Stability of Capillary Waves of Finite Amplitude

2017 ◽  
pp. 51-57
Author(s):  
Alexander Petrov ◽  
Mariana Lopushanski ◽  
Vladimir Vanovskiy
Keyword(s):  
Finite Amplitude ◽  
Capillary Waves

A new family of capillary waves

1980 ◽  
Vol 98 (1) ◽  
pp. 161-169 ◽  
Author(s):  
Jean-Marc Vanden-Broeck ◽  
Joseph B. Keller
Keyword(s):  
Integral Equation ◽  
Nonlinear Integral Equation ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Wave Steepness ◽  
Algebraic Equations ◽  
Infinite Depth ◽  
New Family ◽  
Nonlinear Algebraic Equations ◽  
Finite Set

A new family of finite-amplitude periodic progressive capillary waves is presented. They occur on the surface of a fluid of infinite depth in the absence of gravity. Each pair of adjacent waves touch at one point and enclose a bubble at pressure P. P depends upon the wave steepness s, which is the vertical distance from trough to crest divided by the wavelength. Previously Crapper found a family of waves without bubbles for 0 [les ] s [les ] s* = 0·730. Our solutions occur for all s > s** = 0·663, with the trough taken to be the bottom of the bubble. As s → ∞, the bubbles become long and narrow, while the top surface tends to a periodic array of semicircles in contact with one another. The solutions were obtained by formulating the problem as a nonlinear integral equation for the free surface. By introducing a mesh and difference method, we converted this equation into a finite set of nonlinear algebraic equations. These equations were solved by Newton's method. Graphs and tables of the results are included. These waves enlarge the class of phenomena which can occur in an ideal fluid, but they do not seem to have been observed.

The blockage of gravity and capillary waves by longer waves and currents

1990 ◽  
Vol 217 ◽  
pp. 115-141 ◽  
Author(s):  
Jinn-Hwa Shyu ◽  
O. M. Phillips
Keyword(s):  
Multiple Scale ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Reflected Waves ◽  
Dominant Wave ◽  
Constant Vorticity ◽  
Waves And Currents ◽  
Uniform Solution ◽  
Group Velocities

Surface waves superimposed upon a larger-scale flow are blocked at the points where the group velocities balance the convection by the larger-scale flow. Two types of blockage, capillary and gravity, are investigated by using a new multiple-scale technique, in which the short waves are treated linearly and the underlying larger-scale flows are assumed steady but can have a considerably curved surface and uniform vorticity. The technique first provides a uniformly valid second-order ordinary differential equation, from which a consistent uniform asymptotic solution can readily be obtained by using a treatment suggested by the result of Smith (1975) who described the phenomenon of gravity blockage in an unsteady current with finite depth.The corresponding WKBJ solution is also derived as a consistent asymptotic expansion of the uniform solution, which is valid at points away from the blockage point. This solution is obviously represented by a linear combination of the incident and reflected waves, and their amplitudes take explicit forms so that it can be shown that even with a significantly varied effective gravity g’ and constant vorticity, wave action will remain conserved for each wave. Furthermore, from the relative amplitudes of the incident and reflected waves, we clearly demonstrate that the action fluxes carried by the two waves towards and away from the blockage point are equal within the present approximation.The blockage of gravity–capillary waves can occur at the forward slopes of a finite-amplitude dominant wave as suggested by Phillips (1981). The results show that the blocked waves will be reflected as extremely short capillaries and then dissipated rapidly by viscosity. Therefore, for a fixed dominant wave, all wavelets shorter than a limiting wavelength will be suppressed by this process. The minimum wavelengths coexisting with the long waves of various wavelengths and slopes are estimated.

Nonlinear evolution of waves on a vertically falling film

1993 ◽  
Vol 250 ◽  
pp. 433-480 ◽  
Author(s):  
H.-C. Chang ◽  
E. A. Demekhin ◽  
D. I. Kopelevich
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Stationary Waves ◽  
Falling Film ◽  
Linear Mode

Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ1travels slower than infinitesimally small waves of the same wavelength while the other family γ2and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ1with wavenumber αs(or α2) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber αmand approaches the wave on γ1with the highest flow rate at low Reynolds numbers. On the fast γ2family, however, multiple bands of near-solitary waves bounded below by αfare found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α2of the slow γ1family. They then approach the long stable waves on the γ2family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.

