Toroidal Droplets: Growth Rates, Dispersion Relations, and Behavior

Article Cite This: Langmuir XXXX, XXX, XXX-XXX

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Toroidal Droplets: Growth Rates, Dispersion Relations, and Behavior in the Thick-Torus Limit Alexandros A. Fragkopoulos,† Ekapop Pairam,‡ Eric Berger,† and Alberto Fernandez-Nieves*,† †

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, United States Department of Food Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

‡

ABSTRACT: Toroidal droplets in a viscous liquid are unstable and transform into single or multiple spherical droplets. For thin tori, this can happen via the Rayleigh−Plateau instability causing the breakup of cylindrical jets. In contrast, for thick tori, this can happpen via the shrinking of the “hole”. In this work, we use the thin-torus limit to directly measure the growth rate associated with capillary disturbances. In the case of toroidal droplets inside a much more viscous liquid, we even obtain the full dispersion relation, which is in agreement with theoretical results for cylindrical jets. For thick tori, we employ particle image velocimetry to determine the flow field of a sinking toroidal drop inside another viscous liquid. We find that the presence of the “hole” greatly suppresses one of the circulation loops expected for sinking cylinders. Finally, using the flow field of a shrinking toroidal droplet and the time-reversal symmetry of the Stokes equations, we theoretically predict the expected shape deformation of an expanding torus and confirm the result experimentally using charged toroidal droplets.

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INTRODUCTION The torus is a mathematically rich surface. From a topological perspective, it is distinct from a sphere in that it has a handle; as a result, there is no continuous mapping that allows the transformation of a torus into a sphere. From a geometrical perspective, a torus is also very different from a sphere in that it is not a surface with constant curvature. While the principal curvatures of a sphere are identical and equal to the inverse of its radius, for a torus, the values of the two principal curvatures, κmax and κmin, are not constant and change from point to point. Furthermore, they can both be positive or be positive and negative. Interestingly, the geometrical features of a torus affect the stability and fluid mechanics of droplets shaped in this way. The lack of a constant mean curvature, H = (κmax + κmin)/2, for example, implies a nonconstant Laplace pressure, Δp = 2Hγ, where Δp is the pressure jump at the interface resulting from a non-zero interfacial tension, γ. It is the variation of Δp through the surface of a torus what causes toroidal droplets to evolve in remarkable ways. Indeed, toroidal droplets with a sufficiently small aspect ratio, ξ = R0/a0, where R0 is the radius of the central circle of the torus and a0 the tube radius (see Figure 1), shrink towards their center to eventually coalesce onto themselves and become single spherical droplets.1 This shrinking instability is unique to the torus and is not seen for cylindrical jets,2 since for a cylinder the mean curvature H = 1/(2R), where R is the radius of the circular cross section, is constant throughout the surface. The geometric details of the droplet shape thus bring about novel effects that are not present for simpler and perhaps more common situations. © XXXX American Chemical Society

Figure 1. Schematic showing the right cross section of a torus. The solid torus is generated by revolving the cross section around the z-axis. Rin, R0, and Rout represent the inner, central ring, and outer radii of the torus, respectively. a0 represents the tube radius. We also show the polar coordinates (r,θ). Note θ is measured from the inside of the torus.

Because controllably generating droplets with “holes” is not trivial, due to the inherent instability of toroidal droplets, fundamental studies with fluid shapes that are topologically different from the sphere and the cylinder are not that common. This changed after it was experimentally shown that toroidal droplets could be made in a well-controlled fashion by injecting a liquid through a needle inside a rotating viscous bath1 (see Figure 2a). In this case, the viscous drag exerted by the outer fluid over the extruded liquid resulted in the formation of a curved jet (see Special Issue: Early Career Authors in Fundamental Colloid and Interface Science Received: July 1, 2017 Revised: September 18, 2017

A

DOI: 10.1021/acs.langmuir.7b02280 Langmuir XXXX, XXX, XXX−XXX

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the internal circulation in the case of thick droplets compared to sinking cylinders. Finally, guided by the flow field inside toroidal droplets associated with shrinking and employing the timereversal symmetry of Stokes flow, we qualitatively predict that expanding toroidal droplets should elongate horizontally, that is, along the x-axis (see Figure 1); this is subsequently verified experimentally using charged toroidal droplets.

