Elastohydrodynamics of wet bristles, carpets and brushes

Surfaces covered by bristles, hairs, polymers and other filamentous structures arise in
a variety of natural settings in science such as the active lining of many biological
organs, e.g. lungs, reproductive tracts, etc., and have increasingly begun to be used in
technological applications. We derive an effective field theory for the elastohydrodynamics
of ordered brushes and disordered carpets that are made of a large number of elastic
filaments grafted on to a substrate and interspersed in a fluid. Our formulation for the
elastohydrodynamic response of these materials leads naturally to a set of constitutive
equations coupling bed deformation to fluid flow, accounts for the anisotropic properties
of the medium, and generalizes the theory of poroelasticity to these systems. We use the
effective medium equations to study three canonical problems—the normal settling of
a rigid sphere onto a carpet, the squeeze flow in a carpet and the tangential shearing
motion of a rigid sphere over the carpet, all problems of relevance in mechanosensation
in biology with implications for biomimetic devices.

The shape of a long leaf

Long leaves in terrestrial plants and their submarine counterparts,
algal blades, have a typical, saddle-like midsurface and rippled
edges. To understand the origin of these morphologies, we dissect
leaves and differentially stretch foam ribbons to show that these
shapes arise from a simple cause, the elastic relaxation via bending that follows either differential growth (in leaves) or differential
stretching past the yield point (in ribbons). We quantify these different modalities in terms of a mathematical model for the shape
of an initially flat elastic sheet with lateral gradients in longitudinal growth. By using a combination of scaling concepts, stability
analysis, and numerical simulations, we map out the shape space
for these growing ribbons and find that as the relative growth
strain is increased, a long flat lamina deforms to a saddle shape
and/or develops undulations that may lead to strongly localized
ripples as the growth strain is localized to the edge of the leaf. Our
theory delineates the geometric and growth control parameters
that determine the shape space of finite laminae and thus allows
for a comparative study of elongated leaf morphology.

The shape and motion of a ruck in a rug

The motion of a ruck in a rug is used as an analogy to explain the role of dislocations in crystalline solids. We take literally one side of this analogy and study the shape and motion of a bump, wrinkle or ruck in a thin sheet in partial contact with a rough substrate in a gravitational field. Using a combination of experiments, scaling analysis and numerical solutions of the governing equations, we quantify the static shape of a ruck on a horizontal plane. When the plane is inclined, the ruck becomes asymmetric and moves by rolling only when the inclination of the plane reaches a critical angle, at a speed determined by a simple power balance. We find that the angle at which rolling starts is larger than the angle at which the ruck stops; i.e., static rolling friction is larger than dynamic rolling friction. We conclude with a generalization of our results to wrinkles in soft adherent extensible films.

Botanical ratchets

Ratcheting surfaces are a common motif in nature and appear in plant awns and grasses. They are known
to proffer selective advantages for seed dispersion and burial. In two simple model experiments, we show
that these anisotropically toothed surfaces naturally serve as motion rectifiers and generically move in a
unidirectional manner, when subjected to temporally and spatially symmetric excitations of various
origins. Using a combination of theory and experiment, we show that a linear relationship between awn
length and ratchet efficiency holds under biologically relevant conditions. Grass awns can thus efficiently
transform non-equilibrium environmental stresses from such sources as humidity variations into useful
work and directed motion using their length as a fluctuation amplifier, yielding a selective advantage to
these organelles in many plant species.