Electrokinetic effects in the breakup of electrified jets: a Volume-Of-Fluid numerical study

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1 Electrokinetic effects in the breakup of electrified jets: a Volume-Of-Fluid numerical study J. M. Lopez-Herrera 1, A. M. Gañan-Calvo 1, S. Popinet 2, M. A. Herrada 1 1.-University of Sevilla, Spain. 2.-Université Pierre et Marie Curie, Paris, France. 1

2 INDEX 1. Introduction 2. The electrokinetic model 3. A VOF approach 4. An study: The breakup of charged capillary jets 5. Conclusions 2

3 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 3

4 Introduction. Electrohydrodynamics (EHD) deals with fluid motion induced by electric fields. Electric fields 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 Fluid motion Fields: Microfluidic devices Electrospray Electrified liquid bridges 4

5 Introduction. General EHD equations Bulk equations Maxwell Equations r ("E) = q r E + r J = 0 E = rá r ("rá) = q being J = qv + ke Navier-Stokes Equations r v = 0 ½ Dv Dt = rp + r ¹(rv + rv T ) + r T e being T e = "EE "E2 2 I 5

6 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 6

7 LIMITS. cone-jet electrospraying EHD model has been used with success to develop scaling laws Gañan-Calvo PRL 79 (2) (1997) 7

8 LIMITS. cone-jet electrospraying Classic EHD can not explain the different behavior observed when the polarity is changed H.H. Kim et al. JAS 76 (2014) 8

9 LIMITS. Conclusions 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. 9

10 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 10

11 Electrokinetic equations NERNST-PLANCK-POISSON c i t + r (c i u) = r (PNP) EQUATION! i k B T rc i e! i z i c i E r ("E) = r ( "r') = q = X i ez i c i NAVIER-STOKES EQUATIONS r u = 0; + u ru = rp + r T v + F e + ¾ ± s n T v = 2¹D = ¹(ru + ru T ) T e = " µee E2 2 I F e = r T e = qe 1 2 E2 r": 11

12 Equations Dimensionless parameter Basis: ½ e, c o, L, ¾ and " o. ² Dimensionless ion di usivities, D i =!i k B T LU c being U c = (¾=½ e L) 1=2. P e i = 1=D i. ² Ratio of characteristic electric elds, i = E c Lez i =(k B T ), with E c = (¾=" o L) 1=2. ion speci c conductivities, i = D i i. ² The Ohnesorge number, C ¹ = ¹ e = p ¾L½ e. ² The dimensionless q Debye parameter, K i = L= D, D = is the Debye length. "e k B T 2e 2 (z i ) 2 c o ² Ratios of the relevant uid properties, R = ½ o =½ e ; M = ¹ o =¹ e and S = " e =" o. 12

13 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 13

14 A VOF approach. Conservation of ionic species. c i t + r (c i u) = r D i rc i + c i i r' Á t + r (Áu) = 0 (c i Á) t + r (c i Áu) = r ÁD i rc i + r i Ác i r' 14

15 A VOF approach. Conservation of ionic species. (c i Á) t + r (c i Áu) = r ÁD i rc i + r i Ác i r' ² The concentration c i is advected using the geometrical Volume-Of-Fluid technique. Z dt r (Ác i u) = X c f u f t C faces ² The concentration face value, c f is calculated from cell concentration and slope-limited concentration gradients. ² Gerris wording: GfsVariableVOFConcentration 15

16 A VOF approach. Conservation of ionic species. (c i Á) t + r (c i Áu) = r ÁD i rc i + r i Ác i r' The factor Á in the migration terms is as a weighted di usivity/conductivity ( ¹ D i / ¹ i ) that nulli es the migration uxes across boundaries out of the solvent phase. 16

17 Conservation of ionic species: A VOF approach (IV) r (ÁD i rc i ) = r [D i r(c i Á)] r (D i c i rá) ; (c i Á) t + r (c i Áu) = r [D i r(c i Á)] r (D i c i rá) r i Ác i E ; {z } {z } {z } term A term B term C Z Z h r [D i r(c i Á)] = D i r(c i Á) n = X (D i ) f r f (c i Á); f GERRIS wording: term A: SourceDiffusion term B: SourceDiffusionExplicit term C: SourceDiffusionExplicit 17

18 Validation BC: Slip velocity 18

19 Validation Ca E = 0:025 To the dielectric limit To the leaky-dielectric limit d = 9 (S 1) ns 2 K 2 =(K coth K 1) o K 2 S Ca E 16 [2(S 1) + SK 2 =(K coth K 1)] 2 corresponds to R = 1, C ¹ = 1, M = 1, D + = D = 1 Zholkovskij et al. J. Fluid Mech. 472 (2002) 19

20 Validation Level d rate of convergence relative error (%) K = 0: { K = { 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 low value of d for K = 0:1 (dj K=0:1 = 7: ). 20

21 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 21

22 DESCRIPTION OF THE PROBLEM Breakup of a liquid charged capillary column Initial conditions f(z; 0) = 1 + ² sin( w z) c + (z; r; 0) = B + c (z; r; 0) = B 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 R 1 =A; and (4) for the uid properties, C ¹, S, R and M. 22

23 RESULTS Breakup of a liquid charged capillary column We focus mainly in electrokinetic effects. So the following parameters are kept fixed. ² = 0:1, w = 0:6283, C ¹ = 0:05, R = M = 10 2 and S = 10. In order to have a more pronounced effect cation is smaller than anion We investigate the influence of polarity, Positive polarity: B + = 1:01 and B + = 0:99 Negative polariy: B + = 0:99 and B + = 1:01. The only free parameter is K= L/λ D. The level of electrification is kept fixed, Ca E = Then γ is calculated from D + = 7 and D = 1. 23

24 RESULTS Breakup of a liquid charged capillary column 24

Results The validity of the homogeneous conductivity

D + rc + D + E c t + r (c u) = r D rc + D E µ q t +

25 Results The validity of the homogeneous conductivity assumption Classic EHD Model: From PNP equation: q t + r (qu) = r ( SE) Dimensionless homogeneous conductivity q = SK2 2 (c+ c ) c + t + r (c + u) = r D + rc + D + E c t + r (c u) = r D rc + D E µ q t + r (qu) = K2 S D 2 r (D+ rc + D rc + c + + D c ) r K 2 SE 2 25

26 Results The validity of the homogeneous conductivity assumption Relative bulk conductivity = Positive polarity K=0.5 K=20 26

27 Results The validity of the homogeneous conductivity assumption 27

28 INDEX 1. Introduction 2. The electrokinetic model 3. A VOF approach 4. An study: The breakup of charged capillary jets 5. Conclusions 28

29 Conclusions 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. 29

30 Electrokinetic effects in the breakup of electrified jets: a Volume-Of-Fluid numerical study J. M. Lopez-Herrera 1, A. M. Gañan-Calvo 1, S. Popinet 2, M. A. Herrada 1 1.-University of Sevilla, Spain. 2.-Université Pierre et Marie Curie, Paris, France. 30

A charge-conservative approach for simulating electrohydrodynamic two-phase flows using Volume-Of-Fluid

A charge-conservative approach for simulating electrohydrodynamic two-phase flows using Volume-Of-Fluid J. M. López-Herrera,, S. Popinet b, M. A. Herrada a Dept. Ingeniería Aerospacial y Mecánica de Fluidos,