Solitary waves at arbitrary density ratios
We will construct solitary wave solutions of the system (3.13) and find the range of parameters for which they exist. Looking for travelling wave solutions of () of the form
$$\begin{aligned} S=S(\xi ),\quad W=W(\xi ),\quad \xi =X+cT, \end{aligned}$$
(4.1)
where c is the positive wavespeed, yields the coupled ordinary differential equations
$$\begin{aligned}&-\,cS_\xi +2(S^2W)_\xi =2W_\xi , \end{aligned}$$
(4.2a)
$$\begin{aligned}&-\,c(W-\alpha S W)_\xi -(\alpha S^2 W^2 -2SW^2 + \alpha W^2)_\xi \nonumber \\&\quad =\frac{\alpha }{F} S_\xi - \frac{\sigma }{1+\rho }S_{\xi \xi \xi } - \frac{E_b}{2(1+\rho )}\left[ \epsilon _+ \left( V^+_{Y}\right) ^2- \epsilon _-\left( V^-_{Y}\right) ^2\right] _\xi . \end{aligned}$$
(4.2b)
where c is the wave-speed. It is trivial to solve the first equation for W in terms of S, thereby eliminating W from the problem; we find
$$\begin{aligned} W=-\frac{1}{2} \frac{D+cS}{1-S^2}, \end{aligned}$$
(4.3)
where D is a constant of integration. In order to integrate (4.2b), we will need
$$\begin{aligned} \int \left[ \epsilon _+ \left( V^+_{Y}\right) ^2- \epsilon _-\left( V^-_{Y}\right) ^2\right] \mathrm{{d}}S= \frac{\epsilon _+}{\epsilon +1 +S(\epsilon -1)}+C_1, \end{aligned}$$
(4.4)
where \(C_1\) is a constant-this follows by use of the expressions (4.3). Next, we integrate (4.2b) with respect to \(\xi \), multiply the result by \(S_\xi \), carry out an additional \(\xi \) integration using (4.4), and yield the following dynamical system for S:
$$\begin{aligned} \frac{1}{2} \gamma (S_\xi )^2&= \frac{\alpha }{2F} S^2 +GS-\frac{1}{4}\alpha c^2 S+ H -\frac{1}{8} \frac{(D+c)^2(1-\alpha )}{1-S}\nonumber \\&\quad -\frac{1}{8} \frac{(D-c)^2(1+\alpha )}{1+S} -\frac{\delta }{2} \frac{1}{\epsilon +1+S(\epsilon -1)}, \end{aligned}$$
(4.5)
where G and H are constants and the parameters
$$\begin{aligned} \gamma =\frac{\sigma }{1+\rho }>0,\quad \delta =\frac{E_b \epsilon _+}{1+\rho }, \end{aligned}$$
(4.6)
measure surface tension and electric field strength, noting that \(\gamma >0\) and \(\delta \ge 0\). By requiring that as \(|\xi | \rightarrow \infty \), S and all its derivatives tend to 0 (thus restricting our solutions to solitary waves), we see from (4.3) that D is the jump in horizontal velocity across the interface, i.e. it is the undisturbed vortex sheet strength. Equation (4.5) becomes, after appropriate selection of G and H to achieve the desired decay at infinity,
$$\begin{aligned} \gamma (S_\xi )^2=S^2 \left[ \frac{\alpha }{F} + \frac{\alpha S -1}{2(1-S^2)}\left( D^2+c^2-2cD \frac{S-\alpha }{\alpha S -1}\right) \right] - \delta \frac{S^2(\epsilon -1)^2}{(\epsilon +1)^2(\epsilon +1+S(\epsilon -1))}. \end{aligned}$$
(4.7)
Note that for solitary waves, \(\gamma \) can be scaled to unity by a redefinition of \(\xi \). We can see that this equation implies that \(\alpha >0\) is a necessary condition for solitary waves to exist, hence excluding Rayleigh–Taylor unstable configurations as would be expected. Note that this condition is the same as in the non-electrified case considered by [16], consistent from the fact that a vertical electric field is destabilising. In order to analyse the properties of the solitary waves that may be present in the channel, it is useful to rewrite Eq. (4.7) in the form
$$\begin{aligned} \gamma (S_\xi )^2=\frac{S^2}{(1-S^2)\left[ \epsilon +1 +S(\epsilon -1)\right] }\,p_3(S), \end{aligned}$$
(4.8)
where \(p_3(S)\) is a cubic polynomial given by
$$\begin{aligned} p_3(S)=jS^3+kS^2 + lS+m. \end{aligned}$$
(4.9)
The constants j, k, l and m are given by
