explains why stalactites and icicles are so similar in form despite the vastly different physics and chemistry of their formation. In addition, a curious link is noted between the shape so calculated and that found through consideration of only the thin coating water layer.

More recently, we have studied in a more controlled fashion the melting of a cylinder of ice in warm air in order to quantify the heat-transfer mechanisms controlling the evolution of its shape, which are inherent in a range of phenomena involving phase change and fluid flow. Motivated by the initial melting at the top of a flat-topped cylinder of ice, we analyse laminar, natural convection above a cooled, finite, horizontal plate (or below a heated, finite, horizontal plate) and show that, to a very good approximation, the partial-differential, boundary-layer equations can be separated with self-similar vertical profiles scaled by the boundary-layer thickness. We find that the horizontal evolution of the boundary-layer thickness is governed by equations describing a steady, viscous gravity current fed by diffusive entrainment, and therefore describe such flows as diffusive gravity currents. We first use the predictions of our model to examine previous experimental results in two dimensions. Our experimental results relating to the melting of ice in air are then compared with predictions based on our analysis of the axisymmetric thermal boundary layer. This comparison confirms the vertical thermal structure and shows that melting is governed in roughly equal measure by heat transfer from the air, the latent heat of condensation of water vapour, and the net radiative heat transfer from the surroundings to the ice.