TY - JOUR
T1 - Liquid toroidal drop in compressional flow with arbitrary drop-to-ambient fluid viscosity ratio
AU - Zabarankin, Michael
N1 - Publisher Copyright:
© 2016 The Author(s).
PY - 2016/3
Y1 - 2016/3
N2 - Existing experiments show that a sufficiently fat toroidal drop freely suspended in another liquid shrinks towards its centre to form a spherical drop. However, recent simulations reveal that if a liquid torus with circular cross section is embedded in a compressional same-viscosity flow that acts to expand the torus, then depending on the torus radius R and a capillary number Ca characterizing the balance between the viscous forces and the interfacial tension, the torus may either coalesce, expand indefinitely or attain a stationary shape. For each Ca less than 0.2, there is a single value of R, called the critical radius, for which the torus attains the stationary shape. Here, the drop-to-ambient fluid viscosity ratio, λ, is assumed to be arbitrary. The corresponding two-phase Stokes flow problem is solved for a liquid toroidal drop with circular cross section in terms of stream functions in the toroidal coordinates. When λ=1, the stream functions admit a closed-form integral representation for a drop of arbitrary axisymmetric shape. 'Stationary' circular tori minimize a certain measure of the normal velocity over the interface, and as in the case of λ=1, their radii are expected to predict the critical ones for arbitrary λ and Ca in a certain range (e.g. for Ca<0.2 when λ=1). Streamlines about 'stationary' circular tori are analysed for various Ca and λ.
AB - Existing experiments show that a sufficiently fat toroidal drop freely suspended in another liquid shrinks towards its centre to form a spherical drop. However, recent simulations reveal that if a liquid torus with circular cross section is embedded in a compressional same-viscosity flow that acts to expand the torus, then depending on the torus radius R and a capillary number Ca characterizing the balance between the viscous forces and the interfacial tension, the torus may either coalesce, expand indefinitely or attain a stationary shape. For each Ca less than 0.2, there is a single value of R, called the critical radius, for which the torus attains the stationary shape. Here, the drop-to-ambient fluid viscosity ratio, λ, is assumed to be arbitrary. The corresponding two-phase Stokes flow problem is solved for a liquid toroidal drop with circular cross section in terms of stream functions in the toroidal coordinates. When λ=1, the stream functions admit a closed-form integral representation for a drop of arbitrary axisymmetric shape. 'Stationary' circular tori minimize a certain measure of the normal velocity over the interface, and as in the case of λ=1, their radii are expected to predict the critical ones for arbitrary λ and Ca in a certain range (e.g. for Ca<0.2 when λ=1). Streamlines about 'stationary' circular tori are analysed for various Ca and λ.
KW - Analytical solution
KW - Compressional flow
KW - Stationary shape
KW - Stokes flow
KW - Stream function
KW - Toroidal drop
UR - http://www.scopus.com/inward/record.url?scp=84963558673&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84963558673&partnerID=8YFLogxK
U2 - 10.1098/rspa.2015.0737
DO - 10.1098/rspa.2015.0737
M3 - Article
AN - SCOPUS:84963558673
SN - 1364-5021
VL - 472
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2187
M1 - 20150737
ER -