2.2.1. Rescaling the governing equations

Despite (2.3) already being in dimensionless form, in view of the small thickness and length of the transition region predicted for the small $Ca_{{B}}$ limit, Bretherton (Reference Bretherton1961) found it convenient to rescale it. The analysis proceeds by defining a rescaled film thickness in the thin film region $J^{\ast } = h^{\ast } / (3 Ca_\textrm {B})^{2/3}$, this value being a priori unknown. A rescaled coordinate along the film $\zeta = Z / ((3 Ca_\textrm {B})^{1/3}J^{\ast })$, and a rescaled thickness in the transition region $G = h/((3 Ca_\textrm {B})^{2/3} J^{\ast })$ are also defined. The problem to be considered here involves an extra dimensionless parameter, denoted $\chi '$, which is defined (Grassia Reference Grassia2019) as the ratio between the nominal film thickness based on the scaling of Bretherton (Reference Bretherton1961) and the Debye screening length

(2.4)\begin{equation} \chi'=(3Ca_{{B}}^{2/3})/\epsilon_{2}. \end{equation}

The larger the value of $\chi '$, the more charges are screened. Equation (2.3) now becomes

(2.5)\begin{equation} G_{\zeta\zeta\zeta} = \frac{{G - 1}}{G^{3}} - \frac{1}{\Gamma^{\ast}} \left(T_{\mathit{EO}}(G\,\chi^{\ast}) \right)_{G} G_{\zeta}, \end{equation}

which now contains (as yet unknown) parameters $\Gamma ^{\ast }$ and $\chi ^{\ast }$ defined as

(2.6)\begin{gather} \Gamma^{\ast} = \bar{\Gamma}/J^{\ast} \end{gather}

(2.7)\begin{gather} \chi^{\ast} = \chi' J^{\ast}. \end{gather}

Observe that $h/\epsilon _{2}\equiv G\,\chi ^{\ast }$. Note also the curious fact via (2.6) that decreasing the film thickness $J^{\ast }$ actually decreases the factor $1/\Gamma ^{\ast }$, which in (2.5) governs the relative importance of electro-osmotic terms, despite the fact that electro-osmotic tensions themselves increase as the film thins. The reason is that the Poiseuille flow due to those electro-osmotic tensions decreases more rapidly with decreasing film thickness than the viscous flow carried along by the moving walls does. Likewise the Poiseuille flow due to the electro-osmotic tensions decreases more rapidly with decreasing film thickness than the Poiseuille flow associated with capillary pressures does: this follows from including a factor of $J^{\ast }$ in the conversion between the coordinates $Z$ and $\zeta$, effectively compressing the horizontal length scale, which favours capillary effects.

In the absence of electro-osmotic effects as per Bretherton (Reference Bretherton1961), $T_{\mathit {EO}}\equiv 0$, and the rescaling given in (2.5) is sufficient to solve the system, proceeding as follows. As $\zeta \to -\infty$ approaching the thin film region, we require $G\to 1$. Using this information, we can integrate (2.5) towards $\zeta \to +\infty$ (the capillary static region): the integration needs to be done numerically. According to the above definitions $G_{\zeta \zeta }=J^{\ast }h_{ZZ}$. Since $\lim _{Z\to \infty }h_{ZZ}=1$ (the dimensionless scale for curvature has been chosen to ensure that), it follows that $J^{\ast }$ can be read off as the $\zeta \to \infty$ limit of $G_{\zeta \zeta }$.

In the case of non-zero $T_{\mathit {EO}}$, the situation is a little more complicated (Grassia Reference Grassia2019). If it is assumed that the values of $\Gamma ^{\ast }$ and $\chi ^{\ast }$ are known, we can solve (2.5) and obtain $J^{\ast }$ by the same process as in the classical system of Bretherton (Reference Bretherton1961). Formally $J^{\ast }$ is $\lim _{\zeta \to \infty }G_{\zeta \zeta }$. However, in general what we know are the values of $\bar {\Gamma }$ and $\chi '$ given by (2.2) and (2.4), not $\Gamma ^{\ast }$ and $\chi ^{\ast }$. To find the correct value of $J^{\ast }$, it is necessary to proceed iteratively (Grassia Reference Grassia2019), as explained in the next section.

2.5.1. Differentiating the electro-osmotic tension

Equation (2.5) involves derivatives of electro-osmotic terms of the form $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ or equivalently in our system $|\mathrm {d}p_{\mathit {EO}}/\mathrm {d}G|$. Therefore, to understand how electro-osmotic terms affect (2.5), it is first necessary to appreciate how $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ behaves.

In the case of interest here, although $T_{\mathit {EO}}$ is a negative quantity, it becomes less negative as $G$ increases. Hence $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ (like $p_{\mathit {EO}}$ itself) is positive, as figure 2 shows. For convenience we prefer to work in terms of these positive quantities $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ and $p_{\mathit {EO}}$ (instead of negative quantities such as $\mathrm {d}p_{\mathit {EO}}/\mathrm {d}G$ and $T_{\mathit {EO}}$).

Figure 2.

