The behavior of matter at the mesoscopic/macroscopic scale is a theme of central interest, particularly in understanding how matter is shaped and how it flows. This leads naturally to questions of the organization and self-organization of matter in space and time as manifest in the rich range of patterns that surround us, from the ripples seen on the surface of a moving liquid to the dynamics of drapery, from the settling of yogurt under its weight to the cracking of drying mud, from the mechanics of plate subduction to the flow of sand on a beach. We use a combination of experiential, experimental, analytical and computational approaches to study these sometimes frivolous and sometimes serious (the difference is often not clear a priori !) questions with the aim (although it is only rarely realized!) of stripping the complexity of the underlying phenomenon to its minimal essence. Indeed a goal is thus get at a qualitative understanding using quantitative methods.
How can one not be fascinated by the flow of fluids ? From the dripping faucet to meandering rivers, from a viscous thread of honey to the flow of blood, fluids are in us and around us, at every imaginable scale. Our interest in flow is deliberately unfocused: we let interesting unexplained phenomena lead us to new vistas. For example, we have explored a deep analogy between creeping flows and linear elasticity, but accounting for geometric nonlinearities, to explain such oddities as the coiling and folding of viscous fluids (which are relevant to food and materials processing on the one hand, and geological phenomena on the other). We have also explored the dynamics of the dripping faucet in some detail with the goal of building a simple, but rationally derived qualitative model for the chaotic transitions that are known to exist in such systems. Our current interests in hydrodynamics have started to move towards understanding how we might control complex flows, particularly those with free boundaries in such instances as acoustics.
Hydro-dynamical models for the dripping faucet,
P. Coullet, L. Mahadevan, and C. Riera, Journal of Fluid Mechanics, 526, 1, 2005.
Abstract: We give a hydrodynamical explanation for the chaotic behaviour of a dripping faucet using the results of the stability analysis of a static pendant drop and a proper orthogonal decomposition (POD) of the complete dynamics. We find that the only relevant modes are the two classical normal forms associated with a Saddle-Node-Andronov bifurcation and a Shilnikov homoclinic bifurcation. This allows us to construct a hierarchy of reduced order models including maps and ordinary differential equations which are able to qualitatively explain prior experiments and numerical simulations of the governing partial differential equations and provide an explanation for the complexity in dripping. We also provide a new mechanical analogue for the dripping faucet and a simple rationale for the transition from dripping to jetting modes in the flow from a faucet.
Some other representative publications:
Folding of viscous filaments and sheets,
Skorobogatiy, M., and L. Mahadevan, Europhysics Letters, 52, 532-38, 2000.
Rippling of a collapsing bubble,
R. da Silveira, S. Chaieb, and L. Mahadevan, Science, 287, 1468, 2000.
Peeling, healing and bursting in lubricated elastic sheets,
Hosoi, A. and L. Mahadevan, Physical Review Letters, 93, 137802, 2004.
Soft lubrication: the elastohydrodynamics of non-conforming and conforming contacts,
J. Skotheim and L. Mahadevan, Physics of Fluids, 17, 092101, 2005.
Fluid-flow induced flutter of a flag,
Argentina, M. and L. Mahadevan, Proceedings of the National Academy of Sciences (USA), 102, 1829-34, 2005.
Characterizing the shape of a solid is inherently a geometrical problem in that one is interested in defining the distances, angles (and changes therein) of material points near and far, relative to each other. In the past, we have studied aspects of this problem in the context of the low-dimensional behavior of filaments and membranes in the context of their equilibrium, and nonequilibrium behavior to understand how these objects fold, wrinkle, and pattern often driven by simple packing constraints. The rich experimental phenomenology combined with the host of mathematical questions that these lead to have kept us occupied for longer than one might think. The contrast is self-evident in an experiment that is the result of many a failed calculation: how easy it is to crumple a piece of paper, and yet how hard it is to understand it ! We continue to work on variants of these problems with applications to a range of problems, from origami to tectonics.
Geometry and physics of wrinkling,
Cerda, E. and L. Mahadevan, Physical Review Letters, 90 (7) 074302, 2003 (Physical Review Focus Article).
Abstract: The ubiquitous 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 for the wavelength of the wrinkles and the amplitude of the wrinkles. Our results could form the basis of a highly sensitive quantitative wrinkling assay for the mechanical characterization of thin solid membranes.
Some other representative publications:
Rings, rackets and kinks in filamentous assemblies,
Cohen, A. and L. Mahadevan, Proceedings of the National Academy of Sciences (USA), 100, 12141-46, 2003.
Popliteal instability of bent multi-walled elastic tubes,
Mahadevan, L., J. Bico and G. McKinley, Europhysics Letters, 65 (3), 323-29, 2004.