$$\begin{aligned}&j=\frac{\alpha }{F}(1-\epsilon ), \end{aligned}$$
(4.10a)
$$\begin{aligned}&k=\delta \frac{(\epsilon -1)^2}{(\epsilon +1)^2} +\alpha \left[ -\frac{\epsilon +1}{F} + \frac{1}{2}(\epsilon -1)\left( D^2+c^2-\frac{2cD}{\alpha }\right) \right] ,\end{aligned}$$
(4.10b)
$$\begin{aligned}&l=\frac{\alpha }{2}(\epsilon +1)\left( D^2+c^2-\frac{2cD}{\alpha }\right) +(\epsilon -1)\left( \frac{\alpha }{F}-\frac{D^2+c^2}{2}+cD\alpha \right) ,\end{aligned}$$
(4.10c)
$$\begin{aligned}&m=(\epsilon +1)\left( \frac{\alpha }{F}- \frac{D^2+c^2}{2} +\alpha D c \right) - \delta \frac{(\epsilon -1)^2}{(\epsilon +1)^2}. \end{aligned}$$
(4.10d)
Inspection of (4.8) and noting the facts \(|S|<1\) and \(\epsilon +1+S(\epsilon -1)>0\), implies that the sign of \(p_3\) determines the existence of solitary waves. First, we must have \(p_3(S)>0\) for small |S| so that \(S_\xi ^2\) is positive. By continuity, we require \(p_3(0)=m>0\) which gives the following necessary condition for the existence of solitary waves:
$$\begin{aligned} (\epsilon +1)\left( \frac{\alpha }{F}- \frac{D^2+c^2}{2} +\alpha Dc \right) - \delta \frac{(\epsilon -1)^2}{(\epsilon +1)^2}>0. \end{aligned}$$
(4.11)
Solitary waves emerge when we can connect \(S=0\) to another root(s) \(S_r\), say, as long as \(|S_r|<1\). For this to happen, we need at least one real root of \(p_3(S)\) to lie in \(-1<S<1\). Now,
$$\begin{aligned} p_3(-1)=-(\alpha +1)(D-c)^2< 0, \quad p_3(1)=\epsilon (\alpha -1)(D+c)^2< 0, \end{aligned}$$
(4.12)
with \(p_3(1)=0\) when \(\alpha =1\), i.e. \(\rho =0\). This is possible when the upper fluid is passive – this is considered in more detail later—see also [16]. Further special cases arise when \(D=\pm c\), in which case \(p_3\) has roots at either \(S=-1\) or \(S=1\), and in both cases, it is possible to have zero, one, or two solitary waves produced depending on the other parameters—this has been confirmed numerically, but details are not included for brevity. As these cases do not correspond to a meaningful physical limit since the interface touches the wall, they have not been investigated further.
From (4.12), we conclude that if it is assumed that \(p_3(S)\) has real roots in \(-1<S<1\) (something that is necessary for solitary waves to exist), then it must have exactly two roots or a repeated double root; otherwise, it is impossible to satisfy all the inequalities of (4.11), (4.12). The latter scenario is inadmissible because it forces \(S_\xi ^2<0\) in the vicinity of \(S=0\) even if the repeated root is at \(S=0\). The former scenario is again inadmissible unless the roots have opposite signs, and this is the generic situation for the existence of solitary waves in this rather general case. Interestingly, we establish that solitary waves come in pairs, an elevation wave corresponding to \(1>S_r^+>0\) and a depression (or dark) solitary wave for the root \(-1<S_r^-<0\). An illustration of this is provided in Fig. 2, in which regions where \((S_\xi )^2<0\) are shown for mathematical completeness despite this not being physical. Integration of (4.8) provides integrals for the solutions analogous to (4.19) constructed below.
Solitary waves with upper fluid of zero density (\(\alpha =1\))
When the upper fluid has negligible density compared to the lower fluid the Atwood ratio \(\alpha =1\). The equations simplify in this case, and in particular are not prone to a Kelvin–Helmholtz instability since the system is one-sided for the hydrodynamics. We begin with a general dielectric fluid in the lower region and then consider the additional limit of a perfectly conducting lower liquid first studied by Melcher [5] (Chapter 7).
