30 research outputs found
Multilayer asymptotic solution for wetting fronts in porous media with exponential moisture diffusivity
We study the asymptotic behaviour of sharp front solutions arising from the
nonlinear diffusion equation \theta_t = (D(\theta)\theta_x)_x, where the
diffusivity is an exponential function D({\theta}) = D_o exp(\beta\theta). This
problem arises for example in the study of unsaturated flow in porous media
where {\theta} represents the liquid saturation. For physical parameters
corresponding to actual porous media, the diffusivity at the residual
saturation is D(0) = D_o << 1 so that the diffusion problem is nearly
degenerate. Such problems are characterised by wetting fronts that sharply
delineate regions of saturated and unsaturated flow, and that propagate with a
well-defined speed. Using matched asymptotic expansions in the limit of large
{\beta}, we derive an analytical description of the solution that is uniformly
valid throughout the wetting front. This is in contrast with most other related
analyses that instead truncate the solution at some specific wetting front
location, which is then calculated as part of the solution, and beyond that
location the solution is undefined. Our asymptotic analysis demonstrates that
the solution has a four-layer structure, and by matching through the adjacent
layers we obtain an estimate of the wetting front location in terms of the
material parameters describing the porous medium. Using numerical simulations
of the original nonlinear diffusion equation, we demonstrate that the first few
terms in our series solution provide approximations of physical quantities such
as wetting front location and speed of propagation that are more accurate (over
a wide range of admissible {\beta} values) than other asymptotic approximations
reported in the literature.Comment: 28 pages, 9 figures, 1 tabl
Projected SO(5) Hamiltonian for Cuprates and Its Applications
The projected SO(5) (pSO(5)) Hamiltonian incorporates the quantum spin and
superconducting fluctuations of underdoped cuprates in terms of four bosons
moving on a coarse grained lattice. A simple mean field approximation can
explain some key feautures of the experimental phase diagram: (i) The Mott
transition between antiferromagnet and superconductor, (ii) The increase of T_c
and superfluid stiffness with hole concentration x and (iii) The increase of
antiferromagnetic resonance energy as sqrt{x-x_c} in the superconducting phase.
We apply this theory to explain the ``two gaps'' problem found in underdoped
cuprate Superconductor-Normal- Superconductor junctions. In particular we
explain the sharp subgap Andreev peaks of the differential resistance, as
signatures of the antiferromagnetic resonance (the magnon mass gap). A critical
test of this theory is proposed. The tunneling charge, as measured by shot
noise, should change by increments of Delta Q= 2e at the Andreev peaks, rather
than by Delta Q=e as in conventional superconductors.Comment: 3 EPS figure
A model for reactive porous transport during re-wetting of hardened concrete
A mathematical model is developed that captures the transport of liquid water
in hardened concrete, as well as the chemical reactions that occur between the
imbibed water and the residual calcium silicate compounds residing in the
porous concrete matrix. The main hypothesis in this model is that the reaction
product -- calcium silicate hydrate gel -- clogs the pores within the concrete
thereby hindering water transport. Numerical simulations are employed to
determine the sensitivity of the model solution to changes in various physical
parameters, and compare to experimental results available in the literature.Comment: 30 page
Simulation of the Response of the Inner Hair Cell Stereocilia Bundle to an Acoustical Stimulus
Mammalian hearing relies on a cochlear hydrodynamic sensor embodied in the inner
hair cell stereocilia bundle. It is presumed that acoustical stimuli induce a
fluid shear-driven motion between the tectorial membrane and the reticular
lamina to deflect the bundle. It is hypothesized that ion channels are opened by
molecular gates that sense tension in tip-links, which connect adjacent stepped
rows of stereocilia. Yet almost nothing is known about how the fluid and bundle
interact. Here we show using our microfluidics model how each row of stereocilia
and their associated tip links and gates move in response to an acoustical input
that induces an orbital motion of the reticular lamina. The model confirms the
crucial role of the positioning of the tectorial membrane in hearing, and
explains how this membrane amplifies and synchronizes the timing of peak tension
in the tip links. Both stereocilia rotation and length change are needed for
synchronization of peak tip link tension. Stereocilia length change occurs in
response to accelerations perpendicular to the oscillatory fluid shear flow.
Simulations indicate that nanovortices form between rows to facilitate diffusion
of ions into channels, showing how nature has devised a way to solve the
diffusive mixing problem that persists in engineered microfluidic devices
Spread of Coxiella burnetii between dairy cattle herds in an enzootic region: modelling contributions of airborne transmission and trade
Advection-Diffusion Equation with Absorbing Boundary
We consider a spatially homogeneous advection–diffusion equation in which the diffusion tensor and drift velocity are time-independent, but otherwise general. We derive asymptotic expressions, valid at large distances from a steady point source, for the flux onto a completely permeable boundary and onto an absorbing boundary. The absorbing case is treated by making a source of antiparticles at the boundary: this method is more general than the method of images. In both cases there is an exponential decay as the distance from the source increases; we find that the exponent is the same for both boundary conditions