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Electrokinetic effects in the breakup of electrified jets:
a Volume-Of-Fluid numerical study J. M. Lopez-Herrera1, A. M. Gañan-Calvo1, S. Popinet2, M. A. Herrada1
1.-University of Sevilla, Spain. 2.-Université Pierre et Marie Curie, Paris, France.
Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain. 1
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
INDEX
1. Introduction 2. The electrokinetic model 3. A VOF approach 4. An study: The breakup of charged capillary jets 5. Conclusions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
INDEX
1. Introduction a. The EHD problem b. The EHD equations c. Limits
2. The electrokinetic model 3. A VOF approach 4. An study: The breakup of charged capillary jets 5. Conclusions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Electrohydrodynamics (EHD) deals with fluid motion induced by electric fields.
Introduction.
Electric fields
Fluid motion
The fluid motion modifies in turn the existing electric field
by changing either the geometry and the distribution of charges
Electric forces affects motion They can appears in the bulk and/or in
the interfaces
Microfluidic devices Electrospray Electrified liquid bridges …
Fields:
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Introduction. General EHD equations
r ¢ ("E) = qr£E = 0
@q
@t+r ¢ J = 0
being J = qv + kE
Maxwell Equations
Navier-Stokes Equations r ¢ v = 0
½Dv
Dt= ¡rp +r ¢
£¹(rv +rvT )
¤+r ¢Te
being
Te = "EE¡ "E2
2I
Bulk equations
r ¢ ("rÁ) = ¡q
E = ¡rÁ
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
The EHD model assumes two main simplifications: Condense in all the charged species. Is a more detailed description necessary? Conductivity is homogeneous and constant. True?
Depends on the problem. For example in cone-jet electrospraying
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Gañan-Calvo PRL 79 (2) (1997)
LIMITS. cone-jet electrospraying
EHD model has been used with success to develop scaling laws
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
H.H. Kim et al. JAS 76 (2014)
Classic EHD can not explain the different behavior observed when the polarity is changed
LIMITS. cone-jet electrospraying
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Therefore, to descend to a more detailed physical description of the charged species that form the electric charge is desirable.
The study of the evolution, distribution, etc, of this charged species is the object of the electrokinetic theory.
LIMITS. Conclusions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
INDEX
1. Introduction 2. The electrokinetic model
a. Electrokinetic equations 3. A VOF approach 4. An study: The breakup of charged capillary jets 5. Conclusions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
cit +r ¢ (ciu) = r ¢¡!ikBTrci ¡ e!izi ciE
¢
r ¢ ("E) = r ¢ (¡"r') = q =X
i
ezici
Electrokinetic equations NERNST-PLANCK-POISSON (PNP) EQUATION
NAVIER-STOKES EQUATIONS r ¢ u = 0;
½
µ@u
@t+ u ¢ ru
¶= ¡rp +r ¢ Tv + Fe + ¾·±sn
Tv = 2¹D = ¹(ru +ruT )
Te = "
µEE¡ E2
2I
¶
Fe = r ¢ Te = qE¡ 1
2E2r":
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Equations Dimensionless parameter
Basis: ½e, co, L, ¾ and "o.
² Dimensionless ion di®usivities,
Di = !ikBTLUc
being Uc = (¾=½eL)1=2 . Pei = 1=Di.
² Ratio of characteristic electric ¯elds,°i = EcLezi=(kBT ), with Ec = (¾="oL)1=2.ion speci¯c conductivities, ¤i = Di°i.
² The Ohnesorge number,C¹ = ¹e=
p¾L½e.
² The dimensionless Debye parameter,
Ki = L=¸D, ¸D =q
"ekBT2e2(zi)2co
is the Debye length.
² Ratios of the relevant °uid properties,R = ½o=½e; M = ¹o=¹e and S = "e="o.
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
INDEX
1. Introduction 2. The electrokinetic model 3. A VOF approach
a. conservation equation of ionic species b. Validation
4. An study: The breakup of charged capillary jets 5. Conclusions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
A VOF approach. Conservation of ionic species.
cit +r ¢ (ciu) = r ¢¡Dirci + ci¤ir'
¢
(ciÁ)t +r ¢ (ciÁu) = r ¢¡ÁDirci
¢+r ¢
¡¤iÁcir'
¢Át +r ¢ (Áu) = 0
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
(ciÁ)t +r ¢ (ciÁu) = r ¢¡ÁDirci
¢+r ¢
¡¤iÁcir'
¢
² The concentration ci is ad-vected using the geometricalVolume-Of-Fluid technique.
dt
Z
Cr¢(Áciu) =
X
faces
cfuf ¢t
² The concentration face value,cf is calculated from cell con-centration and slope-limitedconcentration gradients.
² Gerris wording:GfsVariableVOFConcentration
A VOF approach. Conservation of ionic species.
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
(ciÁ)t +r ¢ (ciÁu) = r ¢¡ÁDirci
¢+r ¢
¡¤iÁcir'
¢
The factor Á in the migration terms isas a weighted di®usivity/conductivity( ¹Di /¹¤i) that nulli¯es the migration°uxes across boundaries out of the sol-vent phase.