Perfect dielectric lower fluid
In order for this model to be physical we will assume that the ratio of permittivities is \(\epsilon <1\). In this special case, \(S=1\) is a root of \(p_3\) so we can simplify (4.8) to
$$\begin{aligned} \gamma (S_\xi )^2=\frac{S^2}{(1+S)(\epsilon +1+S(\epsilon -1))} p_2(S), \end{aligned}$$
(4.13)
where
$$\begin{aligned} p_2(S)&=S^2 \frac{1}{F}(\epsilon -1)+S \left( \frac{2\epsilon }{F}-\delta \frac{(\epsilon -1)^2}{(\epsilon +1)^2} -\frac{1}{2}(\epsilon -1)(D-c)^2 \right) \nonumber \\&\quad - \,\delta \frac{(\epsilon -1)^2}{(\epsilon +1)^2}+\frac{1}{F}(\epsilon +1)-\frac{1}{2}(\epsilon +1)(D-c)^2. \end{aligned}$$
(4.14)
Evaluating \(p_2\) at \(S=0\) and requiring \(p_2(0)>0\), it can be seen that in this case, the necessary condition for solitary waves to be produced is
$$\begin{aligned} \frac{2}{F}-2\delta \frac{(\epsilon -1)^2}{(\epsilon +1)^3}-(D-c)^2>0. \end{aligned}$$
(4.15)
As in the previous section, we have that \(p_2(-1)=-(D-c)^2<0\), so for the existence of solitary waves, we require one root of \(p_2\) is \(-1<S<0\). However, the right-hand side of (4.13) is now finite as \(S\rightarrow 1\) and can take either a positive or negative value. Assuming that there is one root of \(p_2\) in the range \(-1<S<0\), if the second root is in \(0<S<1\), then there will be a second solitary wave produced. This extra condition is equivalent to requiring \(p_2(1)=\frac{4\epsilon }{F}-\frac{2\delta (\epsilon -1)^2}{(\epsilon +1)^2}-\epsilon (D-c)^2<0\). So given that there is one root in \(-1<S<0\), if \(p_2(1)\) is positive we have only one depression solitary wave, if it is negative, we have an additional elevation solitary wave. An example of how the value of \(p_2(1)\) depends on \(\epsilon \) is given in Fig. 3b, with \(\epsilon = 0.15\, 0.2\) giving \(p_2(1)<1\) so producing two solitary waves. From (4.15) and the expression for \(p_2(1)\), we have the explicit condition for the existence of two solitary waves:
$$\begin{aligned} \frac{4}{F}-2\delta \frac{(\epsilon -1)^2}{\epsilon (\epsilon +1)^2}<(D-c)^2<\frac{2}{F}-2\delta \frac{(\epsilon -1)^2}{(\epsilon +1)^3}. \end{aligned}$$
(4.16)
For one solitary wave, we must have
$$\begin{aligned} (D-c)^2< \frac{2}{F}-2\delta \frac{(\epsilon -1)^2}{(\epsilon +1)^3}\quad \mathrm{and}\quad (D-c)^2<\frac{4}{F}-2\delta \frac{(\epsilon -1)^2}{\epsilon (\epsilon +1)^2}. \end{aligned}$$
(4.17)
The dependence on the number of roots on the ratio of permitivitties \(\epsilon \) can be seen in Fig. 3a for a typical set of parameter values given in the caption. These can be found explicitly from (4.14), and they are plotted when they are real as functions of \(\epsilon \). In region 1, there are no admissible solitary waves since the roots are both negative. For \(\epsilon >0.07\) we have one positive and one negative root, admitting two solitary waves. Region 2 terminates at \(\epsilon \approx 0.23\) since the positive root now equals unity, i.e. touches the wall and must be excluded. Beyond this value we have region 3 which supports a single depression solitary wave. The region 3 solutions are consistent with the results of [16] who found only depression solitary waves in the case of no electric field equivalent to the \(\epsilon =1\) case here. Example phase planes for certain values of \(\epsilon \) are given in Fig. 3b.
We can also numerically solve Eq. (4.13) for S and construct solitary waves. In the present case, we have either one or two roots of \(p_2(S)\) in the physical range \(-1<S<1\). As discussed above, we require exactly one root, denoted by \(S_-\), to be in the range \(-1<S_-<0\). Denoting the other root of \(p_2(S)\) to be \(S_0\) (note that we do not assume \(0<S_0<1\)), we can write (4.13) in the form
$$\begin{aligned} \gamma (S_\xi )^2=\left( \frac{\epsilon -1}{F}\right) \frac{S^2(S-S_-)(S-S_0)}{(1+S)(1+\epsilon +S(\epsilon -1))}. \end{aligned}$$
(4.18)
Separating variables and integrating from the trough \(S=S_-\) to an elevation height S yields
$$\begin{aligned} \int _{S_-}^S\left[ \left( \frac{F}{\epsilon -1}\right) \frac{(1+z)(1+\epsilon +z(\epsilon -1))}{(z-S_-)(z-S_0)}\right] ^{\frac{1}{2}} \frac{\mathrm{{d}}z}{z}=\frac{\xi }{\sqrt{\gamma }}. \end{aligned}$$
(4.19)
The positive square root was taken in (4.19) to produce half the wave—the negative root gives the other symmetric half. Numerically solving this equation for varying parameters produces the lower (red) solitary wave depicted in Fig. 4. A depression solitary wave is always produced if the necessary condition (4.15) for solitary waves is met. If in addition (4.16) is met so that \(0<S_0<1\), then a second elevation solitary wave is supported having the same speed as its depression counterpart. In the example of Fig. 4 we also show the elevation wave in blue.
