Small Deformation Analysis for Stationary Toroidal Drops in a Compressional Flow

Abstract

It is known that a toroidal drop freely suspended in a quiescent ambient fluid shrinks and forms a simply connected drop. However, if it is embedded in a compressional flow and has an initially circular cross section, then for a certain value of the major radius, it may attain a stationary toroidal shape. This value is called critical major radius $R_{cr}$ and depends on capillary number Ca that defines the ratio of viscous forces to surface tension: the smaller Ca, the larger $R_{cr}$. It is relatively insensitive to the drop-to-ambient fluid viscosity ratio $\lambda$ particularly for small Ca. For Ca $\lesssim$ 0.16 (or $R_{cr} \gtrsim 1$), a stationary shape is close to a torus with an elliptical cross section, whose major radius $R$ and flattening $\Delta$ depend on Ca. In fact, for small Ca, any of the three Ca, $R$, and $\Delta$ can be considered an independent variable and the other two its functions. This work obtains asymptotic behavior of Ca($R$) and $\Delta(R)$ as $R\rightarrow\infty$ and, as a result, of $\Delta$(Ca) as Ca $\rightarrow$ 0. Those analytical relationships are in a good agreement with the existing numerical results for Ca $\lesssim$ 0.06 (or $R_{cr} \gtrsim 1.5$) for various values of $\lambda$ and play the role similar to that in the well-known small deformation theories for spherical drops. The central part of the presented analytical analysis is a novel boundary-integral equation for the axisymmetric velocity field of the corresponding two-phase Stokes flow problem. The equation was derived based on the Cauchy integral formula for generalized analytic functions.