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1. Introduction
The concept of integrating biochemical analysis with microelectromechanical systems (MEMS) is involved in the new field of bioMEMS, which is undergoing tremendous growth in a multitude of applications [1], including technology that enable scientific discovery, detection, diagnostics, and therapy and span the fields of biology, chemistry, and medicine [2]. In this context, the terms lab on a chip (LOC) and micro total analysis system (µTAS) are a subset of MEMS dedicated to chemical and biological analyses and discoveries [2]; here, the study of fluid flow in these submillimeter-sized systems is cover by microfluidics and nanofluidics [3].
In the mentioned systems can be applied different strategies to pump fluids as electrohydrodynamic and magnetohydrodynamic effects [4,5], electrothermal effects [6], acoustic and ultrasonic effects [6, 7], pressure-driven effects by syringe, peristaltic or rotary pumps [8], among others [6]. However, a common method for pumping fluids is the use of the electrokinetic phenomenon called electroosmosis, whose basic principle is the movement of an electrolytic solution relative to a stationary charged surface when an electric field is applied [9].
Focusing on electroosmotic flows, these have been studied extensively by the scientific community since many years, carrying investigations about the transport of homogeneous single-phase fluids based in electrolytic solutions both steady-state and transient-state, and using conduits formed with cylindrical shape [10-12], annular channels [13-15], parallel flat plates [16-18] and rectangular channels [19-21]; all of them, under the consideration of the Debye-Hückel approximation for the volumetric free charge density within the electric double layer.
The research about electroosmotic flows, have been extended to the study of the pumping of two parallel and immiscible fluids as a technique of transport by viscous drag. In this direction, Liu et al. [22] refer that non-conducting liquids and certain biofluids cannot be pumped directly by electroosmosis, but still can be dragged along by shear forces of a neighboring conducting liquid which is driven by electroosmosis. Their study reports an analytical solution for the velocity profiles into a two-phase electroosmotic-viscous pump in a circular microchannel. About this theme, other investigations under electroosmotic effects to pumping a nonconducting fluid by viscous drag due to electroosmotic effects over a conducting liquid in cylindrical channels can be reviewed in the works conducted by Movahed et al. [23], Moghadam [24] and Matías et al. [25]; moreover, in parallel flat plates by Huang et al. [26] and Afonso et al. [27], and rectangular channels by Gao et al. [28]; all of them in steadystate. Concerning the transient-state analysis of pumping of non-conducting liquids by viscous drag induced by electroosmotic effects, we can cite the investigation realized by Gao et al. [29] in a rectangular channel.
All the studies cited in the above paragraph consider that one of the fluids in the two-layer arrangement is electrically conducting and the other not, which yields partially or null employment of the Maxwell electric stresses at the liquidliquid interfaces between fluid layers. However, following the study of the electrochemistry, if two solutions of immiscible electrolytes are in contact and develop a polarizable and impermeable interface to charged particles, there arises, in equilibrium, an electrical potential difference, whose structure trough diffuse layers and a compact inner layer in an electrical double layer, has been studied by Volkov et al. [30], Wandlowsky et al. [31], Samec et al. [32] and Cui et al. [33]. In the mentioned works, the electrical potential difference is determined by the distribution of charged and dipolar particles near the interface. However, Masliyha and Bhattacharjee [9], propose a simpler contact structure between the immiscible media, where the thin compact layers or the presence of electrical dipoles are not considered; therefore, the potential difference can be considered as a constant and when this constant is zero the electrical potential continuity is reached.
Another important aspect that emerges in the analysis of the structure of the electric double layer in polarizable, and non-polarizable interfaces, is the study of capacitance, which is a measure of the penetration of ions in the electric double layer at the interface. From the capacitance and the potential difference, the surface charge density at the interface can be determined [32, 33]; therefore, the potential difference and surface charge density have a direct correspondence when two immiscible electrolyte solutions are in contact through an interface.
Considering the previous paragraph about certain concepts involved within the double electric layer generated at the interface between two immiscible electrolyte solutions, the studies of Choi et al. [34], Su et al. [35] and Jian et al. [36] carry out the analysis of the transient electroosmotic flow of two-immiscible fluid layers in slit channels formed by parallel flat plates. Here, both fluids are electrolytes, and this condition leads to the Maxwell electric and shear stresses be included in each fluid phase when the total stresses balance is established at the liquid-liquid interface. The flow field experiments important changes in their velocity profiles when the electric stresses are present at the interface concerning to the case when only one fluid is conductive, due to the presence of a potential difference and a surface charge density in the interfacial boundary conditions. Extending the mentioned works, Shit et al. [37] realized a study in steady-state about the two-layer fluid flow and heat transfer in a hydrophobic microchannel formed by parallel flat plates using a combination of a pressure gradient and electroosmotic forces. Their results indicate that the effects of the interfacial zeta potential are significant on the fluid velocity; moreover, the electroosmotic flow has a finite jump the interface between two fluid layers.
