## Abstract

In the absence of external forces, a liquid toroidal drop freely suspended in another fluid shrinks towards its centre. It is shown that if the two phases are slightly conducting viscous incompressible fluids with the drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities to be 1/*R*, *Q* and *λ*, respectively, then the toroidal drop with volume 4*π*/3 and having major radius *ρ* can become almost stationary when subjected to a uniform electric field aligned with the drop’s axis of symmetry. In this case, *Q* and electric capillary number *Ca*_{E} that defines the ratio of electric stress to surface tension, are functions of *R*, *ρ* and *λ* and are found analytically. Those functions are relatively insensitive to *λ*, and for *ρ*≥3, they admit simple approximations, which coincide with those obtained recently for *λ*=1. Streamlines inside and outside the toroidal drop for the same *R* and *ρ* but different *λ* are qualitatively similar.

## 1. Introduction

A liquid drop, being freely suspended in another fluid, deforms under the presence of an electric field [1,2]. If both phases are slightly conducting (leaky dielectrics) viscous incompressible fluids with the drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities to be 1/*R*, *Q* and *λ*, respectively, then Taylor's small-deformation theory [2] predicts that an initially spherical drop becomes prolate, oblate or remains spherical if _{E} that defines the ratio of electric stress to surface tension. If Ca_{E} exceeds a critical value, which is a function *R*, *Q* and *λ*, the drop either deforms indefinitely or disintegrates. See reviews [4–6] on deformation of spheroidal drops.

A number of natural processes can result in the formation of toroidal drops. They include but are not limited to drops in rotating fluids [7,8], free fall of drops in immiscible fluids [9,10], impact of drops with a solid surface [11], drop sedimentation [12,13] and drops subjected to electric and magnetic fields [14–16]. In fact, in the absence of any external forces, a toroidal drop freely suspended in another fluid shrinks towards its centre [8,17,18]. However, it can become stationary when embedded in a compressional flow [19,20] and under certain conditions can become ‘qualitatively’ (almost) stationary^{1} if subjected to an electric field uniform at infinity when *λ*=1 [21]. In the latter case, *Q* and Ca_{E} are functions of *R* and torus’ major radius *ρ*. For large *ρ*, e.g. *ρ*≥3 or even *ρ*≥2, the asymptotic behaviour of such *Q* and Ca_{E} is given by Zabarankin [21, (4.9)]:
*Q*=(*R*^{2}+*R*+1)/(3*R*^{2}) is close to the curves _{E}, whereas a toroidal drop is ‘stationary’ for certain Ca_{E}, which depends on *R*, *ρ* and *λ*— this is because the toroidal drop will collapse in the absence of external forces (electric field).

The goal of this work is to obtain conditions on *R*, *Q*, Ca_{E} and *ρ* for arbitrary *λ* under which a toroidal drop freely suspended in another fluid remains almost stationary (undeformed) when subjected to a uniform electric field aligned with the drop's axis of revolution. The flow inside and outside the drop should satisfy the velocity and stress boundary conditions, and the drop is stationary when the interface has zero normal velocity, which is called the kinematic condition, i.e. there are five scalar conditions in total. For an arbitrary fixed shape of the drop and, in particular, for toroidal, only four out of the five conditions can be satisfied exactly. When *λ*=1, the velocity field governed by the Stokes equations inside and outside the drop and satisfying the velocity and stress boundary conditions admits a closed-form integral representation for any shape. With this representation, Zabarankin [21] minimized a square error of the normal velocity over a toroidal surface with respect to *Q* and Ca_{E}. In this case, the scale of the normal velocity ranges from 10^{−6} to 10^{−2} for different *R* and *ρ*, and the cross section of the toroidal surface in the meridional plane coincides visually with a streamline. This shows that, for the optimal *Q* and Ca_{E}, a toroidal shape is qualitatively stationary—a true stationary toroidal shape is expected to have almost circular cross section. For arbitrary *λ*, there are two approaches: (I) satisfy the velocity and stress boundary conditions in toroidal coordinates exactly and minimize a measure of the normal velocity over the surface with respect to *Q* and Ca_{E}, and (II) interchange the roles of the normal stress boundary condition and the kinematic condition. For a true stationary shape, both approaches should yield the same solution. Otherwise, their solutions will be different but should be close if corresponding error measures (of the kinematic condition and of the normal stress boundary condition) are sufficiently small. Each approach has advantages. Approach I is consistent with quasi-stationary simulations of drop dynamics, and the velocity field can be checked against the said closed-form integral representation for *λ*=1. Approach I was used for finding ‘stationary’ spheroidal and toroidal shapes in [19–25]—in those cases, a measure of the kinematic condition was minimized with respect to shape parameters for given *R*, *Q* and Ca_{E}. Approach II is simpler for analytical treatment and was used to study a pair of spheroidal drops under the presence of an electric field [26] and stationary spindle-shaped drops in an extensional flow [27].