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THIN TOROIDAL DROPLETS: GROWTH RATES AND DISPERSION RELATION Thin toroidal droplets break into multiple droplets, if κ ≈ 1/20 (see Figure 3a−c), or into a single droplet, if κ ≈ 1/30000 (see

Figure 2. (a) Main components of the experimental apparatus for generating toroidal droplets. (b) Curved jet resulting from the drag force exerted by the rotating viscous bath. (c) The toroidal droplet forms when the jet makes one full revolution. Scale bars are 4 mm. Figure 3. Snapshots for the time evolution of a toroidal droplet for (a− c) κ ≈ 1/20 and (d−f) κ ≈ 1/30000. The scale bar is 5 mm.

Figure 2b), which eventually closed on itself to form a toroidal droplet (see Figure 2c). In these experiments, the typical liquid injected is either water with 60 mM sodium dodecyl sulfate (SDS) or glycerol, and the continuous phase is a 3 × 104 cSt silicone oil. The viscosity ratio of the inner to outer liquid (κ = ηi/ ηo) and the interfacial tension are ≈1/30000 and ≈10 mN/m, respectively, for the case of water as the inner liquid and ≈1/20 and ≈27 mN/m, respectively, for the case of glycerol. After generation, it was observed that thin toroidal droplets evolved via the classical Rayleigh−Plateau instability causing the breakup of long fluid cylinders, or jets, into droplets, and that the measured wavelength associated with the breakup process was in agreement with the theoretically predicted wavelength of the fastest unstable mode, which is the mode that among the many possible ones, grows the fastest. This result illustrates that thin toroidal droplets can be used to study jet instabilities. Remarkably, for sufficiently thick tori, no unstable modes of the Rayleigh−Plateau type can develop. In this case, shrinking is the only way for the torus to become a sphere. Note that shrinking is always present irrespective of the aspect ratio of the torus. However, it becomes the only possible instability for sufficiently thick toroidal droplets to transform into a spherical droplet. As a result of its inherent relation to curvature, the shrinking instability has become the subject of both theoretical and numerical studies,2,3 as well as of additional experimental work.4 Interestingly, shrinking toroidal droplets elongate vertically, that is, along the z-axis (see Figure 1), as they shrink.3,4 In this paper, we study several aspects related to toroidal droplets. First, we quantify the growth rate associated with the unstable mode that causes the breakup of thin toroidal drops. We do this by monitoring the breakup process as a function of time for two viscosity ratios. Importantly, for one of them, we are able to obtain the full dispersion relation, and not just the growth rate associated with the fastest unstable mode; this is hard to do with cylinders, but as we will show, it is very natural with toroidal droplets. Our results conform to what was theoretically predicted for cylindrical jets in a viscous fluid with an identical viscosity contrast. Second, we perform particle image velocimetry to determine the flow field inside toroidal droplets associated with sinking and show that the presence of the “hole” greatly affects

Figure 3d−f). To understand why this is the case and rationalize various aspects associated with these breakup processes, we consider the theory for the breakup of cylindrical jets. From linear-stability analysis, we expect that the radius of the jet will grow as α(t ) = α0 + εe σt + ikz

(1)

where α0 is the unperturbed radius and ε, σ, and k = 2π/λ are the initial amplitude, growth rate, and wavenumber of the perturbation, respectively, with λ being the wavelength. For the case of a viscous jet immersed in another viscous liquid, Tomotika calculated the dispersion relation, which relates the dimensionless wavenumber, kα0, of an unstable mode to its γ associated growth rate.5 He found that σ = η a f (kα0 , κ ) and, o 0

hence, that the growth rate depends on both the dimensionless wavenumber of the mode and the viscosity contrast of the two liquids. Note that experimentally, the mode observed is usually the mode that corresponds to the maximum growth rate, as this is the fastest mode and the one that dominates over the other slower modes.6 For toroidal droplets, because of the inherent periodic boundary conditions, we can fit only a natural number of wavelengths of this mode. Hence:1 2πR 0 = nλ

