Korteweg–de Vries solitons on electrified liquid jets

The propagation of axisymmetric waves on the surface of a liquid jet under the action of a radial electric field is considered. The jet is assumed to be inviscid and perfectly conducting, and a field is set up by placing the jet concentrically inside a perfectly cylindrical tube whose wall is maintained at a constant potential. A nontrivial interaction arises between the hydrodynamics and the electric field in the annulus, resulting in the formation of electrocapillary waves. The main objective of the present study is to describe nonlinear aspects of such axisymmetric waves in the weakly nonlinear regime, which is valid for long waves relative to the undisturbed jet radius. This is found to be possible if two conditions hold: the outer electrode radius is not too small, and the applied electric field is sufficiently strong. Under these conditions long waves are shown to be dispersive and a weakly nonlinear theory can be developed to describe the evolution of the disturbances. The canonical system that arises is the Kortweg–de Vries equation with coefficients that vary as the electric field and the electrode radius are varied. Interestingly, the coefficient of the highest-order third derivative term does not change sign and remains strictly positive, whereas the coefficient $\alpha$ of the nonlinear term can change sign for certain values of the parameters. This finding implies that solitary electrocapillary waves are possible; there are waves of elevation for $\alpha >0$ and of depression for $\alpha <0$. Regions in parameter space are identified where such waves are found.

Figure 1

Schematic of the problem showing the liquid thread placed concentrically inside a cylindrical electrode of radius $d$.

Figure 2

Linear dispersion relation Eq. (8) showing stabilization of long waves due to the presence of an electric field; the outer electrode is at $d=5a$. In the absence of a field, $E_b=0$, the classical Rayleigh instability is found (thick solid curve), while the waves become dispersively stable as $E_b$ increases from 5 to 20 as shown.

Figure 3

Different regions in the $E_b\text{−}D$ plane where admissible solitary waves can be found. The curves indicate where $c^2=0$ (solid), $\beta =0$ (dashed), and $\alpha =0$ (dotted); the corresponding values of the parameters are positive above the curves and negative below the curves. The open circle indicates the point $lnD=1.75,E_b=12.5$ [i.e., $\alpha =1.649,\beta =6.5874$ in Eq. (48)], and the square corresponds to $lnD=1.4,E_b=8$ [i.e., $\alpha =-1.0841,\beta =3.5630$ in Eq. (48)]. The corresponding solitary waves are plotted in Fig. 4.

Figure 4

Elevation (top) and depression (bottom) of solitary waves corresponding to the open circle and square, respectively, in Fig. 3; the parameter values are $lnD=1.17,E_b=12.5$ and $lnD=1.4,E_b=8$.