This work follows approach II with the following steps: (i) finding the electric field for a slightly conducting toroidal drop and ambient fluid (the electric field is uniform at infinity and is aligned with the drop's axis of symmetry), (ii) finding the velocity field governed by the Stokes equations and satisfying the velocity and the tangential stress boundary conditions and the kinematic condition, (iii) finding the pressure inside and outside the drop to be used in the normal stress boundary condition, and (iv) finding *Q* and Ca_{E} by minimizing a quadratic error of the normal stress boundary condition over the drop surface. Each step is treated analytically in the toroidal coordinates and yields corresponding closed-form representations. In particular, in step (ii), the velocity field inside and outside a toroidal drop is represented by stream functions expanded into Fourier series in the toroidal coordinates, and coefficients in the series expansions are found analytically in a non-recursive form. As a result, asymptotic behaviour of optimal *Q* and Ca_{E} for large *ρ* can be readily established and is shown to coincide with (1.2) for *λ*=1. This is a considerable advantage over approach I, in which finding of the velocity field is reduced to a system of high-order difference equations to be truncated and solved numerically [20].

The rest of the work is organized into four sections and an appendix. Section 2 formulates the problem of a drop freely suspended in another fluid and subjected to an electric field uniform at infinity. Section 3 addresses outlined steps (i)–(iv) and establishes the asymptotic behaviour of optimal *Q* and Ca_{E} for large *ρ*. Section 4 presents numerical results and §5 concludes the work. Appendix A presents a reformulation of the stress boundary conditions to be used in §3.

## 2. Problem formulation

Let a liquid toroidal drop of volume 4*πl*^{3}/3, where *l* is the radius of an equal-volume sphere, be freely suspended in an unbounded ambient fluid—the drop interface *S* divides *r*,*φ*,*z*) with the basis (**e**_{r},**e**_{φ},**e**_{z}) and its axis of revolution coincides with the *z*-axis (figure 2). Suppose the phases in *z*-axis, and suppose they are slightly conducting (leaky dielectrics) viscous incompressible fluids with corresponding dielectric constants *ε*^{±}, electric conductivities *σ*^{±} and viscosities *μ*^{±}, whose ratios are given by
*R*, *Q*, *λ* and strength of the electric field is the toroidal drop with major radius *ρ* stationary or nearly stationary? The formulation of this problem is identical to that in §2(a)–(c) in [21] and is restated below in a concise form.

Given that there are no free electric charges and magnetic field, a stationary electric field **E**^{±} satisfies curl**E**^{±}=0 and div**E**^{±}=0 in **E**^{±} on *S* is continuous, whereas, under the assumption that the effect of surface charge convection can be neglected^{2} (see [21, §2(c), §4(c))], its normal component *Φ*^{±} are electric potentials in *S* with outward normal vector *q* is determined by *ε*_{0} (see, e.g. [21, Eq. (2.2)]), and the difference in electric properties of the phases implies that the electric stress has a jump over *S* from *S* and where *S*; see [6,21,23] for details.