(2)

where n ∈ . If we now substitute the jet with the tube radius of the torus, we find that the dimensionless wavenumber associated with the mode that causes breakup is ka0 = n/ξ

(3)

Experimentally, we measure the aspect ratio as ξ = (Rout + Rin)/ (Rout − Rin), where Rin and Rout are defined in Figure 1, which equals R0/a0 if the cross section of the torus is circular. Earlier experiments confirmed that the values of ka0 for κ values of both ≈1/20 and ≈1/30000 were in good agreement with the values expected from the classical linear-stability analysis.5 We show B


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Langmuir here that the associated values of the growth rate are also consistent with these expectations. To measure σ, we monitor the swell radius, as(t) = a0 + εeσt, defined at the location where the tube radius is maximal, and the neck radius, an(t) = a0 − εeσt, defined at the location where the tube radius is minimal, as a function of time (see Figure 4a for κ ≈

Figure 5. Dispersion relation for toroidal glycerol (empty circles) and water (filled circles) droplets in 30000 cSt silicone oil. Each point represents the breakup of one toroidal droplet. Tomotika’s calculations for cylindrical jets are plotted for κ = 1/20 (−·−), κ → 0 (), and κ = 1/ 30000 (−−−). γ ≈ 27 mN/m for toroidal glycerol droplets, and γ ≈ 10 mN/m for toroidal water droplets.

progressively smaller wavelength. By measuring the associated growth rates, we are thus able to obtain a wide range of the full dispersion relation, as shown in Figure 5. The fact that the fastest unstable form is the one with the largest wavelength also explains why for this viscosity ratio we almost always see breakup in a single spherical droplet, as shown in Figure 4c. If we now consider the case in which κ ≈ 1/20, because the fastest unstable mode has a smaller λ, we can always fit multiple wavelengths of this mode inside the ring of the torus. Hence, irrespective of ξ, it is always this mode that causes breakup. In this case, ξ determines the number of wavelengths that grow and hence the number of droplets that result from the breakup process; this number is 5 in the example shown in Figure 4a.

Figure 4. (a) Snapshot of a breaking toroidal droplet indicating the locations where as and an are measured, for κ ≈ 1/20. (b) Corresponding logarithmic plot of (squares) as − an, (triangles) as − a0, and (circles) a0 − an as a function of time, with the slope of the lines being the associated growth rates. (c) Snapshot of a breaking toroidal droplet indicating the locations where as and an are measured, for κ ≈ 1/30000. (d) Corresponding logarithmic plot of (squares) as − an, (triangles) as − a0, and (circles) a0 − an as a function of time, with the slope of the lines being the associated growth rates. (e) Time evolution of the inner radius, Rin, for the torus depicted in panel c. Scale bars are 4 mm.

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THICK TOROIDAL DROPLETS: SHRINKING, SINKING, AND EXPANSION BEHAVIOR The logarithmic plots in panels b and d of Figure 4 indirectly show the influence of shrinking, as we briefly discussed above. We can more directly confirm this by monitoring the time evolution of Rin; we see that it decreases over time, as shown in Figure 4e, emphasizing that shrinking and breaking are coupled.1 Note, however, that the decrease in Rin is more pronounced at shorter times, which is then when shrinking is most influential; this causes an overall increase in tube radius. As a result, the growth of ln[as(t) − a0] is enhanced, while the growth of ln[a0 − an(t)] is diminished; this can be seen in panels b and d of Figure 4 with triangles and circles, respectively. These two effects tend to cancel each other if we consider the time evolution of as(t) − an(t), as also shown in panels b and d of Figure 4 with squares. We also emphasize that the shrinking of a torus is also key in allowing toroidal droplets for κ ≈ 1/20 to always fit an integer number of wavelengths of the fastest unstable mode. Because this condition will not be met for an arbitrary ξ, shrinking reduces ξ up to the point where this can happen; it is then when Rayleigh− Plateau instabilities take over and the toroidal droplet breaks. Importantly, shrinking is the only instability that can happen whenever an unstable mode of the Rayleigh−Plateau type cannot fit along the torus. Experimentally, it was found that for ξ < 2, no breakup occurs. The existence of a limiting aspect ratio below which no unstable mode can grow is reminiscent of the Plateau criterion for the breakup of cylindrical threads,9 which states that no breakup can happen if the length of the cylinder is smaller than 2πα0. The threshold ξ = 2 then corresponds to the Plateau

