A filament of an incompressible highly viscous fluid that is supported at its ends sags under the influence of gravity. Its instantaneous shape resembles that of a catenary, but evolves with time. At short times, the shape is dominated by bending deformations. At intermediate times, the effects of stretching become dominant everywhere except near the clamping boundaries where bending boundary layers persist. Finally, the filament breaks off in finite time via strain localization and pinch-off.

When a cylindrical tool cuts through a thin sheet of a relatively brittle material, it leaves behind a visually arresting crack street in its wake, reminiscent of a vortex street in the wake of a cylinder moving through a fluid. We show that simple geometrical arguments based on the interplay of in-plane stretching and out-of-plane bending suffice to explain the cycloidal morphology of the curved crack. The coupling between geometry and dynamics also allows us to explain the ‘‘stick-slip’’-like behavior of tearing and suggests that these oscillations should occur generically in the brittle fracture of thin solid films.

The edge of a torn plastic sheet forms a complex three-dimensional fractal shape. We have found that the shape results from a simple elongation of the sheet in the direction along its edge. Natural growth processes in some leaves, flowers and vesicles could lead to a similar elongation and hence to the generation of characteristic wavy shapes. We used rectangular plastic sheets pulled from the sides (in the y-direction) to generate a steadily travelling crack (in the xdirection). The high stresses near the crack tip produce an irreversible plastic deformation of the sheet and, as they are relieved, the deformed sheet is free to relax and to adopt a new shape in space.

Insects provide examples of many cunning stratagems to cope with the challenges of living in a world dominated by surface forces. Despite being the current masters of the land environment, they are at constant risk of being entrapped in liquids, which they prevent by having waxy and hairy surfaces. The problem is particularly acute in an enclosed space, such as a plant gall. Using secreted wax to efficiently parcel and transport their own excrement, aphids were able to solve this problem 200 Myr ago. Here, we report on the physical and physiological significance of this ingenious solution. The secreted powdery wax has three distinct roles: (i) it is hydrophobic, (ii) it creates a microscopically rough inner gall surface made of weakly compacted wax needles making the gall ultra-hydrophobic, and (iii) it coats the honeydew droplets converting them into liquid marbles, that can be rapidly and efficiently moved.

We consider the statics of compound droplets made of two immiscible fluids on a rigid substrate, in the limit when gravity is dominated by capillarity. In particular, we show that the merging of four phases along a single contact line is a persistent and robust phenomenon from a mechanical and thermodynamic perspective; it can and does occur for a range of interfacial energies and droplet volumes. We give an interpretation for this in the context of the macroscopic Young–Laplace law and its microscopic counterpart due to van der Waals, and show that the topological transitions that result can be of either a continuous or discontinuous type depending on the interfacial energies in question.

We study the formation of shocks on the surface of a granular material draining through an orifice at the bottom of a quasi-two-dimensional silo. At high flow rates, the surface is observed to deviate strongly from a smooth linear inclined profile, giving way to a sharp discontinuity in the height of the surface near the bottom of the incline, the typical response of a choking flow such as encountered in a hydraulic jump in a Newtonian fluid like water. We present experimental results that characterize the conditions for the existence of such a jump, describe its structure and give an explanation for its occurrence.

Much to the consternation of adults, a broken mercury thermometer is a source of delight to a curious child absorbed by the spectacle of the silvery beads that elude capture. The epithet ‘quicksilver’ aptly describes the droplets, which seem to roll rapidly on the surface much like a solid marble.This unusual behaviour elicits a host of questions that are relevant to scientists in diverse fields: from chemists and engineers interested in the dynamics of drops, to astronomers interested in the stability of rotating stars and planets. How and when can liquid droplets actually roll on a surface? How fast can they move? Can they be controlled? And to what end?

When a bubble of air rises to the top of a highly viscous liquid, it forms a dome-shaped protuberance on the free surface. Unlike a soap bubble, it bursts so slowly as to collapse under its own weight simultaneously, and folds into a wavy structure. This rippling effect occurs for both elastic and viscous sheets, and a theory for its onset is formulated. The growth of the corrugation is governed by the competition between gravitational and bending (shearing) forces and is exhibited for a range of densities, stiffnesses (viscosities), and sizes—a result that arises less from dynamics than from geometry, suggesting a wide validity. A quantitative expression for the number of ripples is presented, together with experimental results that support the theoretical predictions.

