Instabilities and Singularities in Hele-Shaw Flow
with Raymond Goldstein and Adriana Pesci, in Physics
of Fluids 10, p. 2701, 1998
Abstract
A mechanism by which smooth initial conditions evolve towards a topological
reconfiguration of fluid interfaces is studied in the context of Darcy's
law. In the case of thin fluid layers, nonlinear PDEs for the local thickness
are derived from an asymptotic limit of the vortex sheet representation. A
particular example considered is the Rayleigh-Taylor instability of stratified
fluid layers, where the instability of the system is controlled by a Bond
number B. It is proved that, for a range of B and initial data ``subharmonic''
to it, interface pinching must occur in at least infinite time. Numerical
simulations suggest that ``pinching'' singularities occur generically when
the system is unstable, and in particular immediately above a bifurcation
point to instability. Near this bifurcation point an approximate analytical
method describing the approach to a finite-time singularity is developed.
The method exploits the separation of time scales that exists close to the
first instability in a system of finite extent, with a discrete spectrum of
modes. In this limit, slowly growing long-wavelength modes entrain faster
short-wavelength modes, and thereby, allow the derivation of a nonlinear evolution
equation for the amplitudes of the slow modes. The initial-value problem
is solved in this slaved dynamics, yielding the time and analytical structure
of a singularity that is associated with the motion of zeros in the complex
plane, suggesting a general mechanism of singularity formation in this system.
The discussion emphasizes the significance of several variational principles,
and comparisons are made between the numerical simulations and the approximate
theory.
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Domain of Convergence of Perturbative Solutions for Hele-Shaw Flow near
Interface Collapse
with Raymond Goldstein and Adriana Pesci, in Physics
of Fluids 11, p. 2809, 1999
Abstract
Recent work [Phys. Fluids 10, 2701 (1998)] has shown that for Hele-Shaw
flows sufficiently near a finite-time pinching singularity, there is a breakdown
of the leading-order solutions perturbative in a small parameter e controlling the large-scale
dynamics. To elucidate the nature of this breakdown we study the structure
of these solutions at higher order. We find a finite radius of convergence
that yields a new length scale exponentially small in e. That length scale defines
a ball in space and time, centered around the incipient singularity, inside
of which perturbation theory fails. Implications of these results for a
possible matching of outer solutions to inner scaling solutions are discussed.
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Singularity Formation in Thin Jets with Surface Tension
with Mary Pugh, in Communications in Pure and Applied
Mathematics 51, p. 733, 1998.
Abstract
We derive and study asymptotic models for the dynamics of a thin jet of
fluid that is separated from an outer immiscible fluid by fluid interfaces
with surface tension. Both fluids are assumed to be incompressible, inviscid,
irrotational, and density matched. One such thin jet model is a coupled system
of PDEs with nonlocal terms -- Hilbert transforms -- that result from expansion
of a Biot-Savart integral. In order to make the asymptotic model well-posed,
the Hilbert transforms act upon time derivatives of the jet thickness, making
the system implicit. Within this thin jet model, we demonstrate numerically
the formation of finite-time pinching singularities, where the width of
the jet collapses to zero at a point. These singularities are driven by
the surface tension, and are very similar to those observed previously by
Hou, Lowengrub, and Shelley in large-scale simulations of the Kelvin-Helmholtz
instability with surface tension, and in other related studies. Dropping
the nonlocal terms of the model, we also study a much simpler local model.
For this local model we can preclude analytically the formation of certain
types of singularities, though not those of pinching type. Surprisingly,
we find that this local model forms pinching singularities of a very similar
type to those of the nonlocal thin jet model.
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M.P. was supported by an NSF post-doctoral fellowship while at Courant
and by the Ambrose Monell Foundation while at the Institute for Advanced
Study. M.J.S. acknowledges support from Department of Energy grant DE-FG02-88ER25053,
National Science Foundation grants DMS-9396403 (PYI) and DMS-9404554, and
the Exxon Educational Foundation.