Electro-osmotic tension $T_{\mathit {EO}}$ (i.e. the negative of electro-osmotic pressure $p_{\mathit {EO}}$) vs $G\,\chi ^{\ast }\equiv h/\epsilon _{2}$ (the ratio between aqueous layer thickness at an arbitrary point of the transition region and Debye screening length): see (2.10). Opposite and equal charges are assumed with $s^{\ast }=10$, $\sigma _{c0}=1$, $\sigma _{o0}=-0.1$. Although $T_{\mathit {EO}}$ is negative, $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ is positive. Asymptotic formulae for $G\,\chi ^{\ast }\ll 1$ and $G\,\chi ^{\ast }\gg 1$, respectively (2.12) and (2.11), are also plotted.

Although $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ is indeed positive, (2.11) and (2.12) indicate that its value is small for both $G\,\chi ^{\ast }\gg 1$ and $G\,\chi ^{\ast }\ll 1$. Electro-osmotic effects within (2.5) are therefore weak in both limits. The result when $G\,\chi ^{\ast }\gg 1$ is unsurprising, since $p_{\mathit {EO}}$ itself decays exponentially to zero in that case. The result for $G\,\chi ^{\ast }\ll 1$ is slightly more surprising. Despite $p_{\mathit {EO}}\equiv -T_{\mathit {EO}}$ approaching a non-zero value in that limit, the variation of $p_{\mathit {EO}}$ about this limit is second order in $G \,\chi ^{\ast }$ as (2.12) shows.

The largest $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ values, and hence the largest impact of electro-osmotic terms in (2.5) occurs for intermediate values of $G\,\chi ^{\ast }$. The maximum $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ is

(2.13)\begin{equation} \mathrm{d}T_{\mathit{EO}}/\mathrm{d}G|_{max} \approx 0.384 |s^{\ast}\sigma_{c0}\sigma_{o0}|\chi^{\ast} \quad \textrm{when } G \approx 1.316/\chi^{\ast}. \end{equation}

Note that within (2.5) we are only usually interested in values of $G$ satisfying $G\ge 1$, since $G\equiv 1$ corresponds to the thin film region, with the transition region being thicker. Hence, there is a qualitative difference between the behaviour of $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ with respect to $G$ according to whether $\chi ^{\ast }\ge 1.316$ or $\chi ^{\ast }<1.316$. In the former case, $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ is a monotonically decreasing function as $G$ increases above $G=1$. In the latter case, $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ first increases then decreases as $G$ increases.

2.5.2. Differentiating the electro-osmotic tension in the limit $G\to 1$

Later on in our analysis, it turns out to be convenient to define a quantity $F$ via

(2.14)\begin{equation} F \equiv \mathrm{d}T_{\mathit{EO}}/\mathrm{d}G |_{G=1}.\end{equation}

The value of $F$ is easy to evaluate for any general $\chi ^{\ast }$ via differentiating (2.10). For $\chi ^{\ast }\gg 1$ it is clear from (2.11) that $F$ is small, given by

(2.15)\begin{equation} F \approx 4 |s^{\ast}\sigma_{c0}\sigma_{o0}| \chi^{\ast} \exp(-\chi^{\ast}). \end{equation}

Via (2.12), we note that for $\chi ^{\ast }\ll 1$,

(2.16)\begin{equation} F\approx |s^{\ast}\sigma_{c0}\sigma_{o0}| \chi^{\ast\,2}/2, \end{equation}

hence is also small when $\chi ^{\ast }\ll 1$. The maximum of $F$, denoted $F_{max}$ occurs for intermediate values of $\chi ^{\ast }$. It turns out that $F_{max}\approx 0.639 |s^{\ast }\sigma _{c0}\sigma _{o0}|$ and occurs when $\chi \approx 2.019$.

Note the difference between maximising $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ and maximising $F$. If we use the symbol $T'_{\mathit {EO}}$ to denote the derivative of $T_{\mathit {EO}}$ with respect to its argument, in the first instance we are maximising $T'_{\mathit {EO}}(G\,\chi ^{\ast })\,\chi ^{\ast }$ with respect to $G$, which is equivalent simply to finding the maximum of $T'_{\mathit {EO}}$. In the second instance, however, we are maximising $T'_{\mathit {EO}}(\chi ^{\ast })\,\chi ^{\ast }$ with respect to $\chi ^{\ast }$: we are not simply maximising $T'_{\mathit {EO}}$. Despite these subtle differences, the overall picture for $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ and for $F$ is similar. The former quantity is small for $G\,\chi ^{\ast }\ll 1$, significant for $G\,\chi ^{\ast }$ of the order of unity, and small again for $G\,\chi ^{\ast }\gg 1$. The latter quantity is small for $\chi ^{\ast }\ll 1$, significant for $\chi ^{\ast }$ of the order of unity, and small again for $\chi ^{\ast }\gg 1$.

