93,196 research outputs found
A Class of Free Boundary Problems with Onset of a new Phase
A class of diffusion driven Free Boundary Problems is considered which is
characterized by the initial onset of a phase and by an explicit kinematic
condition for the evolution of the free boundary. By a domain fixing change of
variables it naturally leads to coupled systems comprised of a singular
parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even
though the one dimensional case has been thoroughly investigated, results as
basic as well-posedness and regularity have so far not been obtained for its
higher dimensional counterpart. In this paper a recently developed regularity
theory for abstract singular parabolic Cauchy problems is utilized to obtain
the first well-posedness results for the Free Boundary Problems under
consideration. The derivation of elliptic regularity results for the underlying
static singular problems will play an important role
Directed percolation in aerodynamics: resolving laminar separation bubble on airfoils
In nature, phase transitions prevail amongst inherently different systems,
while frequently showing a universal behavior at their critical point. As a
fundamental phenomenon of fluid mechanics, recent studies suggested
laminar-turbulent transition belonging to the universality class of directed
percolation. Beyond, no indication was yet found that directed percolation is
encountered in technical relevant fluid mechanics. Here, we present first
evidence that the onset of a laminar separation bubble on an airfoil can be
well characterized employing the directed percolation model on high fidelity
particle image velocimetry data. In an extensive analysis, we show that the
obtained critical exponents are robust against parameter fluctuations, namely
threshold of turbulence intensity that distinguishes between ambient flow and
laminar separation bubble. Our findings indicate a comprehensive significance
of percolation models in fluid mechanics beyond fundamental flow phenomena, in
particular, it enables the precise determination of the transition point of the
laminar separation bubble. This opens a broad variety of new fields of
application, ranging from experimental airfoil aerodynamics to computational
fluid dynamics.Comment: 8 pages, 11 figure
On well-posedness of variational models of charged drops
Electrified liquids are well known to be prone to a variety of interfacial
instabilities that result in the onset of apparent interfacial singularities
and liquid fragmentation. In the case of electrically conducting liquids, one
of the basic models describing the equilibrium interfacial configurations and
the onset of instability assumes the liquid to be equipotential and interprets
those configurations as local minimizers of the energy consisting of the sum of
the surface energy and the electrostatic energy. Here we show that,
surprisingly, this classical geometric variational model is mathematically
ill-posed irrespectively of the degree to which the liquid is electrified.
Specifically, we demonstrate that an isolated spherical droplet is never a
local minimizer, no matter how small is the total charge on the droplet, since
the energy can always be lowered by a smooth, arbitrarily small distortion of
the droplet's surface. This is in sharp contrast with the experimental
observations that a critical amount of charge is needed in order to destabilize
a spherical droplet. We discuss several possible regularization mechanisms for
the considered free boundary problem and argue that well-posedness can be
restored by the inclusion of the entropic effects resulting in finite screening
of free charges.Comment: 18 pages, 2 figure
Trapping and Steering on Lattice Strings: Virtual Slow Waves, Directional and Non-propagating Excitations
Using a lattice string model, a number of peculiar excitation situations
related to non-propagating excitations and non-radiating sources are
demonstrated. External fields can be used to trap excitations locally but also
lead to the ability to steer such excitations dynamically as long as the
steering is slower than the field's wave propagation. I present explicit
constructions of a number of examples, including temporally limited
non-propagating excitations, directional excitation and virtually slowed
propagation. Using these dynamical lattice constructions I demonstrate that
neither persistent temporal oscillation nor static localization are necessary
for non-propagating excitations to occur.Comment: 16 pages, 5 figures, RevTex4, references added, figure captions
improved, to appear in Physical Review
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