1. Scaling of F-actin rheology to probe single filament elasticity and dynamics, Gardel, M., J. Shin, F. Mackintosh, L. Mahadevan, P. Matsudaira and D. Weitz,  Physical Review Letters , 93, 188102, 2004.
    [View PDF] [Download PDF] The linear and nonlinear viscoelastic response of networks of cross-linked and bundled cytoskeletal filaments demonstrates remarkable scaling with both frequency and applied prestress, which helps elucidate the origins of the viscoelasticity. The frequency dependence of the shear modulus reflects the underlying single-filament relaxation dynamics for 0:1–10 rad=sec. Moreover, the nonlinear strain stiffening of such networks exhibits a universal form as a function of prestress; this is quantitatively explained by the full force-extension relation of single semiflexible filaments.
  2. Peeling from a patterned thin elastic film,  Ghatak, A., L. Mahadevan, J. Yun, M. Chaudhury and V. Shenoy,  Proceedings of the Royal Society of London (A) , 460, 2725-35 (2004).
    [View PDF] [Download PDF] Inspired by the observation that many naturally occurring adhesives arise as textured thin films, we consider the displacement-controlled peeling of a flexible plate from an incision-patterned thin adhesive elastic layer. We find that crack initiation from an incision on the film occurs at a load much higher than that required to propagate it on a smooth adhesive surface; multiple incisions thus cause the crack to propagate intermittently. Microscopically, this mode of crack initiation and propagation in geometrically confined thin adhesive films is related to the nucleation of cavitation bubbles behind the incision which must grow and coalesce before a viable crack propagates. Our theoretical analysis allows us to rationalize these experimental observations qualitatively and quantitatively and suggests a simple design criterion for increasing the interfacial fracture toughness of adhesive films
  3. Elastic behavior of cross-linked and bundled actin networks,  Gardel, M., J. Shin, F. Mackintosh, L. Mahadevan, P. Matsudaira and D. Weitz,  Science,  304, 1301-5, 2004.
    [View PDF] [Download PDF] Networks of cross-linked and bundled actin filaments are ubiquitous in the cellular cytoskeleton, but their elasticity remains poorly understood. We show that these networks exhibit exceptional elastic behavior that reflects the mechanical properties of individual filaments. There are two distinct regimes of elasticity, one reflectingbendingof single filaments and a second reflectingstretchingof entropic fluctuations of filament length. The mechanical stiffness can vary by several decades with small changes in cross-link concentration, and can increase markedly upon application of external stress. We parameterize the full range of behavior in a state diagram and elucidate its origin with a robust model.
  4. Capillarity-induced zippering of a flexible train floating on an air-water interface, Vella, D., H-Y. Kim and L. Mahadevan,  Journal of Fluid Mechanics , 502, 89-98, 2004.
    [View PDF] [Download PDF] We consider the dynamics of capillary attraction between an articulated train of rigid rods floating at a liquid–gas interface and a nearby wall. We then explain some of the phenomena that are a result of the strong anisotropy and the extended nature of the system, such as the lining up next to the walling in a ‘zippering’ motion that is observed and compare our results qualitatively with those of experiments.
  5. Bending stiffness of a crystalline actin bundle,  Shin, J., L. Mahadevan, P.T. So and P. Matsudaira,  Journal of Molecular Biology , 337, 255-61, 2004.
    [View PDF] [Download PDF] The acrosomal process of the sperm of the horseshoe crab (Limulus polyphemus) is a unique crystalline actin bundle, consisting of multiple actin filaments cross-linked by the actin-bundling protein, scruin. For successful fertilization, the acrosomal bundle must penetrate through a 30 mm thick jelly coat surrounding the egg and thus it must be sufficiently stiff. Here, we present two measurements of the bending stiffness of a single crystalline bundle of actin. Results from these measurements indicate that the actin:scruin composite bundle has an average elastic modulus of 2 GPa, which is similar to that of a single actin filament, and a bending stiffness that is more than two orders of magnitude larger than that of a bundle of uncross-linked actin filaments due to stiffening by the scruin matrix.