A VOF approach. Conservation of ionic species.
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Conservation of ionic species: A VOF approach (IV)
r ¢ (ÁDirci) = r ¢ [Dir(ciÁ)]¡r ¢ (DicirÁ) ;
(ciÁ)t +r ¢ (ciÁu) = r ¢ [Dir(ciÁ)]| {z }term A
¡r ¢ (DicirÁ)| {z }term B
¨r ¢¡¤iÁciE
¢| {z }
term C
;
h
Z
Cr ¢ [Dir(ciÁ)] =
Z
@CDir(ciÁ) ¢ n =
X
f
(Di)frf (ciÁ);
term A: SourceDiffusion term B: SourceDiffusionExplicit term C: SourceDiffusionExplicit
GERRIS wording:
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Validation
Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
BC: Slip velocity
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To the dielectric limit
To the leaky-dielectric limit
Validation
Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
CaE = 0:025
Zholkovskij et al. J. Fluid Mech. 472 (2002)
d
CaE=
9
16£
(S ¡ 1)n
S£2¡K2=(K cothK ¡ 1)
¤2 ¡ 1o
+ K2S
[2(S ¡ 1) + SK2=(K coth K ¡ 1)]2
corresponds to R = 1, C¹ = 1, M = 1, D+ = D¡ = 1
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Validation
Level d rate of convergence relative error (%)
K = 0:18 0.00531 { 33.039 0.00721 1.9 10¡3 9.0410 0.00736 1.5 10¡4 7.21
K = 68 0.01223 { 10.449 0.01332 1.1 10¡3 2.4710 0.01365 3.4 10¡4 0.0
² The rate of convergence with the grid is similar for K = 0:1 and K = 6.
² The relative error is larger for lower K values, Artifact due to very lowvalue of d for K = 0:1 (djK=0:1 = 7:93 10¡3).
Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain. 20
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
INDEX
1. Introduction 2. The electrokinetic model 3. A VOF approach 4. An study: The breakup of charged capillary jets
a. Description of the problem b. Results
5. Conclusions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
DESCRIPTION OF THE PROBLEM Breakup of a liquid charged capillary column
Dimensionless governing parameters (1) for the perturbation, ² and ·w;
(2) for the charged species, D+, D¡, ° and K;
(3) for the electrical conditions, B+, B¡ and R1=A; and
(4) for the °uid properties, C¹, S, R and M .
f(z; 0) = 1 + ² sin(·w z)c+(z; r; 0) = B+
c¡(z; r; 0) = B¡
Initial conditions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
RESULTS Breakup of a liquid charged capillary column We focus mainly in electrokinetic effects. So the following parameters are kept fixed.
In order to have a more pronounced effect cation is smaller than anion
We investigate the influence of polarity,
The only free parameter is K= L/λD.
The level of electrification is kept fixed, CaE = 0.125
Then γ is calculated from
² = 0:1, ·w = 0:6283, C¹ = 0:05, R = M = 10¡2 and S = 10.
D+ = 7 and D¡ = 1.
Positive polarity: B+ = 1:01 and B+ = 0:99Negative polariy: B+ = 0:99 and B+ = 1:01.
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
RESULTS Breakup of a liquid charged capillary column
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Results The validity of the homogeneous conductivity assumption
c+t +r ¢ (c+u) = r ¢
¡D+rc+ ¡D+°E
¢
c¡t +r ¢ (c¡u) = r ¢¡D¡rc¡ + D¡°E
¢
qt +r ¢ (qu) =K2S
2°r ¢ (D+rc+ ¡D¡rc¡)¡r ¢
µD+c+ + D¡c¡
2K2SE
¶
q =SK2
2°(c+ ¡ c¡)
qt +r ¢ (qu) = ¡r ¢ (® SE)
Classic EHD Model:
Dimensionless homogeneous conductivity From PNP equation:
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Relative bulk conductivity =
Results The validity of the homogeneous conductivity assumption
26
K=0.5
K=20
Positive polarity
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Results The validity of the homogeneous conductivity assumption
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
INDEX
1. Introduction 2. The electrokinetic model 3. A VOF approach 4. An study: The breakup of charged capillary jets 5. Conclusions
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Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain.
Conclusions
29
1. A general electrokinetic model and numerical scheme has been presented and validated.
2. The numerical scheme is available through GERRIS.
3. In the cases when thermal diffusion, electrosmotic motion and singularities compete EHD model can yield different results than the present electrokinetic model.
4. Method can be improved restricting the resolution of the diffusion term to the domain of interest.
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Electrokinetic effects in the breakup of electrified jets:
a Volume-Of-Fluid numerical study J. M. Lopez-Herrera1, A. M. Gañan-Calvo1, S. Popinet2, M. A. Herrada1
1.-University of Sevilla, Spain. 2.-Université Pierre et Marie Curie, Paris, France.
Meeting on Numerical challenges in two-phase flows 27 - 28 October 2014, Seville, Spain. 30