In the present study, a fractional-step-based multiphase lattice Boltzmann (LB) method coupled with a solution of a magnetic field evolution is developed to predict the interface behavior in magnetic multiphase flows. The incompressible Navier–Stokes equations are utilized for the flow field, while the Cahn–Hilliard equation is adopted to track the interface, and these governing equations are solved by reconstructing solutions within the LB framework with the prediction–correction step based on a fractional-step method. The proposed numerical model inherits the excellent performance of kinetic theory from the LB method and integrates the good numerical stability from the fractional-step method. Meanwhile, the macroscopic variables can be simply and directly calculated by the equilibrium distribution functions, which saves the virtual memories and simplifies the computational process. The proposed numerical model is validated by simulating two problems, i.e., a bubble rising with a density ratio of 1000 and a viscosity ratio of 100 and a stationary circular cylinder under an external uniform magnetic field. The interfacial deformations of a ferrofluid droplet in organic oil and an aqueous droplet in ferrofluid under the external magnetic field are, then, simulated, and the underlying mechanisms are discussed. Moreover, the rising process of a gas bubble in the ferrofluid is investigated, which shows that the rising velocity is accelerated under the effect of the external magnetic field. All the numerical examples demonstrate the capability of the present numerical method to handle the problem with the interfacial deformation in magnetic multiphase flows.
REFERENCES
1.
F.
Bitter
, “On inhomogeneities in the magnetization of ferromagnetic materials
,” Phys. Rev.
38
(10
), 1903
(1931
).2.
W. C.
Elmore
, “Ferromagnetic colloid for studying magnetic structures
,” Phys. Rev.
54
(4
), 309
(1938
).3.
W. C.
Elmore
, “The magnetization of ferromagnetic colloids
,” Phys. Rev.
54
(12
), 1092
(1938
).4.
J. L.
Neuringer
and R. E.
Rosensweig
, “Ferrohydrodynamics
,” Phys. Fluids
7
, 1927
(1964
).5.
M. D.
Cowley
and R. E.
Rosensweig
, “The interfacial stability of a ferromagnetic fluid
,” J. Fluid Mech.
30
, 671
(1967
).6.
A. B.
Guimarães
, F. R.
Cunha
, and R. G.
Gontijo
, “The influence of hydrodynamic effects on the complex susceptibility response of magnetic fluids undergoing oscillatory fields: New insights for magnetic hyperthermia
,” Phys. Fluids
32
, 012008
(2020
).7.
H.
Massana-Cid
, F.
Meng
, D.
Matsunaga
, R.
Golestanian
, and P.
Tierno
, “Tunable self-healing of magnetically propelling colloidal carpets
,” Nat. Commun.
10
, 2444
(2019
).8.
C.
Kumar
, M.
Hejazian
, C.
From
, S. C.
Saha
, E.
Sauret
, Y.
Gu
, and N.-T.
Nguyen
, “Modeling of mass transfer enhancement in a magnetofluidic micromixer
,” Phys. Fluids
31
(6
), 063603
(2019
).9.
X.
Li
, P.
Yu
, X.
Niu
, H.
Yamaguchi
, and D.
Li
, “Non-contact manipulation of nonmagnetic materials by using a uniform magnetic field: Experiment and simulation
,” J. Magn. Magn. Mater.
497
, 165957
(2020
).10.
T.
Hayat
, M. I.
Khan
, M.
Imtiaz
, A.
Alsaedi
, and M.
Waqas
, “Similarity transformation approach for ferromagnetic mixed convection flow in the presence of chemically reactive magnetic dipole
,” Phys. Fluids
28
, 102003
(2016
).11.
H.
Yamaguchi
and T.
Bessho
, “Long distance heat transport device using temperature sensitive magnetic fluid
,” J. Magn. Magn. Mater.
499
, 166248
(2020
).12.
Lord Rayleigh
, “XX. On the theory of surface forces. II. Compressible fluids
,” Philos. Mag.
33
, 209
(1892
).13.
J. D.
van der Waals
, “Thermodynamische theorie der kapillarität unter voraussetzung stetiger dichteänderung
,” Z. Phys. Chem.
