Exact Local Solutions for the Formation of Singularities on the Free Surface of an Ideal Fluid

Зубарев, Н. М.; Karabut, E. A.

doi:10.1134/s0021364018070135

Cited by 35 publications

(37 citation statements)

References 19 publications

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“…Some exceptional cases like found in Refs. Karabut & Zhuravleva (2014); Zubarev & Karabut (2018) have only a finite number of sheets of Riemann surface (these solutions however have diverging values of V and R at w → ∞).…”

Section: Kelvin Theorem For Phantom Hydrodynamics and Global Analysismentioning

confidence: 99%

Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

Dyachenko

Lushnikov

et al. 2019

J. Fluid Mech.

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We address a problem of potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensional geometry. We admit a presence of 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 show the existence of solutions with an arbitrary finite number N of complex poles in z w (w, t) and Π w (w, t) which are the derivatives of z(w, t) and Π(w, t) over w. We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for nonzero surface tension. We find that the residues of z w (w, t) at these N points are new, previously unknown constants of motion, see also Ref. V. E. Zakharov and A. I. Dyachenko, arXiv:1206.2046(2012 for the preliminary results. All these constants of motion commute with each other in the sense of underlying Hamiltonian dynamics. In absence of both gravity and surface tension, the residues of Π w (w, t) are also the constants of motion while nonzero gravity g ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both z w (w, t) and Π w (w, t) at each poles position reveals an existence of additional integrals of motion for poles of the second order. 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 the second order poles we found 6N motion integral for zero gravity and 6N − 1 for nonzero gravity. We suggest that the existence of these nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.Potential motion means that a velocity v of fluid is determined by a velocity potential Φ(r, t) as v = ∇Φ with ∇ ≡ ( ∂ ∂x , ∂ ∂y ). The incompressibility condition ∇ · v = 0 implies the Laplace equation(1.5) inside fluid, i.e. Φ is the harmonic function inside fluid. Eq. (1.5) is supplemented with a decaying boundary condition (BC) at infinity, ∇Φ → 0 for |x| → ∞ or y → −∞.(1.6)

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Section: Kelvin Theorem For Phantom Hydrodynamics and Global Analysismentioning

confidence: 99%

Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

Dyachenko

Lushnikov

et al. 2019

J. Fluid Mech.

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“…Theoretical and simulation studies building on the rigid-dipole approximation have shown that steric and dipolar interactions can significantly modify the susceptibility and magnetoviscosity [2,7,9,[35][36][37]. The diffusion-jump model proposed here offers the possibility to extend these works to efficiently include Néel relaxation by incorporating magnetization reversals as a Poisson jump process satisfying the detailed balance condition (11). For weakly interacting particles and in the absence of an external field, for example, the rates can be approximated by λ ≈ λ 0 and therefore the Poisson processes are independent.…”

Section: Discussionmentioning

confidence: 99%

“…[6,26]. In order to calculate re −e·h e we first note that this quantity vanishes in equilibrium, re −e·h e eq = 0, due to the detailed balance condition (11). Next, due to uniaxial symmetry we have re −e·h ee eq = R 1ĥĥ + R 2 I,…”

Section: Appendix C: Estimate Of Néel Relaxation Timementioning

confidence: 99%

“…(i) for smaller or magnetically weak particles where the energy barrier for Néel relaxation can be overcome on relevant time scales, or (ii) when particle rotation is severely hindered or nearly suppressed. Particles which fall under category (i) are usually ignored in the modeling since they are either considered irrelevant for viscous properties [11] or are treated as a magnetic background [12]. In both cases, however, internal relaxation needs to be accounted for when considering e.g.…”

Section: Introductionmentioning

confidence: 99%

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Diffusion-jump model for the combined Brownian and Néel relaxation dynamics of ferrofluids in the presence of external fields and flow

Ilg

2019

Phys. Rev. E

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ArticleAccepted Version Ilg, P. (2019) Diffusionjump model for the combined Brownian and Neel relaxation dynamics of ferrofluids in the presence of external fields and flow. Physical Review E, 100 (2). 022608.Relaxation of suspended magnetic nanoparticles occurs via Brownian rotational diffusion of the particle as well as internal magnetization dynamics. The latter is often modeled by the stochastic Landau-Lifshitz equation, but its numerical treatment becomes prohibitively expensive in many practical applications due to a time-scale separation between fast, Larmor-type precession and slow, barrier-crossing dynamics. Here, a diffusion-jump model is proposed to take advantage of the timescale separation and to approximate barrier-crossings as thermally activated jump processes that occur alongside rotational diffusion. The predictions of our diffusion-jump model are compared to reference results obtained by solving the stochastic Landau-Lifshitz equation coupled to rotational Brownian motion. Good agreement is found in the regime of high energy barriers where Néel relaxation can be considered a thermally activated rare event. While many works in the field have neglected Néel relaxation altogether, our approach opens the possibility to efficiently include Néel relaxation also into interacting many-particle models.

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“…Our experience with the Stokes wave in [38] suggests that generally the number of sheets is infinite. Some exceptional cases like found in [29,30] have only a finite number of sheets of Riemann surface (these solutions however have diverging values of V and R at w → ∞). We suggest that the detailed study of such many- and infinite-sheet Riemann surfaces is one of the most important goal in free-surface hydrodynamics.…”

Section: Short Branch Cut Approximation and Square Root Singularity Solutionsmentioning

confidence: 99%

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

Dyachenko

Lushnikov

et al. 2021

Proc. R. Soc. A.

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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.

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Exact Local Solutions for the Formation of Singularities on the Free Surface of an Ideal Fluid

Cited by 35 publications

References 19 publications

Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

Diffusion-jump model for the combined Brownian and Néel relaxation dynamics of ferrofluids in the presence of external fields and flow

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

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