  6. Modeling DNA loops using continuum and statistical mechanics Balaeff, A., C. Koudella, L. Mahadevan and K. Schulten,  Philosophical Transactions of the Royal Society of London (A) , 362, 1355-71, 2004.
    [View PDF] [Download PDF] The classical Kirchhoff elastic-rod model applied to DNA is extended to account for sequence-dependent intrinsic twist and curvature, anisotropic bending rigidity, electrostatic force interactions, and overdamped Brownian motion in a solvent. The zero-temperature equilibrium rod model is then applied to study the structural basis of the function of the lac repressor protein in the lac operon of Escherichia coli. The structure of a DNA loop induced by the clamping of two distant DNA operator sites by lac repressor is investigated and the optimal geometries for the loop of length 76 bp are predicted. Further, the mimicked binding of catabolite gene activator protein (CAP) inside the loop provides solutions that might explain the experimentally observed synergy in DNA binding between the two proteins. Finally, a combined Monte Carlo and Brownian dynamics solver for a worm-like chain model is described and a preliminary analysis of DNA loop-formation kinetics is presented.
  7. Biomimetic ratcheting motion of lubricated hydrogel filaments, Mahadevan, L., S. Daniel and M. Chaudhury,  Proceedings of the National Academy of Sciences (USA) , 101, 23-26, 2004.
    [View PDF] [Download PDF] Inspired by the locomotion of terrestrial limbless animals, we study the motion of a lubricated rod of a hydrogel on a soft substrate. We show that it is possible to mimic observed biological gaits by vibrating the substrate and by using a variety of mechanisms to break longitudinal and lateral symmetry. Our simple theory and experiments provide a unified view of the creeping, undulating, and inchworming gaits observed in limbless locomotion on land, all of which originate as symmetry-breaking bifurcations of a simple base state associated with periodic longitudinal oscillations of a slender gel. These ideas are therefore also applicable to technological situations that involve moving small, soft solids on substrates.
  8. Elements of Draping Cerda, E., L. Mahadevan and J. Passini,  Proceedings of the National Academy of Sciences (USA) , 101 (7), 1806-10, 2004.
    [View PDF] [Download PDF] We consider the gravity-induced draping of a 3D object with a naturally flat, isotropic elastic sheet. As the size of the sheet increases, we observe the appearance of new folded structures of increasing complexity that arise because of the competition between elasticity and gravity. We analyze some of the simpler 3D structures by determining their shape and analyzing their response and stability and show that these structures can easily switch between a number of metastable configurations. For more complex draperies, we derive scaling laws for the appearance and disappearance of new length scales. Our results are consistent with commonplace observations of drapes and complement large-scale computations of draping by providing benchmarks. They also yield a qualitative guide to fashion design and virtual reality animation.
  9. Relating microstructure to rheology of a bundled and cross-linked F-actin network in-vitro Shin, J., M. Gardel, L. Mahadevan, P. Matsudaira and D. Weitz,  Proceedings of the National Academy of Sciences (USA) , 101(26), 9636-41 (2004).
    [View PDF] [Download PDF] The organization of individual actin filaments into higher-order structures is controlled by actin-binding proteins (ABPs). Although the biological significance of the ABPs is well documented, little is known about how bundling and cross-linking quantitatively affect the microstructure and mechanical properties of actin networks. Here we quantify the effect of the ABP scruin on actin networks by using imaging techniques, cosedimentation assays, multiparticle tracking, and bulk rheology. We show how the structure of the actin network is modified as the scruin concentration is varied, and we correlate these structural changes to variations in the resultant network elasticity.
  10. Elasticity of interfacial particle rafts, Vella, D., P. Aussillous and L. Mahadevan,  Europhysics Letters , 68 (2), 212-18, 2004.
