2. Formulation

A cylindrical immiscible ‘drop’ (radius $a$ , dielectric permittivity $\bar{{\it\epsilon}}$ , conductivity $\bar{{\it\sigma}}$ , viscosity $\bar{{\it\mu}}$ ) is suspended in another liquid (dielectric permittivity ${\it\epsilon}$ , electric conductivity ${\it\sigma}$ , viscosity ${\it\mu}$ ) and subjected to a uniform electric field of magnitude $E$ . Our goal is to calculate the resulting steady-state flow, assuming that surface tension is strong enough to retain an approximately circular drop cross-section. We normalize electric fields and potentials by $E$ and $aE$ , respectively, the surface-charge density by ${\it\epsilon}E$ and the velocity field by Taylor’s velocity scale

We employ cylindrical coordinates ( $r,{\it\theta}$ ) with $r=0$ at the drop axis and ${\it\theta}=0$ in the direction of the applied field, see figure 1. The velocity field in the suspending liquid is $\boldsymbol{u}=\hat{\boldsymbol{e}}_{r}u+\hat{\boldsymbol{e}}_{{\it\theta}}v$ , while that in the drop is $\bar{\boldsymbol{u}}=\hat{\boldsymbol{e}}_{r}\bar{u}+\hat{\boldsymbol{e}}_{{\it\theta}}\bar{v}$ .

The electric potentials ${\it\varphi}$ and $\bar{{\it\varphi}}$ are governed by Laplace’s equation, regularity at $r=0$ and the approach to the uniformly applied field,

At $r=1$ they satisfy the continuity condition, ${\it\varphi}=\bar{{\it\varphi}}$ , as well as the charging condition

in which $R=\bar{{\it\sigma}}/{\it\sigma}$ . The surface-charge density appearing in (2.3) is provided by the jump in electric displacement,

in which $S={\it\epsilon}/\bar{{\it\epsilon}}$ . The essential difference from the classical description of Taylor (Reference Taylor1966) is the incorporation in (2.3) of charge convection. This nonlinear mechanism is quantified by the electric Reynolds number (Melcher & Taylor Reference Melcher and Taylor1969),

constituting the ratio of the charge-relaxation time ${\it\epsilon}/{\it\sigma}$ to the convective time $a/\mathscr{U}$ .

The velocity fields $\boldsymbol{u}$ and $\bar{\boldsymbol{u}}$ are governed by the continuity and Stokes equations. At $r=1$ they satisfy the impermeability condition, $u=\bar{u}=0$ , the continuity condition (tacitly used in (2.3)), $v=\bar{v}$ , and the shear-stress balance,

in which $M={\it\mu}/\bar{{\it\mu}}$ . At $r=0$   $\bar{\boldsymbol{u}}$ is regular, while at large $r$ $\boldsymbol{u}$ decays.

3. Symmetry arguments and vanishing of net charge

Even with incorporation of surface convection, the governing equations and boundary conditions still possess an underlying fore–aft symmetry about the midplane ${\it\theta}=\pm {\rm\pi}/2$ similar to that in the classical (3D) problem of Taylor (Reference Taylor1966). This may appear to suggest that, under the reflection ${\it\theta}\rightarrow {\rm\pi}-{\it\theta}$ , the solutions transform as

However, the inherent nonlinearity in the surface convection mechanism now allows for symmetry breaking, with solutions that are not fore–aft-symmetric. (Given the experimental and numerical evidence for spontaneous rotation, we actually anticipate a solution that is not fore–aft-symmetric at sufficiently large $\mathit{Re}$ .) Furthermore, the symmetry of the above nonlinear problem with respect to the direction of the applied field necessitates that the ‘mirror image’ of any solution $(\boldsymbol{u},q)$ (where $q$ and the velocity components are regarded as functions of $r$ and ${\it\theta}$ ), namely

is also a solution.

Allowing for asymmetric solutions, and noting that the slowest decay rate in 2D Stokes flow (Leal Reference Leal2007) is that associated with an irrotational vortex, the far-field decay of $\boldsymbol{u}$ implies

(For fore–aft-symmetric flows, where ${\it\omega}=0$ , the decay rate is faster.) By the above-mentioned mirror-image symmetry, a solution involving drop rotation with angular velocity ${\it\omega}$ implies the existence of a solution with angular velocity $-{\it\omega}$ .

