Confinement effects on electrohydrodynamics of two-dimensional miscible liquid drops

Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

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Confinement effects on electrohydrodynamics of two-dimensional miscible liquid drops Ali Behjatian, Asghar Esmaeeli ∗ Department of Mechanical Engineering & Energy Processes, Southern Illinois University, Carbondale, IL 62901, United States

h i g h l i g h t s

g r a p h i c a l

a b s t r a c t

• Electrohydrodynamics of a miscible drop in confined domains is investigated. • The confinement may strengthen or weaken the electric field stresses. • The confinement weakens the flow field and changes the flow structure. • Below a critical domain size, necessary condition for oblate deformation is reversed.

a r t i c l e

i n f o

Article history: Received 29 April 2013 Received in revised form 18 July 2013 Accepted 29 August 2013 Available online 10 September 2013

E0

a b s t r a c t The effect of domain confinement on the electrohydrodynamics of a two-dimensional miscible drop is investigated analytically, as a prototypical problem to explore confinement effect in two applications; namely, continuous flow electrophoresis and pattern formation in suspension of inhomogeneous colloidal dispersions. It is shown that the domain confinement may strengthen or weaken the electric field and leads to weakening of the flow field and change of the flow structure. A characteristic function is found to determine the sense of drop elongation and it is shown that below a critical domain size, the necessary condition for elongation in the direction normal to the field will be opposite to that for an unbounded domain. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The interaction of electric field with fluid interfaces has been a problem of long-standing interest, dating back to the 16th century when Gilbert [1] reported deformation of a water droplet to a cone in the presence of charged piece of amber. Not many follow up studies seem to have been performed in the intervening period between Gilbert’s observation and Rayleigh’s [2] theoretical work concerning the stability of charged drops some 250 years later. Similarly, the century-long period between the Rayleigh’s study and the first half of the 20th century can be also marked with low level of research activities in the field. The second half of the 20th century, however, witnessed a dramatic growth in research

∗ Corresponding author. Tel.: +1 618 453 7001; fax: +1 618 453 7001. E-mail address: [email protected] (A. Esmaeeli). 0927-7757/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.colsurfa.2013.08.080

in this area, driven mostly by the desire to understand and exploit the electric-induced force in practical applications; this trend has continued unabated to this date. The use of electric field for manipulation of fluid interfaces is appealing because the field strength can be easily adjusted and the electric field can act from a distance. Examples of widespread use of electric field/fluid interface interaction include transport of minuscule amount of fluids in microfluidics [3], micro- and nano-encapsulation [4] for food processing and targeted drug delivery, electrospraying [5] for particle deposition and variety of other purposes [6], continuous flow electrophoresis [7], and electric field-induced pattern formation in colloidal structures [8], to name a few. Motivated by applications in (i) continuous flow electrophoresis (CFE) and (ii) pattern formation in suspension of inhomogeneous colloidal dispersions, which are discussed in the next two paragraphs, our goal is to improve the theoretical understanding of interface deformation by electrohydrodynamic (EHD) forces in

A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

“confined domains”. To this end, we study a prototypical problem in which a system of two miscible fluids, comprised a twodimensional liquid droplet that is immersed in another fluid and confined in a cylindrical container, is exposed to a uniform electric field. Here the mismatch between the dielectric properties (i.e., electric conductivity and permittivity) of the fluids leads to “net” normal and tangential interfacial stresses, which set the fluids in motion and tends to deform the interface. We seek an analytical solution for this problem in the framework of the Taylor–Melcher “leaky-dielectric theory” [9,10]. The theoretical foundation and the mathematical formulation of the theory have been well described in the review articles [10–12], as well as in the books [13,14]. Briefly, the essence of the model is to assume that fluids have finite electric conductivities and that the time scale of charge relaxation due to conduction from the bulk to the interface to be much shorter than any process time of interest. The first assumption allows for the accumulation of free charges at the interface, and therefore, fluid flow. The second assumption leads to substantial simplification in the mathematical formulation as the electric field equations will be decoupled from the momentum equation and reduce to quasi-steady state laws. CFE is a preparative technique that is used in health industry to provide large quantities of protein and pure biological materials. In this process, a thin stream of dilute colloidal particles (called sample) is forced to move coaxially with a buffer fluid through an elongated parallelpiped chamber, where a uniform electric field is applied normal to the stream axis (Fig. 1a). This leads to separation of the colloidal particles into distinct streams of purified products, according to their differences in electrophoretic mobilities. The fractionated samples are subsequently collected at the exit of the chamber. The efficiency of sample separation, however, is generally hindered by various sample deformation problems due to sedimentation, thermal convection, and electroosmosis [15]. When all these effects were minimized in experiments by optimization of the geometry and experimental conditions [16] the sample deformation still remains. To unravel the cause of the residual deformation, in a noteworthy study, Rhodes et al. [7] modeled this process as a two-dimensional problem in a plane perpendicular to the sample. The authors ignored the axial shear (between the sample and the buffer) and the wall effects and considered a two-dimensional liquid drop in a homogeneous buffer of infinite extension (Fig. 1c). Accordingly, they solved the electrohydrodynamic equations for creeping flows in the framework of leaky-dielectric theory. Since the sample and the buffer were miscible, they ignored the surface tension. However, they treated the interface as a sharp discontinuity. The authors showed that the sample deformation was induced by electrohydrodynamic forces, and its sense depended on the relative importance of the dielectric properties of the sample and the buffer, according to the sign of a characteristic function. The results of this study were similar to those of Taylor’s [9] seminal study for EHD-driven deformation of a spherical drop. This study sparked a wave of interest on this subject and followed by other investigators; see, for example, [17,18]. The idea of electric field-induced pattern formation of inhomogeneous colloidal dispersions was initiated by Trau et al. [8] in the context of materials processing, where they aimed to bypass the intrinsic limitations of mechanical forming in manipulation of colloidal structures by resorting to electric forces. To this end, the authors performed an experimental study in conjunction with a theoretical modeling. In their experiments, they examined the deformation of a bolus (comprised nanometer-sized spherical barium titanate particles) in castor oil under an applied external electric field (Fig. 1b). In one case, the electric conductivity of the bolus was adjusted to be much higher than the ambient fluid and the bolus elongated in the direction of the field. In the other case, the conductivity mismatch was reversed and the bolus elongated