To understand the mathematical structure of the constructed solitary waves we consider the evolution PDEs (3.15) with \(\alpha =1\). (Note that in general (3.15) recover the equations in [5] when \(\sigma =0\).) The equations are
$$\begin{aligned} \begin{pmatrix} S \\ \Delta \end{pmatrix}_T+ \begin{pmatrix} -2 \Delta &{} -2(1+S) \\ - \frac{1}{F} +2 \delta \frac{(1-\epsilon )^2}{(\epsilon +1+S(\epsilon -1))^3} &{} - 2 \Delta \end{pmatrix} \begin{pmatrix} S \\ \Delta \end{pmatrix}_X=\begin{pmatrix} 0 \\ -\sigma S_{XXX}, \end{pmatrix} \end{aligned}$$
(4.20)
where in this case, we have that for solitary waves,
$$\begin{aligned} \Delta =-\frac{1}{2} \left( \frac{D+cS}{1+S}\right) . \end{aligned}$$
(4.21)
If the eigenvalues of the nonlinear flux matrix in (4.20) are denoted by \(\mu (S, \Delta )\), then we find that they are determined through the equation:
$$\begin{aligned} (\mu +2\Delta )^2 +2(1+S)\left[ \frac{2\delta (1-\epsilon )^2}{(\epsilon +1+S(\epsilon -1))^3}-\frac{1}{F}\right] =0. \end{aligned}$$
(4.22)
We find that complex eigenvalues of the flux matrix in (4.20) arise when
$$\begin{aligned} S>\frac{1+\epsilon -\left[ 2\delta F (1-\epsilon )^2\right] ^{1/3}}{1-\epsilon }:=S_{tr}. \end{aligned}$$
(4.23)
When we have \(\mu \) is complex for a range of S we call such a region elliptic, and when \(\mu \in \Re \), we refer to the region as hyperbolic, so \(S_{tr}\) is the transition point where the system changes from hyperbolic to elliptic or vice versa. An immediate consequence of (4.23) is that when \(\epsilon \) is near 1 we can make \(S_{tr}>1\), i.e. elevation waves can be found which are what we term hyperbolic; this is in contrast to the \(\epsilon =0\) case discussed in Sect. 4.2.2. We also note that the bound (4.23) only makes physical sense when the parameters are such that additionally \(|S|<1\). In the case of two solitary waves, we have from the inequality: (4.16)
$$\begin{aligned} \frac{\epsilon (\epsilon +1)^3}{(\epsilon -1)^2}<\delta F < \frac{(\epsilon +1)^3}{(\epsilon -1)^2}. \end{aligned}$$
(4.24)
Using (4.24) in (4.23) provides the following range of values for the transition boundary \(S=S_{tr}\):
$$\begin{aligned} \frac{(1-2^{1/3})(1+\epsilon )}{1-\epsilon }<S_{tr}<\frac{(1+\epsilon )(1-2^{1/3}\epsilon ^{1/3})}{1-\epsilon }. \end{aligned}$$
(4.25)
This inequality in turn predicts that the solitary waves produced when \(\alpha =1\), \(0<\epsilon <1\), can potentially have parts where the equations are locally elliptic and other parts where they are locally hyperbolic, in addition to wholly elliptic or hyperbolic. It is quite easy to check numerically the upper and lower bounds of \(S_{tr}\) in (4.25) for \(0<\epsilon <1\), and to confirm that transitions occur for both depression and elevation waves when \(0<\epsilon \lessapprox 0.587\). Such analytical estimates were used prior to searching for transitional solitary waves like those in Fig. 6.
