TY - JOUR
T1 - Small deformation analysis for stationary toroidal drops in a compressional flow
AU - Zabarankin, Michael
N1 - Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Boundary-integral equation
KW - Compressional flow
KW - Generalized analytic function
KW - Small deformation analysis
KW - Stokes flow
KW - Toroidal drop
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U2 - 10.1137/19M1242070
DO - 10.1137/19M1242070
M3 - Article
AN - SCOPUS:85077382055
SN - 0036-1399
VL - 79
SP - 2150
EP - 2167
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 5
ER -