Uncategorized – Page 4 – Mahadevan Natural Philosophy

Periodic folding of thin sheets

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.

Rings, rackets and kinks in filamentous assemblies

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

Stored elastic energy powers the 60-micron extension of the Limulus polyphemus sperm actin bundle

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.

How aphids lose their marbles

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.

Tumbling cards

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
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Folding of viscous filaments and sheets

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

Rippling instability of a collapsing Bubble

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.

Axial instability of a free-surface front in a partially-filled horizontal rotating cylinder

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!#.

Shocks in sand flowing in a silo

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.

Four-phase merging in compound drops

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.