776 research outputs found
Singularities in the classical Rayleigh-Taylor flow: Formation and subsequent motion
The creation and subsequent motion of singularities of solution to classical Rayleigh-Taylor flow (two dimensional inviscid, incompressible fluid over a vacuum) are discussed. For a specific set of initial conditions, we give analytical evidence to suggest the instantaneous formation of one or more singularities at specific points in the unphysical plane, whose locations depend sensitively on small changes in initial conditions in the physical domain. One-half power singularities are created in accordance with an earlier conjecture; however, depending on initial conditions, other forms of singularities are also possible. For a specific initial condition, we follow a numerical procedure in the unphysical plane to compute the motion of a one-half singularity. This computation confirms our previous conjecture that the approach of a one-half singularity towards the physical domain corresponds to the development of a spike at the physical interface. Under some assumptions that appear to be consistent with numerical calculations, we present analytical evidence to suggest that a singularity of the one-half type cannot impinge the physical domain in finite time
Global existence for a translating near-circular Hele-Shaw bubble with surface tension
This paper concerns global existence for arbitrary nonzero surface tension of
bubbles in a Hele-Shaw cell that translate in the presence of a pressure
gradient. When the cell width to bubble size is sufficiently large, we show
that a unique steady translating near-circular bubble symmetric about the
channel centerline exists, where the bubble translation speed in the laboratory
frame is found as part of the solution. We prove global existence for symmetric
sufficiently smooth initial conditions close to this shape and show that the
steady translating bubble solution is an attractor within this class of
disturbances. In the absence of side walls, we prove stability of the steady
translating circular bubble without restriction on symmetry of initial
conditions. These results hold for any nonzero surface tension despite the fact
that a local planar approximation near the front of the bubble would suggest
Saffman Taylor instability.
We exploit a boundary integral approach that is particularly suitable for
analysis of nonzero viscosity ratio between fluid inside and outside the
bubble. An important element of the proof was the introduction of a weighted
Sobolev norm that accounts for stabilization due to advection of disturbances
from the front to the back of the bubble
Prandtl–Batchelor flow past a flat plate with a forward-facing flap
Two-dimensional steady inviscid flow past an inclined flat plate with a forward-facing flap attached to the rear edge is considered for the case when a vortex sheet separates from the leading edge of the flat plate and reattaches at the leading edge of the flap, with uniform vorticity distributed between the vortex sheet and the body. Solutions are found for a particular geometry and a range of values of the vorticity. The method used to calculate the flow is an extension of a free-streamline method widely used in cases where the velocity is a constant on the separating streamline
The touching pair of equal and opposite uniform vortices
The shape and speed of a pair of touching finite area vortices are calculated and an error in previous work corrected
Proof of the Dubrovin conjecture and analysis of the tritronqu\'ee solutions of
We show that the tritronqu\'ee solution of the Painlev\'e equation , which is analytic for large with is pole-free in a region containing the full sector and the disk . This proves in
particular the Dubrovin conjecture, an open problem in the theory of Painlev\'e
transcendents. The method, building on a technique developed in Costin, Huang,
Schlag (2012), is general and constructive. As a byproduct, we obtain the value
of the tritronqu\'ee and its derivative at zero within less than 1/100 rigorous
error bounds
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