Abstract

This research delves into the intricate interplay of fluid rheology, characterized by the power-law model, and density ratio ρr=ρl/ρg in the context of droplet collision dynamics. The power-law index (n) is systematically varied within the range of 0.5–1.5, while the density ratio spans 2 orders of magnitude, ranging from 101to103. Comprehensive investigations are conducted across various impact parameters (B = 0–0.75) and Weber numbers (We = 40–160). A noteworthy finding is the cessation of droplet coalescence at elevated Weber numbers (We = 160), revealing a critical threshold beyond which coalescence is no longer sustained. The impact of fluid rheology on internal fluid flow dynamics within the complex droplet structure is substantial. The variation in viscous dissipation with (n) contributes to observable changes in the critical wavelength of the complex droplet rim structure, consequently influencing the size of child droplets. Furthermore, the density ratio is a pivotal factor influencing the deformation rate during collision events. A decrease in density ratio correlates with a reduction in the deformation ratio, shedding light on the significant role of density ratio in shaping the dynamics of droplet collisions.

References

1.
Lichty
,
L. C.
,
1929
, “
Combustion at High Pressures
,”
ASME Trans. Am. Soc. Mech. Eng.
,
51
(
2
), pp.
37
44
.10.1115/1.4059012
2.
Ishii
,
E.
,
Ishikawa
,
M.
,
Sukegawa
,
Y.
, and
Yamada
,
H.
,
2011
, “
Secondary-Drop-Breakup Simulation Integrated With Fuel-Breakup Simulation Near Injector Outlet
,”
ASME J. Fluids Eng.
,
133
(
8
), p.
081302
.10.1115/1.4004764
3.
Pawar
,
S. K.
,
Henrikson
,
F.
,
Finotello
,
G.
,
Padding
,
J. T.
,
Deen
,
N. G.
,
Jongsma
,
A.
,
Innings
,
F.
, and
Kuipers
,
J. A. M. H.
,
2016
, “
An Experimental Study of Droplet-Particle Collisions
,”
Powder Technol.
,
300
, pp.
157
163
.10.1016/j.powtec.2016.06.005
4.
Jacobs
,
M. I.
,
Davies
,
J. F.
,
Lee
,
L.
,
Davis
,
R. D.
,
Houle
,
F.
, and
Wilson
,
K. R.
,
2017
, “
Exploring Chemistry in Microcompartments Using Guided Droplet Collisions in a Branched Quadrupole Trap Coupled to a Single Droplet, Paper Spray Mass Spectrometer
,”
Anal. Chem.
,
89
(
22
), pp.
12511
12519
.10.1021/acs.analchem.7b03704
5.
Malet
,
J.
,
Mimouni
,
S.
,
Foissac
,
A.
,
Malet
,
J.
,
Mimouni
,
S.
, and
Feuillebois
,
F.
,
2010
, “
Binary Water Droplet Collision Study in Presence of Solid Aerosols in Air
,” 7th International Conference on Multiphase Flow (
ICMF
), Tampa, FL, May 30–June 4, pp.
1
8
.https://www.researchgate.net/publication/289745977_Binary_water_droplet_collision_study_in_presence_of_solid_aerosols_in_air
6.
Jafari
,
S. M.
,
Assadpoor
,
E.
,
He
,
Y.
, and
Bhandari
,
B.
,
2008
, “
Re-Coalescence of Emulsion Droplets During High-Energy Emulsification
,”
Food Hydrocolloids
,
22
(
7
), pp.
1191
1202
.10.1016/j.foodhyd.2007.09.006
7.
Post
,
S.
,
Iyer
,
V.
, and
Abraham
,
J.
,
2000
, “
A Study of Near-Field Entrainment in Gas Jets and Sprays Under Diesel Conditions
,”
ASME J. Fluids Eng.
,
122
(
2
), pp.
385
395
.10.1115/1.483268
8.
Cao
,
J.