2.6.1. Selecting values for $\bar {\Gamma }$

A base case value for $\bar {\Gamma }$ was selected (Grassia Reference Grassia2019) as being of the order of approximately 10, for a ‘typical’ waterflooding operation in a ‘typical’ width pore or channel ($R\approx 10^{-6}$ m was chosen as the channel half-width, $\gamma \approx 0.02$ N m$^{-1}$ was chosen as the interfacial tension and,following Malmberg & Maryott (Reference Malmberg and Maryott1956), $\varepsilon _{w}$ was taken as 56 times the permittivity of free space at the temperature of interest (Wilmott et al. Reference Wilmott, Breward and Chapman2018) close to 100 $^{\circ }$C). Some variation in the exact value of $\bar {\Gamma }$ is expected owing to its dependence on channel width (see (2.2)); however, the biggest source of uncertainty is the value of $s_{c}^{\ast }$, the density of charge adsorption sites on the pore wall. This was estimated for kaolinite clays (using literature data of Li & Xu (Reference Li and Xu2008) as $10^{16}$ m$^{-2}$. However, the values of $s_{c}^{\ast }$ and hence $\bar {\Gamma }$ are likely to be sensitive to clay chemistry.

A detailed review of different clay chemistries is beyond the scope of the present work, but in an effort to explore sensitivity to clay chemistry, Grassia (Reference Grassia2019) considered a decrease in $s_{c}^{\ast }$ relative to the base case by up to a factor of 10. This weakened the electro-osmotic effects and led to a 100-fold increase in $\bar {\Gamma }$. In the present work, by contrast, we will consider up to a 10-foldincrease in $s_{c}^{\ast }$, strengthening electro-osmotic effects and leading to a 100-fold decrease in $\bar {\Gamma }$ relative to the base case. It is for these lower $\bar {\Gamma }$ values that electro-osmotic effects might be able to replace capillary effects leading to a visco-electro-osmotic balance as hypothesised in the introduction. The domain of $\bar {\Gamma }$ of interest therefore will be $0.1\le \bar {\Gamma }\le 10$, but with a special focus on the lower end of the domain.

2.6.2. Selecting values for $\chi '$

We now turn to the parameter $\chi '$. As seen in (2.4) this is the ratio between $(3 Ca_{{B}})^{2/3}$ and $\epsilon _{2}$. A base case value of $Ca_{{B}}$ was estimated to be (Grassia Reference Grassia2019) approximately $5\times 10^{-7}$ in rough order of magnitude terms. This corresponds to a flow velocity in the pores of the order of $10^{-5}$ m s$^{-1}$, a viscosity of the order of $10^{-3}$ Pa s, and an interfacial tension as mentioned earlier of 0.02 N m$^{-1}$. Since $Ca_{{B}}$ is directly proportional to flow velocity, we assume $Ca_{{B}}$ could be up to 10 times larger than the base case (for a fast waterflood) or 10 times smaller (a slower waterflood). Hence the domain of $Ca_{{B}}$ of interest is assumed to lie between values $5\times 10^{-8}$ and $5 \times 10^{-6}$.

Meanwhile, the value of $\epsilon _{2}$ (which denotes here Debye length non-dimensionalised by channel half-width) is given by (Grassia Reference Grassia2019)

(2.17)\begin{equation} \epsilon_{2} = (\varepsilon_{w}k_{B}T/2 C^{+}q^{2}R^{2})^{1/2}, \end{equation}

where $\varepsilon _{w}$ is the permittivity of the aqueous phase, $k_{B}$ is Boltzmann's constant, $T$ is absolute temperature, $C^{+}$ is the number density of monovalent ions, $q$ is electronic charge and $R$ is channel half-width. There is a slight change to the above expression for $\epsilon _{2}$ if a mixture of ions of different valencies is present (Grassia Reference Grassia2019), but this is neglected here since the change is minor and is not material to the argument to follow. According to (2.17) then, for a number density of ions of the order of $10^{27}$ m$^{-3}$ (Wilmott et al. Reference Wilmott, Breward and Chapman2018) and a channel half-width $10^{-6}$ m, and at the temperature mentioned earlier (100 $^{\circ }$C or 373 K) a ‘base case’ value of $\epsilon _{2}$ would be approximately $2\times 10^{-4}$.

The value of $\epsilon _{2}$ is, however, sensitive to ion concentration, and the value above corresponds to a high salinity flood. There have been reports in the literature (Yildiz & Morrow Reference Yildiz and Morrow1996; McGuire et al. Reference McGuire, Chatham, Paskvan, Sommer and Carini2005; Lager et al. Reference Lager, Webb, Black, Singleton and Sorbie2008; Ligthelm et al. Reference Ligthelm, Gronsveld, Hofman, Brussee, Marcelis and van der Linde2009; Austad et al. Reference Austad, Rezaei Doust and Puntervold2010; Rezaei Doust et al. Reference Rezaei Doust, Puntervold and Austad2011) of performing waterfloods with water that is far less saline than seawater, with ion number densities around $10^{25}$ m$^{-3}$ being typical. This gives a 10-fold increase in $\epsilon _{2}$ relative to the aforementioned base case, up to $2\times 10^{-3}$ instead of $2\times 10^{-4}$.

Given these domains for $Ca_{{B}}$ and $\epsilon _{2}$, we estimate $\chi '$ given by (2.4) could vary approximately between 0.02 and 5, with the lower end value corresponding to a low speed and low salinity waterflood, and the upper end being for high speed and high salinity.