117

Sample Stream Buffer Flow

Buffer Flow

E0

(a)

E0

(b)

E∞

r

θ a (c)

Fig. 1. (a) A schematic diagram depicting the continuous flow electrophoresis (CFE) process, (b) a model system depicting pattern formation in “inhomogeneous” colloidal dispersions, and (c) the prototypical model used to study processes (a) and (b) by Refs. [7,8], respectively.

in the direction perpendicular to the field. The authors argued that the mechanism behind deformation of boluses was due to the electric body force lodged in the suspension, and therefore, it was significantly different from that of the previously known pattern formation in “homogeneous” colloidal dispersions (e.g., formation of chains in electrorheological and ferrofluids [19–22]), which is generally believed to be the result of dipole–dipole interactions between the suspended particles in a “perfect dielectric” liquid. Trau et al. [8] modeled this problem by considering a twodimensional fluid drop immersed in a pool of another liquid of infinite extension (Fig. 1c) and developed a semi-analytical formulation, building on the analysis of Saville [17], who suggested to treat the interface as a thin diffuse layer. Their results concerning the deformation of the bolus were qualitatively the same as those

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A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

of Ref. [7]. In particular, they recovered the deformation characteristic function of Ref. [7] in the limit of small thickness of the diffuse layer. In the theoretical modeling of the two processes that we just described, the authors ignored the wall effects; this has been also true for the follow up studies. While the electrohydrodynamics of a two-dimensional liquid drop in an unbounded domain is reasonably well understood, not much is known about the effect of domain confinement on the behavior of the drop. This understanding, however, finds relevance in microfluidic applications where the dimensions of the channel are typically of the same order as the drop size. Although from the solutions for unbounded domain one can find an estimate of the minimum distance at which the confinement effect becomes insignificant, to understand the manner by which the results will be affected and for quantification purposes, one must incorporate the wall effects into the solution. The goal of this study is, therefore, to explore the effect of confinement on the electrohydrodynamics of a two-dimensional liquid drop. To this end, we build on Rhodes et al. solution [7] by considering a sharp interface, where the material properties change discontinuously from one fluid to the other, and solve the governing electrohydrodynamic equations analytically for a circular drop in a confined domain. We note that a more refined approach would be to consider the interface as a diffuse layer [17]; however, this model has not been confirmed by experiments and requires an ad hoc assumption regarding the thickness of the layer and introduction of an unknown concentration profile. Furthermore, the sharp interface approximation has been adapted by other investigators [18] as it is deemed to capture the essential aspects of the phenomenon. It should be emphasized that the result of this study represents only the initial stage of the interface deformation, since in the absence of the restoring force of surface tension the interface deforms continuously, eventually becoming a ribbon. The organization of the paper is as follows. In Section 2, we present the problem setup and governing nondimensional numbers. The governing equations, their solutions, and validations are presented in Section 3. Section 4 contains the results including the effect of confinement on the electric and flow fields and the drop deformation. We conclude with a discussion of the new findings in Section 5.

2. Problem setup and nondimensional parameters The problem setup is shown in Fig. 2, depicting a liquid drop of radius a, immersed in another liquid and confined in a rigid container of radius b. The drop and the container are concentric and the origin of the polar coordinate system (r, ), which is used to represent this problem, is at the center of the drop. Here we ignore buoyant effect. This is because in CFE the gravity is normal to the direction of the electric field and under the two-dimensional simplification used here the gravity does not come to the picture. In the inhomogeneous colloidal suspension, since the density of the two media is close, the buoyant effects are again negligible. The EHD-induced fluid shear does not lead to a net motion of the drop because of the symmetry. Therefore, the center of mass of the drop remains at the origin of the coordinate. A uniform electric field E0 in the vertical direction is imposed at the wall. This boundary condition in essence is imposing a nonuniform electric potential () ∼ cos() at the wall, which can be materialized in practice by using segmented electrodes and resistance bridges to constitute the outer wall. Smith and Melcher [24], for instance, used the same technique to create a nonuniform electric potential of the form (x) = 0 cos(kx) on a flat electrode. Under the assumptions of creeping flow and negligible interface deformation, the governing nondimensional numbers of this problem are the electric

y

o,

r

θ

i,

σo , μ o

σi , μ i a

b

Fig. 2. The geometric setup, depicting a two-dimensional fluid drop of radius a, enclosed by another fluid and confined by a cylindrical container of radius b. A uniform electric field E0 is imposed at the wall.

conductivity ratio R =  i / o , the electric permittivity ratio S = i /o , ˜ = i /o , and the ratio of the radius of the the viscosity ratio  drop to that of the container  = a/b. Here, subscripts i and o denote quantities associated with the inner and the outer fluids, respectively. It should be pointed out that the use of a concentric rigid cylinder (or sphere) as a prototypical model to explore the effect of domain confinement has precedent in analytical solution of particulate flows. For instance, to estimate the effect of particle/particle interactions and domain confinement in a suspension of particles, several investigators have modeled the problem by considering motion of a solitary spherical particle of radius a inside a concenn v )1/3 , where n, tric spherical rigid container of radius b = a(V/˙i=1 i vi , and V are the number of particles, volume of each particle, and the volume of the container, respectively. See, for example, pages 130–135 of Ref. [23].