1/20 and Figure 4c for κ ≈ 1/30000). We extract the value of σ by measuring the slope of the logarithmic plot of either the relative swell radius, ln[as(t) − a0], the relative neck radius, ln[a0 − an(t)], or the swell radius relative to the neck radius, ln[as(t) − an(t)], versus time.7,8 Note, however, that because toroidal droplets always shrink because of their inherent geometry,2 these plots are not linear at early times, where shrinking is most important. As a result, we measure the growth rates at longer times, where our plots are linear (see Figure 4b for κ ≈ 1/20 and Figure 4d for κ ≈ 1/30000). The value of σ is taken to be the average of the values obtained from the slopes of the three logarithmic curves. We then plot the dimensionless growth rate, σηoa0/γ, as a function of ka0, which we can rewrite as n/ξ using eq 3; the result is shown in Figure 5. The lines are Tomotika’s theoretical predictions for cylindrical jets for κ → 0 (solid line), κ = 1/30000 (dashed line), and κ = 1/20 (dashed−dotted line). Our results compare well with the theory. Importantly, for κ ≈ 1/30000, our data span a wide range of dimensionless wavenumbers. In this case, the fastest unstable mode is the one with the largest wavelength that can fit inside the toroidal ring, since the maximum in the dispersion relation occurs for a low ka0. As a result, we can effectively select different unstable modes by changing the aspect ratio of the torus. As ξ decreases, ka0 increases, and we effectively force breakup by a mode of a C


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Langmuir criterion, provided the relevant length of the torus is 2πRin. In this case, breakup cannot happen if 2πRin < 2πa0, that is, if Rin < a0; this is observed experimentally, as shown in Figure 6. The

Figure 7. Toroidal coordinates (η,χ,ϕ) and unit vectors êη and êχ. A torus is obtained by revolving around the z-axis. af is the radius of the focal circle and η0 defines a toroidal surface with aspect ratio R0/a0. η > η0 corresponds to the inside of the torus. Figure 6. “Phase” diagram for the evolution of toroidal droplets in terms of a0 and Rin. The line corresponds to a0 = Rin and separates regions where toroidal droplets shrink (empty symbols) and break (filled symbols).

⎛ ∂x ⎞2 ⎛ ∂y ⎞2 ⎛ ∂z ⎞2 ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ is the diagonal component of the ⎝ ∂qi ⎠ ⎝ ∂qi ⎠ ⎝ ∂qi ⎠ metric tensor, with i = η,χ,ϕ, are given by gii =

hη = hχ =

existence of a Plateau criterion for the breakup of toroidal droplets also explains why we could not probe dimensionless wavenumbers in Figure 5 above 1/2. In this case, because the growing unstable mode has ka0 = 1/ξ, and below ξ = 2 no breakup can happen, the largest dimensionless wavenumber we can probe against breakup must necessarily be 1/2. Recently, particle image velocimetry has been performed on shrinking toroidal droplets.4 In this work, the droplet was both shrinking and sinking. However, by exploiting the linearity of the Stokes equations and the symmetries of the velocity flow field, the authors successfully isolated the shrinking components. Here, we discuss the sinking components and emphasize the differences with the cylindrical case. We will then discuss the flow field of charged toroidal droplets, which are known to expand rather than shrink, if they are sufficiently charged.10 Theoretical Overview. Given the value of the density, ρ, of our liquids, and the values of the other experimental parameters, the Ohnesorge number Oh ≡ ηo / ρa0γ ≈ 100 ≫ 1. Thus, the Navier−Stokes equations reduce to the Stokes equations. In this case η∇2 v ⃗ = ∇p

