Nonlinear Modes of Liquid Drops as Solitary Waves

Ludu, Andrei; Draayer, J. P.

doi:10.1103/physrevlett.80.2125

Cited by 54 publications

(44 citation statements)

References 19 publications

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“…Busse [19] has calculated ω for small-amplitude oscillations, treating small rotation rates as a perturbation and Radyakin [23] investigated the frequencies of the l = 2, m = ±2, ±1, 0 spherical harmonics at higher rotation rates using numerical methods, but the calculations for the |m| = 3 modes have yet to be attempted. Ludu and Draayer have developed a non-linear theory of large-amplitude traveling waves, on the scale observed in our experiments, for non-rotating droplets [24], but the case of rapid rotation has not been studied. For comparison, our measured critical frequencies are approximately 30% smaller than those predicted by Busse.…”

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Nonaxisymmetric Shapes of a Magnetically Levitated and Spinning Water Droplet

Hill

Eaves

2008

Phys. Rev. Lett.

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The shape of a weightless spinning liquid droplet is governed by the balance between the surface tension and centrifugal forces. The axisymmetric shape for slow rotation becomes unstable to a non-axisymmetric distortion above a critical angular velocity, beyond which the droplet progresses through a series of 2-lobed shapes. Theory predicts the existence of a family of 3-and 4-lobed equilibrium shapes at higher angular velocity. We investigate the formation of a triangular-shaped magnetically levitated water droplet, driven to rotate by the Lorentz force on an ionic current within the droplet. We also study equatorial traveling waves which give the droplet 3, 4 and 5-fold symmetry.In experiments published between 1863 and 1865, Plateau devised a way to study the behavior of liquids in the absence of gravity by suspending olive oil in a density-matched water/alcohol mixture [1]. In studies of a spinning oil droplet driven by a rotating shaft, he observed that the droplet shape, spherical at rest, became flattened at the poles as the droplet gained speed, whilst the equatorial diameter increased. Beyond a critical angular velocity, the droplet evolved into a spinning non-axisymmetric shape resembling a triaxial ellipsoid, which developed into a two-lobed shape with increasing speed. Plateau's experiments were inspired by the idea that the action of surface tension on the spinning droplet could model the influence of self-gravitation on the shape of spinning astronomical bodies [2]. It was later realized that the droplet model could provide insights into the behavior of atomic nuclei [2,3]. The analogies between the behavior of a liquid droplet and fundamental physics at both the astronomical and nuclear length scales has inspired some of the greatest mathematicians and physicists to study the instabilities of a rotating droplet (see [2]). Such analogies still attract great interest, as demonstrated by recent studies of the equilibrium shapes of relativistic rotating stellar masses [4,5] and of atomic nuclei [6].Although the simplicity of Plateau's technique for studying weightless fluids on Earth is attractive, comparison of the results with theory is complicated by shape deformations due to the viscous drag of the surrounding fluid. Here we avoid the problem of drag by using a strong magnetic field B to diamagnetically levitate centimetersized spinning water droplets in air. We have developed a 'liquid electric motor' technique to spin the droplet [7] and to realize a stable equilibrium droplet shape that has striking triangular symmetry in the plane of rotation.Brown and Scriven investigated the equilibrium shapes of a rigidly-rotating droplet theoretically using finite element analysis, grouping the shapes into 'families' according to their symmetry [8]. The 2-lobed shape family, which includes the ellipsoid-like shapes observed in Plateau's experiments and 2-lobed 'peanut' shapes observed in orbiting spacecraft [9,10] and in rolling 'liquid marbles' [11], branches from the axisymmetric family when the dimens...

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Nonaxisymmetric Shapes of a Magnetically Levitated and Spinning Water Droplet

Hill

Eaves

2008

Phys. Rev. Lett.