3. Governing equations and their solutions Electrohydrodynamics deals with interactions of electric field and fluid flow. As such, the laws concerning the fluid dynamics and electric field and their coupling should be considered. Here, the governing equations are the conservations of mass and momentum, and simplified Maxwell’s electromagnetic equations. For leaky dielectric fluids with constant properties and net zero charge in the bulk, it can be shown that the electric field equations are decoupled from the fluid flow equations, but the fluid flow equations are coupled to the electric field equations through the momentum jump conditions [10,12]. This decoupling allows one to solve the electric field and hydrodynamic equations sequentially.

A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

where qs∞ is the free surface charge density in an unbounded domain.

3.1. Solution of the electric field The charge conservation equation, along with the fact that the electric field E is irrotational and divergence free, leads to the Laplace equation for the electric potential

∇ 2  = 0.

(1)

Eq. (1) is valid for fluids in the drop and the shell, and is solved using the following boundary conditions: (1) continuity of electric potential at the phase boundary, (2) finiteness of the electric potential at r = 0, (3) continuity of the normal component of electric current density at the phase boundary, (4) and the known form of the electric potential at the wall, (b, ) = E0 b cos , where E0 = |E0 |. More details about these boundary conditions can be found in [26]. Solution of Eq. (1) results in the electric field potentials in the drop i∞ 2 = E0 a R+1

i = i∞ ;

and in the shell o∞ = E0 a

o = o∞ ;

r  a

  r a

cos ,

R−1 − R+1

(2)

 a 2  r

cos ,

(3)

R+1 (R + 1) − 2 (R − 1)

(4)

is a correction factor that takes into account the confinement effect on the electric field. Here, i∞ and o∞ represent the electric potentials in an unbounded domain. The jumps in the tangential and normal electric stresses at the interface are the drivers behind the fluid flow circulation and interface deformation. To compute these stresses, it is more useful to express them in terms of a tangent-normal coordinate system (i.e., t − n) and to customize the resulting expressions to other coordinates afterwards. Doing so, the stresses will be valid for a general interface in any coordinate system. Here the symbol 冀Q 冁 = Qo − Qi denotes the jump in a physical parameter (such as Q) at the interface. The jumps in the normal and tangential stresses are [26], respectively, e 冀 nn 冁=

o 2



1−

S R2



En2o + (S − 1)Et2



(5)

and

  S e 冀 nt 冁 = o Eno Et 1 − = qs Et .

In Eq. (6), qs is the free surface charge density. For the problem at hand where n ≡ r and t ≡ , evaluation of Eno ≡ Ero and E ≡ Eti = Eto using Eqs. (2) and (3) in conjunction with E = − ∇  and substitution of the resulting expressions in Eqs. (5) and (6) yield

冀 冁= 冀 冁 2

e rr ∞;

冀 rre 冁∞ o E02

=

The governing equations for steady state, incompressible, and creeping flows are the conservations of mass, ∇ · u = 0, and momentum − ∇ p +  ∇ 2 u + Fe = 0, where Fe = qv E − (1/2)E · E∇ is the electric force density [Nm−3 ]. It should be noted that the electric force density comprises three force, the electrophoretic (Coulomb) force qv E, the dielectrophoretic force −(1/2)E · E ∇ , and the electrostriction force. For incompressible flows, the electrostriction force can be grouped with the pressure [12], and we have done so. For leaky dielectric fluids with constant properties, Fe is zero in the fluid bulk, since qv = 0 and ∇ = 0. As such, the electric force enters the picture only through the momentum jump conditions. Here it is possible to derive an equation for the streamfunction , which satisfies the mass and momentum conservation equations

2[(R2 + 1 − 2S)cos2  + S − 1] (R + 1)2

= 0,

(10)

where ∇ 4 is the biharmonic operator. The velocities are related to the streamfunction through ur = − (1/r)(∂ /∂) and u = ∂ /∂r. Eq. (10) is solved using the following boundary and jump conditions: (1) finiteness of the velocity field at r = 0, (2) continuity of the velocity at the interface, (3) balance of tangential stress at the h 冁 + 冀 e 冁 = 0, (4) balance of the normal stress phase boundary, 冀 r r

h 冁 − 冀p冁 + 冀 e 冁 = 0, and (5) no-slip and at the phase boundary, 冀 rr rr h and h no-through flow boundary conditions at the wall. Here r rr are the tangential and normal hydrodynamic stresses, respectively. The details of the solution of Eq. (10) can be found in Refs. [26,7]. Briefly, the suggested solution for streamfunction (r, ) = rn sin 2, where n is a real constant to be deteris mined [26]. Substitution for using the above expression into Eq. (10) and solving the resulting ordinary differential equation yield n = 0, 2, −2, and 4. Thus, o = (A + Br2 + Cr−2 + Dr4 ) sin 2 2 −2 + Hr4 ) sin 2, where A − H are unknown and i = (E + Fr + Gr constants. Application of boundary condition (1) results in E = 0 and G = 0. To proceed, we need to calculate the pressure first. This is done by substituting for the velocity field in the momentum equation, − ∇ p +  ∇ 2 u = 0, and integration of the resulting equation in terms of p. Subsequently, applications of the boundary and jump conditions (2)–(5) yield the streamfunction in the drop