=

af cosh(η) − cos(χ )

⎤ hϕ ⎡ ∂ ⎛ hχ ∂ ⎞ ∂ ⎛ h η ∂ ⎞⎥ ⎢ ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ hηhχ ⎢⎣ ∂η ⎝ hηhϕ ∂η ⎠ ∂χ ⎝ hχ hϕ ∂χ ⎠⎥⎦

E2 =

(6)

(7)

This fourth-order differential equation has been solved analytically, and the solution is given by2,11 Ψ=

a f sinh(η) [cosh(η) − cos(χ )]3/2

+∞

∑

{sin(nχ ), cos(nχ )}

n =−∞

{Pn1− 1/2[cosh(η)], Q n1− 1/2[cosh(η)]}

Pmn

(8)

and Qmn are

where the associated Legendre polynomials of the first and second kind, respectively, and the curly brackets denote a linear superposition of the arguments. Considering that Qmn and Pmn diverge at infinity and at the central circle of the torus, respectively, for us to obtain physically sound solutions, we must associate Qmn and Pmn to the flow field inside and outside the torus, respectively. In addition, it has been shown that the streamfunction for sin(nχ) corresponds to shrinking or expanding flow fields, while the streamfunction for cos(nχ) corresponds to sinking flow fields.2 From the streamfunction, the velocity can be obtained:12

where p is the pressure and v⃗ is the velocity. Note that we have also used at the outset the fact that the liquid is incompressible and thus that ∇·v ⃗ = 0. By taking the curl of eq 4, we derive the vorticity equation: (5)

where ω⃗ =∇ × v ⃗ is the vorticity. We will consider this equation in the case of toroidal geometries. Hence, we will use a set of toroidal coordinates (qη,qχ,qϕ) = (η,χ,ϕ),11 where ϕ is the azimuthal angle (see Figure 1). The angle χ ∈ (−π,π] spans the tube of the torus along a direction that is opposite to polar angle θ (see Figure 1) and is defined with respect to a focal circle with 2

sinh(η)

For shrinking, sinking, or expanding tori, there is azimuthal symmetry. As a result, the expected flow field is independent of ϕ. In this case, we can obtain the flow field from a streamfunction,11 Ψ, which satisfies a fourth-order differential equation, E2E2Ψ = 0, where E2 is a second-order differential operator given by11

(4)

∇2 ω⃗ = 0

hϕ

vη =

1 ∂Ψ hχ hϕ ∂χ

vχ = −

2

radius a f = R 0 − a0 on the xz plane, as shown in Figure 7. Coordinate η defines a toroidal surface and is related to the aspect ratio of the corresponding surface, ξ = cosh(η). The scale factors associated with these coordinates, hi = gii , where

1 ∂Ψ hηhϕ ∂η

(9)

(10)

where vη and vχ are the velocities along the unit vectors, êη and êχ, associated with the coordinates η and χ, respectively (see Figure 7). Finally, we can write these velocities in Cartesian coordinates as D


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Langmuir vx = [vηeη̂ + vχ eχ̂ ]·ex̂

