490 research outputs found
Motility of active fluid drops on surfaces
Drops of active liquid crystal have recently shown the ability to
self-propel, which was associated with topological defects in the orientation
of active filaments [Sanchez {\em et al.}, Nature {\bf 491}, 431 (2013)]. Here,
we study the onset and different aspects of motility of a three-dimensional
drop of active fluid on a planar surface. We analyse theoretically how motility
is affected by orientation profiles with defects of various types and
locations, by the shape of the drop, and by surface friction at the substrate.
In the scope of a thin drop approximation, we derive exact expressions for the
flow in the drop that is generated by a given orientation profile. The flow has
a natural decomposition into terms that depend entirely on the geometrical
properties of the orientation profile, i.e. its bend and splay, and a term
coupling the orientation to the shape of the drop. We find that asymmetric
splay or bend generates a directed bulk flow and enables the drop to move, with
maximal speeds achieved when the splay or bend is induced by a topological
defect in the interior of the drop. In motile drops the direction and speed of
self-propulsion is controlled by friction at the substrate.Comment: 11 pages, 8 figure
Knotted Defects in Nematic Liquid Crystals
We show that the number of distinct topological states associated to a given
knotted defect, , in a nematic liquid crystal is equal to the determinant of
the link . We give an interpretation of these states, demonstrate how they
may be identified in experiments and describe the consequences for material
behaviour and interactions between multiple knots. We show that stable knots
can be created in a bulk cholesteric and illustrate the topology by classifying
a simulated Hopf link. In addition we give a topological heuristic for the
resolution of strand crossings in defect coarsening processes which allows us
to distinguish topological classes of a given link and to make predictions
about defect crossings in nematic liquid crystals.Comment: 10 pages, 4 figure
The many-body reciprocal theorem and swimmer hydrodynamics
We present a reinterpretation and extension of the reciprocal theorem for
swimmers, extending its application from the motion of a single swimmer in an
unbounded domain to the general setting, giving results for both swimmer
interactions and general hydrodynamics. We illustrate the method for a squirmer
near a planar surface, recovering standard literature results and extending
them to a general squirming set, to motion in the presence of a ciliated
surface, and expressions for the flow field throughout the domain. Finally, we
present exact results for the hydrodynamics in two dimensions which shed light
on the near-field behaviour.Comment: 6 pages, 6 figure
Umbilic Lines in Orientational Order
Three-dimensional orientational order in systems whose ground states possess
non-zero, chiral gradients typically exhibits line-like structures or defects:
lines in cholesterics or Skyrmion tubes in ferromagnets for example.
Here we show that such lines can be identified as a set of natural geometric
singularities in a unit vector field, the generalisation of the umbilic points
of a surface. We characterise these lines in terms of the natural vector
bundles that the order defines and show that they give a way to localise and
identify Skyrmion distortions in chiral materials -- in particular that they
supply a natural representative of the Poincar\'{e} dual of the cocycle
describing the topology. Their global structure leads to the definition of a
self-linking number and helicity integral which relates the linking of umbilic
lines to the Hopf invariant of the texture.Comment: 14 pages, 9 figure
Maxwell's Theory of Solid Angle and the Construction of Knotted Fields
We provide a systematic description of the solid angle function as a means of
constructing a knotted field for any curve or link in . This is a
purely geometric construction in which all of the properties of the entire
knotted field derive from the geometry of the curve, and from projective and
spherical geometry. We emphasise a fundamental homotopy formula as unifying
different formulae for computing the solid angle. The solid angle induces a
natural framing of the curve, which we show is related to its writhe and use to
characterise the local structure in a neighborhood of the knot. Finally, we
discuss computational implementation of the formulae derived, with C code
provided, and give illustrations for how the solid angle may be used to give
explicit constructions of knotted scroll waves in excitable media and knotted
director fields around disclination lines in nematic liquid crystals.Comment: 20 pages, 9 figure
Straight Round the Twist: Frustration and Chirality in Smectics-A
Frustration is a powerful mechanism in condensed matter systems, driving both
order and co plexity. In smectics, the frustration between macroscopic
chirality and equally spaced layers generates textures characterised by a
proliferation of defects. In this article, we study several different ground
states of the chiral Landau-de Gennes free energy for a smectic liquid crystal.
The standard theory finds the twist grain boundary (TGB) phase to be the ground
state for chiral type II smectics. However, for very highly chiral systems, the
hierarchical helical nanofilament (HN) phase can form and is stable over the
TGB.Comment: 9 pages, 3 figures, submitted to J. Interface Focu
Active Nematic Multipoles: Flow Responses and the Dynamics of Defects and Colloids
We introduce a general description of localised distortions in active
nematics using the framework of active nematic multipoles. We give the
Stokesian flows for arbitrary multipoles in terms of differentiation of a
fundamental flow response and describe them explicitly up to quadrupole order.
We also present the response in terms of the net active force and torque
associated to the multipole. This allows the identification of the dipolar and
quadrupolar distortions that generate self-propulsion and self-rotation
respectively and serves as a guide for the design of arbitrary flow responses.
Our results can be applied to both defect loops in three-dimensional active
nematics and to systems with colloidal inclusions. They reveal the
geometry-dependence of the self-dynamics of defect loops and provide insights
into how colloids might be designed to achieve propulsive or rotational
dynamics, and more generally for the extraction of work from active nematics.
Finally, we extend our analysis also to two dimensions and to systems with
chiral active stresses.Comment: 24 pages, 10 figure
Exact solutions for hydrodynamic interactions of two squirming spheres
We provide exact solutions of the Stokes equations for a squirming sphere
close to a no-slip surface, both planar and spherical, and for the interactions
between two squirmers, in three dimensions. These allow the hydrodynamic
interactions of swimming microscopic organisms with confining boundaries, or
each other, to be determined for arbitrary separation and, in particular, in
the close proximity regime where approximate methods based on point singularity
descriptions cease to be valid. We give a detailed description of the circular
motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary
or flat free surface at close separation, finding that the circling generically
has opposite sense at free surfaces and at solid boundaries. While the
asymptotic interaction is symmetric under head-tail reversal of the swimmer, in
the near field microscopic structure can result in significant asymmetry. We
also find the translational velocity towards the surface for a simple model
with only the lowest two squirming modes. By comparing these to asymptotic
approximations of the interaction we find that the transition from near- to
far-field behaviour occurs at a separation of about two swimmer diameters.
These solutions are for the rotational velocity about the wall normal, or
common diameter of two spheres, and the translational speed along that same
direction, and are obtained using the Lorentz reciprocal theorem for Stokes
flows in conjunction with known solutions for the conjugate Stokes drag
problems, the derivations of which are demonstrated here for completeness. The
analogous motions in the perpendicular directions, i.e. parallel to the wall,
currently cannot be calculated exactly since the relevant Stokes drag solutions
needed for the reciprocal theorem are not available.Comment: 27 pages, 7 figure
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