We propose a simple model for the chaotic dripping of a faucet in terms of a return map constructed by analyzing the stability of a pendant drop. The return map couples two classical normal forms, an Andronov saddle-node bifurcation, and a Shilnikov homoclinic bifurcation. The former corresponds to the initiation of the instability when the drop volume exceed a critical value set by the balance between surface tension and gravity, while the latter models the global reinjection associated with pinch-off that eventually return the drop to a state close to its original unstable configuration. The results obtained using the return map are consistent with those of numerical simulations of the governing PDEs and prior experiments, and show periodic and quasi-periodic dripping at low and high flow rates, and chaotic behavior at intermediate flow rates.

We propose a simple model for the chaotic dripping of a faucet in terms of a return map constructed by analyzing the stability of a pendant drop. The return map couples two classical normal forms, an Andronov saddle-node bifurcation, and a Shilnikov homoclinic bifurcation. The former corresponds to the initiation of the instability when the drop volume exceed a critical value set by the balance between surface tension and gravity, while the latter models the global reinjection associated with pinch-off that eventually return the drop to a state close to its original unstable configuration. The results obtained using the return map are consistent with those of numerical simulations of the governing PDEs and prior experiments, and show periodic and quasi-periodic dripping at low and high flow rates, and chaotic behavior at intermediate flow rates.

We consider the nonlinear folding behavior of a viscous filament or a sheet under the influence of an external force such as gravity. Everyday examples of this phenomenon are provided by the periodic folding of a sheet of honey as it impinges on toast, or the folding of a stream of shampoo as it falls on one’s hand. To understand the evolution of a fold, we formulate and solve a free-boundary problem for the phenomenon, give scaling laws for the size of the folds and the frequency with which they are laid out, and verify these experimentally

A crumpled piece of paper is made up of cylindrically curved or nearly planar regions folded along line-like ridges, which themselves pivot about point-like peaks; most of the deformation and energy is focused into these localized objects. Localization of deformation in thin sheets is a diverse phenomenon1±6, and is a consequence of the fact7 that bending a thin sheet is energetically more favourable than stretching it. Previous studies8±11 considered the weakly nonlinear response of peaks and ridges to deformation. Here we report a quantitative description of the shape, response and stability of conical dislocations, the simplest type of topological crumpling deformation. The dislocation consists of a stretched core, in which some of the energy resides, and a peripheral region dominated by bending. We derive scaling laws for the size of the core, characterize the geometry of the dislocation away from the core, and analyse the interaction between two conical dislocations in a simple geometry. Our results show that the initial stages of crumpling (characterized by the large deformation of a few folds) are dominated by bending only. By considering the response of a transversely forced conical dislocation, we show that it is dynamically unstable above a critical load threshold. A similar instability is found for the case of two interacting dislocations, suggesting that a cascade of related instabilities is responsible for the focusing of energy to progressively smaller scales during crumpling.

When a rigid circular cylinder or sphere is placed on a rough inclined plane it will roll down the plane. When the experiment is repeated with a rigid cube it will slide down the plane. If the object is deformable a variety of motions become possible; the motion of elastic bodies and fluid drops depends on the interfacial energies of the materials, the roughness of the interfaces, the size of the objects, etc. This is because a deformable body maintains contact with the surface over a finite area. For a viscous fluid droplet, two possible motions may ensue. If the droplet partially wets the surface it slides along it, while if the droplet is nonwetting, it can roll on the surface, much like an elastic body when viewed from the exterior. Here we consider the motion of a small nonwetting droplet forced by a weak gravitational field. A classic example of this motion is exhibited by a droplet of mercury on an inclined plane and is probably the origin of the name quicksilver, after the Latin Argentum Vivum for the swiftly moving droplet of the silvery liquid.

We use the theory of elasticity to compute the shape of the DNA loop bridging the gap in the crystal structure of the lac repressor-DNA complex. The Kirchhoff system of equations with boundary conditions derived from the crystal structure is solved using a continuation method. This approach can be applied effectively to find coarse-grained conformational minima of DNA loops.