This completes our estimates of the key parameters $\bar {\Gamma }$ and $\chi '$. In what follows, however, often we prefer to work with (2.5), which involves parameters $\Gamma ^{\ast }$ and $\chi ^{\ast }$. Estimates for these parameters are discussed next.

2.6.3. Selecting values for $\Gamma ^{\ast }$ and $\chi ^{\ast }$

The parameters $\Gamma ^{\ast }$ and $\chi ^{\ast }$ are related to $\bar {\Gamma }$ and $\chi '$ via (2.6) and (2.7), but the parameter $J^{\ast }$ in those equations is a priori unknown, and must be found to satisfy (2.9). As a result the domain of interest for parameters $\bar {\Gamma }$ and $\chi ^{\ast }$ is a priori unknown.

For large $\bar {\Gamma }$ and/or large $\chi '$ (Grassia Reference Grassia2019), $J^{\ast }$ is found to approach the known value $J^{\ast }_{{B}}\approx 0.643$. Clearly, $J^{\ast }_{{B}}$ is an order-unity value, and on the basis of this we will, in the first instance, explore $\Gamma ^{\ast }$ values over the same domain as $\bar {\Gamma }$ (values down to 0.1) and $\chi ^{\ast }$ values over the same domain as $\chi '$ (between 0.02 and 5). Later, however, we will discover that for smaller $\bar {\Gamma }$ and/or $\chi '$ values, the value of $J^{\ast }$ can be significantly smaller than $J^{\ast }_{{B}}$. Via (2.6) and (2.7) the system thereby selects a $\Gamma ^{\ast }$ that is rather larger than $\bar {\Gamma }$ and likewise selects a $\chi ^{\ast }$ that is rather smaller than $\chi '$. The consequences of this in terms of how it affects the shape of the droplet will be described towards the end of the paper.

3.1.1. Governing equations for visco-electro-osmotic model

It is clear that if $\Gamma ^{\ast }\gg 1$, electro-osmotic terms in (2.5) are never important, and hence a visco-electro-osmotic balance requires having $\Gamma ^{\ast }$ less than or equal to order unity. Secondly, it is clear from (2.11) that if $G\,\chi ^{\ast }\gg 1$ then electro-osmotic terms decay exponentially, so a visco-electro-osmotic balance requires $G\,\chi ^{\ast }$ of order unity or less. Since we expect $G \ge 1$ always, it follows that $\chi ^{\ast }$ must itself be at most order unity for an electro-osmotic balance to apply. The smaller the value of $\chi ^{\ast }$ therefore, the larger the possible domain for $G$ over which such a balance could apply.

Suppose we introduce a new coordinate variable $\zeta '$ via $\zeta ' \equiv \Gamma ^{\ast } \zeta / F$, where $F$ is defined by (2.14). Written in terms of $\zeta '$, (2.5) becomes

(3.1)\begin{equation} \left(\frac{\Gamma^{\ast}}{F}\right)^{3} G_{\zeta'\zeta'\zeta'} = \frac{G-1}{G^{3}} - \frac{T'_{\mathit{EO}}(G \,\chi^{\ast})}{T'_{\mathit{EO}}(\chi^{\ast})} G_{\zeta'}.\end{equation}

We are interested here in situations with $\Gamma ^{\ast }\ll 1$. Provided $\Gamma ^{\ast }\ll F$, the capillary term on the left-hand side of (3.1) can be neglected, and we obtain

(3.2)\begin{equation} \frac{\mathrm{d}G}{\mathrm{d}\zeta'} \approx \frac{{G-1}}{G^{3}} \frac{T'_{\mathit{EO}}(\chi^{\ast})}{T'_{\mathit{EO}}(G\chi^{\ast})} \equiv \frac{{G-1}}{G^{3}} \frac{F}{\mathrm{d}T_{\mathit{EO}}/\mathrm{d}G}. \end{equation}

For $\Gamma ^{\ast } \ll F$ to apply, (2.15) and (2.16) imply that $\chi ^{\ast }$ must satisfy both $\chi ^{\ast }\le O(1)$ and $\Gamma ^{\ast }/|s^{\ast }\sigma _{c0}\sigma _{o0}|\ll \chi ^{\ast \,2}$ (with $|s^{\ast }\sigma _{c0}\sigma _{o0}|=1$ here). Thus the visco-electro-osmotic limit applies for small $\Gamma ^{\ast }$ and small to moderate $\chi ^{\ast }$.

Equation (3.2) can also be converted back to the original set of variables giving

(3.3)\begin{equation} \frac{\mathrm{d}G}{\mathrm{d}\zeta} \approx \Gamma^{\ast}\frac{{G-1}}{G^{3}} \frac{1}{\chi^{\ast} T'_{\mathit{EO}}(G\,\chi^{\ast})} \equiv \Gamma^{\ast}\frac{{G-1}}{G^{3}} \frac{1}{\mathrm{d}T_{\mathit{EO}}/\mathrm{d}G}.\end{equation}

Via quadrature, solutions of (3.2) or (3.3) can be obtained in implicit form ($\zeta '$ or $\zeta$ as functions of $G$). Features of these solutions are discussed in what follows.