(11)

vz = [vηeη̂ + vχ eχ̂ ]·eẑ

(12)

torus exhibits up-down symmetry, these symmetry properties of the flow field remain unchanged irrespective of the aspect ratio of the torus and can be used to obtain the velocities associated with sinking. We first find the centroid associated with the cross section of the torus, which we take as the z = 0 position (see Figure 8b) and then calculate the antisymmetric part of vx as vx,A = [vx(z) − vx(−z)]/2 and the symmetric part of the symmetric part of vz as vz,S = [vz(z) + vz(−z)]/2. The velocities associated with sinking are obtained as vx,Ax̂ + vz,Sẑ; these are shown in Figure 9b. To obtain them in the reference frame of the cross section, we subtract the sinking velocity, which we equate to the average value of the z-component of the velocity, vsink = ⟨vz⟩; the result is shown in Figure 9c. Comparing this result with that expected for thin tori, as schematically shown in Figure 9a, we see that the internal circulation at the outer part of the torus dominates the flow field; the internal circulation at the inside part of the torus is thus greatly diminished, as highlighted in the enlarged image in Figure 9c. Hence, the toroidal geometry tends to suppress the internal circulation at the inside of the torus. This is reasonable because as ξ decreases, the “hole” of the torus becomes smaller, causing an increase in the drag experienced by the outer liquid as it flows through. To theoretically obtain the internal flow field, we consider the streamfunction for the sinking problem:

Sinking Toroidal Droplets. For these experiments, we use a 6 × 104 cSt silicone oil as the outer liquid and a solution of 0.1% (w/w) PEG 8000 in ultrapure water as the inner liquid.4 The typical total volume of the toroidal drops we use is 20−25 mL. The inner liquid is seeded with polystyrene particles with an average diameter of 16.2 μm, to introduce the optical heterogeneities needed to perform particle image velocimetry. We use a laser sheet to illuminate the cross section of the toroidal droplet, as shown schematically in Figure 8a.

Figure 8. (a) Schematic of the setup for particle image velocimetry used to determine the flow field inside toroidal droplets. A laser sheet is created using a cylindrical lens, and illuminates the center of the cross section of the torus; for our analysis, we use only the right cross section. (b) Flow field within this cross section for a shrinking and sinking toroidal droplet with ξ ≈ 1.4. The intersection of the two solid lines defines the centroid of the cross section used to exploit the symmetries in the problem.

Ψsink =

a f sinh(η) [cosh(η) − cos(χ )]3/2

+∞

∑ n =−∞

Dn cos(nχ )Q n1− 1/2[cosh(η)]

(13)

where Dn are coefficients that need to be determined. To do this, we first determine the radial and tangential components of the velocity at the interface of the torus in the experiments, vr⃗ = vrr ̂ ⎯v = v θ ,̂ as a function of θ. We define θ using the previously and→ t t calculated centroid of the cross section of the torus (see Figure 9c). We find that both vr and vt exhibit extrema around θ = π/3 and θ = 5π/3, corresponding to the locations where the velocity experiences rapid changes due to the approach to the inner smaller circulation (see Figure 9c). In addition, vt has another significant extremum at θ = π. We then obtain the values of Dn by performing a simultaneous least-squares fit of vr and vt as a function of θ, leaving the coefficients as free parameters. We consider only the first five modes in the streamfunction (n = −2, −1, 0, 1, and 2). We obtain D−2 = (0.0 ± 0.1)vsink, indicating that n = −2 is negligible. For the

We find that the resultant flow field does not have the up− down symmetry expected for a sinking or shrinking toroidal droplet, indicating that both these processes are at play in our experiments; this is shown in Figure 8b for ξ ≈ 1.4. However, because the Stokes equations are linear, we can exploit the symmetries of the flow field associated with sinking to extract the corresponding velocities. To illustrate the symmetries, let us consider the case of a sinking liquid cylinder, which we associate with the very thin torus limit. In this case, the expected flow field is characterized by a double circulation, as schematically shown in Figure 9a; note that this is in the reference frame of the cross section. We then see that the x-component of the velocity is updown antisymmetric, vx(z) = −vx(−z), and that the z-component of the velocity is up-down symmetric, vz(z) = vz(−z). Since a

Figure 9. (a) Schematic of the flow field associated with a sinking liquid cylinder, with a sinking speed vsink. We also indicate the respective up−down symmetries of such a flow field: vx is antisymmetric, while vz is symmetric. Using these symmetries we obtain the flow field for sinking in (b) the lab frame and (c) the drop frame, with polar angle θ defined using the centroid. The color code has been scaled with vsink. E