    [View PDF] [Download PDF] We study the collective behaviour of a close-packed monolayer of non-Brownian particles at a fluid-liquid interface. Such a particle raft forms a two-dimensional elastic solid and can support anisotropic stresses and strains, e.g. it buckles in uniaxial compression and cracks in tension. We characterise this solid in terms of Young’s modulus and Poisson ratio derived from simple theoretical considerations and show the validity of these estimates by using an experimental buckling assay to deduce Young’s modulus.
  11. Popliteal instability of bent multi-walled elastic tubes, Mahadevan, L., J. Bico and G. McKinley,  Europhysics Letters , 65 (3), 323-29, 2004.
    [DOI] [View PDF] [Download PDF] Soft slender structures are ubiquitous in natural and artificial systems, in active and passive settings and across scales, from polymers and flagella, to snakes and space tethers. In this paper, we demonstrate the use of a simple and practical numerical implementation based on the Cosserat rod model to simulate the dynamics of filaments that can bend, twist, stretch and shear while interacting with complex environments via muscular activity, surface contact, friction and hydrodynamics. We validate our simulations by solving a number of forward problems involving the mechanics of passive filaments and comparing them with known analytical results, and extend them to study instabilities in stretched and twisted filaments that form solenoidal and plectonemic structures. We then study active filaments such as snakes and other slender organisms by solving inverse problems to identify optimal gaits for limbless locomotion on solid surfaces and in bulk liquids.
  12. Multiscale methods for modeling protein-DNA complexes, Villa, E. , Balaeff, A., L. Mahadevan and K. Schulten,  SIAM Multiscale Modeling and Simulation , 2, 527-553 (2004).
    [View PDF] [Download PDF] We present a multiresolution approach to modeling complexes between protein and DNA that contain looped or coiled DNA. The approach combines a coarse-grained model of the DNA loop, based on the classical theory of elasticity, with an atom level model of proteins and proteinDNA interfaces based on molecular dynamics. The coarse-grained DNA description is controlled through the atom level protein description and vice versa. The feasibility of the resulting multiscale modeling approach is demonstrated for a protein-DNA complex in which a protein called the E. coli lac repressor forces DNA into a 76 base pair loop. The required simulation involves 230,000 atoms, a number that would triple if both protein and DNA loops were described at the atomic level.
  13. Structural model for cooperative DNA binding by CAP and Lac repressor, Balaeff, A., L. Mahadevan and K. Schulten,  Structure , 12, 123-32, 2004.
    [DOI] [View PDF] [Download PDF] Thin adhesive pads used to attach objects to each other often fail catastrophically. Here we consider the nature of failure of such a pad under loading parallel to the adhesive substrate. To determine the modes of failure of the pad and to understand what limits its load bearing capacity, we conduct experiments with finite pads composed of a soft adhesive layer with a stiff backing and load them parallel to the surface of adhesion. We find that two different peeling mechanisms emerge as a function of the slenderness of the adhesive pad: an interfacial peeling mechanism that starts close to the pulling end for very long pads, and an unstable curling mechanism that starts at the opposite end for relatively short pads. A minimal theoretical framework allows us to explain our observations and reveals the adhesive bond stiffness as a dominant parameter in defining the peeling mode. A phase diagram that delineates the different regimes of peeling modes brings our experiments and theory together. Our results suggest that unstable peeling by curling may be more common than previously thought, and could perhaps occur naturally in such examples as the gecko foot.
  14. Dynamics of poroelastic filaments,
    Skotheim, J. and L. Mahadevan,  Proceedings of the Royal Society of London (A) , 460, 1995-2020 (2004).  
    [View PDF] [Download PDF]
  15. Stored elastic energy powers the 60-micron extension of the Limulus polyphemus sperm actin bundle, Shin, J., L. Mahadevan, G. Waller, K. Langsmo and P. Matsudaira,  Journal of Cell Biology , 162(7), 1183-88, 2003.