Certainly, the development of techniques to pumping immiscible fluids by electroosmotic effects in miniaturized systems has reached the handling of three layers of fluids [38] and also any number of fluid layers [39], when the flowfocusing effect is required. Therefore, the aim of the present work is addressing the study about the transport of multilayer immiscible fluids in a cylindrical capillary under pressure and electroosmotic effects. The parametric study on the flow field is focused on the interfacial phenomena via a potential difference, the Gauss’s law, and Maxwell stress at each liquid-liquid interface between fluid layers. Because the interface conditions established here for the star-up of a multilayer flow within a capillary have not been treated yet, this paper also has the purpose of extending the theoretical knowledge about the application of the electrokinetic phenomena in fluids transport.
2. Mathematical formulation
2.1. Physical model
In the present work, we realize the transient analysis of the transport of multi-layer immiscible fluids in a narrow capillary. Therefore, to establish the fluid phenomenon that will be studied here, we start describing the physical model with a cylindrical coordinate system (r,z), as is shown in Fig. 1 on the centerline of a capillary. The fluids, which fill the conduit with radius R, are layers of symmetrical electrolytes consider with a Newtonian behavior. Each liquid-liquid interface is placed in an r n position; here, the subscript n = 1,2,3,...,i represents the number of the fluid layer, and i is the fluid layer in contact with the wall of the capillary. Because the fluids are immiscible and electrically conductive, and also the interfaces between them are polarizable and impermeable to charged particles, a surface electric charge density q s and a potential difference ∆ψ appears at the liquid-liquid interfaces; moreover, the wall of the capillary is also polarizable and acquire a surface electric charge represented by the zeta potential ζ w . The fluids movement is due to two factors; firstly, to the ends of the conduit, and is subject to an electric potential generated by a pair of electrodes, that gives rise to a uniform electric field E z inducing electroosmotic effects. Secondly, by a pressure gradient p z , that also could be applied along the z−direction.
2.2 General governing equations
The flow field of the multi-layer immiscible fluids is governed by the Poisson-Boltzmann equation for the electric potential distribution
where Φ is the total electric potential, ρ e is the volumetric free charge density and (( is the dielectric permittivity. Also, with the continuity equation for incompressible fluids as
being v the velocity vector. And the momentum governing equation
where ρ is the density, t is the time, p is the pressure, µ is the viscosity, g is the gravitational acceleration, and E is the electric field vector.
2.3 Simplified mathematical model
The mathematical model to solve this multi-layer flow based on the previous general governing equations can be simplified, taking into account the following assumptions:
Incompressible fluids.
The fluid’s properties are independent of the local electric field, ion concentration, and temperature [22,40]. Complementary to this consideration, we assume a net change in the temperature of the fluids is less to 10 K [41, 42] and the value of the external electric field is less to 10 kVm−1 to despise any Joule heating effect [43].
The interfaces between the fluids represent either sharp boundaries with zero-thickness, impermeable, and ideally polarizable.
There is a planar interface between fluids layers [34, 44]. The mentioned can be assumed by considering that we have: i) laminar flow for low Reynold’s numbers, being
The gravitational forces in the system are negligible [44].
The capillary is sufficiently long, and the analysis is a focus in a region far from the ends of the capillary neglecting inlet and outlet effects.
For a long capillary, the total electric potential Φ at any location in the system is given by a linear superposition of the potential in the electric double layer and by the externally applied potential as follows [9,46,47]
Here, ψ is the electric potential distribution within the electric double layer,
-
The ionic distribution into the electric double layers follow the Boltzmann distribution as
where
The Debye-Hückel approximation for low enough potentials (≤ 25 mV) at the solid-liquid [9,47] and liquidliquid interfaces [35,36] is considered.
Creeping flow (low Reynolds number) and constant pressure gradient on the z-direction.
The Debye lengths into electric double layers do not overlap.