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Langmuir other four coefficients, we find D−1 = (0.31 ± 0.05)vsink, D0 = (−0.4 ± 0.1)vsink, D1 = (1.0 ± 0.2)vsink, and D2 = (−0.25 ± 0.05) vsink. Note that with these modes we are able to qualitatively reproduce the features seen in the experimental vr and vt curves (see Figure 10a,b). Note that in our experiments, the shrinking

Figure 10. Experimental (a) radial, vr, and (b) tangential, vt, components of the velocity at the interface as a function of θ (). The dashed lines correspond to the least-squares fit considering only the n = −2, −1, 0, 1, and 2 modes in the streamfunction. (c) Reconstructed flow field inside a sinking toroidal droplet using the coefficients determined from the fits in panels a and b. The color code has been scaled with vsink.

Figure 11. Flow field (drop frame) associated with the (a) n = 0, (b) n = 1, (c) n = −1, and (d) n = 2 modes. The color code in all panels has been scaled with vsink.

velocity, vsh, is significantly larger than vsink. As a result, shrinking is more prominent than sinking, complicating an accurate determination of vsink and likely explaining why we are unable to quantitatively reproduce the experimental results. Using the values of the coefficients, we calculate Ψsink everywhere inside the torus and, from that, the velocity field using eqs 9−12. We find that the reconstructed flow field is qualitatively very similar to that obtained experimentally; in particular, it exhibits a dominant internal circulation at the outside of the torus and a greatly diminished internal circulation at the inside of the torus. To further understand our results, we take a closer look at each of the four modes that contribute to the sinking flow profile. For n = 0 or 1, the associated flow field has a single internal circulation that spans the entire toroidal tube, as shown in panels a and b of Figure 11. In contrast, we find that the n = −1 mode features a double circulation, with one circulation dominating over the other (see Figure 11c). This is in agreement with our experiments, as the circulation at the inside of the torus is greatly suppressed compared to the circulation at the outside of the torus. We note that if we plot the flow field associated with this mode for a higher aspect ratio, say ξ ≥ 10, both circulations are nearly the same size; in this limit, we recover the flow field of a sinking liquid cylinder. Finally, the flow field associated with n = 2 is characterized by a vr that points outward at θ ≈ π/4 and θ ≈ 5π/4 and that points inward at θ ≈ 3π/4 and θ ≈ 7π/4. Interestingly, this kind of flow field favors a shape evolution that breaks the up−down symmetry of the circular cross section of the torus; this also contributes to the uncertainty in the experimental determination of the sinking velocities and, in particular, of vr and vt, because in experiments, the cross section is not circular but vertically elongated, compromising the up-down symmetries used to obtain them. This is not the case for a shrinking torus; even for a deformed cross section, the up−down symmetry of the flow field is preserved.4 The errors in obtaining vr and vt in this last case are thus expected to be much smaller than those in the sinking problem.

Expanding Toroidal Droplets. Shrinking is the natural response of a toroidal droplet to minimize its surface area. The flow field associated with such evolution can be obtained from the streamfunction Ψshrink =

a f sinh(η) [cosh(η) − cos(χ )]3/2 +∞

∑

Cn sin(nχ )Q n1− 1/2[cosh(η)]

(14)

