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 PIP_I

We show that the tritronqu\'ee solution of the Painlev\'e equation $\P1$, $y"=6y^2+z$ which is analytic for large $z$ with $\arg z \in (-\frac{3\pi}{5}, \pi)$ is pole-free in a region containing the full sector ${z \ne 0, \arg z \in [-\frac{3\pi}{5}, \pi]}$ and the disk ${z: |z| < 37/20}$. 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