3.1.2. Analytical approximations to the visco-electro-osmotic model

In the limit of $\chi ^{\ast }$ rather smaller than unity, there is a range of $G$ values (those values satisfying $G\,\chi ^{\ast }\ll 1$), such that $F/(\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G)\approx G^{-1}$, as follows from (2.12) and (2.16). Equation (3.2) then simplifies to

(3.4)\begin{equation} \mathrm{d}G/\mathrm{d}\zeta' = (G - 1) / G^{4}. \end{equation}

This has exponential growth for $G$ when $G-1$ is small, but the behaviour soon departs from exponential. In fact, on the basis of (3.4), it is possible to show that $G$ vs $\zeta '$ has an inflection point at $G=\frac {4}{3}$. An exponential of course has no such inflection.

The solution of (3.4) is

(3.5)\begin{align} \zeta' = G^{4}/4 + G^{3}/3 + G^{2}/2+ G+ \log((G-1)/\varepsilon) -\varepsilon^{4}/4 -4 \varepsilon^{3}/3 -3 \varepsilon^{2} -4 \varepsilon- 25/12. \end{align}

Moreover, if $G\gg 1$ (but still assuming $G\,\chi ^{\ast }\ll 1$), this simplifies to

(3.6)\begin{equation} G \approx (4 \zeta')^{1/4}. \end{equation}

Note that this implies that the growth of $G$ becomes increasingly slow as $\zeta '$ increases.

The value of $G$ can only start to increase very significantly (overcoming the aforementioned increasingly slow growth) when $G\,\chi ^{\ast }$ exceeds order unity, at which point (3.6) breaks down, since $F/(\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G)$ ceases to be approximated by $G^{-1}$. If we use (3.6) to provide a rough estimate of when $G\,\chi ^{\ast }$ might first attain values of order unity, and hence $G$ might attain values of order $1/\chi ^{\ast }$, we determine $\zeta ' \sim O((\chi ^{\ast })^{-4})$. This result can also be expressed in terms of $\zeta$ giving $\zeta \equiv F \zeta '/\Gamma ^{\ast } \sim O((\chi ^{\ast })^{-2}\, (\Gamma ^{\ast })^{-1})$ where $F$ has been estimated via (2.16). In the limit of small $\chi ^{\ast }$, this predicts extremely large $\zeta '$ and/or $\zeta$ values for $G\,\chi ^{\ast }$ to exceed unity. Growth in $G$ with respect to the coordinate $\zeta '$ and/or $\zeta$ is therefore very slow in this small $\chi ^{\ast }$ limit. The asymptotic picture whereby the aqueous thin film joins up with a capillary static region across a compact transition region might be no longer valid in that case.

3.1.3. Divergence of the visco-electro-osmotic model

So far discussion of the visco-electro-osmotic model has considered values of $G\,\chi ^{\ast }$ up to order unity. It is interesting to examine how the model breaks down once $G\,\chi ^{\ast }$ becomes significantly larger than unity, so that $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ decays exponentially via (2.11). According to (3.2), $\mathrm {d}G/\mathrm {d}\zeta '$ then becomes so large that $G$ diverges to infinity at finite $\zeta '$. The divergence happens so rapidly after $G\,\chi ^{\ast }$ exceeds order unity, that the estimates $\zeta ' \sim O((\chi ^{\ast })^{-4})$ and $\zeta \sim O((\chi ^{\ast })^{-2}\, (\Gamma ^{\ast })^{-1})$ given in § 3.1.2 can be used to estimate not only where $G\,\chi ^{\ast }$ attains order unity, but also where the divergence occurs. Of course, before $G$ diverges, (3.2) has ceased to be a good approximation to (2.5). The electro-osmotic term is now so weak, it is no longer tenable to retain electro-osmotic effects but neglect capillary ones. Another regime in which capillary effects make an essential contribution to the shape of the profile is discussed next.

3.2.1. Capillary, electro-osmotic balance

Suppose we define $\hat \zeta = (\chi ^{\ast }/\Gamma ^{\ast })^{1/2}\,\zeta$ and $\hat G=G\,\chi ^{\ast }$. Equation (2.5) now becomes

(3.7)\begin{equation} \hat G_{{\hat\zeta}{\hat\zeta}{\hat\zeta}} = \left({\Gamma^{\ast}\,\chi^{\ast}}\right)^{3/2} \frac{\hat G-\chi^{\ast}}{ \hat G^{3}} - T'_{\mathit{EO}} (\hat G) \hat G_{\hat\zeta}. \end{equation}

If the product $\Gamma ^{\ast }\,\chi ^{\ast } \equiv \bar \Gamma \,\chi '$ is small, the viscous term is negligible, and a capillary, electro-osmotic balance results.

Converting back to the original variable set, gives

(3.8)\begin{equation} G_{\zeta\zeta\zeta}\approx-\frac{\mathrm{d}T_{\mathit{EO}}/\mathrm{d}G}{\Gamma^{\ast}} \, G_{\zeta}. \end{equation}

This corresponds to an augmented Young–Laplace equation (Teletzke et al. Reference Teletzke, Davis and Scriven1988), but differs from augmented Young–Laplace systems previously discussed by Grassia (Reference Grassia2019).

