48 research outputs found
Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation
We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth-order and destabilising second-order term. It has previously been shown that blow up is only possible when the exponent in the second-order term is above a certain critical value (dependent on the exponent in the fourth-order term). We show that in the limit that the critical value is approached from above, the primary branch of similarity profiles exhibits a well-defined structure consisting of a peak near the origin, and a thin, algebraically decaying tail, connected by an inner region equivalent (to leading order) to a generalised version of the Landau–Levich ‘drag-out’ problem in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between a parameter representing the height of the peak and the distance from the criticality threshold. The asymptotic results are supported by numerical computations found using continuation
Corner and finger formation in Hele--Shaw flow with kinetic undercooling regularisation
We examine the effect of a kinetic undercooling condition on the evolution of
a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We
present analytical and numerical evidence that the bubble boundary is unstable
and may develop one or more corners in finite time, for both expansion and
contraction cases. This loss of regularity is interesting because it occurs
regardless of whether the less viscous fluid is displacing the more viscous
fluid, or vice versa. We show that small contracting bubbles are described to
leading order by a well-studied geometric flow rule. Exact solutions to this
asymptotic problem continue past the corner formation until the bubble
contracts to a point as a slit in the limit. Lastly, we consider the evolving
boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The
boundary may either form corners in finite time, or evolve to a single long
finger travelling at constant speed, depending on the strength of kinetic
undercooling. We demonstrate these two different behaviours numerically. For
the travelling finger, we present results of a numerical solution method
similar to that used to demonstrate the selection of discrete fingers by
surface tension. With kinetic undercooling, a continuum of corner-free
travelling fingers exists for any finger width above a critical value, which
goes to zero as the kinetic undercooling vanishes. We have not been able to
compute the discrete family of analytic solutions, predicted by previous
asymptotic analysis, because the numerical scheme cannot distinguish between
solutions characterised by analytic fingers and those which are corner-free but
non-analytic
Saffman-Taylor fingers with kinetic undercooling
The mathematical model of a steadily propagating Saffman-Taylor finger in a
Hele-Shaw channel has applications to two-dimensional interacting streamer
discharges which are aligned in a periodic array. In the streamer context, the
relevant regularisation on the interface is not provided by surface tension,
but instead has been postulated to involve a mechanism equivalent to kinetic
undercooling, which acts to penalise high velocities and prevent blow-up of the
unregularised solution. Previous asymptotic results for the Hele-Shaw finger
problem with kinetic undercooling suggest that for a given value of the kinetic
undercooling parameter, there is a discrete set of possible finger shapes, each
analytic at the nose and occupying a different fraction of the channel width.
In the limit in which the kinetic undercooling parameter vanishes, the fraction
for each family approaches 1/2, suggesting that this 'selection' of 1/2 by
kinetic undercooling is qualitatively similar to the well-known analogue with
surface tension. We treat the numerical problem of computing these
Saffman-Taylor fingers with kinetic undercooling, which turns out to be more
subtle than the analogue with surface tension, since kinetic undercooling
permits finger shapes which are corner-free but not analytic. We provide
numerical evidence for the selection mechanism by setting up a problem with
both kinetic undercooling and surface tension, and numerically taking the limit
that the surface tension vanishes.Comment: 10 pages, 6 figures, accepted for publication by Physical Review
Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow
We study a model for the evolution of an axially symmetric bubble of inviscid
fluid in a homogeneous porous medium otherwise saturated with a viscous fluid.
The model is a moving boundary problem that is a higher-dimensional analogue of
Hele-Shaw flow. Here we are concerned with the development of pinch-off
singularities characterised by a blow-up of the interface curvature and the
bubble subsequently breaking up into two; these singularities do not occur in
the corresponding two-dimensional Hele-Shaw problem. By applying a novel
numerical scheme based on the level set method, we show that solutions to our
problem can undergo pinch-off in various geometries. A similarity analysis
suggests that the minimum radius behaves as a power law in time with exponent
just before and after pinch-off has occurred, regardless of the
initial conditions; our numerical results support this prediction. Further, we
apply our numerical scheme to simulate the time-dependent development and
translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman
bubbles in a cylindrical tube, highlighting key similarities and differences
with the well-studied two-dimensional cases.Comment: 16 pages, 16 figure
Discrete Self-Similarity in Interfacial Hydrodynamics and the Formation of Iterated Structures
The formation of iterated structures, such as satellite and sub-satellite
drops, filaments and bubbles, is a common feature in interfacial hydrodynamics.
Here we undertake a computational and theoretical study of their origin in the
case of thin films of viscous fluids that are destabilized by long-range
molecular or other forces. We demonstrate that iterated structures appear as a
consequence of discrete self-similarity, where certain patterns repeat
themselves, subject to rescaling, periodically in a logarithmic time scale. The
result is an infinite sequence of ridges and filaments with similarity
properties. The character of these discretely self-similar solutions as the
result of a Hopf bifurcation from ordinarily self-similar solutions is also
described.Comment: LaTeX, 5 pages, replaced with minor changes, accepted for publication
in Physical Review Letter
Two-interface and thin filament approximation in Hele--Shaw channel flow
For a viscous fluid trapped in a Hele--Shaw channel, and pushed by a pressure
difference, the fluid interface is unstable due to the Saffman--Taylor
instability. We consider the evolution of a fluid region of finite extent,
bounded between two interfaces, in the limit the interfaces are close, that is,
when the fluid region is a thin liquid filament separating two gases of
different pressure. In this limit, we derive a geometric flow rule that
describes the normal velocity of the filament centreline, and evolution of the
filament thickness, as functions of the thickness and centreline curvature. We
show that transverse flow along the filament is necessary to regularise the
instability. Numerical simulation of the thin filament flow rule is shown to
closely match level-set computations of the complete two-interface model, and
solutions ultimately evolve to form a bubble of increasing radius and
decreasing thickness
Onset of convective instability in an inclined porous medium
The diffusion of a solute from a concentrated source into a horizontal,
stationary, fluid-saturated porous medium can lead to a convective motion when
a gravitationally unstable density stratification evolves. In an inclined
porous medium, the convective flow becomes intricate as it originates from a
combination of diffusion and lateral flow, which is dominant near the source of
the solute. Here, we investigate the role of inclination on the onset of
convective instability by linear stability analyses of Darcy's law and mass
conservation for the flow and the concentration field. We find that the onset
time increases with the angle of inclination () until it reaches a
cut-off angle beyond which the system remains stable. The cut-off angle
increases with the Rayleigh number, . The evolving wavenumber at the onset
exhibits a lateral velocity that depends non-monotonically on and
linearly on . Instabilities are observed in gravitationally stable
configurations () solely due to the non-uniform base
flow generating a velocity shear commonly associated with Kelvin-Helmholtz
instability. These results quantify the role of medium tilt on convective
instabilities, which is of great importance to geological CO sequestration.Comment: 18 pages, 7 figure