Liquid toroidal drop in compressional Stokes flow

Abstract

The deformation of an immiscible toroidal drop embedded in axisymmetric compressional Stokes flow is analysed via the boundary integral formulation in the case of equal viscosity. Numerical simulations are performed for the drop having initially the shape of a torus with circular cross-section. The quasi-stationary dynamic simulations reveal that, when the viscous forces, proportional to the intensity of the flow, are relatively weak compared with the surface tension (the ratio of these forces is characterized by the capillary number, Ca), three different scenarios of drop evolution are possible: indefinite expansion of the liquid torus, contraction to the centre and a stationary toroidal shape. When the intensity of the flow is low, the stationary shapes are shown to be close to circular tori. Once the outer flow strengthens, the cross-section of the stationary torus assumes first an elliptic and then an egg-like shape. For the capillary number greater than a critical value, Ca cr , toroidal stationary shapes were not found. Remarkably, Ca cr is close to the critical capillary number found previously for a simply connected drop flattened in compressional flow. Thus, a new example of non-uniqueness of stationary drop shape in viscous flow is obtained. Approximate stationary solutions in the form of tori with circular and elliptic cross-sections are obtained by minimizing the normal velocity over the drop interface. They are shown to be in good agreement with the stationary shapes from quasi-dynamic simulations for the corresponding intervals of the capillary number.

Shrinking instability of toroidal droplets

“…This result is qualitatively similar to what was seen in computer simulations (10). We then use PIV to determine the flow field inside the torus.…”

“…Assuming that the cross-section of the torus remains circular during the process and that the velocity at the interface is radial, in the reference frame of the circular cross-section, calculations of the shrinking speed were consistent with experimental observations (9). However, recent simulations have found that the cross-section does not remain circular but that it rather deforms significantly as the torus shrinks (10). Despite this difference with the theory, which brings about other additional differences in the flow fields, the simulated shrinking speed was also consistent with the experimental results.…”

Liquid toroidal drop in compressional flow with arbitrary drop-to-ambient fluid viscosity ratio

“…1 The theoretical work of Zabarankin et al [12] showed that when the toroidal drop and compressional flow have equal viscosities and the drop's initial cross section in the meridional rz-half plane is a circle with major radius R as shown in figure 1b, then for each capillary number Ca, there is a critical radius R cr such that the drop collapses towards its centre if R < R cr , expands indefinitely if R > R cr , and attains a stationary toroidal shape when R = R cr . Deformation of the toroidal drop was analysed via quasi-stationary simulation of drop dynamics under the assumptions that the both phases were viscous incompressible fluids with the same viscosity and that inertia was negligible, i.e.…”

“…In such a simulation, a drop freely suspended in another liquid satisfies the velocity and stress boundary conditions, and its shape is advanced by using the normal velocity of the interface as a shape gradient (see [12,[15][16][17] …”

Charge-Induced Saffman-Taylor Instabilities in Toroidal Droplets

“…For low enough voltage the torus either shrinks and coalesces onto itself to become a single spherical droplet [6,[8][9][10], or expands before breaking due to Rayleigh-Plateau instabililities [3]; an example of the latter is shown in Fig. 2(a-c) for a torus with an aspect ratio ξ = R 0 /a 0 ≈ 3.5 and at V = 500 V. For a similar torus at V = 800 V, we observe that the interface slightly distorts, eventually resulting in the formation of fingers; this evolution is exemplified in Fig.…”