Furthermore, it is readily seen that the vanishing of the total drop charge,

which trivially holds for fore–aft-symmetric solutions (see (3.1)), is satisfied in general. Indeed, from Laplace’s equation within the drop domain it follows that the integral of $\partial \bar{{\it\varphi}}/\partial r$ vanishes at $r=1$ . Integration of (2.3) over the unit circle $r=1$ implies the same for the integral of $\partial {\it\varphi}/\partial r$ . (Since $qv$ is necessarily single-valued on the unit circle, it is evident that the integral of the surface-divergence term on the right-hand side of (2.3) vanishes.) We then get (3.4) from integration of (2.4). Physically, (3.4) may be considered a consequence of the Ohmic nature of leaky-dielectric liquids: the net outward flux of electric field associated with a non-zero total drop charge would be accompanied by a net outward charge flux, which is incompatible with the assumed steadiness.

Given the vanishing of the drop’s net charge and the far-field approach (2.2) to a uniform field, the drop does not experience any net electric force (Yariv Reference Yariv2006). The drop is accordingly force-free. It is not, however, necessarily torque-free.

4. Exact solutions

Following Feng (Reference Feng2002) one can solve the exact 2D problem comprising (2.2)–(2.6) and (3.3) for arbitrary $\mathit{Re}$ . Indeed, as the only nonlinearities in this system are introduced in conditions (2.3) and (2.6), the electric potentials and stream functions are provided by the standard eigenfunction expansions for Laplace’s equation and the biharmonic equation, respectively. Substituting these expansions into the exact problem yields the flow field, and, in particular, the angular speed ${\it\omega}$ appearing in (3.3). As in Feng (Reference Feng2002), we find a bifurcation from a fore–aft-symmetric solution (with ${\it\omega}=0$ ) for a sufficiently small $\mathit{Re}$ to an asymmetric solution (with non-zero ${\it\omega}$ ) at a sufficiently large $\mathit{Re}$ . Provided that $RS<1$ , this trend holds for any combination of $R$ , $S$ and $M$ . This bifurcation is illustrated in figure 2, showing the variation of ${\it\omega}$ with $\mathit{Re}$ for the viscosity ratio $M=0.4$ and three sets of liquid pairs represented by the indicated values of $R$ and $S$ .

Figure 2.

Variation with $\mathit{Re}$ of the angular velocity ${\it\omega}$ , as obtained from an eigenfunction-expansion solution of the exact problem for the indicated values of $R$ and $S$ . The viscosity ratio is $M=0.4$ .

5. Large $\mathit{Re}$

Our interest is in the asymptotic limit $\mathit{Re}\gg 1$ . Naively, it appears from (2.3) that at leading order $qv$ is uniform with ${\it\theta}$ at $r=1$ ; however, constraint (3.4) implies that the continuous field $q$ must vanish at some value of ${\it\theta}$ , therefore giving the contradictory result $qv\equiv 0$ . We therefore need a non-trivial scaling where Ohmic charging balances convection in (2.3).

Because of the far-field approach to a uniform unity field, the gradients of both ${\it\varphi}$ and $\bar{{\it\varphi}}$ are $O(1)$ . This may appear to suggest that $q$ is $O(1)$ as well. But then (2.3) implies an $O(1/\mathit{Re})$ velocity field, whereby (2.6) is unbalanced. This impasse is resolved by noting that even with $O(1)$ electric-field distributions, it is possible that these distributions ‘conspire’ to have a vanishing surface-charge density at this order. We accordingly postulate $O(1)$ electric fields and $O({\it\delta})$ surface-charge density where ${\it\delta}\ll 1$ remains to be found. With these scalings, the charging condition (2.3) implies that the velocity is of order $({\it\delta}\,\mathit{Re})^{-1}$ . The shear balance (2.6) then implies that

which is indeed $\ll 1$ . We therefore employ the following scalings:

where $\boldsymbol{U}=\hat{\boldsymbol{e}}_{r}U+\hat{\boldsymbol{e}}_{{\it\theta}}V$ and $\bar{\boldsymbol{U}}=\hat{\boldsymbol{e}}_{r}\bar{U}+\hat{\boldsymbol{e}}_{{\it\theta}}\bar{V}$ . The rescaled fields $\boldsymbol{U}$ and $\bar{\boldsymbol{U}}$ represent a normalization by the velocity scale (cf. (2.1))

while surface-charge density $Q$ represents a normalization by the scale

which is remarkably independent of the magnitude of the applied field.