If inertial, gravitational and thermal effects are neglected, the velocity field **u**^{±} and pressure *p*^{±} of a steady-state viscous incompressible flow in **Δ****u**^{±}≡grad div **u**^{±}−curl curl **u**^{±}, and **u**^{−} and *p*^{−} should vanish at infinity:
*γ* is the interfacial tension, *S* and **F**_{E} is determined by (2.2).

The drop is stationary when the phase interface does not move (kinematic condition)

Further, let the dimensions (*r* and *z*), **u**^{±} and *p*^{±} be rescaled by *l*, *U* and *μ*^{−}*U*/*l*, respectively, where *π*/3, and its rescaled major and minor radii are denoted by *ρ* and *a*, respectively. The ratio of the electric stress to the surface tension in (2.5b) is the electric capillary number
*R*, *Q*, *λ* and Ca_{E}.

## 3. ‘Stationary’ toroidal shapes

For the toroidal drop, whose axis of revolution coincides with the *z*-axis (figure 2), problem (2.1)–(2.6) is considered to be axisymmetric, i.e. *Φ*^{±}, **u**^{±} and *p*^{±} do not depend on the angular coordinate *φ*:

In fact, (2.1)–(2.6) can be treated analytically in toroidal coordinates (*ξ*,*η*) related to *r* and *z* by
*c* is a metric parameter. In the meridional *rz*-half-plane, the cross section of a torus (circle) is a level curve *ξ*=*ξ*_{0} for some *ξ*_{0}—let it be denoted by ℓ, and the interior and exterior of the torus correspond to *ξ*∈[0,*ξ*_{0}), respectively. In this case, *π*^{2}*ρa*^{2}=4*π*/3, has only one free parameter—let it be *ρ*. Then *a*, *ξ*_{0} and *c* are expressed in terms of *ρ* as

In this case, the electric potentials *Φ*^{±} satisfying (2.1) in the toroidal coordinates were found analytically in [21], §3—the solution is restated here in §3a, whereas the velocity field **u**^{±}, governed by the Stokes equations (2.3), can be determined analytically in terms of stream functions that satisfy exactly the velocity and tangential stress boundary conditions and the kinematic condition in the toroidal coordinates. Then, for given *R*, *ρ* and *λ*, a square error of the normal stress boundary condition over *S* can be minimized with respect to *Q* and Ca_{E}—optimal *Q* and Ca_{E} as functions of *R*, *ρ* and *λ* represent conditions under which a toroidal drop with major radius *ρ* is as close to being stationary as possible (it will be referred to as ‘stationary’ toroidal drop) and which play the role similar to the condition

Further, let *k*=0, the superscript is omitted), and for brevity, let

### (a) Electric potentials for leaky dielectric toroidal drop

In the toroidal coordinates, electric potentials *Φ*^{±} can be represented by Fourier series [28, §6]:
*X*_{m} and *Z*_{m} determined recursively by
*ρ*=1 and *ρ*=2 for *R*=0.01, 0.5 and 100, and fig. 4 in [21] for charge distribution on *S* for *ρ*=1 in two cases: *R*=0.01 and *R*=100.

### (b) Stream functions for toroidal drop

The axisymmetric velocity fields **u**^{±} can be represented in terms of bi-one-harmonic stream functions *Ψ*^{±}=*Ψ*^{±}(*r*,*z*):
_{k} is the *k*-harmonic operator defined by
_{k} takes the form

In terms of *Ψ*^{±}, the scalar vorticity functions *ω*^{±}=**e**_{φ}⋅curl **u**^{±} are determined by

The velocity boundary conditions (2.5a) and the kinematic condition (2.6) yield^{3}

Appendix A shows that if *Ψ*^{±}|_{S}=0, then in the axisymmetric case, the tangential and normal stress boundary conditions, i.e. (2.5b) projected onto *λ*^{+}=*λ*, *λ*^{−}=1 and [*a**]|^{+}_{−}=*a*^{+}−*a*^{−}, with ‘*’ indicating variables and functions that have a jump across *S*. In the toroidal coordinates, the normal and tangential unit vectors for *S*, i.e. *S* are given by

