45 research outputs found
A new ghost cell/level set method for moving boundary problems:application to tumor growth
In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth
Laminar drag reduction in surfactant-contaminated superhydrophobic channels
While superhydrophobic surfaces (SHSs) show promise for drag reduction
applications, their performance can be compromised by traces of surfactant,
which generate Marangoni stresses that increase drag. This question is
addressed for soluble surfactant in a three-dimensional laminar channel flow,
with periodic SHSs on both walls. We assume that diffusion is sufficiently
strong for cross-channel concentration gradients to be small. Exploiting a
long-wave theory that accounts for a rapid transverse Marangoni-driven flow, we
derive a one-dimensional model for surfactant evolution, which allows us to
predict the drag reduction across the parameter space. The system exhibits
multiple regimes, involving competition between Marangoni effects, bulk and
interfacial diffusion, advection and shear dispersion. We map out asymptotic
regions in the high-dimensional parameter space, deriving approximations of the
drag reduction in each region and comparing them to numerical simulations. Our
atlas of maps provides a comprehensive analytical guide for designing
surfactant-contaminated channels with SHSs, to maximise the drag reduction in
applications
Unsteady evolution of slip and drag in surfactant-contaminated superhydrophobic channels
Recognising that surfactants may impede the drag reduction resulting from
superhydrophobic surfaces (SHSs), and that surfactant concentrations can
fluctuate in space and time, we examine the unsteady transport of soluble
surfactant in a laminar pressure-driven channel flow bounded between two SHSs.
The SHSs are periodic in the streamwise and spanwise directions. We assume that
the channel length is much longer than the streamwise period, the streamwise
period is much longer than the channel height and spanwise period, and bulk
diffusion is sufficiently strong for cross-channel concentration gradients to
be small. By combining long-wave and homogenisation theories, we derive an
unsteady advection-diffusion equation for surfactant flux transport over the
length of the channel, which is coupled to a quasi-steady advection-diffusion
equation for surfactant transport over individual plastrons. As diffusion over
the length of the channel is typically small, the leading-order surfactant flux
is governed by a nonlinear advection equation that we solve using the method of
characteristics. We predict the propagation speed of a bolus of surfactant and
describe its nonlinear evolution via interaction with the SHS. The propagation
speed can fall significantly below the average streamwise velocity as the
surfactant adsorbs and rigidifies the plastrons. Smaller concentrations of
surfactant are therefore advected faster than larger ones, so that
wave-steepening effects can lead to shock formation in the surfactant-flux
distribution. These findings reveal the spatio-temporal evolution of the slip
velocity and enable prediction of the dynamic drag reduction and effective slip
length in microchannel applications