## 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 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.

Original language | English |
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Pages (from-to) | 2150-2167 |

Number of pages | 18 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 79 |

Issue number | 5 |

DOIs | |

State | Published - 2019 |

## ASJC Scopus subject areas

- Applied Mathematics

## Keywords

- Boundary-integral equation
- Compressional flow
- Generalized analytic function
- Small deformation analysis
- Stokes flow
- Toroidal drop