# Dynamics of fully nonlinear capillary–gravity solitary waves under normal electric fields

## Abstract

Two-dimensional capillary–gravity waves travelling under the effect of a vertical electric field are considered. The fluid is assumed to be a dielectric of infinite depth. It is bounded above by another fluid which is hydrodynamically passive and perfectly conducting. The problem is solved numerically by time-dependent conformal mapping methods. Fully nonlinear waves are presented, and their stability and dynamics are studied.

## Keywords

Surface wave Solitary wave Wave interactions## 1 Introduction

Water waves propagating on the interface between two fluids have been studied intensively using either analytical or numerical methods. Many different mathematical methods have been introduced to study the steady or time-dependent solutions both in shallow and deep waters (for review, see, e.g. [1, 2] and the references therein). In the case of deep water, it is well acknowledged that there exist two families of capillary–gravity solitary waves bifurcating from the minimum of phase speed-denoted elevation and depression waves. In [3], the stability of these waves was studied using a numerical spectral analysis. It was found that depression waves with single-valued profiles were stable, whereas there was a stability exchange on the branch of elevation waves. These results were later verified numerically by Milewski et al. [1]. Recently, the problem of dynamics and stability was investigated systematically by Wang [4] where the depression waves with overhanging structure were proved to be also stable. On the experimental side, early experiments on three-dimensional capillary–gravity waves in a wind-wave tank were carried out by Zhang [5]. Fully localised lumps were observed. Later wavepacket solitary waves were generated in deep water by Diorio et al. [6].

In the presence of electric fields, this topic attracted much attention because it has many physical and industrial applications such as cooling systems and coating processes. In [7, 8], capillary waves on a fluid sheet under the effects of horizontal electric fields were studied. Fully nonlinear solutions were computed using a boundary integral equation method, and weakly nonlinear solutions were studied analytically by assuming the long-wave approximation limit. The effect of vertical electric fields was investigated in [9, 10, 11] where an asymptotic model equation for long waves was derived. Once again fully nonlinear solutions were computed by a boundary integral equation method and compared to the ones produced by the long-wave equation. To our knowledge, there have been, so far, no studies of time-dependent fully nonlinear water waves under the influence of electric fields.

The present work considers a two-dimensional dielectric fluid of infinite depth bounded above by a perfectly conducting gas such as plasma. The electric field is applied vertically throughout the space. This particular configuration converts a two-layer problem to the one of a one layer. Therefore, all the mathematical techniques which are used to solve the capillary–gravity problem can be inherited. Future work will involve cases where the upper layer also plays an active role in the dynamics.

## 2 Formulation

*y*-axis directed vertically upwards and \(y=0\) at the undisturbed level (Fig. 1). The gravity

*g*and the surface tension \(\sigma \) are both included in the formulation. The deformation of the free surface is denoted by \(\zeta (x,t)\). A vertical electric field with voltage potential

*v*is applied. We assume that \(v \sim V_0y\) as \(y\rightarrow -\infty \), where \(V_0\) is a constant. Since the fluid motion can be described by a velocity potential function \(\phi (x,y,t)\), introducing dimensionless variables by choosing

*c*is the phase velocity. Saffman’s work was based on the Hamiltonian formulation (see [12]) where only gravity is considered. It can be generalised to include surface tension and electric fields due to the Hamiltonian structure (12), (13). Since all the perturbations are superharmonic for solitary waves, this argument is particularly useful in our numerical studies of stability.

*x*-direction. The dispersion relation admits a minimum of

*c*at \(k=1\) whenever \(0\le \beta <2\). This minimum phase speed is denoted by \(c^*\). Wave-packet like solitary waves with decaying tails bifurcate from that point. We have immediately

*A*of which has the explicit solution:

*C*is a constant. If \(\beta >2\), the electric field destabilises the system therefore no solitary waves can exist. When \(\beta \) approaches 2, the coefficients of \(A_{\chi \chi }\) and \(|A|^2 A\) in (22) both tend to infinity. The variable

*T*is required to be of order \(O( {\sqrt{2-\beta }})\) to balance equation (22) which violates the initial assumptions when \(\sqrt{2-\beta }=o(1)\). Consequently the amplitude and horizontal length need to be rescaled and for \(\beta \) near 2 the envelope becomes very broad and small (we note that \(\omega \rightarrow 0\) when \(\beta \rightarrow 2\)). It is expected that the solutions approach linear sinusoidal waves in the limit \(\beta \rightarrow 2\).

## 3 Numerical scheme

*PV*demotes the principal value of the integral. We note that \(V_\xi =0\) as

*v*is identically zero everywhere on the free surface. Next we follow [1] to derive the time-evolution equations

*N*terms. The coefficients \(a_n\) and \(b_n\) are the unknowns to be found. In particular for symmetric waves, we choose \(\xi =0\) to be the axis of symmetry, i.e. the coefficients \(b_n\) are zero. The wavelength 2

*L*and the number of modes

*N*are chosen to be sufficiently large so that the solutions do not change when

*L*and

*N*are further increased. We discretise the domain \([-L,L)\) in the mapped plane with a uniform mesh. The dynamic boundary condition (34) is satisfied on these grid points. The resultant system is solved by Newton’s method for the unknown Fourier coefficients. In most computations, we use 4096 or 6144 collocation points. The iterations are stopped when the \(l^\infty \)-norm of residual errors are less than \(10^{-10}\). This numerical approach has been successfully used in [1, 4, 18] for capillary–gravity waves and in [19, 20, 21] for flexural–gravity waves.

