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Trapping and displacement of liquid collars and plugs in rough-walled tubes
A liquid film wetting the interior of a long circular cylinder redistributes
under the action of surface tension to form annular collars or occlusive plugs.
These equilibrium structures are invariant under axial translation within a
perfectly smooth uniform tube and therefore can be displaced axially by very
weak external forcing. We consider how this degeneracy is disrupted when the
tube wall is rough, and determine threshold conditions under which collars or
plugs resist displacement under forcing. Wall roughness is modelled as a
non-axisymmetric Gaussian random field of prescribed correlation length and
small variance, mimicking some of the geometric irregularities inherent in
applications such as lung airways. The thin film coating this surface is
modelled using lubrication theory. When the roughness is weak, we show how the
locations of equilibrium collars and plugs can be identified in terms of the
azimuthally averaged tube radius; we derive conditions specifying equilibrium
collar locations under an externally imposed shear flow, and plug locations
under an imposed pressure gradient. We use these results to determine the
probability of external forcing being sufficient to displace a collar or plug
from a rough-walled tube, when the tube roughness is defined only in
statistical terms
A Helson-Szeg\"o theorem for subdiagonal subalgebras with applications to Toeplitz operators
We formulate and establish a noncommutative version of the well known
Helson-Szego theorem about the angle between past and future for subdiagonal
subalgebras. We then proceed to use this theorem to characterise the symbols of
invertible Toeplitz operators on the noncommutative Hardy spaces associated to
subdiagonal subalgebras
Quantum critical points of Helical Fermi Liquids
Following our previous work, we study the quantum phase transitions which
spontaneously develop ferromagnetic spin order in helical fermi liquids which
breaks continuous spin-space rotation symmetry, with application to the edge
states of 3d topological band insulators. With finite fermi surface, the
critical point has both z = 3 over-damped and z = 2 propagating quantum
critical modes, and the z = 3 mode will lead to non-fermi liquid behavior on
the entire fermi surface. In the ordered phase, the Goldstone mode is
over-damped unless it propagates along special directions, and quasiparticle is
ill defined on most parts of the fermi surface except for special points.
Generalizations of our results to other systems with spin-orbit couplings are
also discussed.Comment: 5 pages, 2 figure
Drop spreading with random viscosity
We examine theoretically the spreading of a viscous liquid drop over a thin
film of uniform thickness, assuming the liquid's viscosity is regulated by the
concentration of a solute that is carried passively by the spreading flow. The
solute is assumed to be initially heterogeneous, having a spatial distribution
with prescribed statistical features. To examine how this variability
influences the drop's motion, we investigate spreading in a planar geometry
using lubrication theory, combining numerical simulations with asymptotic
analysis. We assume diffusion is sufficient to suppress solute concentration
gradients across but not along the film. The solute field beneath the bulk of
the drop is stretched by the spreading flow, such that the initial solute
concentration immediately behind the drop's effective contact lines has a
long-lived influence on the spreading rate. Over long periods, solute swept up
from the precursor film accumulates in a short region behind the contact line,
allowing patches of elevated viscosity within the precursor film to hinder
spreading. A low-order model provides explicit predictions of the variances in
spreading rate and drop location, which are validated against simulations
Asymptotic Stability for Equilibria of Nonlinear Semiflows with Applications to Rotating Viscoelastic Rods, Part I
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