n =−∞ 4

where Cn are coefficients that have been found before. An interesting property of Stokes flow is that it exhibits time-reversal symmetry under velocity and pressure exchanges (v⃗ → −v ⃗ and p → −p). This means that if we consider the case of an expanding torus instead of a shrinking torus, the velocity field will be reversed. Then, using the previously determined values of Cn, we can obtain the expected flow field for an expanding torus; this is shown in Figure 12a in the lab frame and in Figure 12b in the reference frame of the droplet. Now recall that for shrinking toroidal droplets, the flow field determines how the interface deforms, because the interface will simply follow the fluid flow. In this case, we found that the interface elongated vertically.4 Similarly, by considering the flow field in the frame of reference of the cross section for an expanding toroidal droplet (see Figure 12b), we infer that the interface will move inward around θ ≈ ±π/4 and slightly outward around θ ≈ π, and remains stationary around θ = 0. This suggests that the cross section of the toroidal droplet will elongate horizontally, in marked contrast to what is observed for a shrinking toroidal droplet. To experimentally study this expansion, we charge the torus by applying a constant voltage across the toroidal surface and the rotating stage.10 In this case, the electrostatic stresses at the interface, which oppose the capillary stresses, for a sufficiently high voltage, can cause the torus to expand. We use the same liquids and volume that were used for the study of sinking toroidal droplets and find that the cross section of the torus elongates horizontally (see Figure 12c), consistent with our F


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Langmuir

relation can be accurately determined by changing the aspect ratio of the droplet. In the case of thick toroidal droplets, the geometry of the torus has a significant effect on the flow field associated with sinking. By performing particle image velocimetry, we find that the typical double circulation expected for a liquid cylinder exists, but that the inner circulation is greatly suppressed because of the presence of the “hole”, which inhibits flow through the torus. Finally, using the time-reversal symmetry of Stokes flow, we predicted that an expanding toroidal droplet should elongate horizontally, which we then confirmed experimentally. However, the experimental flow field deviates significantly from the predicted flow field, most likely because of the significant deformation of the cross section, which is far from circular as the theory assumes.

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Figure 12. Theoretical prediction for the flow field of an expanding torus with ξ = 2 in the (a) lab frame and (b) the frame of reference of the cross section. (c) Image of a 25 mL toroidal droplet with ξ ≈ 2.1 and V = 6 kV that is expanding. The white spots are the optical inhomogeneities produced from the polystyrene particles suspended in the inner liquid. Flow field associated with expansion, after using the symmetries in the problem, in (d) the lab frame and (e) the frame of reference of the cross section. The apparent up-down symmetry of the cross-sections in (d) and (e) results from the analysis, since we are only considering points in v for both z and −z. the cross-section in (c) for which we can measure → The color code of both panels has been scaled with the expansion velocity. The scale bar in panel c is 1 cm.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Alberto Fernandez-Nieves: 0000-0002-1286-9809 Notes

The authors declare no competing financial interest.

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REFERENCES

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expectations. However, we note that the shape of the toroidal cross section does not have up−down symmetry. We believe the reason for this is that the free surface at the top of the cuvette, which is ∼1 cm above the torus in these experiments, generates an image charge due to the difference in dielectric constants between the silicone oil and the air. As a result, this image charge repels the interface of the torus, which is charged. However, because the torus has the highest suface charge density at θ = π and the lowest surface charge density at θ = 0,10 the electrostatic stresses due to this image charge will be higher at θ = π than at θ = 0, causing the exterior of the torus to shift downward a larger amount compared to the interior of the torus, as seen experimentally in Figure 12c. Despite this asymmetry, the experiment still shows the expected deformation of the interface. In addition, we can perform particle image velocimetry and isolate the velocity field that corresponds to expansion. We do this by using the symmetries in the problem, which are the opposite of those of the sinking flow field. In this case, vx(z) = vx(−z) and vz(z) = −vz(−z). The resultant flow field is shown in Figure 12d in the lab frame and in Figure 12e in the droplet frame, which we obtain after subtracting the expansion velocity. We observe that the flow field is qualitatively similar to the flow field around the central region that is expected for droplets with a circular section (see Figure 12b). However, additional comparisons between the two are unrealistic given the significant deviation of the experimental cross-section from a circle.

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CONCLUSIONS Toroidal droplets are a rich system to study. In the thin-torus limit, they behave like cylindrical jets. Indeed, we find that the fastest unstable mode and its growth rate are in agreement with theoretical predictions for viscous jets inside viscous liquids. Furthermore, for small values of κ, a wide range of the dispersion G


Toroidal Droplets: Growth Rates, Dispersion Relations, and Behavior

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