That earlier work (Grassia Reference Grassia2019) focussed primarily on a case with repulsive charges such that $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ was large and negative in the small $\chi ^{\ast }$ limit. That gave monotonic growth of $G$ vs $\zeta$, whereas here (with $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G >0$ as per figure 2) oscillatory behaviour is anticipated. As was mentioned in § 1.1, oscillatory behaviour is seen in the classical system (without electro-osmotic effects), but only at the back of a droplet (Bretherton Reference Bretherton1961; Wilmott et al. Reference Wilmott, Breward and Chapman2018), not at the front, as we are currently considering. It is known also that adding repulsive electro-osmotic effects to the back of the droplet actually removes the oscillatory behaviour (Wilmott et al. Reference Wilmott, Breward and Chapman2018). Here, on the other hand, we are talking about adding attractive electro-osmotic effects to the front of the droplet, and this potentially can introduce oscillations even though there were none originally.

Equation (3.8) is the equation of an oscillator, but since $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ is variable (varying in the fashion described in § 2.5.1), it is a nonlinear oscillator. The wavelength of the oscillations will therefore vary with $G$, and tends to be smaller in a parameter range where $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ is larger: as mentioned in § 2.5.1, $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ is maximal when $G$ is order $1/\chi ^{\ast }$, i.e. $\hat G$ is order unity. It is clear, however, from (3.8) that the smaller the value of $\Gamma ^{\ast }$, the smaller the wavelength of the oscillations tends to be. Short wavelength oscillations help to ensure that capillary terms (which correspond to the highest-order spatial derivatives in the system) can become non-negligible.

3.2.2. Transition from oscillatory to non-oscillatory behaviour

Although (3.8) does describe an oscillator, we know that if ever $G\,\chi ^{\ast }$ manages to grow significantly larger than unity, electro-osmotic effects are screened in an exponential fashion, the value of $\mathrm {d}T_{\mathit {EO}}/\mathrm {d}G$ decays, and the oscillations must increase in wavelength and eventually stop. Either a capillary static state must then be attained, or more generally, a classical capillary, viscous balance (Bretherton Reference Bretherton1961) might then ensue. Either way, in such a system, at sufficiently large $\zeta$, the value of $G_{\zeta \zeta }$ should approach a well-defined limit $G_{\zeta \zeta ,\infty }$. Using this fact it is possible to describe in more detail how the system behaves as it transitions from an oscillatory to a non-oscillatory state.

It is possible to write (2.5) in the form

(3.9)\begin{equation} G_{\zeta\zeta} - \frac{1}{\Gamma^{\ast}} p_{\mathit{EO}}(G\,\chi^{\ast}) =G_{\zeta\zeta,\infty} - \int_{\zeta}^{\infty}\frac{{G-1}}{G^{3}} \,\mathrm{d}\zeta. \end{equation}

Like (3.8), this describes a nonlinear oscillator, and moreover provided $\Gamma ^{\ast }$ is small (as we suppose here), oscillations have short wavelength as mentioned earlier. Notice, however, that (3.9) has a right-hand side that evolves, being an increasing function of $\zeta$, albeit with the evolution of that right-hand side being slow compared to the typical wavelength of each oscillation. This then is a multiple scales problem, and we anticipate (Kuzmak Reference Kuzmak1959) that the value of $G$ averaged over each oscillation cycle undergoes some slow secular change with coordinate $\zeta$.

We assume also that $\chi ^{\ast }$ is small, $\chi ^{\ast }\ll 1$, and consider a situation whereby during each oscillation $G$ varies between a value of order unity, and a much larger value of order $O(1/\chi ^{\ast })$, any larger amplitude oscillation being impossible due to $p_{\mathit {EO}}$ decaying at large $G\,\chi ^{\ast }$. Any part of the oscillation cycle during which $G$ is $O(1/\chi ^{\ast })$ should make a negligible contribution to the integrated term on the right-hand side of (3.9). The only non-negligible contributions to that integrated term arise intermittently, i.e. once per oscillation cycle, around the point at which $G$ has its $O(1)$ minimum. This then is what drives secular change. As the secular change in $G$ accumulates, eventually the oscillations cease, for the reason pointed out above, i.e. $p_{\mathit {EO}}$ decays.

Note also that at the minimum of each oscillation cycle for $G$, the value of $p_{\mathit {EO}}$ is a maximum. According to (2.12), this maximal $p_{\mathit {EO}}$ should be roughly $|s^{\ast }\sigma _{c0}\sigma _{o0}|$, and should, moreover, barely change from one oscillation cycle to the next given that (2.12) exhibits only second-order variation when $G\,\chi ^{\ast }\ll 1$. It follows also that $G_{\zeta \zeta }$ on the left-hand side of (3.9) is a maximum when $G$ is a minimum. Moreover, since any secular change in the right-hand side of (3.9) happens around the point in the cycle where $G$ is minimal, we should see secular changes in the maximum $G_{\zeta \zeta }$ value that is attained from one oscillation cycle to the next. Each oscillation should thereby give an increment to $G_{\zeta \zeta }$ and eventually, for $\zeta$ sufficiently large, $G_{\zeta \zeta }$ should attain some final value $G_{\zeta \zeta ,\infty }$.

