Formation of singularities on the free surface of an ideal fluid

Кузнецов, Е. А.; Spector, M.D.; Захаров, В. Е.

doi:10.1103/physreve.49.1283

Cited by 60 publications

(88 citation statements)

References 7 publications

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“…On the whole, such behavior is similar to that of the velocity potential of an ideal liquid in the absence of external forces [10,11,12]. In Sec.…”

Section: Introductionmentioning

confidence: 75%

“…[10,11,12], that these relations describe (if we require that v be analytic) the formation of a singularity in a finite time. The problem of finding the explicit solution reduces to analyzing the map x → x ′ specified by Eq.…”

Section: Formation Of a Branch Point Of The Root Typementioning

confidence: 99%

“…More than that, if we replace x by x → x + it, we arrive at the equation derived in Refs. [10,11,12] as a result of a discussion of the nonlinear dynamics of a free surface of ideal liquid in the absence of external forces, where it describes the temporal evolution of the complex-valued velocity. The solution of this first-order partial differential equation can be found by using the method of characteristics:…”

Section: Formation Of a Branch Point Of The Root Typementioning

confidence: 99%

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Formation of singularities on the surface of a liquid metal in a strong electric field

Зубарев

1998

J. Exp. Theor. Phys.

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The nonlinear dynamics of the free surface of an ideal conducting liquid in a strong external electric field is studied. It is establish that the equations of motion for such a liquid can be solved in the approximation in which the surface deviates from a plane by small angles. This makes it possible to show that on an initially smooth surface for almost any initial conditions points with an infinite curvature corresponding to branch points of the root type can form in a finite time.

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“…On the whole, such behavior is similar to that of the velocity potential of an ideal liquid in the absence of external forces [10,11,12]. In Sec.…”

Section: Introductionmentioning

confidence: 75%

Section: Formation Of a Branch Point Of The Root Typementioning

confidence: 99%

Section: Formation Of a Branch Point Of The Root Typementioning

confidence: 99%

See 1 more Smart Citation

Formation of singularities on the surface of a liquid metal in a strong electric field

Зубарев

1998

J. Exp. Theor. Phys.

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“…It should be mentioned that such a behavior of the charged surface is similar to the behavior of a free surface of an ideal fluid in the absence of external forces [5,6], though the singularities are of a different nature (in the latter case the singularity formation is connected with inertial forces).…”

mentioning

confidence: 99%

“…Kuznetsov for attracting my attention to Refs. [5,6]. This work was supported by Russian Foundation for Basic Research, Grant No.…”

mentioning

confidence: 99%

Formation of root singularities on the free surface of a conducting fluid in an electric field

Зубарев

1998

Physics Letters A

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The formation of singularities on a free surface of a conducting ideal fluid in a strong electric field is considered. It is found that the nonlinear equations of two-dimensional fluid motion can be solved in the small-angle approximation. This enables us to show that for almost arbitrary initial conditions the surface curvature becomes infinite in a finite time.Electrohydrodynamic instability of a free surface of a conducting fluid in an external electric field [1,2] plays an essential role in a general problem of the electric strength. The interaction of strong electric field with induced charges at the surface of the fluid (liquid metal for applications) leads to the avalanche-like growth of surface perturbations and, as a consequence, to the formation of regions with high energy concentration which destruction can be accompanied by intensive emissive processes.In this Letter we will show that the nonlinear equations of motion of a conducting fluid can be effectively solved in the approximation of small perturbations of the boundary. This allows us to study the nonlinear dynamics of the electrohydrodynamic instability and, in particular, the most physically meaningful singular solutions.Let us consider an irrotational motion of a conducting ideal fluid with a free surface, z = η(x, y, t), that occupies the region −∞ < z ≤ η(x, y, t), in an external uniform electric field E. We will assume the influence of gravitational and capillary forces to be negligibly small, which corresponds to the conditionwhere g is the acceleration of gravity, α is the surface tension coefficient, and ρ is the mass density. The potential of the electric field ϕ satisfies the Laplace equation, ∆ϕ = 0, with the following boundary conditions, ϕ → −Ez, z → ∞,

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Singularity formation in thin jets with surface tension

Pugh

Shelley

1998

Comm. Pure Appl. Math.

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We derive and study asymptotic models for the dynamics of a thin jet of fluid that is separated from an outer immiscible fluid by fluid interfaces with surface tension. Both fluids are assumed to be incompressible, inviscid, irrotational, and density‐matched. One such thin jet model is a coupled system of PDEs with nonlocal terms—Hilbert transforms—that result from expansion of a Biot‐Savart integral. In order to make the asymptotic model well‐posed, the Hilbert transforms act upon time derivatives of the jet thickness, making the system implicit. Within this thin jet model, we demonstrate numerically the formation of finite‐time pinching singularities, where the width of the jet collapses to zero at a point. These singularities are driven by the surface tension and are very similar to those observed previously by Hou, Lowengrub, and Shelley in large‐scale simulations of the Kelvin‐Helmholtz instability with surface tension and in other related studies. Dropping the nonlocal terms, we also study a much simpler local model. For this local model we can preclude analytically the formation of certain types of singularities, though not those of pinching type. Surprisingly, we find that this local model forms pinching singularities of a very similar type to those of the nonlocal thin jet model. © 1998 John Wiley & Sons, Inc.

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Formation of singularities on the free surface of an ideal fluid

Cited by 60 publications

References 7 publications

Formation of singularities on the surface of a liquid metal in a strong electric field

Formation of singularities on the surface of a liquid metal in a strong electric field

Formation of root singularities on the free surface of a conducting fluid in an electric field

Singularity formation in thin jets with surface tension

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