,
2002
, “
On the Theoretical Prediction of Fuel Droplet Size Distribution in Nonreactive Diesel Sprays
,”
ASME J. Fluids Eng.
,
124
(
1
), pp.
182
185
.10.1115/1.1445140
9.
Qian
,
L.
,
Lin
,
J.
,
Xiong
,
H.
, and
Leung Chan
,
T.
,
2011
, “
Theoretical Investigation of the Influence of Liquid Physical Properties on Effervescent Atomization Performance
,”
ASME J. Fluids Eng.
,
133
(
10
), p.
101205
.10.1115/1.4004256
10.
Weiwei
,
E.
,
Pope
,
K.
, and
Duan
,
X.
,
2020
, “
Droplet Coalescence in Liquid/Liquid Separation
,”
ASME J. Fluids Eng.
,
142
(
11
), p.
111402
.10.1115/1.4047796
11.
Biswas
,
G.
, and
Sahu
,
K. C.
,
2021
, “
Recent Advances in Free Surface Flows
,”
Mechanical Sciences: The Way Forward
, U. S. Dixit and S. K. Dwivedy, eds.,
Springer Singapore, Singapore
, pp.
121
144
.
12.
Blaszczuk
,
A.
,
Nowak
,
W.
, and
Krzywanski
,
J.
,
2017
, “
Effect of Bed Particle Size on Heat Transfer Between Fluidized Bed of Group b Particles and Vertical Rifled Tubes
,”
Powder Technol.
,
316
, pp.
111
122
.10.1016/j.powtec.2016.12.027
13.
Islamova
,
A. G.
,
Kerimbekova
,
S. A.
,
Shlegel
,
N. E.
, and
Strizhak
,
P. A.
,
2022
, “
Droplet-Droplet, Droplet-Particle, and Droplet-Substrate Collision Behavior
,”
Powder Technol.
,
403
, p.
117371
.10.1016/j.powtec.2022.117371
14.
Ashgriz
,
N.
, and
Poo
,
J. Y.
,
1990
, “
Coalescence and Separation in Binary Collisions of Liquid Drops
,”
J. Fluid Mech.
,
221
, pp.
183
204
.10.1017/S0022112090003536
15.
Qian
,
J.
, and
Law
,
C. K.
,
1997
, “
Regimes of Coalescence and Separation in Droplet Collision
,”
J. Fluid Mech.
,
331
, pp.
59
80
.10.1017/S0022112096003722
16.
Estrade
,
J. P.
,
Carentz
,
H.
,
Lavergne
,
G.
, and
Biscos
,
Y.
,
1999
, “
Experimental Investigation of Dynamic Binary Collision of Ethanol Droplets—A Model for Droplet Coalescence and Bouncing
,”
Int. J. Heat Fluid Flow
,
20
(
5
), pp.
486
491
.10.1016/S0142-727X(99)00036-3
17.
Brenn
,
G.
,
Valkovska
,
D.
, and
Danov
,
K. D.
,
2001
, “
The Formation of Satellite Droplets by Unstable Binary Drop Collisions
,”
Phys. Fluids
,
13
(
9
), pp.
2463
2477
.10.1063/1.1384892
18.
Brenn
,
G.
, and
Kolobaric
,
V.
,
2006
, “
Satellite Droplet Formation by Unstable Binary Drop Collisions
,”
Phys. Fluids
,
18
(
8
), p.
087101
.10.1063/1.2225363
19.
Kuschel
,
M.
, and
Sommerfeld
,
M.
,
2013
, “
Investigation of Droplet Collisions for Solutions With Different Solids Content
,”
Exp. Fluids
,
54
(
2
), p.
1440
.10.1007/s00348-012-1440-z
20.
Sommerfeld
,
M.
, and
Kuschel
,
M.
,
2016
, “
Modelling Droplet Collision Outcomes for Different Substances and Viscosities
,”
Exp. Fluids
,
57
(
12
), p.
187
.10.1007/s00348-016-2249-y
21.
Tang
,
C.
,
Zhang
,
P.
