We investigate the minimal yield-stress required in order to hold static an ellipsoidal Newtonian droplet inside a yield-stress liquid. This critical limit (Yc) is influenced by the droplet aspect ratio (χ), the interfacial tension (γ), and the viscosity ratio (M) between the droplet and the surrounding liquid, as well as the ratio of the yield-stress to the buoyancy stress (Y). The droplet will remain trapped by the liquid yield-stress for . Our study bridges the gap in the published results between those calculated for bubbles ( ) and the solid rigid particles ( ), being of practical use for those estimating the design of stable yield-stress emulsions. In general, the critical yield number increases with the interfacial tension and the droplet aspect ratio and will decrease with the droplet viscosity. For spherical droplets, our results computed for yield numbers below Yc suggest that the spherical shaped droplet may propagate in steady motion.
REFERENCES
1.
J.
Hadamard
, “
Slow permanent movement of a viscous liquid sphere in a viscous liquid
,” C. R. Acad. Sci.
152
, 1735
–1738
(1911
).2.
W.
Rybczynski
, “
On the progressive motion of a liquid sphere in a viscous medium
,” Bull. Acad. Sci. Cracovie A
1
, 40
–46
(1911
).3.
R. P.
Chhabra
, Bubbles, Drops, and Particles in non-Newtonian Fluids
(
CRC Press
, 2006
).4.
G.
Dazhi
and
R.
Tanner
, “
The drag on a sphere in a power-law fluid
,” J. Non-Newtonian Fluid Mech.
17
, 1
–12
(1985
).5.
R. I.
Tanner
, “
Stokes paradox for power-law flow around a cylinder
,” J. Non-Newtonian Fluid Mech.
50
, 217
–224
(1993
).6.
R. W.
Ansley
and
T. N.
Smith
, “
Motion of spherical particles in a Bingham plastic
,” AIChE J.
13
, 1193
–1196
(1967
).7.
K.
Adachi
and
N.
Yoshioka
, “
On creeping flow of a visco-plastic fluid past a circular cylinder
,” Chem. Eng. Sci.
28
, 215
–226
(1973
).8.
A.
Beris
,
J.
Tsamopoulos
,
R.
Armstrong
, and
R.
Brown
, “
Creeping motion of a sphere through a Bingham plastic
,” J. Fluid Mech.
158
, 219
–244
(1985
).9.
B. T.
Liu
,
S. J.
Muller
, and
M. M.
Denn
, “
Interactions of two rigid spheres translating collinearly in creeping flow in a Bingham material
,” J. Non-Newtonian Fluid Mech.
113
, 49
–67
(2003
).10.
L.
Jossic
and
A.
Magnin
, “
Drag and stability of objects in a yield stress fluid
,” AIChE J.
47
, 2666
–2672
(2001
).11.
M.
Beaulne
and
E.
Mitsoulis
, “
Creeping motion of a sphere in tubes filled with Herschel–Bulkley fluids
,” J. Non-Newtonian Fluid Mech.
72
, 55
–71
(1997
).12.
J.
Blackery
and
E.
Mitsoulis
, “
Creeping motion of a sphere in tubes filled with a Bingham plastic material
,” J. Non-Newtonian Fluid Mech.
70
, 59
–77
(1997
).13.
E.
Mitsoulis
, “
On creeping drag flow of a viscoplastic fluid past a circular cylinder: Wall effects
,” Chem. Eng. Sci.
59
, 789
–800
(2004
).14.
N.
Roquet
and
P.
Saramito
, “
An adaptive finite element method for Bingham fluid flows around a cylinder
,” Comput. Meth. Appl. Mech. Eng.
192
, 3317
–3341
(2003
).15.
D.
Tokpavi
,
A.
Magnin
, and
P.
Jay
, “
Very slow flow of Bingham viscoplastic fluid around a circular cylinder
,” J. Non-Newtonian Fluid Mech.
154
, 65
–76
(2008
).16.
J.
Tsamopoulos
,
Y.
Dimakopoulos
,
N.
Chatzidai
,
G.
Karapetsas
, and
M.
Pavlidis
, “
Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment
,” J. Fluid Mech.
601
, 123
–164
(2008
).17.
Y.
Dimakopoulos
,
M.
Pavlidis
, and
J.
Tsamopoulos
, “
Steady bubble rise in Herschel–Bulkley fluids and comparison of predictions via the augmented Lagrangian method with those via the Papanastasiou model
,” J. Non-Newtonian Fluid Mech.
200
, 34
–51
(2013
).18.
M.
Tripathi
,
K.
Sahu
,
G.
Karapetsas
, and
O.
Matar
, “
Bubble rise dynamics in a viscoplastic material
,” J. Non-Newtonian Fluid Mech.
222
, 217
–226
(2015
).19.
G.
Karapetsas
,
D.
Photeinos
,
Y.
Dimakopoulos
, and
J.
Tsamopoulos
, “
Dynamics and motion of a gas bubble in a viscoplastic medium under acoustic excitation
,” J. Fluid Mech.
865
, 381
–413
(2019
).20.
L.
Deoclecio
,
E.
Soares
,
H.