An expression for $G_{\zeta \zeta ,\infty }$ can be obtained by evaluating (3.9) in the limit $\zeta \to -\infty$, which corresponds to $G\to 1$ and $G_{\zeta \zeta }\to 0$. Hence (using also (2.6))

(3.10)\begin{equation} G_{\zeta\zeta,\infty} = \int_{-\infty}^{\infty} \frac{{G-1}}{G^{3}} \,\mathrm{d}\zeta - \frac{p_{\mathit{EO}}(\chi^{\ast})}{\Gamma^{\ast}} = \int_{-\infty}^{\infty} \frac{{G-1}}{G^{3}} \,\mathrm{d}\zeta - \frac{J^{\ast} p_{\mathit{EO}}(\chi^{\ast})}{\bar{\Gamma}}. \end{equation}

As noted above, (2.12) shows that $p_{\mathit {EO}}(\chi ^{\ast })$ for small $\chi ^{\ast }$ should approach $|s^{\ast } \sigma _{c0}\sigma _{o0}|$, with only second-order variation with $\chi ^{\ast }$. Under these circumstances, variation in $G_{\zeta \zeta ,\infty }$ as a function of $\chi ^{\ast }$ therefore depends on variation in the integrated term within (3.10). Each oscillation in $G$ should contribute to this integrated term, and hence the more oscillatory is the function $G$, the larger is the value of $G_{\zeta \zeta ,\infty }$ that we expect.

3.3.1. Methodology for estimating $J^{\ast }$ for small $\chi '$

We proceed as follows. Small $\chi '$ (and hence small $\chi ^{\ast }$ via (2.7) tends to imply small second-order variation with respect to $G$ of the electro-osmotic pressure via (2.12), even though $p_{\mathit {EO}}$ itself might be non-zero. As a first approximation we integrate (2.5) ignoring electro-osmotic pressure gradient terms over a subdomain from $G=1$ to a $G$ value satisfying $1\ll G \ll 1/\chi ^{\ast }$. Since the final $G$ value in the subdomain of interest satisfies $G\gg 1$, and we solve a classical capillary, viscous problem there (Bretherton Reference Bretherton1961) it follows that $G_{\zeta \zeta }$ attains a value approximately $J^{\ast }_{{B}}\approx 0.643$.

Our next task is to continue integrating (2.5) over a new subdomain from a value $1\ll G\ll 1/\chi ^{\ast }$ to a value $G\gg 1/\chi ^{\ast }$ at which point $G_{\zeta \zeta }$ has changed from $J^{\ast }_{{B}}$ to some final limiting value $G_{\zeta \zeta ,\infty }$. On the grounds that $G\gg 1$ in this new subdomain, we assume that the viscous term in (2.5) can be neglected, or equivalently the integral term in (3.9) can be neglected. Gradients of electro-osmotic terms, however, are relevant when $G\,\chi ^{\ast }$ is order unity, with the change in $G_{\zeta \zeta }$ across the subdomain reflecting the change in the electro-osmotic pressure, which from (2.11) and (2.12) is $|s^{\ast } \sigma _{c0}\sigma _{o0}|$ between the limits $G\,\chi ^{\ast }\ll 1$ and $G\,\chi ^{\ast }\gg 1$. Hence (2.5) gives

(3.11)\begin{equation} {G_{\zeta\zeta,\infty} \approx J^{\ast}_{{B}} - {|s^{\ast} \sigma_{c0}\sigma_{o0}|}{/\Gamma^{\ast}} \equiv J^{\ast}_{{B}} -J^{\ast}{|s^{\ast} \sigma_{c0}\sigma_{o0}|}{/\bar{\Gamma}}}\end{equation}

where (2.6) has also been used. The above result can also be derived directly from (3.10) recognising that the integral term in that equation is now contributed almost entirely from the first subdomain, and evaluates to $J^{\ast }_{{B}}$ since a classical capillary, viscous balance applies in that subdomain. Observe that $G_{\zeta \zeta ,\infty }\le J^{\ast }_{{B}}$. The implication is that $G_{\zeta \zeta }$ is no longer a monotonically increasing function of $\zeta$, by contrast with what happens in the classical system of Bretherton (Reference Bretherton1961). Rather $G_{\zeta \zeta }$ is an increasing function of $\zeta$ on the first subdomain, but a decreasing function on the second subdomain.

We now determine $J^{\ast }$ by imposing the condition within (3.11) that $G_{\zeta \zeta ,\infty }=J^{\ast }$ thereby enforcing (2.9). Rearranging (3.11) leads to

(3.12)\begin{equation} {J^{\ast}|_{\chi'\to 0} \approx J^{\ast}_{{B}} \, {\bar{\Gamma}}{/(\bar{\Gamma} + |s^{\ast}\sigma_{c0}\sigma_{o0}|)}.} \end{equation}

Here, recall that $|s^{\ast }\sigma _{c0}\sigma _{o0}|=1$ (see § 2.4) and hence, if $\bar {\Gamma }\ll 1$, then $J^{\ast }\ll J^{\ast }_{{B}}$.

