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Complex Approximations, Orthogonal Polynomials and Applications Workshop
June 10, 2021 16:30–16:55, Section II, Sochi
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Dynamics of complex singularities in fluid dynamics and analytical continuation by rational approximants
P. M. Lushnikovab a University of New Mexico
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
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Abstract:
A motion of ideal incompressible fluid with a free surface is considered
in two-dimensional geometry. A time-dependent conformal mapping of the
lower complex half-plane of the auxiliary complex variable $w$ into the
area filled with fluid is performed with the real line of $w$ mapped into
the free fluid's surface. The fluid dynamics is fully characterized by the
motion of the complex singularities in $w$ plane obtained by the
analytical continuation into area outside of fluid.
We consider both the
exact dynamics and the short branch cut approximation of the dynamics with
the small parameter being the ratio of the length of the branch cut to the
distance between its center and the real line of $w$.
The exact dynamics
is shown to conserve the infinite number of integrals of motion which are
residues of poles in auxiliary complex variables as well as it results in
the motion of power law branch points.
Fluid dynamics in short branch cut
approximation is reduced to the complex Hopf equation for the complex
velocity coupled with the complex transport equation for the conformal
mapping. These equations are fully integrable by characteristics producing
the infinite family of solutions, including the pairs of moving square
root branch points.
The solutions are compared with the simulations of the
full Eulerian dynamics giving excellent agreement. The numerical
continuation is performed into the complex plane using the rational
approximants. It allows to compare the dynamics of singularities with the
analytical predictions including the existence of multiple integrals of
motion.
We also analyze the dynamics of singularities and finite time
blow up of Constantin–Lax–Majda equation which corresponds to non-potential
effective motion of non-viscous fluid with competing convection and
vorticity stretching terms.
A family of exact solutions is found together
with the different types of complex singularities approaching the real
line in finite times. These singularities are also recovered by numerical
analytical continuation.
Language: English
* Zoom conference ID: 861 852 8524 , password: caopa |
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