33,879 research outputs found
Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution
Mathematically, it takes an infinite amount of time for the transient
solution of a diffusion equation to transition from initial to steady state.
Calculating a \textit{finite} transition time, defined as the time required for
the transient solution to transition to within a small prescribed tolerance of
the steady state solution, is much more useful in practice. In this paper, we
study estimates of finite transition times that avoid explicit calculation of
the transient solution by using the property that the transition to steady
state defines a cumulative distribution function when time is treated as a
random variable. In total, three approaches are studied: (i) mean action time
(ii) mean plus one standard deviation of action time and (iii) a new approach
derived by approximating the large time asymptotic behaviour of the cumulative
distribution function. The new approach leads to a simple formula for
calculating the finite transition time that depends on the prescribed tolerance
and the th and th moments () of the distribution.
Results comparing exact and approximate finite transition times lead to two key
findings. Firstly, while the first two approaches are useful at characterising
the time scale of the transition, they do not provide accurate estimates for
diffusion processes. Secondly, the new approach allows one to calculate finite
transition times accurate to effectively any number of significant digits,
using only the moments, with the accuracy increasing as the index is
increased.Comment: 17 pages, 2 figures, accepted version of paper published in Physical
Review
Rear-surface integral method for calculating thermal diffusivity from laser flash experiments
The laser flash method for measuring thermal diffusivity of solids involves
subjecting the front face of a small sample to a heat pulse of radiant energy
and recording the resulting temperature rise on the opposite (rear) surface.
For the adiabatic case, the widely-used standard approach estimates the thermal
diffusivity from the rear-surface temperature rise history by calculating the
half rise time: the time required for the temperature rise to reach one half of
its maximum value. In this article, we develop a novel alternative approach by
expressing the thermal diffusivity exactly in terms of the area enclosed by the
rear-surface temperature rise curve and the steady-state temperature over time.
Approximating this integral numerically leads to a simple formula for the
thermal diffusivity involving the rear-surface temperature rise history. Using
synthetic experimental data we demonstrate that the new formula produces
estimates of the thermal diffusivity - for a typical test case - that are more
accurate and less variable than the standard approach. The article concludes by
briefly commenting on extension of the new method to account for heat losses
from the sample.Comment: 7 pages, 1 figure, accepted versio
Kinematic Self-Similar Cylindrically Symmetric Solutions
This paper is devoted to find out cylindrically symmetric kinematic
self-similar perfect fluid and dust solutions. We study the cylindrically
symmetric solutions which admit kinematic self-similar vectors of second,
zeroth and infinite kinds, not only for the tilted fluid case but also for the
parallel and orthogonal cases. It is found that the parallel case gives
contradiction both in perfect fluid and dust cases. The orthogonal perfect
fluid case yields a vacuum solution while the orthogonal dust case gives
contradiction. It is worth mentioning that the tilted case provides solution
both for the perfect as well as dust cases.Comment: 22 pages, accepted for publication in Int. J. of Mod. Phys.
Dynamics of kinks in the Ginzburg-Landau equation: Approach to a metastable shape and collapse of embedded pairs of kinks
We consider initial data for the real Ginzburg-Landau equation having two
widely separated zeros. We require these initial conditions to be locally close
to a stationary solution (the ``kink'' solution) except for a perturbation
supported in a small interval between the two kinks. We show that such a
perturbation vanishes on a time scale much shorter than the time scale for the
motion of the kinks. The consequences of this bound, in the context of earlier
studies of the dynamics of kinks in the Ginzburg-Landau equation, [ER], are as
follows: we consider initial conditions whose restriction to a bounded
interval have several zeros, not too regularly spaced, and other zeros of
are very far from . We show that all these zeros eventually disappear
by colliding with each other. This relaxation process is very slow: it takes a
time of order exponential of the length of
Accurate and efficient calculation of response times for groundwater flow
We study measures of the amount of time required for transient flow in
heterogeneous porous media to effectively reach steady state, also known as the
response time. Here, we develop a new approach that extends the concept of mean
action time. Previous applications of the theory of mean action time to
estimate the response time use the first two central moments of the probability
density function associated with the transition from the initial condition, at
, to the steady state condition that arises in the long time limit, as . This previous approach leads to a computationally convenient
estimation of the response time, but the accuracy can be poor. Here, we outline
a powerful extension using the first raw moments, showing how to produce an
extremely accurate estimate by making use of asymptotic properties of the
cumulative distribution function. Results are validated using an existing
laboratory-scale data set describing flow in a homogeneous porous medium. In
addition, we demonstrate how the results also apply to flow in heterogeneous
porous media. Overall, the new method is: (i) extremely accurate; and (ii)
computationally inexpensive. In fact, the computational cost of the new method
is orders of magnitude less than the computational effort required to study the
response time by solving the transient flow equation. Furthermore, the approach
provides a rigorous mathematical connection with the heuristic argument that
the response time for flow in a homogeneous porous medium is proportional to
, where is a relevant length scale, and is the aquifer
diffusivity. Here, we extend such heuristic arguments by providing a clear
mathematical definition of the proportionality constant.Comment: 22 pages, 3 figures, accepted version of paper published in Journal
of Hydrolog
New homogenization approaches for stochastic transport through heterogeneous media
The diffusion of molecules in complex intracellular environments can be
strongly influenced by spatial heterogeneity and stochasticity. A key challenge
when modelling such processes using stochastic random walk frameworks is that
negative jump coefficients can arise when transport operators are discretized
on heterogeneous domains. Often this is dealt with through homogenization
approximations by replacing the heterogeneous medium with an
homogeneous medium. In this work, we present a new class
of homogenization approximations by considering a stochastic diffusive
transport model on a one-dimensional domain containing an arbitrary number of
layers with different jump rates. We derive closed form solutions for the th
moment of particle lifetime, carefully explaining how to deal with the internal
interfaces between layers. These general tools allow us to derive simple
formulae for the effective transport coefficients, leading to significant
generalisations of previous homogenization approaches. Here, we find that
different jump rates in the layers gives rise to a net bias, leading to a
non-zero advection, for the entire homogenized system. Example calculations
show that our generalized approach can lead to very different outcomes than
traditional approaches, thereby having the potential to significantly affect
simulation studies that use homogenization approximations.Comment: 9 pages, 2 figures, accepted version of paper published in The
Journal of Chemical Physic
Rear-surface integral method for calculating thermal diffusivity: finite pulse time correction and two-layer samples
We study methods for calculating the thermal diffusivity of solids from laser
flash experiments. This experiment involves subjecting the front surface of a
small sample of the material to a heat pulse and recording the resulting
temperature rise on the opposite (rear) surface. Recently, a method was
developed for calculating the thermal diffusivity from the rear-surface
temperature rise, which was shown to produce improved estimates compared with
the commonly used half-time approach. This so-called rear-surface integral
method produced a formula for calculating the thermal diffusivity of
homogeneous samples under the assumption that the heat pulse is instantaneously
absorbed uniformly into a thin layer at the front surface. In this paper, we
show how the rear-surface integral method can be applied to a more physically
realistic heat flow model involving the actual heat pulse shape from the laser
flash experiment. New thermal diffusivity formulas are derived for handling
arbitrary pulse shapes for either a homogeneous sample or a heterogeneous
sample comprising two layers of different materials. Presented numerical
experiments confirm the accuracy of the new formulas and demonstrate how they
can be applied to the kinds of experimental data arising from the laser flash
experiment.Comment: 10 pages, 3 figures, accepted versio
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