(6)

R

e rr

3.2. Solution of the flow field

∇4

where  (, R) =

i

aus

=



2

e 冁 冀 r ∞

o E02

=

2(S − R) (R + 1)2

sin 2,

o

aus

=

qs = qs∞ ;

qs∞ 2(S − R) = cos , o E0 R+1

(9)

a

+ H

 r 4  a

sin 2,

(11)



2 12˘(R + 1)2

× 3A + 3B

e e冁 where 冀 rr ∞ and 冀 r 冁∞ are the jumps in electric stresses in an unbounded domain. The effect of domain confinement on the free surface charge qs is of key interest because the strength and sense surface charge control the electrohydrodynamics of the drop. In terms of t − n coordinate it can be shown that qs = o Eno (1 − S/R) [26], which upon the substitution for Eno = −∂o /∂r|r=a yields

 r 2

and in the shell

(7)

(8)

3F

12˘(R + 1)2

 r 2

and e e 冀 r 冁 =  2 冀 r 冁∞ ;

119

a

+ C

 a 2 r

+ D

 r 4  a

sin 2,

(12)

where we have recast coefficients A − D, F, and G in terms of new coefficients A − D , G , and H , respectively. These coefficients, as well as ˘, are given in Appendix A. Here, us = o E02 a/o is a velocity scale, which can be found by balancing the electric and e and h , at the interface. The velochydrodynamic shear stresses, r r ities are readily determined: uri us ui us

=−

=



2

3F

6˘(R + 1)2 2

6˘(R + 1)2

 2H

r  a

 r 3 a

+ H

 r 3 

+ 3F

a

 r  a

cos 2,

(13)

sin 2,

(14)

120

A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

uro 2 =− us 6˘(R + 1)2



× C

 a 3 r

1.8

+ 3A

a r

+ 3B

r  a

+ D

 r 3  a

R = 10−4 R = 0.9 R = 104

cos 2, (15) 1.4

 −C

 a 3 r

+ 3B

r  a

+ 2D

 r 3  a

sin 2. 1

(16)

3.3. Validation We performed a number of tests to validate the solution. Specifically: (1) In the limit of  → 0, we fully recover the solution of Ref. [7] for unbounded domain (Appendix B). At this limit,  = 1, and Eqs. (2), (3) and (7)–(9) readily converge to the corresponding ones for unbounded domain, and the coefficients used in the streamfunction solution (Appendix A) converge to the corresponding ones for unbounded domain (Appendix B). (2) For finite domain, the results are physically plausible; the evolution of the streamlines, the strength of the flow field, and the change in the drop deformation all are justified from the fluid dynamics and electrohydrodynamics point of views. (3) Preliminary computer simulations using our immersed boundary technique [25], for a range of nondimensional parameters yielded the results (fluid flow and deformation) that were qualitatively the same as those predicted by the analytical solution.

0.6 0 3

0.2

0.4 λ

0.6

0.8

λ =0.8 λ =0.6 λ =0.4

2 Γ

uo 2 = us 6˘(R + 1)2

Γ

and

1

−3

10

0

10 R

3

10

Fig. 3. Variations of the wall correction factor for the electric field ( ) with nondimensional domain size  = a/b (top panel) and the conductivity ratio R =  i / o (bottom panel).

4. Results and discussion 4.1. Effect of confinement on electric potential, stresses, and surface charge

4.2. Effect of confinement on the flow field

The confinement leads to scaling of ∞ , E∞ , and qs∞ by  , and e e冁 2 scaling of 冀 rr ∞ and 冀 r 冁∞ by  . However, it does not alter the structure of any of these entities. Fig. 3 shows the variations of  with  = a/b and R =  i / o . A few observations can be made about this figure. First, with an increase in ,  increases for R > 1 and decreases for R < 1. This suggests that when the fluid in the drop is more (less) conducting than the shell fluid, the confinement effect yields higher (lower) electric potential. Second, in the two opposing limits of R  1 and R 1,  is asymptotic to  0 ≡ 1/(1 + 2 ) and  ∞ ≡ 1/(1 − 2 ), respectively. Since  ∞ / 0 > 1, it is expected that the confinement effect to be more pronounced when the fluid in the drop is more conducting than the fluid in the shell (R > 1) compared to the opposite case. Fig. 4 depicts schematically the relative importance of R and S in setting the distribution of the free electric surface charge and the sense of net shear electric stresses. For R < S, the upper half is induced with positive charge, while the lower half is induced with negative charge. Here, E > 0 for 0 <  < , and E < 0 for − <  < 0. As e 冁∼E q is in the positive -direction such, the net shear stress 冀 r  sf for 0 <  < /2 and <  < 3 /2, and is negative otherwise. For R > S, e 冁 is reversed since the direction of the net electric shear stress 冀 r the sign of the charge qsf is reversed in the two halves, while the direction of the tangential electric field strength E remains in tact.

