Small deformation analysis for stationary toroidal drops in a compressional flow

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 Rcr and depends on capillary number Ca that defines the ratio of viscous forces to surface tension: the smaller Ca, the larger Rcr. It is relatively insensitive to the drop-to-ambient fluid viscosity ratio λ particularly for small Ca. For Ca . 0.16 (or Rcr & 1), a stationary shape is close to a torus with an elliptical cross section, whose major radius R and flattening ∆ depend on Ca. In fact, for small Ca, any of the three Ca, R, and ∆ can be considered an independent variable and the other two its functions. This work obtains asymptotic behavior of Ca(R) and ∆(R) as R → ∞ and, as a result, of ∆(Ca) as Ca → 0. Those analytical relationships are in a good agreement with the existing numerical results for Ca . 0.06 (or Rcr & 1.5) for various values of λ 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.