Deka
, and
J.-L.
Pierson
, “
Bubble entrapment condition in Bingham materials
,” J. Non-Newtonian Fluid Mech.
295
, 104616
(2021
).21.
P.
Moschopoulos
,
A.
Spyridakis
,
S.
Varchanis
,
Y.
Dimakopoulos
, and
J.
Tsamopoulos
, “
The concept of elasto-visco-plasticity and its application to a bubble rising in yield stress fluids
,” J. Non-Newtonian Fluid Mech.
297
, 104670
(2021
).22.
A.
Kordalis
,
D.
Pema
,
S.
Androulakis
,
Y.
Dimakopoulos
, and
J.
Tsamopoulos
, “
Hydrodynamic interaction between coaxially rising bubbles in elastoviscoplastic materials: Equal bubbles
,” Phys. Rev. Fluids
8
, 083301
(2023
).23.
A.
Potapov
,
R.
Spivak
,
O. M.
Lavrenteva
, and
A.
Nir
, “
Motion and deformation of drops in Bingham fluid
,” Ind. Eng. Chem. Res.
45
, 6985
–6995
(2006
).24.
J.
Singh
and
M.
Denn
, “
Interacting two-dimensional bubbles and droplets in a yield-stress fluid
,” Phys. Fluids
20
, 040901
(2008
).25.
S.
Goel
and
A.
Ramachandran
, “
The suppression of droplet-droplet coalescence in a sheared yield stress fluid
,” J. Colloid Interface Sci.
492
, 199
–206
(2017
).26.
H.
Deka
,
J.-L.
Pierson
, and
E. J.
Soares
, “
Retraction criteria of viscoplastic drops and sheets: Long-wave approximations
,” J. Non-Newtonian Fluid Mech.
284
, 104352
(2020
).27.
L. H.
Deoclecio
,
E. J.
Soares
, and
S.
Popinet
, “
Drop rise and interfacial coalescence initiation in Bingham materials
,” J. Non-Newtonian Fluid Mech.
319
, 105075
(2023
).28.
N.
Dubash
and
I.
Frigaard
, “
Conditions for static bubbles in viscoplastic fluids
,” Phys. Fluids
16
, 4319
–4330
(2004
).29.
A.
Pourzahedi
,
E.
Chaparian
,
A.
Roustaei
, and
I. A.
Frigaard
, “
Flow onset for a single bubble in a yield-stress fluid
,” J. Fluid Mech.
933
, A21
(2022
).30.
A.
Putz
and
I.
Frigaard
, “
Creeping flow around particles in a Bingham fluid
,” J. Non-Newtonian Fluid Mech.
165
, 263
–280
(2010
).31.
E.
Chaparian
and
I.
Frigaard
, “
Yield limit analysis of particle motion in a yield-stress fluid
,” J. Fluid Mech.
819
, 311
–351
(2017
).32.
E.
Chaparian
and
I.
Frigaard
, “
Cloaking: Particles in a yield-stress fluid
,” J. Non-Newtonian Fluid Mech.
243
, 47
–55
(2017
).33.
A.
Wachs
and
I. A.
Frigaard
, “
Particle settling in yield stress fluids: Limiting time, distance and applications
,” J. Non-Newtonian Fluid Mech.
238
, 189
–204
(2016
).34.
J.
Iglesias
,
G.
Mercier
,
E.
Chaparian
, and
I.
Frigaard
, “
Computing the yield limit in three-dimensional flows of a yield stress fluid about a settling particle
,” J. Non-Newtonian Fluid Mech.
284
, 104374
(2020
).35.
F.
Hecht
, “
New development in FreeFEM++
,” J. Numer. Math.
20
, 251
–266
(2012
).36.
A.
Roustaei
,
T.
Chevalier
,
L.
Talon
, and
I.
Frigaard
, “
Non-Darcy effects in fracture flows of a yield stress fluid
,” J. Fluid Mech.
805
, 222
–261
(2016
).37.
E.
Chaparian
,
D.
Izbassarov
,
F.
De Vita
,
L.
Brandt
, and
O.
Tammisola
, “
Yield-stress fluids in porous media: A comparison of viscoplastic and elastoviscoplastic flows
,” Meccanica
55
, 331
–342
(2020
).38.
E.
Chaparian
and
O.
Tammisola
, “
Sliding flows of yield-stress fluids
,” J. Fluid Mech.
911
, A17
(2021
).39.
E.
Chaparian
and
O.
Tammisola
, “
An adaptive finite element method for elastoviscoplastic fluid flows
,” J. Non-Newtonian Fluid Mech.
271
, 104148
(2019
).40.
E.
Chaparian
,
C. E.
Owens
, and
G. H.
McKinley
, “
Computational rheometry of yielding and viscoplastic flow in vane-and-cup rheometer fixtures
,” J. Non-Newtonian Fluid Mech.
307
, 104857
(2022
).41.
42.
M.
Brizard
,
M.
Megharfi
,
E.
Mahe
, and
C.
Verdier
, “
Design of a high precision falling-ball viscometer
,” Rev. Sci. Instrum.
76
, 025109
(2005
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