, and
Law
,
C. K.
,
2012
, “
Bouncing, Coalescence, and Separation in Head-On Collision of Unequal-Size Droplets
,”
Phys. Fluids
,
24
(
2
), p.
022101
.10.1063/1.3679165
22.
Kumar
,
M.
,
Bhardwaj
,
R.
, and
Sahu
,
K. C.
,
2020
, “
Coalescence Dynamics of a Droplet on a Sessile Droplet
,”
Phys. Fluids
,
32
(
1
), p.
012104
.10.1063/1.5129901
23.
Chaitanya
,
G. S.
,
Sahu
,
K. C.
, and
Biswas
,
G.
,
2021
, “
A Study of Two Unequal-Sized Droplets Undergoing Oblique Collision
,”
Phys. Fluids
,
33
(
2
), p.
022110
.10.1063/5.0038734
24.
Nobari
,
M. R.
,
Jan
,
Y.‐J.
, and
Tryggvason
,
G.
,
1996
, “
Head‐On Collision of Drops—A Numerical Investigation
,”
Phys. Fluids
,
8
(
1
), pp.
29
42
.10.1063/1.868812
25.
Rieber
,
M.
, and
Frohn
,
A.
,
1997
, “
Navier–Stokes Simulation of Droplet Collision Dynamics
,”
7th Int. Symp. On Comp. Fluid Dynamics
, Citeseer, University Park, PA, pp.
520
525
.https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=70287571992fa7e6785cde41c324ace44ed230d3
26.
Schelkle
,
M.
,
Prochazka
,
M. E.
, and
Frohn
,
A.
,
1992
, “
Droplet Formation Simulated With an Immiscible Lattice Gas Model
,”
J. Aerosol Sci.
,
23
(
Suppl. 1
), pp.
245
248
.10.1016/0021-8502(92)90395-C
27.
Inamuro
,
T.
,
Tajima
,
S.
, and
Ogino
,
F.
,
2004
, “
Lattice Boltzmann Simulation of Droplet Collision Dynamics
,”
Int. J. Heat Mass Transfer
,
47
(
21
), pp.
4649
4657
.10.1016/j.ijheatmasstransfer.2003.08.030
28.
Tanguy
,
S.
, and
Berlemont
,
A.
,
2005
, “
Application of a Level Set Method for Simulation of Droplet Collisions
,”
Int. J. Multiphase Flow
,
31
(
9
), pp.
1015
1035
.10.1016/j.ijmultiphaseflow.2005.05.010
29.
Pan
,
Y.
, and
Suga
,
K.
,
2005
, “
Numerical Simulation of Binary Liquid Droplet Collision
,”
Phys. Fluids
,
17
(
8
), p.
082105
.10.1063/1.2009527
30.
Gotaas
,
C.
,
Havelka
,
P.
,
Jakobsen
,
H. A.
,
Svendsen
,
H. F.
,
Hase
,
M.
,
Roth
,
N.
, and
Weigand
,
B.
,
2007
, “
Effect of Viscosity on Droplet-Droplet Collision Outcome: Experimental Study and Numerical Comparison
,”
Phys. Fluids
,
19
(
10
), p.
102106
.10.1063/1.2781603
31.
Finotello
,
G.
,
Kooiman
,
R. F.
,
Padding
,
J. T.
,
Buist
,
K. A.
,
Jongsma
,
A.
,
Innings
,
F.
, and
Kuipers
,
J. A. M.
,
2018
, “
The Dynamics of Milk Droplet–Droplet Collisions
,”
Exp. Fluids
,
59
(
1
), p.
17
.10.1007/s00348-017-2471-2
32.
Finotello
,
G.
,
Padding
,
J. T.
,
Deen
,
N. G.
,
Jongsma
,
A.
,
Innings
,
F.
, and
Kuipers
,
J. A. M.
,
2017
, “
Effect of Viscosity on Droplet-Droplet Collisional Interaction
,”
Phys. Fluids
,
29
(
6
), p.
067102
.10.1063/1.4984081
33.