According to the previous considerations, the set of Eqs. (1)-(3) can be rewritten in the following form, yielding the Poisson-Boltzmann and the momentum equations respectively as
and
where r is the radial coordinate, v
z
and p
z
= ∂p/∂z are the velocity and the pressure gradient on the z− direction, respectively; and
2.4. Initial and boundary conditions
To solve the governing equations given in Eqs. (6) and (7), we consider the following boundary conditions for the electric potential and velocity. At the centerline of the capillary at r = 0 for layer n = 1, we have the symmetry boundary conditions as
In the case of each liquid-liquid interface at r = r n=1,2,3,...,i−1 , we have the following four boundary conditions; we firstly consider a potential difference as
and the Gauss’s law for the electric potential respectively as
And secondly, for each liquid-liquid interface, we consider a velocity continuity
and a total stresses balance that includes Maxwell stresses and viscous shear stresses, also called electro-viscous stresses as
Additionally, at the solid-liquid interface between the wall of the capillary and fluid layer n = i at r = R, are established a specific zeta potential value as
and the no-slip boundary condition respectively as
Finally, the initial condition at t = 0 to solve the momentum Eq. (7) is
2.5 Dimensionless mathematical model
The mathematical model given in Secs. 2.3. and 2.4., is normalized with the following dimensionless variables
where t c = ρ ref R 2 /µ ref and ψ c = ζ w , are the characteristic magnitude of the time and electric potential, respectively. Therefore, by replacing Eq. (16) in Eqs. (6)-(15) we have the dimensionless version of the Poisson-Boltzmann equation
momentum equation
together with the boundary conditions at the centerline of the capillary at
in each liquid-liquid interface at
and
in the solid-liquid interface at
and the initial condition for the flow field at
The dimensionless parameters that appear in previous equations are defined as follows
Where
3. Solution methodology
3.1. Electric potential distribution
The Poisson-Boltzmann equation expressed by Eq. (17) has the form of a modified Bessel differential equation, and it has a well-known form solution as
where C
2n−1
and C
2
n are constant coefficients, and I
0 and K
0 are the modified Bessel functions of first and second kind, respectively, and both of zero order. The Eq. (27) describes the behavior of dimensionless electric potential and has the same form for any value of n-layers of fluid. In this context, we have two constants C per fluid layer. To know the constants C
2n−1
and C
2n
, we have to apply the appropriate boundary conditions. Firstly, by replacing the corresponding term of Eq. (19) into Eq. (27) for the innermost layer with n = 1, we deduce that the value of C
2 = 0. Secondly, and with the aid of Eq. (27), we apply the boundary conditions for the electric potential given by Eqs. (20), (21) and (24), for each liquid-liquid interface located from
which is a set of linear algebraic equations that contains the same number of variables as equations. The constants C are solved by the matrix inverse method [48].
3.2. Transient velocity field
To solve the velocity field, we employ the Laplace transform as
Therefore, by taking the Laplace transform of the momentum equation given by Eq. (18) together with the initial condition from Eq. (25), we have
and with their corresponding boundary conditions from Eqs. (19) and (22)-(24), we have respectively at
at
and
and at
By replacing the expression for dimensionless electric potential distribution given in Eq. (27) into Eq. (30), yields
Equation (35) is a non-homogenous ordinary differential equation of undetermined coefficients, and its solution can be expressed by the sum of a general solution corresponding to the homogeneous equation and a special solution as
The homogeneous and special solution of the Eq. (35) are defined as follows
and respectively
where A
n
, B
n
, F
n
and G
n
, are constants to be determined, and
moreover, from Eq. (39) are obtained the following problems
and
By replacing Eqs. (40) and (41) into Eq. (39), we can find the following relationships for the modified Bessel functions as
and
from which the constants F n and G n are obtained as
Therefore, from Eq. (36) the dimensionless velocity distribution can be written as
To find the constants A
n
and B
n
, we apply the boundary conditions given in Eqs. (31)-(34) to Eq. (45). In this direction, firstly, by replace Eq. (31) at the centerline of the capillary in
where from is deduced that B
1 = 0. Secondly, from the boundary conditions given in Eqs. (32) and (33) for each liquid-liquid interface at
and respectively
and thirdly, from Eq. (34) for the solid-liquid interface placed in the wall of the capillary at
Once all boundary conditions for velocity have been replaced, we construct from Eqs. (47)-(49) the array of the equation system to solve the constants A n and B n , yielding as follows
where
The Eq. (50) is the set of equations that include the term “s” now called symbolic variable, because when the matrix inverse method be applied to this equation, the coefficients A n and B n are solved in terms of this.
