The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.     

The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.     

A Formulation for Water Waves over Topography

Many interesting free-surface flow problems involve a varying bottom. Examples of such flows include ocean waves propagating over topography, the breaking of waves on a beach, and the free surface of a uniform flow over a localized bump. We present here a formulation for such flows that is general and, from the outset, demonstrates the wave character of the free-surface evolution. The evolution of the free surface is governed by a system of equations consisting of a nonlinear wave-like partial differential equation coupled to a time-independent linear integral equation. We assume that the free-surface deformation is weakly nonlinear, but make no a priori assumption about the scale or amplitude of the topography. We also extend the formulation to include the effect of mean flows and surface tension. We show how this formulation gives some of the well-known limits for such problems once assumptions about the amplitude and scale of the topography are made.     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

Orbital stability of solitary waves on a ferrofluid jet

We prove the orbital stability of small-amplitude axisymmetric solitary waves on the surface of an incompressible, inviscid ferrofluid jet. The ferrofluid surrounds a current-carrying rod and is subject to the azimuthal magnetic field generated by the rod. We show that under appropriate assumptions on the magnitude of the magnetic intensity in the ferrofluid, both the trivial flow and the solitary waves with strong surface tension are conditionally orbitally stable, while the conditional orbital stability of solitary waves with near-critical surface tension can be deduced from properties of the corresponding dispersive PDE model equation. The arguments are based on the recent orbital stability results for internal waves by Chen and Walsh (2022) and an improved version of the Grillakis–Shatah–Strauss method introduced by Varholm et al. (2020).     

Capillary waves with variable surface tension

The classical problem of capillary waves propagating at a constant velocity at the surface of a fluid of infinite depth is reexamined. The surface tension is assumed to vary along the free surface. The problem is solved numerically by series truncation. It is shown that the properties of the waves are qualitatively similar to those of waves with constant surface tension and that there are nonsymmetric waves with variable surface tension.     

The zero surface tension limit two‐dimensional water waves

We consider two‐dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases. © 2005 Wiley Periodicals, Inc.     

Travelling waves of the KP equations with transverse modulations

Kadomtsev-Petviashvili (KP) equations arise genetically in modelling nonlinear wave propagation for primarily unidirectional long waves of small amplitude with weak transverse dependence. In the case when transverse dispersion is positive (such as for water waves with large surface tension) we investigate the existence of transversely modulated travelling waves near one-dimensional solitary waves. Using bifurcation theory we show the existence of a unique branch of periodically modulated solitary waves. Then, we briefly discuss the case when the transverse dispersion is negative (such as for water waves with zero surface tension).     

Whitecapping

The work we describe addresses the process of whitecapping. We first argue that, when the winds are strong enough, the ocean surface must develop an alternative means to dissipate energy when its flux from large to small scales becomes too large. We then show that the resulting Phillips' spectrum, which holds at small or meter length scales, is dominated by sharp crested waves. We next idealize such a sea locally by a family of close to maximum amplitude Stokes waves and show, using highly accurate simulation algorithms based on a conformal map representation, that perturbed Stokes waves develop the universal feature of an overturning plunging jet. We analyze both the cases when surface tension is absent and present. In the latter case, we show the plunging jet is regularized by capillary waves that rapidly become nonlinear Crapper waves in whose trough pockets whitecaps may be spawned. We are careful not to claim this as the definitive mechanism for whitecaps because three‐dimensional effects, although qualitatively discussed, are not included in the analysis.     

Lagrangian motion of fluid particles in gravity–capillary standing waves

A third-order analytical solution for the gravity–capillary standing wave is derived in Lagrangian coordinates through the Lindstedt–Poincare perturbation method. By numerical computation, the dynamical properties of nonlinear standing waves with surface tension in finite water depth, including particle trajectory and surface profile are investigated. We find that the presence of surface tension leads to a change of the crest form. Moreover, we also find that the particle trajectories near the surface oscillate back and forth along the arcs which will change from concave to convex as the inverse Bond number increases. There is no mass transport of the particles in a wave period.     

Capillary-gravity waves produced by a moving pressure distribution

Summary The initial-value problem of surface waves generated by a moving oscillatory pressure distribution is considered and the effect of surface tension on such waves is studied in detail. It is found that the surface tension modifies the critical case in a remarkable way. And also it introduces two extra waves that exist in the upstream side of the pressure distribution.

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Zusammenfassung Das Anfangswertproblem der Oberflächenwellen die durch eine bewegte oszillierende Druckverteilung erzeugt werden, wird behandelt und der Einfluss der Oberflächenspannung wird ausführlich untersucht. Es wird gefunden, dass die Oberflächenspannung den kritischen Fall in bemerkenswerter Weise beeinflusst, und auch zwei besondere Wellen stromaufwärts von der Druckstörung verursacht.