The confinement alters the structure of the flow field, in addition to affecting its strength. This can be readily seen by comparison of Eq. (11) with (B.1) and Eq. (12) with (B.2). To explore the effect of confinement on the flow structure, in Fig. 5 we plot the nondimensional streamlines at four different nondimensional domain sizes of  = a/b = 0, 0.25, 0.5, and 0.67, for a fluid system with R = 0.01, S = 1, ˜ = 1. This fluid system is one of the two fluid systems that and  has been of practical interest in electrohydrodynamic studies of CFE and pattern formation of inhomogeneous colloidal dispersions [7,8]. The property ratios for the other fluid system are R = 100, S = 1, ˜ = 1. Note that the flow strength cannot be discerned from and  this figure as the level contours are not given. For an unbounded domain, the flow field consists of four sets of open streamlines in each quadrants, and the sense of the flow circulation is determined by the relative importance of R and S. For R > S, the outer fluid approaches the interface from sides and turns toward the top and the bottom For R < S, the direction of the flow will be reversed. For the first panel,  = 0 and R < S; thus, the fluid approaches the interface from the top and the bottom, and turns toward the right and the left. Note that the structure of this flow field is the same as that of Refs. [7,8,17]. The direction of flow will be reversed if the other system (R = 100, S = 1), where R < S, is used. The domain confinement weakens the flow field and transforms the open streamlines to closed loops since it forces the ambient velocity to diminish

A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

e τrθ

+ ++ + + + + + + + + + + − − − −

R
−

− − −

−

Eo

− − − −

Eo

− − − −

−

− − −

121

e τrθ

−

− − − −

R>S + + + +

+

+ + +

+

+ + + +

Fig. 4. A schematic picture depicting the effect of the electric field polarity and the relative magnitude of R =  i / o and S = i /o on the distribution of the free electric charges at the interface. For the left frame, R < S, positive charges are induced at the upper half of the column which faces the positive electrode, and negative charges are induced at the lower half which faces the negative electrode. For this case, the electric shear stress tends to drive the flow from the top ( = 0) and the bottom ( = ) to the sides. For the right frame, R > S, and the charge distribution and the direction of electric shear stress are reversed.

abruptly at the wall (second panel). Further confinement, however, leads to the formation of a dividing streamline (i.e., the closed circle inside the drop for which = 0) and four vortices inside the drop (third panel). As the confinement increases further, and therefore, the velocity field becomes weaker, the dividing streamline moves outward and the inner vortices grow further (fourth panel). Though not visible, for the last panel the dividing streamline is in the vicinity of the interface. Fig. 6, which compares the nondimensional radial and tangential components of the velocity field (at  = 0 and  = /4, respectively) for these four cases, demonstrate the weakening effect of the domain confinement.

To characterize the effect of confinement on the strength of the velocity field, we construct a scaling factor by considering the ratio of the interfacial radial velocity at  = 0 for a bounded and an unbounded domain,  2 ( Umax ˜ + 1) 3F + H = . Umax∞ 2˘ 3S − R2 − R − 1

(17)

For both R > 1 and R < 1, Umax /Umax∞ is a monotonically decreasing function of . Thus, confinement always weakens the flow field.

Fig. 5. Nondimensional streamlines for a fluid system with S = 1, R = 0.01, and  ˜ = 1. The nondimensional domain sizes are  = 0 (top-left), 0.25 (top-right), 0.5 (bottom-left), and 0.67 (bottom-right), respectively. The streamlines are drawn at selected level, so the intensity of the flow cannot be discerned. However, the confinement weakens the velocity; i.e., ur (a, 0)/ur∞ (a, 0) = 1, 0.6588, 0.1368, and −0.0161, respectively. The scales of the third and fourth panels have been increased for better visibility.

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0.2

where

λ =0.67 λ =0.50 λ =0.25 λ = 0.00

0

˚ = R2 + 1 − 2S + F(R − S), 2

F=

ur us

−0.2

(2

− 1)

2

 ˜ + (1 − 4 )

(19) .

˚ is a characteristic function that determines the sense of interface deformation and F is a parameter that accounts for confine-

−0.4

d

−0.6

−0.8 0

1

2

r/a

3

0.6

4

5

λ =0.67 λ =0.50 λ =0.25 λ = 0.00

0.4

uθ us

(2 − 1)  ˜ − (4 + 62 − 1)

0.2

0

−0.2 0

1

2

r/a

3

4

5

Fig. 6. Effect of confinement on the strength of the velocity field. Here, ur and u are plotted at  = 0 and  = /4, respectively. The velocities correspond to the four cases shown in Fig. 5.

Note that Umax and Umax∞ can be found from Eqs. (13) and (B.3) by considering uri = Umax cos 2 and uri ∞ = Umax∞ cos 2.

