Motion Of Ideal Incompressible Fluid Research Articles

Short branch cut approximation in two-dimensional hydrodynamics with free surface

A potential motion of ideal incompressible fluid with a free surface and infinite depth 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 can be fully characterized by the motion of the complex singularities in the analytical continuation of both the conformal mapping and the complex velocity. We consider 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 centre and the real line of w . We found that the fluid dynamics in that 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 moving square root branch points and poles. These solutions involve practical initial conditions resulting in jets and overturning waves. The solutions are compared with the simulations of the fully nonlinear Eulerian dynamics giving excellent agreement even when the small parameter approaches about one.

Poles and Branch Cuts in Free Surface Hydrodynamics

We consider the motion of ideal incompressible fluid with free surface. We analyzed the exact fluid dynamics though the time-dependent conformal mapping $z=x+iy=z(w,t)$ of the lower complex half plane of the conformal variable $w$ into the area occupied by fluid. We established the exact results on the existence vs. nonexistence of the pole and power law branch point solutions for $1/z_w$ and the complex velocity. We also proved the nonexistence of the time-dependent rational solution of that problem for the second and the first order moving pole.

НОВЫЕ ИНТЕГРАЛЫ ДВИЖЕНИЯ И НЕКАНОНИЧЕСКАЯ ГАМИЛЬТОНОВА СТРУКТУРА ДЛЯ ДВУМЕРНОЙ ГИДРОДИНАМИКИ СО СВОБОДНОЙ ПОВЕРХНОСТЬЮ

We consider a potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensions with gravity forces and surface tension. A time-dependent conformal mapping z(w, t) of the lower complex half-plane of the variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. We study the dynamics of singularities of both z(w, t) and the complex fluid potential Π(w, t) in the upper complex half-plane of w. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian structure for a pair of the Hamiltonian variables (Dyachenko et al., submitted), the imaginary part of z(w, t) and the real part at Π(w, t) (both evaluated of fluid’s free surface). The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. New Hamiltonian structure is a generalization of the canonical Hamiltonian structure of (Zakharov, 1968) (valid only for solutions for which the natural surface parametrization is single valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface). In contrast, new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations as in (Lushnikov and Zubarev, 2018) with the powerful reductions which allowed to find general classes of particular solutions. In Eulerian case we show the existence of solutions with an arbitrary finite number N of complex poles in zw(w, t) and Πw(w, t) which are the derivatives of z(w, t) and Π(w, t) over w (Dyachenko et al., submitted). These solutions are not purely rational because they generally have branch points at other positions of the upper complex halfplane with generally the infinite number of sheets of the Riemann surface for z(w, t) and Π(w, t) (Lushnikov, 2016). The order of poles is arbitrary for zero surface tension while all orders are even for nonzero surface tension. We find that the residues of zw(w, t) at these N points are new, previously unknown constants of motion. These constants of motion commute with each other with respect to the Poisson bracket. There are more integrals of motion beyond these residues. If all poles are simple then the number of independent real integrals of motion is 4N for zero gravity and 4N-1 for nonzero gravity. For higher order poles the number of the integrals is increasing. These nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics. Work of A. Dyachenko, P. Lushnikov and V. Zakharov was supported by state assignment «Dynamics of the complex materials».