Goyal
,
N.
,
Shaikh
,
J.
, and
Sharma
,
A.
,
2020
, “
Bubble Entrapment During Head-On Binary Collision With Large Deformation of Unequal-Sized Tetradecane Droplets
,”
Phys. Fluids
,
32
(
12
), p.
122114
.10.1063/5.0029179
34.
Deka
,
H.
,
Biswas
,
G.
,
Chakraborty
,
S.
, and
Dalal
,
A.
,
2019
, “
Coalescence Dynamics of Unequal Sized Drops
,”
Phys. Fluids
,
31
(
1
), p.
012105
.10.1063/1.5064516
35.
Cong
,
H.
,
Qian
,
L.
,
Wang
,
Y.
, and
Lin
,
J.
,
2020
, “
Numerical Simulation of the Collision Behaviors of Binary Unequal-Sized Droplets at High Weber Number
,”
Phys. Fluids
,
32
(
10
), p.
103307
.10.1063/5.0020709
36.
Pal
,
A. K.
,
Sahu
,
K. C.
,
De
,
S.
, and
Biswas
,
G.
,
2024
, “
Collision of Two Drops Moving in the Same Direction
,”
Phys. Fluids
,
36
(
1
), p.
012122
.10.1063/5.0189168
37.
Pal
,
A. K.
,
Sahu
,
K. C.
, and
Biswas
,
G.
,
2024
, “
Modeling Binary Collision of Evaporating Drops
,”
Int. J. Heat Mass Transfer
,
221
, p.
125048
.10.1016/j.ijheatmasstransfer.2023.125048
38.
Musehane
,
N. M.
,
Oxtoby
,
O. F.
, and
Reddy
,
B. D.
,
2018
, “
Multi-Scale Simulation of Droplet–Droplet Interaction and Coalescence
,”
J. Comput. Phys.
,
373
, pp.
924
939
.10.1016/j.jcp.2018.07.027
39.
Li
,
X. G.
, and
Fritsching
,
U.
,
2011
, “
Numerical Investigation of Binary Droplet Collisions in All Relevant Collision Regimes
,”
J. Comput. Multiphase Flows
,
3
(
4
), pp.
207
224
.10.1260/1757-482X.3.4.207
40.
Pasandideh‐Fard
,
M.
,
Qiao
,
Y. M.
,
Chandra
,
S.
, and
Mostaghimi
,
J.
,
1996
, “
Capillary Effects During Droplet Impact on a Solid Surface
,”
Phys. Fluids
,
8
(
3
), pp.
650
659
.10.1063/1.868850
41.
Pasandideh-Fard
,
M.
,
Chandra
,
S.
, and
Mostaghimi
,
J.
,
2002
, “
A Three-Dimensional Model of Droplet Impact and Solidification
,”
Int. J. Heat Mass Transfer
,
45
(
11
), pp.
2229
2242
.10.1016/S0017-9310(01)00336-2
42.
Li
,
C.
,
Han
,
S.
,
Fu
,
Y.
,
Li
,
F.
,
Bai
,
W.
, and
Hu
,
C.
,
2024
, “
Heat Transfer Process in the Collision of High-Temperature Aluminum Droplets With an Inclined Wall
,”
Acta Astronaut.
,
215
, pp.
168
177
.10.1016/j.actaastro.2023.12.009
43.
Pan
,
K.-L.
,
Chou
,
P.-C.
, and
Tseng
,
Y.-J.
,
2009
, “
Binary Droplet Collision at High Weber Number
,”
Phys. Rev. E
,
80
(
3
), p.
36301
.10.1103/PhysRevE.80.036301
44.
Burtnett
,
E.
,
Thompson
,
D.
,
Jung
,
S.
, and
Raps
,
D.
,
2012
, “
A Comparison of VOF Simulations With Experimental Data for Droplet Impact on a Dry Surface
,”
AIAA
Paper No. 2012-0092.10.2514/6.2012-0092
45.
Tretola
,
G.