13
, 657
(1894
);J. D.
van der Waals
, “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density
,” J. Stat. Phys.
20
, 200
(1979
) (in English).14.
G. I.
Taylor
, “The formation of emulsions in definable fields of flow
,” Proc. R. Soc. A
146
, 501
–523
(1934
).15.
D. M.
Anderson
, G. B.
McFadden
, and A. A.
Wheeler
, “Diffuse-interface methods in fluid mechanics
,” Annu. Rev. Fluid Mech.
30
, 139
(1998
).16.
S.
Chen
and G. D.
Doolen
, “Lattice Boltzmann method for fluid flows
,” Annu. Rev. Fluid Mech.
30
, 329
–364
(1998
).17.
C. K.
Aidun
and J. R.
Clausen
, “Lattice-Boltzmann method for complex flows
,” Annu. Rev. Fluid Mech.
42
, 439
–472
(2010
).18.
T. W.
Zhang
, J.
Wu
, and X. J.
Lin
, “Numerical investigation on formation and motion of bubble or droplet in quiescent flow
,” Phys. Fluids
32
, 032106
(2020
).19.
X.
Shan
and H.
Chen
, “Lattice Boltzmann model for simulating flows with multiple phases and components
,” Phys. Rev. E
47
(3
), 1815
–1819
(1993
).20.
X.
He
, S.
Chen
, and R.
Zhang
, “A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability
,” J. Comput. Phys.
152
, 642
–663
(1999
).21.
H. W.
Zheng
, C.
Shu
, and Y. T.
Chew
, “A lattice Boltzmann model for multiphase flows with large density ratio
,” J. Comput. Phys.
218
, 353
–371
(2006
).22.
H.
Huang
, J.-J.
Huang
, and X.-Y.
Lu
, “A mass-conserving axisymmetric multiphase lattice Boltzmann method and its application in simulation of bubble rising
,” J. Comput. Phys.
269
, 386
–402
(2014
).23.
Y.
Wang
, C.
Shu
, H. B.
Huang
, and C. J.
Teo
, “Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio
,” J. Comput. Phys.
280
, 404
–423
(2015
).24.
X.-D.
Niu
, Y.
Li
, Y.-R.
Ma
, M.-F.
Chen
, X.
Li
, and Q.-Z.
Li
, “A mass-conserving multiphase lattice Boltzmann model for simulation of multiphase flows
,” Phys. Fluids
30
(1
), 013302
(2018
).25.
H. Z.
Yuan
, Y.
Wang
, and C.
Shu
, “An adaptive mesh refinement-multiphase lattice Boltzmann flux solver for simulation of complex binary fluid flows
,” Phys. Fluids
29
, 123604
(2017
).26.
Z.
Chen
, C.
Shu
, Y.
Wang
, L. M.
Yang
, and D.
Tan
, “A simplified lattice Boltzmann method without evolution of distribution function
,” Adv. Appl. Math. Mech.
9
(1
), 1
–22
(2017
).27.
Z.
Chen
, C.
Shu
, and D.
Tan
, “A truly second-order and unconditionally stable thermal lattice Boltzmann method
,” Appl. Sci.
7
(3
), 277
(2017
).28.
Z.
Chen
, C.
Shu
, and D.
Tan
, “High-order simplified thermal lattice Boltzmann method for incompressible thermal flows
,” Int. J. Heat Mass Transfer
127
, 1
–16
(2018
).29.
Z.
Chen
, C.
Shu
, D.
Tan
, X. D.
Niu
, and Q. Z.
Li
, “Simplified multiphase lattice Boltzmann method for simulating multiphase flows with large density ratios and complex interfaces
,” Phys. Rev. E
98
, 063314
(2018
).30.
Z.
Chen
, C.
Shu
, L.
Yang
, X.
Zhao
, and N.
Liu
, “Immersed boundary–simplified thermal lattice Boltzmann method for incompressible thermal flows
,” Phys. Fluids
32
, 013605
(2020
).31.
X. D.
Niu
, H.
Yamaguchi
, and K.