4.3. Effect of confinement on the deformation of the interface The sense and strength of the drop deformation is of key interest from practical and theoretical stand points. For miscible fluids, the interface deforms continuously since there is no surface tension (as in the case of immiscible fluids) to oppose the deformation. Here, the sense of drop deformation can be discerned from the structure of the flow field. As is evident from the first panel of Fig. 5, the flow field tends to squash the drop at the top and the bottom and stretch it at the sides, thus elongating the drop in the direction normal to the electric field. On this basis, the interfacial radial velocity at one of the one of the four points ( = 0, /2, , 3 /2) can be typically used to determine the sense of drop deformation; i.e., for ur (a, 0) = ur (a, ) > 0, the drop elongates in the direction of the electric field, while for ur (a, 0) = ur (a, ) < 0 it elongates in the normal direction. Likewise, the rate of the interface deformation can be estimated by the magnitude of the radial velocity at the interface. Evaluation of uri or uro at the interface (r = a) leads to d

uri (a, ) = uro (a, )=ura

2 6˘(R + 1)2

˚ cos 2,

(18)

ment effect. For ˚ > 0, uri (a, ) = uro (a, )= ura > 0, and the drop elongates in the direction of the field, while for ˚ < 0, ura < 0, and it elongates in the perpendicular direction. For ˚ = 0 the drop remains circular despite the distorting effect of the electric field, which is possible for leaky dielectric fluids. Inspection of F shows that F is a monotonically decreasing function of . In particular, for an unbounded domain ( → 0), F → F∞ = 1, and ˚ → ˚∞ = R2 + R + 1 −3S, which is the same as the characteristic function found by Ref. [7]. The possible senses of fluid circulation and drop deformation can be presented in a so-called deformation-circulation map, by plotting the ˚ = 0 curve in S − R coordinate [26–28]. Here the ˚ = 0 curve divides the domain into two regions: for ˚ > 0, the ambient fluid “at the drop surface” flows from the sides toward the top and the bottom, and therefore, the drop elongates in the direction of the field, for ˚ < 0, the flow direction is reversed, and therefore, the drop elongates in the traverse direction. From ˚∞ = R2 + R + 1 −3S it can be seen that the necessary condition for elongation of the drop in the normal direction in an unbounded domain (˚∞ < 0) is R < S. Therefore, we also add the line of R = S to this map for future clarification. Note that R = S represents a system of perfect dielectric fluids (i.e., no free surface charge) and for this system always ˚∞ = (S − 1)2 > 0. Fig. 7 shows the deformation-circulation map at three selected  = a/b. Here the viscosity ratio is  ˜ = 1 and P and N stand for interface elongation parallel and normal to the field, respectively. The dashed and solid blue lines show the acceptable roots of the ˚ = 0 equation. For the top panel,  is sufficiently small and the map is essentially the same as that for a drop in an unbounded domain. Here, the ˚ = 0 curve has only one acceptable root; i.e., S = (R2 + 1)/2. As the domain confinement increases, region N starts to shrink (middle panel) and beyond a critical domain size, cr , region N is shifted above the line R = S (bottom panel). This is an interesting outcome, since the necessary condition for having an oblate filament is now R > S, which is the opposite to that for a filament in an unbounded domain [7]. Inspection of Eq. (19) shows that cr can be found by solving for  through setting F = Fcr , where Fcr = −2. For F ≡ Fcr = −2, ˚ = (R − 1)2 , and therefore, the ˚ = 0 curve turns into line R = 1. In summary, for F > 0, there is only one acceptable root for ˚ = 0 equation (top panel), for Fcr < F < 0 there are two acceptable roots, but the region N will be below the R = S line; furthermore, smaller F results in narrower region N. Finally, for F < Fcr < 0, region N shifts above the R = S line. The physical interpretation of the results shown in Fig. 7 is as follows. The sense of drop deformation correlates with the sense of the interfacial radial velocity. For an unbounded domain, the latter is controlled by R and S only. For bounded domains, however, it is also influenced by the domain confinement  and the viscosity ratio . ˜ The change in criterion for oblate deformation from R < S to R > S when  > cr is a reflection of the change of sign of the interfacial radial velocity. This can be best understood by comparing the sense of the interfacial radial velocity from Figs. 5 and 6 with the predicted sense of deformation from Fig. 7. For the first three panels of Fig. 5, the outer fluid approaches the drop from the top and the bottom, and moves toward the right and the left; i.e., ur (a, 0) = ur (a, ) < 0 and ur (a, /2) = ur (a, 3 /2) > 0. As such, the fluid tends to elongate the drop in the direction normal to the field. For the last panel, the

A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

123

2

10

R=S Branch-I

1

10

P

0

N

R

10

−1

10

F =0.9697

(a)

=0.1 −2

10 −1 10

0

10

1

S

2

10

10

0.2

2

10

R=S Branch-I Branch-II

θ = 0π θ= 2 θ=π θ = 3π 2

1

10

ur us

P

0

N

10

R

0

−1

10

−0.2 0

F =- 1.998

(b)

=0.707 −2

10 −1 10

0

10

2

10

1

S

2

10

10

R=S Branch-I Branch-II

0.5

r/a

1

1.5

Fig. 8. The top panel shows magnification of the streamlines, corresponding to the flow in the last panel of Fig. 5. The bottom panel shows the radial velocity (for the same flow) with r/a. The radial velocity changes sign across the dividing streamline ( = 0; the dashed-line quarter circle). Thus, the flow tends to elongate the interface in last panel of Fig. 5 in the direction of the field, which is opposite to that for the first three panels.