     

Nonlinear Stationary Surface Waves Above an Underwater Ridge

Given an ideal incompressible heavy irrotational fluid, we consider the exact statement of the problem on gravitational-capillary surface waves of small amplitude travelling along an underwater ridge. We show that, under some requirements on the shape of the bottom and the surface tension, the equations of an ideal incompressible fluid have smooth solutions periodic in the variable directed along the underwater ridge and decreasing exponentially with a small positive exponent in the perpendicular direction.     

A Model Equation to Study the Effects of Nonlinearity,Surface Tension and Viscosity in Water Waves

The formation of short capillary waves on long, finite amplitude gravity waves is studied by solving numerically a non-linear partial differential equation which models effects of surface tension, viscosity, unsteadiness and finite amplitude.     

圆柱形容器中竖直激励表面波的毛细影响

在竖直振动的圆柱形容器中,利用理想流体中两时间尺度奇摄动展开法,研究了包括表面张力影响的自由面单一表面驻波的运动．通过求解势流方程,获得了一个包含三阶非线性项、外激励及表面张力影响的非线性振幅方程．结果表明当驱动频率较低时,表面张力对表面波模式选择不重要;然而,当驱动频率较高时,表面张力的影响是不可忽略的．说明表面张力具有使得自由面返回到平衡位置的作用．另外,由于考虑了表面张力的影响,使得理论结果比无表面张力时更加接近先前的实验结果．     

Qualitative theory of nonlinear steady water waves

In this paper, the nonlinear boundary problem describing two-dimensional steady waves on the surface of water with finite depth is discussed. The problem is formulated in the conventional statement (the gravity is taken into account, but the surface tension is neglected). The latter one allows discussing the whole class of bounded waves that includes periodic waves, solitary waves, and other types of waves (for instance, almost-periodic waves, although their existence has not been established yet). This fact suggests that the results obtained fall within the domain of the qualitative theory of differential equations (investigation of the properties of solutions without finding them). In this paper, two approaches to the qualitative theory are discussed. The first approach consists in averaging the solution along the vertical sections of the region, and the second approach is based on the authors’ modification of Byatt-Smith’s integro-differential equation. Thus, the paper contains an overview of the results obtained for the problem of nonlinear stationary waves on water with finite depth. Two approaches to this problem form a basis of the qualitative theory of such waves, because there are no constraints imposed on the waves except for the boundedness of their profiles and steepness restrictions.     

Generalized solitary waves in the gravity-capillary Whitham equation

We study the existence of traveling wave solutions to a unidirectional shallow water model, which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between 0 and 1/3) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave.     

A new solitary wave solution for water waves with surface tension

In this paper we show that when the Froude number is less than but close to 1 and the Bond number is greater than but close to 1/3 there exists a new solitary wave solution for surface waves on water with surface tension. An approximate expression for the new solitary wave solution, which satisfies a fourth order ordinary differential equation and represents a wave of depression is presented.     

Center manifold approach to the control of a tethered satellite system

It is well known that in a linearized analysis the in-plane oscillation of a tethered satellite system about the radial earth pointing position decouples from the out-of-plane oscillation. By tension control, therefore, only the in-plane but not the out-of-plane oscillation can be affected. Hence, using tension control linearization of the equations of motion cannot be used and a nonlinear problem must be treated. For a simple mechanical model of a tethered satellite system we show by means of center manifold theory that for the nonlinear system the out-of-plane oscillations can be stabilized by tension control.     

Surfactant Influence on Water Wave Packets

The influence of surfactant on water wave packets is investigated. An envelope equation for a slowly varying wave packet in the potential flow equations with variable Bond number is derived. The properties of this equation depend on the relative phases of the wave packet and the distribution of surface tension. We observe that small variations in the Bond number may change the focusing nature of the envelope equation from that of the constant Bond number problem. Variations in Bond number can thus suppress, or incite, the Benjamin‐Feir instability. The existence of envelope solitary waves depends in a similar way on the Bond number variation. The envelope equation is also derived in a larger class of models.     

Modulational instability in a full‐dispersion shallow water model

We propose a shallow water model that combines the dispersion relation of water waves and Boussinesq equations, and that extends the Whitham equation to permit bidirectional propagation. We show that its sufficiently small and periodic traveling wave is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value, like the Benjamin‐Feir instability of a Stokes wave. We verify that the associated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability to the leading order in the amplitude parameter. We discuss the effects of surface tension. The results agree with those from a formal asymptotic expansion and a numerical computation for the physical problem.