    [View PDF] [Download PDF] During the 5 s of the acrosome reaction of Limulus polyphemus sperm, a 60-m-long bundle of scruin-decorated actin filaments straightens from a coiled conformation and extends from the cell. To identify the motive force for this movement, we examined the possible sources of chemical and mechanical energy and show that the coil releases 1013 J of stored mechanical D strain energy, whereas chemical energy derived from calcium binding is 1015 J. These measurements indicate that the coiled actin bundle extends by a spring-based mechanism, which is distinctly different from the better known polymerization or myosin-driven processes, and that calcium initiates but does not power the reaction.
  16. Rings, rackets and kinks in filamentous assemblies, Cohen, A. and L. Mahadevan,  Proceedings of the National Academy of Sciences (USA) , 100, 12141-46, 2003.
    [View PDF] [Download PDF] Carbon nanotubes and biological filaments each spontaneously assemble into kinked helices, rings, and ‘‘tennis racket’’ shapes due to competition between elastic and interfacial effects. We show that the slender geometry is a more important determinant of the morphology than any molecular details. Our mesoscopic continuum theory is capable of quantifying observations of these structures and is suggestive of their occurrence in other filamentous assemblies as well
  17. The force-velocity relationship for the actin-based motility of Listeria-Monocytogenes, McGrath, J., J. Eungdamrong, C. Fisher, F. Peng, L. Mahadevan, T. Mitchison and S. Kuo,  Current Biology , 13 (1-20), 1-6, 2003.
    [View PDF] [Download PDF]
  18. Geometry and physics of wrinkling,  Cerda, E. and L. Mahadevan,  Physical Review Letters , 90 (7) 074302, 2003 (Physical Review Focus Article).
    [View PDF] [Download PDF] The wrinkling of thin elastic sheets occurs over a range of length scales, from the fine scale patterns in substrates on which cells crawl to the coarse wrinkles seen in clothes. Motivated by the wrinkling of a stretched elastic sheet, we deduce a general theory of wrinkling, valid far from the onset of the instability, using elementary geometry and the physics of bending and stretching. Our main result is a set of simple scaling laws; the wavelength of the wrinkles   K1=4, where K is the stiffness due to an ‘‘elastic substrate’’ effect with a multitude of origins, and the amplitude of the wrinkle A  . These could form the basis of a highly sensitive quantitative wrinkling assay for the mechanical characterization of thin solid membranes.
  19. The viscous catenary Teichman, J. and L. Mahadevan,  Journal of Fluid Mechanics , 478, pp. 71-80, 2003.
    [View PDF] [Download PDF] 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.
  20. Crack street: the cycloidal wake of a cylinder ripping through a thin solid sheet,  Ghatak, A. and L. Mahadevan,  Physical Review Letters . 91, 215507, 2003.
    [View PDF] [Download PDF] 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.
  21. Wrinkling of a stretched elastic sheet, Cerda, E., K. Ravi-Chandar and L. Mahadevan,  Nature , 419, 579, 2002.
    [View PDF] [Download PDF] 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.
  22. How aphids lose their marbles, Pike, N., D. Richard, W. Foster and L. Mahadevan,  Proceedings of the Royal Society of London, Series (B), Biological Sciences , 269, 1211-15, 2002.
    [View PDF] [Download PDF] 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.
  23. Four-phase merging in compound drops, Mahadevan, L., M. Adda Bedia and Y. Pomeau,  Journal of Fluid Mechanics , 451, pp. 411-20, 2002.
    [View PDF] [Download PDF] 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.
  24. Shocks in sand flowing in a silo, Samadani, A., L. Mahadevan and A. Kudrolli,  Journal of Fluid Mechanics , 452, 293-301, 2002.
    [View PDF] [Download PDF] 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.
  25. Non-stick water,  Mahadevan, L.,  Nature , 411, 895-96, 2001.
    [View PDF] [Download PDF] 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?
  26. Rippling instability of a collapsing Bubble,  da Silveira, R., S. Chaieb and L. Mahadevan,  Science , 287, 1468-71, 2000.
    [View PDF] [Download PDF] 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.