Given this (very small) $J^{\ast }$ value, (2.6) now implies a value for $\Gamma ^{\ast }$

(3.13)\begin{equation} \Gamma^{\ast} \approx (\bar{\Gamma} + | s^{\ast}\sigma_{c0}\sigma_{o0}|) / J^{\ast}_{{B}}, \end{equation}

which is never exceedingly small, even if $\bar {\Gamma }\ll 1$. Whereas an exceedingly small $\Gamma ^{\ast }$ might have caused $G$ itself to become oscillatory with $\zeta$ with very short wavelength oscillations, here, however, with $\Gamma ^{\ast }$ being somewhat larger, oscillation wavelengths in $\zeta$ grow (based on § 3.2.1, they must scale as order $(\Gamma ^{\ast }/\chi ^{\ast })^{1/2}$ or equivalently $\bar \Gamma ^{1/2}/(\chi ^{\prime \,1/2}J^{\ast })$), and the role of the electro-osmotic term seems to be to introduce non-monotonicity in $G_{\zeta \zeta }$, but not necessarily in $G$ itself. Another way to reach the same conclusion is to convert back from the coordinate $\zeta$ to the original coordinate $Z$ (see § 2.2.1). Since a factor $J^{\ast }$ appears in the conversion, we conclude that the length scale in $Z$ for secular growth in $G$ is compressed as $J^{\ast }$ decreases so that the film becomes thinner, whereas the length scale for oscillations is relatively insensitive to thickness when expressed in terms of $Z$ as opposed to $\zeta$ (as mentioned above, it scales like $(J^{\ast })^{-1}$ when expressed in terms of $\zeta$). Thus the response for $G$ shifts away from oscillation towards secular growth.

Equations (3.12) and (3.13) are only approximations, and in reality $J^{\ast }$ can only be found for any given $\bar {\Gamma }$ and $\chi '$ by solving (2.9) iteratively. Nonetheless (3.12) does give an indication of where one might start the iteration. The implication for (2.5) then is that, regardless of what the ‘nominal’ strength of electro-osmotic interactions appears to be, i.e. regardless of how small $\bar \Gamma$ is, the system selects a $J^{\ast }$ value and hence a $\Gamma ^{\ast }$ value, such that electro-osmotic interactions are only ever of moderate importance.

3.4.1. Methodology for estimating $J^{\ast }$ for arbitrary $\chi '$

We obtain an estimate for $J^{\ast }$ albeit an ad hoc one via (3.10). We suppose that the integrated term on the right-hand side is weakly perturbed from the classical case of Bretherton (Reference Bretherton1961), and hence integrates roughly to $J^{\ast }_{{B}}$. Likewise, we approximate the term $p_{\mathit {EO}}(\chi ^{\ast })$ on the right-hand side by $p_{\mathit {EO}}(J^{\ast }_{{B}}\,\chi ')$. Imposing $G_{\zeta \zeta ,\infty }=J^{\ast }$ gives

(3.14)\begin{equation} {J^{\ast} \approx {\bar{\Gamma} \, J^{\ast}_{{B}} }{/ (\bar{\Gamma} + p_{\mathit{EO}}(J^{\ast}_{{B}}\chi'))}.} \end{equation}

For large $\bar {\Gamma }$, using $p_{\mathit {EO}}(J^{\ast }_{{B}}\,\chi ')$ in place of $p_{\mathit {EO}}(\chi ^{\ast })$ as we have done here, should be a good approximation since $J^{\ast }$ stays close to $J^{\ast }_{{B}}$ at all $\chi '$. For smaller $\bar {\Gamma }$, however, $J^{\ast }_{{B}}\,\chi '$ overestimates the true $\chi ^{\ast }$, meaning that $p_{\mathit {EO}}(J^{\ast }_{{B}}\,\chi ')$ underestimates $p_{\mathit {EO}}(\chi ^{\ast })$ (and hence (3.14) might be expected to overestimate the true $J^{\ast }$). Overestimating the argument of $p_{\mathit {EO}}$ matters little for small arguments, however, because $p_{\mathit {EO}}$ is very insensitive to its argument in that limit. In a number of limits then (3.14) should perform reasonably well despite the rough assumptions used to derive it.

This completes our analysis of the equations governing the interface shape for a droplet advancing along a capillary channel with conjoining electro-osmotic forces acting between the droplet surface and the capillary wall. We have shown the governing equations admit a visco-electro-osmotic balance (§ 3.1) for suitable choices of the parameters $\Gamma ^{\ast }$ and $\chi ^{\ast }$, but also it is possible to find a regime for which nominally weak capillary terms retain importance via the interface developing short wavelength oscillations (§ 3.2). In a situation in which parameters $\bar {\Gamma }$ and $\chi '$ (rather than $\Gamma ^{\ast }$ and $\chi ^{\ast }$) are specified, the system can adjust itself (see § 3.3) to reduce the importance of electro-osmotic terms relative to capillary ones. It does this by reducing the film thickness $J^{\ast }$, which also tends to reduce (relative to the wavelength of any oscillations), the characteristic longitudinal distance over which secular changes thickness take place moving along the film. An explicit estimate (in § 3.4) predicts how $J^{\ast }$ depends on $\bar {\Gamma }$ and $\chi '$.

Now that we have the above set of predictions, we can check whether they are borne out by solving the system numerically. Results are presented in the next section.