, and
Vogiatzaki
,
K.
,
2021
, “
Numerical Treatment of the Interface in Two Phase Flows Using a Compressible Framework in OpenFOAM: Demonstration on a High Velocity Droplet Impact Case
,”
Fluids
,
6
(
2
), p.
78
.10.3390/fluids6020078
46.
Brenn
,
G.
,
2011
, “
Droplet Collision
,”
Handbook of Atomization and Sprays: Theory and Applications
,
N.
Ashgriz
, ed.,
Springer US
,
Boston
, pp.
157
181
.
47.
Motzigemba
,
M.
,
Roth
,
N.
,
Bothe
,
D.
,
Warnecke
,
H.-J.
,
Prüss
,
J.
,
Wielage
,
K.
, and
Weigand
,
B.
,
2002
, “
The Effect of Non-Newtonian Flow Behaviour on Binary Droplet Collisions: VOF-Simulation and Experimental Analysis
,”
ILASS-Europe
, Zaragoza, Spain, Sept. 9–11, pp.
1
6
.https://www.ilasseurope.org/ICLASS/ilass2002/papers/078.pdf
48.
Finotello
,
G.
,
De
,
S.
,
Vrouwenvelder
,
J. C. R.
,
Padding
,
J. T.
,
Buist
,
K. A.
,
Jongsma
,
A.
,
Innings
,
F.
, and
Kuipers
,
J. A. M.
,
2018
, “
Experimental Investigation of Non-Newtonian Droplet Collisions: The Role of Extensional Viscosity
,”
Exp. Fluids
,
59
(
7
), p.
113
.10.1007/s00348-018-2568-2
49.
Focke
,
C.
, and
Bothe
,
D.
,
2011
, “
Computational Analysis of Binary Collisions of Shear-Thinning Droplets
,”
J. Non-Newtonian Fluid Mech.
,
166
(
14–15
), pp.
799
810
.10.1016/j.jnnfm.2011.03.011
50.
Sun
,
K.
,
Zhang
,
P.
,
Law
,
C. K.
, and
Wang
,
T.
,
2015
, “
Collision Dynamics and Internal Mixing of Droplets of Non-Newtonian Liquids
,”
Phys. Rev. Appl.
,
4
(
5
), p.
54013
.10.1103/PhysRevApplied.4.054013
51.
França
,
H. L.
,
Oishi
,
C. M.
, and
Thompson
,
R. L.
,
2022
, “
Numerical Investigation of Shear-Thinning and Viscoelastic Binary Droplet Collision
,”
J. Non-Newtonian Fluid Mech.
,
302
, p.
104750
.10.1016/j.jnnfm.2022.104750
52.
Hirschler
,
M.
,
Oger
,
G.
,
Nieken
,
U.
, and
Le Touzé
,
D.
,
2017
, “
Modeling of Droplet Collisions Using Smoothed Particle Hydrodynamics
,”
Int. J. Multiphase Flow
,
95
, pp.
175
187
.10.1016/j.ijmultiphaseflow.2017.06.002
53.
Xu
,
X.
,
Cheng
,
J.
,
Peng
,
S.
, and
Yu
,
P.
,
2024
, “
Numerical Simulations of Phan-Thien-Tanner Viscoelastic Fluid Flows Based on the SPH Method
,”
Eng. Anal. Boundary Elem.
,
158
, pp.
473
485
.10.1016/j.enganabound.2023.11.020
54.
Xu
,
X.
,
Tang
,
T.
, and
Yu
,
P.
,
2020
, “
A Modified SPH Method to Model the Coalescence of Colliding Non-Newtonian Liquid Droplets
,”
Int. J. Numer. Methods Fluids
,
92
(
5
), pp.
372
390
.10.1002/fld.4787
55.
Huijgen
,
A. H.
,
García Llamas
,
C.