Substituting the above scalings into the charging condition (2.3) and shear-stress balance (2.6) yields at leading order

Moreover, the presumed absence of an $O(1)$ surface charge implies (see (2.4))

This constitutes an effective boundary condition for the harmonic fields ${\it\varphi}$ and $\bar{{\it\varphi}}$ , which additionally satisfy regularity at $r=0$ , continuity at $r=1$ and the far-field condition (2.2). Determining these fields and substituting into conditions (5.5) and (5.6) yields

The charging condition (5.8) serves as a differential equation governing $QV$ as a function of ${\it\theta}$ on $r=1$ . A solution that is compatible with the requirement (3.4) for zero net charge is

As neither $Q$ nor $V$ can vanish at ${\it\theta}={\rm\pi}/2$ , this solution is incompatible with (3.1) and accordingly lacks fore–aft symmetry (but see § 8). Thus, the limit $\mathit{Re}\gg 1$ automatically leads to an asymmetric solution (in agreement with figure 2).

Substitution of (5.10) into (5.9) yields the nonlinear boundary condition

This condition animates the Stokes flows $\boldsymbol{U}$ and $\bar{\boldsymbol{U}}$ , which otherwise satisfy the homogeneous conditions of impermeability and continuity at $r=1$ and far-field decay. We have therefore obtained a ‘self-contained’ problem governing the flow. It is readily verified that condition (5.11) possesses the mirror-image symmetry mentioned earlier (see (3.2) etc.), which in turn is not restricted to $\mathit{Re}\gg 1$ . A remarkable feature of the present asymptotic problem is that it possesses an additional symmetry to flow reversal. Thus, if $(\boldsymbol{U},\bar{\boldsymbol{U}},Q)$ is a solution then so also is $(-\boldsymbol{U},-\bar{\boldsymbol{U}},-Q)$ .

The left-hand side of (5.11) represents the product of the tangential fluid velocity and the tangential shear stress or, equivalently, the rate of mechanical work introduced into the fluid by the interface; since the fluid is in a steady state this work is transformed into viscous dissipation. Considering the drop domain, we see that the integral of $V(\partial \bar{V}/\partial r-\bar{V})$ must be non-negative; similarly, considering the suspending liquid, the integral of $V(\partial V/\partial r-V)$ must be non-positive. Thus, at large $\mathit{Re}$ a steady solution only exists provided that

6. Universal flow problem

The parameters $R$ , $S$ and $M$ affect the leading-order flow only through the inhomogeneous condition (5.11). Defining the rescaled velocity field

together with a similar definition for $\bar{\boldsymbol{U}}^{\prime }$ , the homogeneous equations and conditions remain intact whereas (5.11) transforms to

which is independent of $R$ and $S$ . The problem has been accordingly reduced to a universal one, which depends (through condition (6.2)) only upon the viscosity ratio $M$ . Defining the rescaled charge density,

it is given by (cf. (5.10))

Using (5.3) and (5.4) we find that the scales of velocity and charge density effectively used in normalizing the universal fields are respectively given by

7. Stream function

It is convenient to solve the universal flow problem using a stream-function formulation, with the stream function ${\it\Psi}^{\prime }$ defined by

(and the drop-phase stream function $\bar{{\it\Psi}}^{\prime }$ similarly defined using $\bar{U}^{\prime }$ and $\bar{V}^{\prime }$ ). Both functions satisfy the biharmonic equation (Leal Reference Leal2007). Solutions of this equation that satisfy impermeability and continuity of tangential velocity at $r=1$ , regularity at $r=0$ and far-field decay (see (3.3)) are

The contributions of the ‘zeroth mode’ in these expansions to the tangential velocities $V^{\prime }$ and $\bar{V}^{\prime }$ are respectively given by the external irrotational vortex ${\it\Omega}/r$ (cf. (3.3)) and the internal rigid-body rotation ${\it\Omega}r$ . Since the contributions of all other modes ( $n\geqslant 1$ ) have zero circumferential average, ${\it\Omega}$ effectively represents the angular speed of drop rotation.

Figure 3.

${\it\Omega}$ versus $M$ . The two insets correspond to the gas-bubble limit $M=\infty$ , respectively showing the streamlines ${\it\Psi}^{\prime }=\text{const.}$ and the variation with ${\it\theta}$ of the universal surface-charge density.

Substitution of (7.2) into (6.2) yields

This infinite system of nonlinear algebraic equations can be solved by comparing Fourier modes. The zeroth-mode balance gives

while for $n>1$ we find

Because of the invariance of the universal problem to the reversal of the flow direction, it suffices to consider only the ‘positive’ branch, where ${\it\Omega}>0$ .