Yoshikawa
, “Lattice Boltzmann model for simulating temperature-sensitive ferrofluids
,” Phys. Rev. E
79
, 046713
(2009
).32.
T.
Zhang
and D.
Che
, “Double MRT thermal lattice Boltzmann simulation for MHD natural convection of nanofluids in an inclined cavity with four square heat sources
,” Int. J. Heat Mass Transfer
94
, 87
–100
(2016
).33.
H.
Sajjadi
, A. A.
Delouei
, M.
Atashafrooz
, and M.
Sheikholeslami
, “Double MRT lattice Boltzmann simulation of 3-D MHD natural convection in a cubic cavity with sinusoidal temperature distribution utilizing nanofluid
,” Int. J. Heat Mass Transfer
126
, 489
–503
(2018
).34.
D.
Shi
, Q.
Bi
, and R.
Zhou
, “Numerical simulation of a falling ferrofluid droplet in a uniform magnetic field by the VOSET method
,” Numer. Heat Transfer, Part A
66
(2
), 144
–164
(2014
).35.
A.
Ghaderi
, M. H.
Kayhani
, and M.
Nazari
, “Numerical investigation on falling ferrofluid droplet under uniform magnetic field
,” Eur. J. Mech.: B/Fluids
72
, 1
–11
(2018
).36.
Y.
Hu
, D. C.
Li
, and X. D.
Niu
, “Phase-field-based lattice Boltzmann model for multiphase ferrofluid flows
,” Phys. Rev. E
98
, 033301
(2018
).37.
J.
Kim
and P.
Moin
, “Application of a fractional-step method to incompressible Navier-Stokes equations
,” J. Comput. Phys.
59
, 308
–323
(1985
).38.
H.
Araseki
and S.
Kotake
, “A self-correcting procedure for computational liquid metal magnetohydrodynamics
,” J. Comput. Phys.
110
, 301
–309
(1994
).39.
J. W.
Cahn
and J. E.
Hilliard
, “Free energy of a nonuniform system. I. Interfacial free energy
,” J. Chem. Phys.
28
(2
), 258
–267
(1958
).40.
D.
Jacqmin
, “Contact-line dynamics of a diffuse fluid interface
,” J. Fluid Mech.
402
, 57
–88
(2000
).41.
Y. H.
Qian
, D.
D’Humières
, and P.
Lallemand
, “Lattice BGK models for Navier-Stokes equation
,” Europhys. Lett.
17
(6
), 479
–484
(1992
).42.
S.
Chapman
and T. G.
Cowling
, The Mathematical Theory of Non-Uniform Gases
(Cambridge University Press
, Cambridge
, 1970
).43.
44.
M. F.
Chen
, X.
Li
, X. D.
Niu
, Y.
Li
, A.
Khan
, and H.
Yamaguchi
, “Sedimentation of two non-magnetic particles in magnetic fluid
,” Acta Phys. Sin.
66
, 164703
(2017
).45.
X.
Li
, X.-D.
Niu
, Y.
Li
, and M.-F.
Chen
, “Self-assembly of silica microparticles in magnetic multiphase flows: Experiment and simulation
,” Phys. Fluids
30
, 040905
(2018
).46.
Y.
Wang
, C.
Shu
, J. Y.
Shao
, J.
Wu
, and X. D.
Niu
, “A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio
,” J. Comput. Phys.
290
, 336
–351
(2015
).47.
C.
Flament
, S.
Lacis
, J.-C.
Bacri
, A.
Cebers
, S.
Neveu
, and R.
Perzynski
, “Measurements of ferrofluid surface tension in confined geometry
,” Phys. Rev. E
53
(5
), 4801
–4806
(1996
).48.
C.
Rigoni
, J.
Fresnais
, D.
Talbot
, R.
Massart
, R.
Perzynski
, J.-C.
Bacri
, and A.
Abou-Hassan
, “Magnetic field-driven deformation, attraction, and coalescence of nonmagnetic aqueous droplets in an oil-based ferrofluid
,” Langmuir
36
, 5048
–5057
(2020
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