,
Durubal
,
P. M.
,
Buist
,
K. A.
,
Kuipers
,
J. A. M.
, and
Baltussen
,
M. W.
,
2024
, “
Numerical Investigation of Non-Newtonian Droplet Collisions: Comparison of Volume of Fluid and the Local Front Reconstruction Method With Experimental Data
,”
Chem. Eng. Sci.
,
299
, p.
120428
.10.1016/j.ces.2024.120428
56.
Ghaffari
,
A.
, and
Hashemabadi
,
S. H.
,
2017
, “
Parameter Study and CFD Analysis of Head on Collision and Dynamic Behavior of Two Colliding Ferrofluid Droplets
,”
Smart Mater. Struct.
,
26
(
3
), p.
035010
.10.1088/1361-665X/aa54a2
57.
Deshpande
,
S. S.
,
Anumolu
,
L.
, and
Trujillo
,
M. F.
,
2012
, “
Evaluating the Performance of the Two-Phase Flow Solver InterFoam
,”
Comput. Sci. Discovery
,
5
(
1
), p.
014016
.10.1088/1749-4699/5/1/014016
58.
Gamet
,
L.
,
Scala
,
M.
,
Roenby
,
J.
,
Scheufler
,
H.
, and
Pierson
,
J.-L.
,
2020
, “
Validation of Volume-of-Fluid OpenFOAM® IsoAdvector Solvers Using Single Bubble Benchmarks
,”
Comput. Fluids
,
213
, p.
104722
.10.1016/j.compfluid.2020.104722
59.
Roenby
,
J.
,
Bredmose
,
H.
, and
Jasak
,
H.
,
2016
, “
A Computational Method for Sharp Interface Advection
,”
R. Soc. Open Sci.
,
3
(
11
), p.
160405
.10.1098/rsos.160405
60.
Albadawi
,
A.
,
Donoghue
,
D. B.
,
Robinson
,
A. J.
,
Murray
,
D. B.
, and
Delauré
,
Y. M. C.
,
2013
, “
Influence of Surface Tension Implementation in Volume of Fluid and Coupled Volume of Fluid With Level Set Methods for Bubble Growth and Detachment
,”
Int. J. Multiphase Flow
,
53
, pp.
11
28
.10.1016/j.ijmultiphaseflow.2013.01.005
61.
van Leer
,
B.
,
1979
, “
Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov's Method
,”
J. Comput. Phys.
,
32
(
1
), pp.
101
136
.10.1016/0021-9991(79)90145-1
63.
Kant
,
K.
, and
Banerjee
,
R.
,
2022
, “
Study of the Secondary Droplet Breakup Mechanism and Regime Map of Newtonian and Power Law Fluids at High Liquid-Gas Density Ratio
,”
Phys. Fluids
,
34
(
4
), p.
043108
.10.1063/5.0088144
64.
Kant
,
K.
, and
Banerjee
,
R.
,
2023
, “
Effect of Density Ratios on Droplet Breakup for Newtonian and Power-Law Fluids
,”
Int. J. Multiphase Flow
,
167
, p.
104561
.10.1016/j.ijmultiphaseflow.2023.104561
65.
Nikolopoulos
,
N.
,
Theodorakakos
,
A.
, and
Bergeles
,
G.
,
2009
, “
Off-Centre Binary Collision of Droplets: A Numerical Investigation
,”
Int. J. Heat Mass Transfer
,
52
(
19–20
), pp.
4160
4174
.10.1016/j.ijheatmasstransfer.2009.04.011
66.
Chowdhary
,
S.
,
Reddy
,
S. R.
, and
Banerjee
,
R.
,
2020
, “
Detailed Numerical Simulations of Unequal Sized Off-Centre Binary Droplet Collisions
,”
Int. J. Multiphase Flow
,
128
, p.
103267
.10.1016/j.ijmultiphaseflow.2020.103267
67.
Celik
,
I. B.
,
Ghia
,
U.
,
Roache
,
P. J.
, &
Freitas
,
C. J.
,
2008
, “
Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications
,”
ASME J. Fluids Eng.
,
130
(
7
), p.
078001
.10.1115/1.2960953
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