The boundary conditions (3.6) and (3.7a) imply that *Ψ*^{±} are odd functions of *η*. In the toroidal coordinates (*ξ*,*η*),*Ψ*^{±} can be represented by Fourier series with respect to *η* (see [28])

With _{ξ0} is the derivative with respect to *ξ*_{0} and

The solution of the system (3.10a)–(3.10c) is given by

### (c) Pressure

Stokes equations (2.3) imply that the pressure *p*^{±} and vorticity *ω*^{±}=curl**u**^{±} are related by grad*p*=−*λ*^{±}curl*ω*^{±}, which in the axisymmetric case, simplifies to the Stokes–Beltrami equations:
*ω*^{±} are determined by (3.5) and where *rz*-half-plane. It follows from (3.5) and (3.8) that
_{0}*p*^{±}=0 in

Then, with
*C*^{+}_{0} is an arbitrary constant. The relationships (3.16) are a particular case of Hilbert formulae for *r*-analytic functions in toroidal regions; see [28, §5].

### (d) Normal stress boundary condition

Now, a square error of the normal stress boundary condition (3.7b) over *S*, determined by *ξ*=*ξ*_{0}, is to be minimized with respect to Ca_{E}, *Q* and *C*^{+}_{0} is put to zero, but this is ‘compensated’ by introducing a constant *C*. Further, (3.17) can be rewritten as

The function *f*(*η*) in (3.18) is even on [−*π*,*π*], so its square error over *S* is determined by
*S*=*r* d*φ* d*s* being the surface area element, in which _{E}, *Q* and *C*, the necessary first-order optimality conditions for a minimum of _{E}, *Q* and *C* are also sufficient [29], Theorem 3.1 and yield a system of linear equations for 1/Ca_{E}, *Q* and *C*:
*R*, *ρ* and *λ*: *Q*=*Q*(*R*,*ρ*,*λ*), Ca_{E}=Ca_{E}(*R*,*ρ*,*λ*) and *C*=*C*(*R*,*ρ*,*λ*).

### (e) Asymptotic behaviour of *Q*, Ca_{E} and *C* for large *ρ*

It is shown in [21] that, for large *ρ*, the coefficients *g*_{n} defined by (3.11):

Then, minimizing _{E}, *Q* and *C* is equivalent to setting the coefficients in (3.21) at

which yields

Observe that (3.22a) does not depend on *λ*, whereas (3.22b) is relatively insensitive to *λ*—figure 4 shows that the term in (3.22b) that involves *λ* and *ρ* does not vary much when *λ* changes from 0 to *λ*=0 and *λ*=0.01 and *λ*=100, respectively. Remarkably, (3.22a) and (3.22b) for *λ*=1 coincide with (1.2) obtained in [23] by approach I, i.e. by satisfying exactly the velocity and stress boundary conditions and then by minimizing a square error of the kinematic condition over *S* with respect to *Q* and Ca_{E}. This agreement validates both approaches.

## 4. Numerical results

Table 1 presents *Q*=*Q*(*R*,*ρ*,*λ*) and Ca_{E}=Ca_{E}(*R*,*ρ*,*λ*) found from (3.20) and also shows corresponding values of the average absolute error of the normal stress boundary condition over ℓ
*ρ*, *R* and *λ*, where *f* is defined in (3.18) and

The results in table 1 for *λ*=1 are close to those in table 1 in [21] but are not identical. This is expected because a toroidal drop with circular cross section cannot satisfy exactly the velocity and stress boundary conditions alone with the kinematic condition, and because *Q*=*Q*(*R*,*ρ*,*λ*) and Ca_{E}=Ca_{E}(*R*,*ρ*,*λ*) found here and in [21] for *λ*=1 minimize error measures of different boundary conditions. The fact that ∥*u*^{+}_{n}∥ in table 1 in [21] is considerably smaller than *λ*=1 does not necessarily indicate that approach I, discussed in the introduction, predicts *Q* and Ca_{E} for *λ*=1 more accurately. However, the fact that corresponding *Q* and Ca_{E} in table 1 for *λ*=1 and in table 1 in [21] are sufficiently close validates both approaches. Figures 5 and 6 show that as functions of *ρ*, *Q*(*R*,*ρ*,*λ*) and Ca_{E}(*R*,*ρ*,*λ*) have the same qualitative behaviour for different *λ* and are approximated by (3.22a) and (3.22b), respectively, quite accurately for *ρ*≥3.