For $M\rightarrow 0$ (highly viscous drop) it is readily found that ${\it\Omega}\rightarrow 1$ while $a_{n}=O(M^{n})$ . In this limiting case the streamlines are concentric circles. For finite values of $M$ we use controlled truncation to solve the algebraic system numerically using iterations, starting from a positive ${\it\Omega}$ . The resulting (rather weak) dependence of angular speed ${\it\Omega}$ upon $M$ is depicted in figure 3. As $M\rightarrow \infty$ (inviscid gas ‘bubble’) we get ${\it\Omega}=0.92669$ . The two insets of figure 3 correspond to this limit. The first depicts the corresponding (non-circular) streamline pattern, and the second portrays the variation with ${\it\theta}$ of the universal surface-charge density $Q^{\prime }$ , obtained from (6.4). This limiting stationary distribution corresponding to an induced dipole (cf. figure 1) perpendicular to the applied field reflects the dynamic balance of surface-charge convection and Ohmic charging at asymptotically large $\mathit{Re}$ .

Given the scaling of the large- $\mathit{Re}$ solution and the additional rescaling employed in (6.1), it is evident that, at large $\mathit{Re}$ ,

where ${\it\Omega}$ is rather insensitive to the value of $M$ , see figure 3. This allows the dependence of ${\it\omega}$ upon $\mathit{Re}$ to be captured in a universal form. Thus, figure 4 presents the variation with $\mathit{Re}$ of the rescaled angular speed $(S+1){\it\omega}/\sqrt{S(1-RS)}$ using the numerical solutions of figure 2, obtained for $M=0.4$ and the indicated values of $R$ and $S$ . Regardless of the values of $R$ and $S$ , the rescaled speed approaches the universal asymptote ${\it\Omega}\mathit{Re}^{-1/2}$ (wherein ${\it\Omega}=0.9919$ , see figure 3), with convergence already taking place at moderately large values of $\mathit{Re}$ . As could be anticipated, following an earlier bifurcation to an asymmetric solution, this convergence becomes more rapid with diminishing values of $RS~({<}1)$ .

Figure 4.

Collapse of the rescaled exact solutions, calculated for the indicated values of $R$ and $S$ , on the universal large- $\mathit{Re}$ approximation. The viscosity ratio is $M=0.4$ .

8. Concluding remarks

In summarizing our results, it is illustrative to go back to dimensional quantities. The present analysis for large electric Reynolds numbers is valid for field magnitudes $E$ that are large relative to $\sqrt{{\it\sigma}{\it\mu}}/{\it\epsilon}$ . At this limit, solutions exist only when charge relaxation within the suspending liquid is faster than that in the drop, ${\it\sigma}/{\it\epsilon}>\bar{{\it\sigma}}/\bar{{\it\epsilon}}$ . These solutions lack fore–aft symmetry and involve drop rotation, the corresponding angular velocity scaling as $E\sqrt{{\it\sigma}/{\it\mu}}$ . For a given set of parameters, the reverse flow is also a solution. The dependence of the scaled angular velocity upon $R$ and $S$ is obtained as a simple closed-form expression.

Numerical solutions of the exact 2D equations indeed predict a bifurcation from a fore–aft-symmetric solution to a rotating-drop solution when the electric Reynolds number is sufficiently high. Unsurprisingly, the Reynolds-number threshold for this bifurcation is lower the faster is the relaxation in the suspending liquid, namely with increasing difference between ${\it\sigma}/{\it\epsilon}$ and $\bar{{\it\sigma}}/\bar{{\it\epsilon}}$ . Regardless of the magnitude of that difference, however, the resulting value of the angular velocity eventually approaches that predicted by the asymptotic solution. We therefore predict a useful universal structure that is independent of both the conductivity and permittivity ratios.

Our interest in drop rotation has naturally directed our attention to asymmetric solutions. It is, however, worthwhile to note that, while (5.10) is incompatible with (3.1) for continuous surface distributions of the scaled fields $Q$ and $V$ , it does not exclude fore–aft-symmetric solutions involving finite (jump) discontinuities at ${\it\theta}=\pm {\rm\pi}/2$ . These weak solutions are reminiscent of the ‘charge shocks’ recently described by Lanauze, Walker & Khair (Reference Lanauze, Walker and Khair2015) in the 3D axisymmetric problem. Thus, with increasing $\mathit{Re}$ the surface-charge density approaches a fore–aft-antisymmetric distribution with a narrow transition layer about the equator. Experiments demonstrate (ibid.) that these solutions becomes unstable at large $\mathit{Re}$ , as manifested in the onset of drop rotation.

A desirable extension of the present work is the corresponding 3D analysis of drop rotation. Study of these issues is currently underway, with preliminary results indicating that the present scaling as $\mathit{Re}^{-1/2}$ is essential to the resolution of the 3D problem as well.