1

10

the radial velocity changes across the dividing streamline, which is the dashed-line circle that has a radius slightly larger than that of the drop. In the analysis so far we did not discuss the detail of distributions of the net normal anisotropic stresses and the role of the isotropic stresses in setting the sense of the interface deformation. Understanding these issues, however, are quite insightful for manipulation of these stresses in practical applications. The sense of interface deformation is set by the balance of the “anisotropic” components of the normal electric and hydrodynamics stresses

P

0

N

R

10

−1

10

F =- 2.857

(c)

=0.75 −2

10 −1 10

0

10

1

S

10

2

10

Fig. 7. The deformation-circulation map for three selected nondimensional domain ˜ = 1 and P and N stand for elongation parallel and normal to the sizes  = a/b. Here,  field, respectively. For the first two panels  < cr , and the necessary condition for elongation normal to the field is R < S. However, for the third panel  > cr , and the necessary condition for having deformation normal to the field changes to R > S.

sense of drop deformation is in the direction of the field according to its corresponding map (not show here). Compared to the first three panels, this signals a change of sign in the interfacial velocities at the four major points at the interface (i.e., top, bottom, sides), and therefore, a reversal of the sense of flow circulation at the interface. Fig. 8, which presents a magnification of the velocity field (top panel) and the variation of the radial velocity with the radial distance (bottom panel), confirms this flow reversal. Here, the sign of

冀 rre 冁 + 冀rrh 冁 = 0, which are the coefficients of cos 2  and cos 2 in Eqs. (7) and (C.4), respectively. The deformation of the interface as set by the anisotropic components does not satisfy the incompressibility condition. On the other hand, the “isotropic” components of the normal stresses provide a uniform compression or expansion of the interface that is just enough to satisfy the incompressibility condition. In summary, the anisotropic stresses yield a displacement of the interface points in the direction of the unit normal vector n at the interface and the isotropic stresses lead to a uniform expansion or contraction of the surface. Fig. 9 demonstrates schematically how the anisotropic and isotropic components of the normal stresses contribute to the deformation of a liquid drop (in an unbounded domain). Here, we consider two particularly interesting regions in the S − R space, where the net normal hydrodynamic and electric stresses oppose each other. For simplicity we rewrite Eqs.

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A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

(a)

(b)

(c)

(d)

(e)

Fig. 9. A schematic picture depicting the distribution of the net normal anisotropic electric and hydrodynamic stresses, frames (a) and (b), respectively, and the interplay of

e these stresses with the isotropic stresses (frame d). Here we consider 冀 rr 冁∞ = Cae cos2  + Cie and 冀rrh 冁∞ = Cah cos2  + Cih . For the first fluid system (top row), Cae > 0 (frame

a), Cah < 0 (frame b), and |Cae | < |Cah |. Therefore, the interface deforms to an oblate (frame c). However, since the volume is not conserved, the extensional isotropic stresses Cie + Cih > 0 (frame d) expand the interface uniformly (frame e). For the second fluid system, again Cae > 0 (frame a), Cah < 0 (frame b), but |Cae | > |Cah |. Therefore, the interface

deforms to a prolate (frame c). Here the compressive isotropic stresses Cie + Cih < 0 (frame d) contract the interface uniformly (frame e) to enforce volume conservation. e 冁 = C e cos2  + C e and 冀 h 冁 = C h cos2  + C h , (7) and (C.4) as 冀 rr a rr ∞ a ∞ i i

Cae

Cah

respectively. For the first fluid system, > 0 (frame a), <0 (frame b), and |Cae | < |Cah |. Therefore, the interface deforms to an oblate (frame c). However, since the volume is not conserved, the extensional isotropic stresses Cie + Cih > 0 (frame d) expand the interface uniformly (frame e). For the second fluid system, again Cae > 0 (frame a), Cah < 0 (frame b), but |Cae | > |Cah |. Therefore, the interface deforms to a prolate (frame c). Here the compressive isotropic stresses Cie + Cih < 0 (frame d) contract the interface uniformly (frame e) to enforce volume conservation. Again, it should be emphasized that this schematic figure depicts the initial stage of the interface deformation, since in the absence of surface tension the interface eventually deforms to a ribbon.

direction. The evolution of the zero-deformation curve (˚ = 0) with the domain confinement was followed in a deformation-circulation map. It was shown that the confinement leads to the reduction in the size of region N of the map, where the drop deforms in the direction normal to the field, and that below a critical domain size bcr = a/cr , determined by setting F = Fcr = −2 in the expression for F (Eq. 19), the necessary condition for elongation in the direction normal to the field will transfer to R > S, which is opposite to that for a drop in an unbounded domain. Appendix A. Coefficients used in the solution of streamfunctions d

5. Conclusions The steady state electrohydrodynamics of a two-dimensional miscible drop under a uniform electric field in confined domains was investigated, in the context of continuous flow electrophoresis and pattern formation in suspension of inhomogeneous colloidal dispersions. The confinement led to the scaling of the electric potential ∞ , the electric field E∞ , and the free electric surface charge qs∞ with a correction factor  , where  > 1 for R > 1 and e 冁 and  < 1 for R < 1. On the other hand, the net electric stresses, 冀 rr ∞ e 冁 , were scaled with  2 . The confinement led to the weakening 冀 r ∞ of the flow field and modified the flow structure. The directions of the interfacial velocity at the top and bottom ( = 0 and ) and/or the sides ( /2 and 3 /2) were used to discern the sense of drop deformation. It was shown that for ur (a, 0) > 0, the drop elongates in the direction of the field, while for ur (a, 0) < 0 it elongates in the normal

d

Defining ˇ=R − 1 and ˛=R − S, the following coefficients are used in the solution of streamfunctions: 6 A = [4˛( ˜ − 1) + ˇ2 (26 − 4 − 1)]