## 4 Numerical results

### 4.1 Travelling waves

*c*and appear almost identical.

In conclusion, numerical evidence shows that the wave speeds are slowed down due to the effect of electric fields as predicted by (36). In addition, the branch of elevation waves split into two separate parts, when \(\beta >\beta ^{\dagger }\), and the lower part disappears when \(\beta >\beta ^{\ddagger }\). Our computations for \(\beta =1\) stop at the pentagram in Fig. 2; however, the problem does not end there. The branch continues to snake back and forth where more turning points are expected. There exist other separate solution branches when \(\beta >\beta ^{\dagger }\). As \(\beta \) approaches 2 with many branches reaching \(c=0\), there only remain two branches which are monotonic in *c*. In the next section, we investigate the stability of the computed solitary wave solutions with time-dependent computation.

### 4.2 Stability analysis

We now investigate numerically the stability of the solitary waves by introducing a small disturbance and solving the initial value problem. A frame of reference moving with the initial wave speed is always chosen, and we evolve the resultant initial condition in time. We recall that in the absence of the electric fields, the depression solitary waves are stable, whereas the elevation waves are unstable near the bifurcation point but become stable after the first turn of the branch of elevation waves at \(c\simeq 1.24\) (see [1, 3, 4]).

We now describe our results with the inclusion of the electric fields. The numerical calculations presented here are carried out with \(\beta =1\). The results are qualitatively similar to those of the capillary–gravity problem. A variety of perturbations with \(\pm 5\%\) of the amplitude of the depression waves show no sign of instabilities. One example is presented in Fig. 6. A depression solitary wave ensues with smaller amplitude and therefore travelling faster than the original waves. This explains why the wave moves rightwards as time increases since the wave speed of the smaller wave is greater than the speed of the reference frame. As there is no stationary point or turning point on the curve of *H*(*c*) for depression waves, no stability exchange occurs on this branch, i.e. the depression branch is stable.

Similarly, we apply a disturbance with \(\pm 1\%\) of the amplitude to elevation waves from the elevation branch on the segment between the bifurcation point and the diamond symbol in Fig. 2. The wave eventually evolves into a depression solitary wave as presented in Fig. 7. On the next segment of elevation waves between the diamond and the pentagram, solitary waves are stable as confirmed by the numerical results in Fig. 8. Computations were carried out for other values of \(\beta \) also and yielded similar results to those presented above.

### 4.3 Collision

From the numerical experiments of collisions, we have observed that larger waves get larger (and slower) and smaller waves get smaller (and faster). This is due to the inelastic nature of the collisions where energy exchange always takes place.

### 4.4 Excitation

*A*is chosen to be 0.02. The forcing is initially placed at \(x=-250\) and later switched off at \(t=100\). Snapshots taken at \(t=25\), 100, 400 are presented in Fig. 18a. Immediately after removing the pressure disturbance at \(t=100\), a depression solitary wave forms on the surface. The long time evolution (up to \(t=400\)) shows that this depression wave continues to propagate without losing its main structure. Hence, the results confirm the stability of the fully localised response. In the next set of numerical experiments, we reduce the strength of electric field progressively by setting

With the presence of electric fields, excitations can be realised with single or multiple moving pressure disturbances at a relatively low speed, compared to the case of capillary–gravity waves. By further turning down the electric field strength, the excited depression and elevation waves both maintain their wave structure, i.e. capillary–gravity waves emerge. We propose this as an alternative possible way to excite capillary–gravity solitary waves.

## 5 Conclusion

The problem of capillary–gravity flows under normal electric fields was considered. By means of a conformal mapping, we found steady and time-dependent solutions for various values of the dimensionless electric field strength \(\beta \). In particular, when \(\beta <\beta ^{\dagger }\approx 1.5265\), the properties of waves are qualitatively similar to those for the capillary–gravity waves. The branch of elevation waves split into two separate parts when \(\beta >\beta ^{\dagger }\), and the lower part disappears when \(\beta >\beta ^{\ddagger }\approx 1.9809\). The stability problem was studied systematically using a time-dependent numerical scheme for \(\beta =1\). Several numerical experiments of head-on and overtaking collisions were performed. The stable depression and elevation waves were excited, respectively, using a single load and multiple loads moving with a speed which is slightly less than the minimum phase speed. Overall, the electrified waves have similar properties to those of classical capillary–gravity waves. The main advantage with the presence of normal electric fields is that the waves can be slowed down while keeping the shape. This could open new scenarios for experimentation.

## Notes

### Acknowledgements

The authors would like to thank Z. Wang (Chinese Academy of Sciences) for insightful discussions. T. Gao was supported by the London Mathematical Society under Grant PMG/16-17/17. The work of P. Milewski was supported by the EPSRC under Grant EP/N018176/1. The work of D.T. Papageorgiou was supported by the EPSRC under Grants EP/K041134/1 and EP/L020564/1. The work of J.-M. Vanden-Broeck was supported by the EPSRC under the Grant EP/N018559/1.

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