Elements of Draping,
Cerda, E., L. Mahadevan and J. Passini, Proceedings of the National Academy of Sciences (USA), 101 (7), 1806-10, 2004.
Solenoids and Plectonemes in stretched and twisted elastomeric filaments,
A. Ghatak and L. Mahadevan, Physical Review Letters , 95, 057801, 2005.
Crack-front instability in a confined elastic film
M. Adda Bedia and L. Mahadevan, Proceedings of the Royal Society of London, series A, 462, 3233, 2006.
|Natural and artificial microstructured materials
Real materials are rarely simple, although it is only recently that we have started to grapple with the complexity inherent in such everyday materials such as the gels, powders, colloids, suspensions and polymers that pass of as the food we eat and indeed are made of. The study of these materials cuts across the traditional boundaries of solids, fluids and gases. Theoretical and experimental approaches to these problems at a macroscopic level use a combination of ideas from continuum and statistical mechanics, physical chemistry and various simulation methods. We are interested in the simple properties of these complex materials with the goal of understanding their qualitative behaviors, manifest as their mechanical and transport properties, and their stability in the presence of external stimuli. A particular interest is the appearance and role of order and/or disorder in determining these properties. More recently, we have become interested in learning from Nature, i.e. in trying to understand the solutions that evolution has stumbled upon, in such instances as clever adhesion mechanisms, locomotory designs, and various self-organized and self-assembled material systems.
Peeling from a biomimetically 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).
Abstract: 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.
Some other representative publications:
Shocks in sand flowing in a silo,
Samadani, A., L. Mahadevan and A. Kudrolli, Journal of Fluid Mechanics, 452, 293-301, 2002.
Dynamics of poroelastic filaments,
Skotheim, J. and L. Mahadevan, Proceedings of the Royal Society of London (A), 460, 1995, 2004.
Transitions to nematic states in homogeneous suspensions of high aspect ratio magnetic rods
A. Gopinath, L. Mahadevan and R.C. Armstrong, Physics of Fluids, 18, 028102, 2006.
Dynamics of surfactant-driven fracture of particle rafts
D. Vella, H-Y Kim, P. Aussillous and L. Mahadevan, Physical Review Letters, 96, 178301, 2006.
Fall and rise of a viscoelastic filament
A. Roy, L. Mahadevan and J-L Thiffeault, Journal of Fluid Mechanics, 563, 283, 2006.
Interfaces and boundaries are distinguished by crises, chaos and creativity. Instabilities are often nucleated at boundaries, as are new phases, and interfaces often have a Janus-like life of their own, unable to completely forget one or the other material that they separate ! Unusual capillary phenomena (of which there seems to be no dearth, two centuries after they were first quantified by Th. Young and P.S. Laplace) have been of long standing interest to us. We got started by thinking about flows that involve the deposition of thin films of liquid onto surfaces as a result of external forces associated with inertia, viscosity, gravity, for example in a horizontally rotating cylinder. We have also studied the unusual statics and dynamics of non-wetting droplets, and uncovered a solution in biology for making perfect non-wetting droplets have been used by insects for more than 200 million years as a means of keeping themselves and their environment clean ! We continue to be interested in various applied aspects of non-wetting droplets in physiology and chemical physics. More recently, we have been looking at elastocapillarity, the physics of soft objects at fluid interfaces, revisiting the problem of capillary rise between soft sheets, the way in which particulate materials aggregate, break and buckle at interfaces, and how filaments fold, and self-pack themselves when placed at interfaces.
The ‘‘Cheerios effect’’ Dominic Vella and L. Mahadevan, American Journal of Physics, 73, 817-825, 2005.
Abstract: Objects that float at the interface between a liquid and a gas interact because of interfacial deformation and the effect of gravity. We highlight the crucial role of buoyancy in this interaction, which, for small particles, prevails over the capillary suction that is often assumed to be the dominant effect. We emphasize this point using a simple classroom demonstration, and then derive the physical conditions leading to mutual attraction or repulsion. We also quantify the force of interaction in some particular instances and present a simple dynamical model of this interaction. The results obtained from this model are then validated by comparison to experimental results for the mutual attraction of two identical spherical particles. We conclude by looking at some of the applications of the effect that can be found in the natural and manmade worlds.
Some other representative publications:
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.
Four-phase merging in compound drops,
Mahadevan, L., M. Adda Bedia and Y. Pomeau, Journal of Fluid Mechanics, 451, pp. 411-20, 2002.
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, 2002.
A. Balasubramaniam, M. Abkarian, L. Mahadevan and H.A. Stone, Nature, 438, 930, 2005.
Capillary rise between elastic sheets
H-Y Kim and L. Mahadevan, Journal of Fluid Mechanics, 548, 141, 2006.