+ ˇ2 (4 − 26 − 1) − 4˛(6 + 1),

(A.1)

B = −2 {[ˇ2 (6 + 2 − 2) + 2˛(6 + 32 − 4)] ˜ + ˇ2 (−6 + 2 − 2) − 2˛(6 − 32 + 4)},

(A.2)

2 C = (− ˜ + 1)[6˛(2 + 1) + ˇ2 (32 + 1)]

− ˇ2 (−34 + 22 − 1) + 6˛(4 + 1),

(A.3)

D = 4 {( ˜ 2 − 1)[12˛ + ˇ2 (2 + 3)] + 12˛(2 − 1) − ˇ2 (4 − 22 + 3)},

(A.4)

A. Behjatian, A. Esmaeeli / Colloids and Surfaces A: Physicochem. Eng. Aspects 441 (2014) 116–126

F = (−2 + 1)[(2˛ ˜ + ˇ2 )(2 − 1)

3

− 2˛(6 − 34 − 32 + 1) − ˇ2 (6 − 34 + 2 + 1)],

(A.5)

冀 rrh 冁∞ ˜ − 1) 2(S − R)( cos 2, = o us /a (1 + )(R ˜ + 1)2 and

冀p冁∞

3

H = (2 − 1)[ˇ ˜ 2 (2 − 1) + 24˛4 2

6

4

o us /a

2

− ˇ ( − 7 + 5 + 1)],

(A.6)

and 4

˘ = 8 + 46 − 64 + 42 + 1 − 2(8 − 1) ˜ + (2 − 1)  ˜ 2.

(A.7)

It should be noted that in the limit of  → 0, the solution in unbounded domain is recovered, and for  → 1 all the ’s are zero; i.e., the flow cease to exist. Appendix B. Summary of the results for hydrodynamics of a miscible liquid drop in an unbounded domain In what follows, we present a summary of Rhodes et al. [7] results for electrohydrodynamics of a liquid column in an unbounded domain. The streamfunctions are i∞

4( ˜ + 1)(R + 1)

2



 r 2

× (2S − R2 − 1)

a

+

 4 

r 1 (R − 1)2 3 a

us a

(B.1)

=



 2

1 2 a (R + 4R + 1 − 6S) 3 r



+ 4S − (R + 1)2

The velocities are uri ∞ 1 =− us 2( ˜ + 1)(R + 1)2



× (2S − R2 − 1)

=



× (2S − R − 1)



+ 4S − (R + 1)

a

r 

+

 r 3 

1 (R − 1)2 3 a



 sin 2.

(B.9)



=



2S − R2 − 1 (R + 1)2

cos 2.

(B.10)

Appendix C. Coefficients used in the solution of streamfunctions For the sake of completeness, as well as validation purposes, we report the hydrodynamic stresses and the pressure:

o E02

=

2 ˘(R + 1)2

(C.1)

[(F + H ) ˜ + (A − B + C − D )] cos 2, (C.2)

cos 2,

(B.3)

=−

a



1 6( ˜ + 1)(R + 1)2

sin 2,

(B.4)

 3

 2

(B.5)

 a 3  r

sin 2. (B.6)

The jumps in tangential and normal (deviatoric part) hydrodynamic stresses, and the pressure are, respectively:

冀 rh 冁∞

=

2(R − S) (R + 1)2

sin 2,

˘(R + 1)2

[ ˜ H − (A + D )] cos 2,

(C.3)

=

2 ˘(R + 1)2

(2S − R2 − 1) cos 2.

(C.4)

h 冁 is opposite in sign but equal in magnitude to Eq. (7), Note that 冀rr as required by the interfacial normal stress balance. In the limit of  → 0, Eqs. (C.1)–(C.4) converge to the corresponding values for an unbounded domain, given in Appendix B.

Supplementary data associated with this article can be found, in the online version, at doi:http://dx.doi.org/10.1016/j.colsurfa. 2013.08.080. References

cos 2,

(R + 4R + 1 − 6S)

2

Appendix D. Supplementary data

1 2 a (R + 4R + 1 − 6S) 3 r

r

=

h 冁 is given by Eq. (B.7). We note that Eq. (C.1) is opposite where 冀 r ∞ in sign but equal in magnitude to Eq. (8), as required by the interfacial tangential stress balance. The jump in the total hydrodynamic h 冁 = 冀 h 冁 − 冀p冁, is normal stress, 冀rr rr

o E02

 3 

2 r + (R − 1)2 3 a

 a 2

and

εo E02

cos 2.

Interestingly, the total normal hydrodynamic stress is independent of the viscosity ratio, and as such, it has been scaled by electric shear scale se = o E02 .

冀rrh 冁

r 

uro∞ 1 =− us 2( ˜ + 1)(R + 1)2

us

o E02

o E02

1 2( ˜ + 1)(R + 1)2 2

uo∞

冀rrh 冁∞

冀p冁

4( ˜ + 1)(R + 1)2

(B.2)

us

˜ + 1)2 (1 + )(R

and 1

×

ui ∞

˜ − 4S − (R + 1)2 (R − 1)2 

Finally, the jump in the total normal hydrodynamic stresses, 冀rrh 冁∞ = 冀 rrh 冁∞ − 冀p冁∞ , is

冀 rrh 冁 sin 2,

and o∞



=

(B.8)

h h 冀 r 冁 =  2 冀 r 冁∞ ,

1

=

125

(B.7)

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