  27. Motility driven by macromolecular springs and ratchets, Mahadevan, L. and P. Matsudaira,  Science , 288, 95-99, 2000.
    [View PDF] [Download PDF] 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.
  28. Chaotic dripping from a faucet, Coullet, P., L. Mahadevan and C. Riera,  Progress in Theoretical Physics Supplement , 139, 507-516, 2000.
    [View PDF] [Download PDF] 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.
  29. Folding of viscous filaments and sheets Skorobogatiy, M., and L. Mahadevan,  Europhysics Letters , 52, 532-38, 2000.
    [View PDF] [Download PDF] 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
  30. Conical dislocations in crumpling Cerda, E., S. Chaieb, F. Melo and L. Mahadevan,  Nature , 401, 46-49, 1999.
    [View PDF] [Download PDF] 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.
  31. Rolling droplets Mahadevan, L., and Y. Pomeau,  Physics of Fluids , 11, 2449-53, 1999.
    [View PDF] [Download PDF] 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.
  32. Elastic model of a DNA loop in the lac operon,  Balaeff, A., L. Mahadevan, and K. Schulten,  Physical Review Letters , 83, 4900-03, 1999.
    [View PDF] [Download PDF] 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.
  33. Propagating fronts on sandpile surfaces, Mahadevan, L. and Y. Pomeau,  Europhysics Letters , 46, 595-601, 1999.
    [View PDF] [Download PDF] 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.
  34. Axial instability of a free-surface front in a partially-filled horizontal rotating cylinder,  Hosoi, A.E., and L. Mahadevan,  Physics of Fluids , 11, 97-106, 1999.
    [View PDF] [Download PDF] 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!#.
  35. Fluid rope trick investigated Mahadevan, L., W. Ryu, and A.D.T. Samuel,  Nature, 391, 140, 1998. Corrigendum; ibid., 403, 502, 2000.
    [View PDF] [Download PDF] 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.
  36.  Tumbling cards Mahadevan, L., W. Ryu, and A.D.T. Samuel,  Physics of Fluids , 11, 1-3, 1999.
    [View PDF] [Download PDF] 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.
  37. Conical surfaces and crescent singularities in crumpled sheets,  Cerda, E., and L. Mahadevan,  Physical Review Letters , 80, 2358-61, 1998.
    [View PDF] [Download PDF] 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.
  38. Experimental study of instabilities in a partially-filled horizontally-rotating cylinder, Thoroddsen, S.T., and L. Mahadevan,  Experiments in Fluids , 23, 1-13, 1997.
    [View PDF] [Download PDF] 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.
  39. Colliding waves in an excitable medium: preservation, annihilation and bifurcation, Argentina, M., P. Coullet, and L. Mahadevan,  Physical Review Letters , 79, 2803-07, 1997.
    [View PDF] [Download PDF] 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
  40. Coiling of flexible ropes Mahadevan, L., and J.B. Keller,  Proceedings of the Royal Society of London, Series A, 1452, 1679-1694, 1996.
    [View PDF] [Download PDF]
  41. Tumbling of a falling card Mahadevan, L. Comptes Rendus de l’Academie des Sciences, Paris, Series II , 323, 729-736, 1996.
  42. Shark-teeth patterns in coating flow inside a horizontally-rotating cylinder Thoroddsen, S.T., and L. Mahadevan,  Physics of Fluids , 8(9), S10, 1996.
    [View PDF] [Download PDF]
  43. Comment on "Behavior of a falling paper," Mahadevan, L., H. Aref, and S.W. Jones,  Physical Review Letters , 75 , 1420, 1995.
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  44. Periodic folding of thin sheets Mahadevan, L., and J.B. Keller,  SIAM Journal on Applied Mathematics , 55(6) , 1609-1624, 1995.
    [View PDF] [Download PDF] 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.
  45. The shape of a Möbius band Mahadevan, L., and J.B. Keller,  Proceedings of the Royal Society of London, Series A , 1440, 149-162, 1993.
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