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.

The dynamics of the deformation of a drop in axisymmetric compressional viscous flow is addressed through analytical and numerical analyses for a variety of capillary numbers, $\mathit{Ca}$, and viscosity ratios, $\lambda $. For low $Ca$, the drop is approximated by an oblate spheroid, and an analytical solution is obtained in terms of spheroidal harmonics; whereas, for the case of equal viscosities ($\lambda = 1$), the velocity field within and outside a drop of a given shape admits an integral representation, and steady shapes are found in the form of Chebyshev series. For arbitrary $Ca$ and $\lambda $, exact steady shapes are evaluated numerically via an integral equation. The critical $\mathit{Ca}$, below which a steady drop shape exists, is established for various $\lambda $. Remarkably, in contrast to the extensional flow case, critical steady shapes, being flat discs with rounded rims, have similar degrees of deformation ($D\sim 0. 75$) for all $\lambda $ studied. It is also shown that for almost the entire range of $\mathit{Ca}$ and $\lambda $, the steady shapes have accurate two-parameter approximations. The validity and implications of spheroidal and two-parameter shape approximations are examined in comparison to the exact steady shapes.

Abstract

We consider the deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid, subject to an axisymmetric extensional flow, and the influence of inertia in the continuous phase. The non-Newtonian fluid inside the drop is described by the simple power-law model and the unsteady deformation of the drop is represented by a single partial differential equation. The steady-state problem is governed by four parameters: the capillary number; the viscosity ratio; the external Reynolds number; and the exponent characterizing the power-law model for the non-Newtonian drop. For Newtonian drops, as inertia increases, drop breakup is facilitated. However, for shear thinning drops, the influence of increasing inertia results first in preventing and then in facilitating drop breakup. Multiple stationary solutions were also found and a stability analysis has been performed in order to distinguish between stable and unstable stationary states.

We report experimental evidence of an effect opposite to the "solidification" of small bubbles in liquid where the surface can become immobile. Namely, it is demonstrated that smooth solid spheres falling in a yield-stress fluid under the action of gravity can behave similar to drops. Particle tracking velocimetry was used to determine the shape of the yielded region around solid spherical particles undergoing slow stationary motion in 0.07% w/w Carbopol gel due to gravity under creeping flow conditions. The flow field inside the yielded region was determined by particle image velocimetry. It was found that the shape of the yielded region and the flow field around slow-moving rough particles is similar to the published results of numerical simulations, whereas those around smooth spheres resemble the experimental results obtained for viscous drops. The effect was explained by a slip of the gel on the smooth surface. Most likely, the slip originated from seepage of clean water from the gel, forming a thin lubricating layer near the solid surface.