The flow of granular matter such as sand is often characterized by the motion of a thin superficial layer near the free surface, while the bulk of the solid remains immobile. A pair of equations called the BCRE equations (Bouchaud J-P., Cates M. E., Ravi Prakash J. and Edwards S. F. J. Phys. 4 (1994) 1383) have been proposed to model these flows and account for the dynamic exchange of mass between moving and stationary grains using the simplest kinematic considerations. We uncover a new conservation law for the BCRE equations and its variants that unifies a variety of recent special solutions and show that these equations support simple waves, and are capable of finite time singularities that correspond to propagating erosion fronts.

We investigate the axial instability of the free-surface front of a viscous fluid in a horizontal cylinder rotating about its longitudinal axis. A simplified model equation for the evolution of the free surface is derived and includes the effects of gravity, capillarity, inertia, and viscosity. This equation is solved numerically to determine the base state with no axial variation, and a numerical linear stability analysis is carried out to examine the onset of unstable axial modes. Various computational results are presented for the wavelength of the axial instability. Inertia is found to play an important role in the onset of the instability and the wavelength of the instability l satisfies the power law l;g1/3, where g is surface tension. Finally some numerical simulations of the simplified evolution equation are presented to show that they can capture the steady shark-teeth patterns observed in recent experiments @R. E. Johnson, in Engineering Science, Fluid Dynamics: A Symposium to Honor T. Y. Wu ~World Scientific, Singapore, 1990!, pp. 435–449; S. T. Thoroddsen and L. Mahadevan, ‘‘Experimental studies of the instabilities in a partially filled horizontal rotating cylinder,’’ Exp. Fluids 23, 1 ~1997!#.

Buckling instabilities can arise from competition between axial compression and bending in slender objects. These are not restricted to solids, but also occur with fluids with free surfaces1–4, in geophysics5 and in materials processing6 . Here we consider a classic demonstration of fluid buckling7 . When honey is poured from a sufficient height, it approaches one’s toast as a thin filament which whirls steadily around the vertical forming a regular helical coil (illustrated with silicone oil in Fig. 1), a behaviour reminiscent of the coiling of a falling flexible rope8 . We derive a scaling law that predicts the coiling frequency in terms of the filament radius and the flow rate.

The purpose of this Letters section is to provide rapid dissemination of important new results in the fields regularly covered by Physics of Fluids. Results of extended research should not be presented as a series of letters in place of comprehensive articles. Letters cannot exceed three printed pages in length, including space allowed for title, figures, tables, references and an abstract limited to about 100 words. There is a three-month time limit, from date of receipt to acceptance, for processing Letter manuscripts. Authors must also submit a brief statement justifying rapid publication in the Letters section.

We analyze the geometry and elasticity of the crescentlike singularity on a crumpled elastic sheet. We give a physical realization of this in terms of a free-boundary contact problem. An analytical solution is given for the universal shape of a developable cone that characterizes the singularity far from the tip, and some of its predictions are qualitatively verified experimentally. We also give a scaling relation for the core size, defined as the region close to the tip of the cone where the sheet is not developable.

We describe a number of different phenomena seen in the free-surface flow inside a partially filled circular cylinder which is rotated about its horizontal axis of symmetry. At low angular velocities the flow settles into a steady two-dimensional flow with a front where the coating film coalesces with the pool at the bottom of the cylinder. This mode becomes unstable at higher angular velocities, initially to a sloshing mode on the rising side of the coating film and then to an axial instability on the front. The undulations that appear on the front grow into large-amplitude stationary patterns with cusp-like features for some parameter values. At still higher angular velocities and volume fractions, a number of different inertial instabilities and patterns appear. We present a phase diagram of the various transitions and characterize some of the more prominent instabilities and patterns in detail, along with some possible mechanisms for the observed behaviour.

We analyze the transition from annihilation to preservation of colliding waves. The analysis exploits the similarity between the local and global phase portraits of the system. The transition is shown to be the infinite-dimensional analog of the creation and annihilation of limit cycles in the plane via a homoclinic Andronov bifurcation, and has parallels to the nucleation theory of first-order phase transitions

When a thin sheet of a flexible material such as paper is fed from a horizontal spool towards a rough horizontal plane below it, the sheet folds on itself in a regular manner. We model this phenomenon as a free boundary problem for a nonlinearly elastic sheet, taking into account the stiffness and weight of the sheet and the height of the spool above the plane. By using a continuation scheme we solve the problem numerically and follow the evolution of one period of the fold for various values of the parameters. The results are found to agree well with observations of the folding of paper sheets.