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“…review [4] and [5][6][7][8] for the recent refs.). However any extension of nonlinear dynamics from 1+1 to 2+1 and 3+1 dimensions meets principal difficulties.…”

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confidence: 99%

Nonlinear evolution of the axisymmetric nuclear surface

2002

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We consider an uniformly charged incompressible nuclear fluid bounded by a closed surface. It is shown that an evolution of an axisymmetric surface Γ(r, t) ≡ σ − Σ(z, t) = 0, r = (σ, φ, z) can be approximately reduced to a motion of a curve in the (σ, z)-plane. A nonlinear integro-diffrerential equation for the contour Σ(z, t) is derived. It is pointed on a direct correspondence between Σ(z, t) and a local curvature, that gives possibility to use methods of differential geometry to analyze an evolution of an axisymmetric nuclear surface. I. MOTIVATIONNonlinear dynamics of a nuclear surface is an object of special interest due to the following reasons. First, nuclear density falls considerably in the surface region, where the density fluctuations and clustering may be important. Second, different types of instability may be developed in the surface region and lead to fragmentation processes at low (fission, nucleon transfer) and high (multifragmentation, break-up etc.) energies. Finally, the liquid drop concept [1] for over a century are intensively used in macro-[2] and micro-physics [3]. Nonlinear dynamics of shapes in any complicated systems enevitably leads to mathematical problem of describing global geometric quantities such as surface and enclosed volume in different dimensions (polymers, cell membranes, 3D droplets).An application of a soliton concept to nonlinear nuclear hydrodynamics has given yet new possibilities in this field (See e.g. review [4] and [5][6][7][8] for the recent refs.). However any extension of nonlinear dynamics from 1+1 to 2+1 and 3+1 dimensions meets principal difficulties. The crucial point is to reduce the dimension of a problem. In paper [9] we considered the simplest two-dimensional nonlinear liquid objects. It was shown that 2D pure vortical motion of inviscid nuclear liquid can be reduced to 1D evolution of the contour bounding this drop. In the next paper [10] the extension to semi-3D geometry was done. The equations of motion describing localized vortex on a spherical nuclear surface -a bounded region of constant vorticity, surrounded by irrotational flux -were reduced to 1D nonlinear evolution of the boundary.In this short report we consider an uniformly charged incompressible nuclear 3D fluid bounded by a closed surface. It is shown that evolution of an axisymmetric surface Γ(r, t) ≡ σ − Σ(z, t) = 0, r = (σ, φ, z) can be approximately reduced to a motion of a curve in the (σ, z)-plane. II. FRAMEWORKThe evolution of one-body Wigner phase-space distribution function is analyzed instead of a full many-body wave function. Integrating the kinetic equationover the momentum space with different polynomial weighting functions of the p-variable one comes to an infinite chain of equations for local collective observables including the density, collective velocity, pressure and an infinite set of tensorial functions of the time and space coordinates, which are defined as moments of the distribution function in the momentum space: -the particle n(r, t) ≡ g dp f (r, p, t), and the mas...

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“…Also, it has been applied in plasma physics, radar, rheology [25,32], optical-fiber communications [37], super conductors [10], etc. Several numerical and analytical methods for solving fractional KdV equations have been introduced.…”

Section: Introductionmentioning

confidence: 99%

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

Sepehrian

Shamohammadi

2018

Arab. J. Math.

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A radial basis function method for solving time-fractional KdV equation is presented. The Caputo derivative is approximated by the high order formulas introduced in Buhman (Proc. Edinb. Math. Soc. 36:319-333, 1993). By choosing the centers of radial basis functions as collocation points, in each time step a nonlinear system of algebraic equations is obtained. A fixed point predictor-corrector method for solving the system is introduced. The efficiency and accuracy of our method are demonstrated through several illustrative examples. By the examples, the experimental convergence order is approximately 4 − α, where α is the order of time derivative.Mathematics Subject Classification 65D05 · 65M06 · 65N06 · 65N35

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Nonlinear Modes of Liquid Drops as Solitary Waves

Cited by 54 publications

References 19 publications

Nonaxisymmetric Shapes of a Magnetically Levitated and Spinning Water Droplet

Nonaxisymmetric Shapes of a Magnetically Levitated and Spinning Water Droplet

Nonlinear evolution of the axisymmetric nuclear surface

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

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