We note that the classification of solitary waves into elliptic and hyperbolic regions presented here does not take into account the dispersive term on the right-hand side of (4.20) so strictly speaking only concerns the conservation laws when \(\sigma =0\). However, this diagnostic tool yields useful information because a change from real to complex eigenvalues of the conservation laws predicts the presence of instabilities that destroy the solitary wave structures, which enables us to predict instability of nonlinear solitary waves without doing any spectral analysis. Analogous analyses and classifications have been used in related viscous multifluid interfacial problems where the regularising terms are diffusive - see for example [17, 18].
Lower fluid a perfect conductor
As mentioned earlier this limit was considered in [5] in the absence of surface tension, and hence the solutions there cannot produce solitary waves. The perfect conductor lower fluid limit is found by sending its permittivity to infinity, \(\epsilon _-\rightarrow \infty \). Hence, we have \(\epsilon =\epsilon _+/\epsilon _-=0\) and Eq. (4.13) reduces to
$$\begin{aligned} \gamma (S_\xi )^2=\frac{S^2}{1-S^2}p_2^*(S), \end{aligned}$$
(4.26)
where \(p_2^*(S)\) is
$$\begin{aligned} p_2^*(S)=-\frac{1}{F}S^2+S\left( \frac{1}{2}(D-c)^2 -\delta \right) -\delta +\frac{1}{F}-\frac{1}{2}(D-c)^2. \end{aligned}$$
(4.27)
Using similar reasoning as before, the necessary condition for solitary waves for this sub-case is \(p_2^*(0)>0\)
$$\begin{aligned} 0<(D-c)^2<\frac{2}{F}-2 \delta , \end{aligned}$$
(4.28)
and consequently solitary waves can only exist if \(F<{1}/{\delta }\), i.e. for sufficiently small Froude numbers, given an electric field strength. We also note that \(p_2^*(-1)=-(D-c)^2<0\), \(p_2^*(1)=-2\delta <0\); hence, a solitary wave of depression and one of elevation coexist in general. Typical pairs of solitary waves are given in Fig. 5 as F varies.
For completeness, we consider the mathematical structure of the waves constructed in this section. The flux matrix in this case now reads
$$\begin{aligned} \begin{pmatrix} S \\ \Delta \end{pmatrix}_T+ \begin{pmatrix} -2 \Delta &{}\quad -2(1+S) \\ - \frac{1}{F} +2 \delta \frac{1}{(1-S)^3} &{}\quad - 2 \Delta \end{pmatrix} \begin{pmatrix} S \\ \Delta \end{pmatrix}_X=\begin{pmatrix} 0 \\ -\sigma S_{XXX} \end{pmatrix} \end{aligned}$$
(4.29)
and we note that for solitary waves (4.21) still holds. Repeating the hyperbolic-elliptic calculation of section 4.2.1, and now setting \(\epsilon =0\), we find that complex eigenvalues of the flux matrix in (4.20) arise when
$$\begin{aligned} 2\delta F-(1-S)^3>0\quad \Rightarrow \quad S>1-(2\delta F)^{1/3}:=S_{a}. \end{aligned}$$
(4.30)
It can be shown that \(S_a\) defined above is always smaller than the amplitude of elevation solitary waves given by the positive root of (4.27). From (4.28), we have \(\delta F<1\); combining this with (4.30) and the physical fact that \(\delta \) and F are non-negative, we arrive at the following general condition that is necessary for the evolution equations to become locally elliptic (we exclude the boundaries where (4.22) gives two equal and real eigenvalues)
$$\begin{aligned} 0<\delta F< 1. \end{aligned}$$
(4.31)
The inequality (4.30) in turn predicts that any solitary wave satisfying \(1>S>1-2^{1/3}\approx -0.26\), can potentially have parts where the equations are locally elliptic and other parts where they are locally hyperbolic, in addition to wholly elliptic or hyperbolic. These properties hold for depression as well as elevation waves, and examples of such mixed behaviour are given in Fig. 6. Figure 6a has Froude number \(F=0.65\), while Fig. 6b has \(F=0.1\), the other parameters being \(\delta =1\), \(\sigma =0.1\) (note that \(\alpha =1\) and \(\epsilon =0\) here). For \(F=0.65\), the depression wave exhibits mixed behaviour as seen in the figure that depicts the elliptic parts in red and the hyperbolic ones in blue. When \(F=0.1\), the elevation wave now supports ellipticity where its amplitude is sufficiently large, as seen in Fig. 6b. We note that if transition takes place in the elevation wave then the depression wave is wholly hyperbolic, and if it takes place in the depression wave then the positive wave is wholly elliptic. This can be inferred from the monotonicity of the sufficient condition for ellipticity \(1>S>1-2^{1/3}\approx -0.26\). Of course, exact diagnostics are calculated directly from the eigenvalues (4.22).