1,304 research outputs found
Long-lived and unstable modes of Brownian suspensions in microchannels
We investigate the stability of the pressure-driven, low-Reynolds flow of
Brownian suspensions with spherical particles in microchannels. We find two
general families of stable/unstable modes: (i) degenerate modes with symmetric
and anti-symmetric patterns; (ii) single modes that are either symmetric or
anti-symmetric. The concentration profiles of degenerate modes have strong
peaks near the channel walls, while single modes diminish there. Once excited,
both families would be detectable through high-speed imaging. We find that
unstable modes occur in concentrated suspensions whose velocity profiles are
sufficiently flattened near the channel centreline. The patterns of growing
unstable modes suggest that they are triggered due to Brownian migration of
particles between the central bulk that moves with an almost constant velocity,
and highly-sheared low-velocity region near the wall. Modes are amplified
because shear-induced diffusion cannot efficiently disperse particles from the
cavities of the perturbed velocity field.Comment: 11 pages, accepted for publication in Journal of Fluid Mechanic
The automatic solution of partial differential equations using a global spectral method
A spectral method for solving linear partial differential equations (PDEs)
with variable coefficients and general boundary conditions defined on
rectangular domains is described, based on separable representations of partial
differential operators and the one-dimensional ultraspherical spectral method.
If a partial differential operator is of splitting rank , such as the
operator associated with Poisson or Helmholtz, the corresponding PDE is solved
via a generalized Sylvester matrix equation, and a bivariate polynomial
approximation of the solution of degree is computed in
operations. Partial differential operators of
splitting rank are solved via a linear system involving a block-banded
matrix in operations. Numerical
examples demonstrate the applicability of our 2D spectral method to a broad
class of PDEs, which includes elliptic and dispersive time-evolution equations.
The resulting PDE solver is written in MATLAB and is publicly available as part
of CHEBFUN. It can resolve solutions requiring over a million degrees of
freedom in under seconds. An experimental implementation in the Julia
language can currently perform the same solve in seconds.Comment: 22 page
On the tilting of protostellar disks by resonant tidal effects
We consider the dynamics of a protostellar disk surrounding a star in a
circular-orbit binary system. Our aim is to determine whether, if the disk is
initially tilted with respect to the plane of the binary orbit, the inclination
of the system will increase or decrease with time. The problem is formulated in
the binary frame in which the tidal potential of the companion star is static.
We consider a steady, flat disk that is aligned with the binary plane and
investigate its linear stability with respect to tilting or warping
perturbations. The dynamics is controlled by the competing effects of the m=0
and m=2 azimuthal Fourier components of the tidal potential. In the presence of
dissipation, the m=0 component causes alignment of the system, while the m=2
component has the opposite tendency. We find that disks that are sufficiently
large, in particular those that extend to their tidal truncation radii, are
generally stable and will therefore tend to alignment with the binary plane on
a time-scale comparable to that found in previous studies. However, the effect
of the m=2 component is enhanced in the vicinity of resonances where the outer
radius of the disk is such that the natural frequency of a global bending mode
of the disk is equal to twice the binary orbital frequency. Under such
circumstances, the disk can be unstable to tilting and acquire a warped shape,
even in the absence of dissipation. The outer radius corresponding to the
primary resonance is always smaller than the tidal truncation radius. For disks
smaller than the primary resonance, the m=2 component may be able to cause a
very slow growth of inclination through the effect of a near resonance that
occurs close to the disk center. We discuss these results in the light of
recent observations of protostellar disks in binary systems.Comment: 21 pages, 7 figures, to be published in the Astrophysical Journa
Numerical solution of differential equations
Imperial Users onl
Structure and Stability of Keplerian MHD Jets
MHD jet equilibria that depend on source properties are obtained using a
simplified model for stationary, axisymmetric and rotating magnetized outflows.
The present rotation laws are more complex than previously considered and
include a Keplerian disc. The ensuing jets have a dense, current-carrying
central core surrounded by an outer collar with a return current. The
intermediate part of the jet is almost current-free and is magnetically
dominated. Most of the momentum is located around the axis in the dense core
and this region is likely to dominate the dynamics of the jet. We address the
linear stability and the non-linear development of instabilities for our models
using both analytical and 2.5-D numerical simulation's. The instabilities seen
in the simulations develop with a wavelength and growth time that are well
matched by the stability analysis. The modes explored in this work may provide
a natural explanation for knots observed in astrophysical jets.Comment: 35 pages, accepted by the Ap
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