Figures 7 and 8 present streamlines inside and outside toroidal drops, plotted by Algorithm (i)–(iii) in [20], §5, for triplets (*R*,*ρ*,*λ*) with *R*=0.01 and 100, *ρ*=1 and 2, and *λ*=0.01, 1 and 100. As the stream functions *Ψ*^{±} satisfy the kinematic condition exactly, the cross section of the drop coincides with a streamline. In this case, the streamlines are not affected by *Q* and Ca_{E} because *Q* and Ca_{E}, being found from the normal stress boundary condition, enter *Ψ*^{±} only through the multiplier 1−*RQ*. The streamlines in figures 7 and 8 for *λ*=1 and in [23], Fig. 8 are close for *ρ*=1 and almost coincide for *ρ*=2 (*R*=0.01 and 100). This also shows that although approaches I and II satisfy exactly different sets of boundary conditions and find *Q* and Ca_{E} from the kinematic condition and from the normal stress boundary condition, respectively, they yield quite similar results.

## 5. Conclusion

It was shown in [21] that a toroidal drop freely suspended in another fluid and subjected to an electric field, which at infinity is uniform and aligned with the drop's axis of revolution, remains qualitatively stationary (almost undeformed) when *Q* and Ca_{E} are certain functions of *R* and of the drop's major radius *ρ*. The two phases were assumed to be slightly conducting viscous incompressible fluids with the same viscosity (*λ*=1), and nonlinear effects of charge convection and inertia were neglected. Under the same assumptions except that the phases’ viscosity ratio *λ* is arbitrary, this work obtains similar functions *Q*=*Q*(*R*,*ρ*,*λ*) and Ca_{E}=Ca_{E}(*R*,*ρ*,*λ*) (from system (3.20)) by a different analytical approach (see §1). These functions are quite close to those in [21] for *λ*=1 and admit simple approximations (3.22a) and (3.22b) for *ρ*≥3 that coincide with (1.2) for *λ*=1. As conjectured in [21], they are relatively insensitive to *λ*. Moreover, the streamlines inside and outside a toroidal drop for the same *R* and same *ρ* but different *λ* are qualitatively similar. This is consistent with the observation made about ‘stationary’ toroidal drops in a compressional flow [20] that *λ* does not significantly alter the optimal capillary number as a function of *ρ* (see [20], fig. 7) and does not change the qualitative behaviour of the velocity field.

## Data accessibility

This work uses no external data.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

I am grateful to the referees for the comments and suggestions, which helped to improve the quality of the manuscript.

## Appendix A. Reformulation of the stress boundary conditions (2.5*b*)

Let *Ψ*^{±}|_{S}=0. Then the kinematic condition (2.6) holds. In the axisymmetric case, the vectors *S*) are related by

## Footnotes

↵1 In this case, the drop satisfies the velocity and stress boundary conditions, whereas its surface has minute normal velocity and visually coincides with a streamline.

↵2 In this case, the electric Reynolds number is assumed to be small. It is defined as the ratio of the time scale of the electric charge redistribution to that of the flow and it appears at the term of surface charge convection in the equation of current conservation at

*S*(see, e.g. [21], §2(c)). Justification of this assumption is discussed in [21], §4(c).↵3 In fact, (2.6) and (3.3) imply that

*Ψ*^{±}=*C*/*r*, where*C*is some constant. However, the stress boundary conditions (2.5b) and (3.3) imply that*Ψ*^{±}are odd functions with respect to*z*, and consequently,*C*=0.

- Received June 1, 2017.
- Accepted August 23, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.