scholarly journals Study of finite amplitude capillary waves stability

2018 ◽  
Vol 211 ◽  
pp. 04008
Author(s):  
Alexander Petrov ◽  
Mariana Lopushanski
Keyword(s):  
Lyapunov Function ◽  
Lyapunov Method ◽  
Capillary Wave ◽  
Momentum Conservation ◽  
Finite Amplitude ◽  
Positive Definite ◽  
Capillary Waves ◽  
Dynamic Equations ◽  
Direct Lyapunov Method ◽  
Energy And Momentum

The direct Lyapunov method is used to study capillary waves. The dynamic equations of the capillary wave are presented in the form of an infinite Euler-Lagrange chain of equations for the Stokes coefficients. The stationary solution found for these equations is the Crapper solution for capillary waves. With the help of energy and momentum conservation laws the Lyapunov function is constructed. It is shown that the Lyapunov function is positive definite with respect to any perturbations of waves surfaces with the period that is a multiple of the wave period.

scholarly journals The stability of capillary waves on fluid sheets

2016 ◽  
Vol 806 ◽  
pp. 5-34 ◽  
Author(s):  
M. G. Blyth ◽  
E. I. Părău
Keyword(s):  
Nonlinear Waves ◽  
Time Integration ◽  
Lower Surface ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Mapping Technique ◽  
Surprising Result ◽  
Periodic Waves ◽  
The Stability ◽  
The Waves

The linear stability of finite-amplitude capillary waves on inviscid sheets of fluid is investigated. A method similar to that recently used by Tiron & Choi (J. Fluid Mech., vol. 696, 2012, pp. 402–422) to determine the stability of Crapper waves on fluid of infinite depth is developed by extending the conformal mapping technique of Dyachenko et al. (Phys. Lett. A, vol. 221 (1), 1996a, pp. 73–79) to a form capable of capturing general periodic waves on both the upper and the lower surface of the sheet, including the symmetric and antisymmetric waves studied by Kinnersley (J. Fluid Mech., vol. 77 (02), 1976, pp. 229–241). The primary, surprising result is that both symmetric and antisymmetric Kinnersley waves are unstable to small superharmonic disturbances. The waves are also unstable to subharmonic perturbations. Growth rates are computed for a range of steady waves in the Kinnersley family, and also waves found along the bifurcation branches identified by Blyth & Vanden-Broeck (J. Fluid Mech., vol. 507, 2004, pp. 255–264). The instability results are corroborated by time integration of the fully nonlinear unsteady equations. Evidence is presented for superharmonic instability of nonlinear waves via a collision of eigenvalues on the imaginary axis which appear to have the same Krein signature.

scholarly journals Dispersion and viscous attenuation of capillary waves with finite amplitude

2017 ◽  
Vol 226 (6) ◽  
pp. 1229-1238 ◽  
Author(s):  
Fabian Denner ◽  
Gounséti Paré ◽  
Stéphane Zaleski
Keyword(s):  
Finite Amplitude ◽  
Capillary Waves

Nonlinear gravity–capillary surface waves in a slowly varying current

1973 ◽  
Vol 57 (4) ◽  
pp. 797-802 ◽  
Author(s):  
Dennis Holliday
Keyword(s):  
Surface Waves ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Eikonal Approximation ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Capillary Surface ◽  
Conservation Equations ◽  
Wave Slope ◽  
Nonlinear Gravity

The propagation of nonlinear gravity–capillary surface waves in a deep slowly varying current is investigated using the conservation equations in the eikonal approximation. Graphical comparisons are made between solutions of the wave- slope and wavenumber equations for infinitesimal waves and finite amplitude waves. Finite amplitude effects are shown to be weaker for small amplitude capillary waves than for gravity waves. The ‘wave barrier’ noted by Gargett & Hughes (1972) for infinitesimal gravity waves on a slowly varying current is seen to be removed by finite amplitude effects.

Steady nonlinear capillary waves on curved sheets

2001 ◽  
Vol 12 (6) ◽  
pp. 689-708 ◽  
Author(s):  
DARREN CROWDY
Keyword(s):  
Flow Field ◽  
Closed Form ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Free Surfaces ◽  
New Class ◽  
Finite Amplitude Waves

This paper presents a new class of solutions for steady nonlinear capillary waves on a curved sheet of fluid in the plane. The solutions are exact in that the free surfaces of the sheet and the associated flow field can be found in closed form. The solutions are generalizations of the classic solutions for finite amplitude waves on fluid sheets [5] to the case where the fluid sheets are curved.

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