Finally, the given constants Fn and Gn in Eq. (44), and the constants A
n
and B
n
derived from Eq. (50), are replaced into Eq. (45), where the inverse Laplace transform is applied to solve the velocity profile of combined electroosmotic and pressure-driven flow analyzed here. To this, the velocity distribution along the capillary radius
where the approximate function
where j are the digits desired, 1 ≤ k ≤ 2M, and M = 14 for all the cases.
3.3. Steady-state velocity
To obtain the steady-state solution for the flow field, Eq. (18) is rewritten as follows
which is integrated twice yielding
where D
n
and E
n
are constants which are determined with the application of the appropriate boundary conditions for velocity. Hence, firstly by applying the symmetry boundary condition from Eq. (19) into Eq. (55), for the innermost fluid layer with n = 1, we deduce that D
1 = 0. Secondly, by using the solution expressions for the electric potential and velocity given by Eqs. (27) and (55), respectively, we apply the boundary condition given by Eq. (23) to each liquid-liquid interface from
for n = 1 to n = i − 1. Finally, to close the general problem for the velocity, we apply the last boundary condition given by Eq. (24), which corresponds to the no-slip boundary condition at the outermost layer for n = i. Here, the last constant E n=i is found, yielding
4. Results and discussion
The dimensionless parameters in the present work have been obtained by a suitable combination of the following parameters ranging of: 0.1≤ R ≤ 10 µm,
4.1. Validation
The validation of the transient solution for velocity from the present work was compared against the investigation done by Keh and Tseng [11] on a fine capillary. To this, are used the following dimensionless parameters Γ = 0,
FIGURE 2 Comparison of the dimensionless velocity profiles in a purely electroosmotic flow for different times between the results presented by Keh and Tseng [11] with n =1, and the present work with
4.2. Parametric study
Figure 3 shows the dimensionless electric potential distributions as a function of the dimensionless radial coordinate of three layers of immiscible fluids within a narrow capillary with interfaces placed in
FIGURE 3 Dimensionless electric potential distributions for three immiscible fluids in a capillary with n =3,
Conversely, negative values of Q
s,n
= −10, yield negative electric potential distributions around each liquid-liquid interface due to the excess of counterions into the electric double layers; while the electric potential discontinuity is maintained via the value of
Figures 4 presents the transient evolution of the dimensionless velocity profiles as a function of the dimensionless radius of a purely electroosmotic flow with Γ = 0, pumping three layers of immiscible fluids. With the aid of the previous results in the electric potential distribution given in Fig. 3, the velocity profiles in Fig. 4(a) corresponds to the condition with Q
s,n
= 10 and
FIGURE 4 Dimensionless velocity profiles of a purely electroosmotic flow for different times with n = 3,
Figure 5 shows the development in the time of dimensionless velocity profiles as a function of the dimensionless radial coordinate of a purely electroosmotic flow of three immiscible fluids, under the influence of the dimensionless dielectric permittivity ratios
FIGURE 5 Dimensionless velocity profiles of a purely electroosmotic flow with n = 3,
In Fig. 7 are presented the dimensionless velocity profile of three immiscible fluids as a function of the radial coordinate and time and under different combinations of viscosity ratios
FIGURE 7 Dimensionless velocity profiles of a purely electroosmotic flow with n = 3,
The effect of an external and constant pressure gradient on the flow field via dimensionless parameter Γ, is shown in Fig. 8; therefore, this graph represents the transient evolution of combined electroosmotic-pressure driven flow. The values of Γ = −2 and Γ = 2 show the influence of pressure forces in favor and contrary to the positive z-direction, respectively; the aforementioned can be demonstrated if the velocity profiles under pressure effects are compared in this graph with the case when Γ = 0, for a purely electroosmotic flow. Also, the invariant electric potential at the time is presented in Figs. 8(a)-(c) as dashed lines for Q
s,n
= 15 and
FIGURE 8 Dimensionless velocity profiles (solid lines) and electric potential distribution (dashed lines) of a combined electroosmotic-pressure driven flow with n = 3,
In Fig. 9, the characteristics of a purely electroosmotic flow under the influence of the electrokinetic parameter
FIGURE 9 (a) Dimensionless electric potential distributions and (b) velocity profiles of a purely electroosmotic flow for four different times with n = 3,
In Fig. 10, the influence of the dimensionless densities ratios on the velocity profiles during the start-up of purely electroosmotic flow is given. Here, the magnitude of the velocity profile is defined by the heaviness of each layer of fluid, being the lightest fluids that move faster in the flow or vice versa. Therefore, the magnitude of the velocity profile in the transient period of the flow depends on the density ratio value in each layer and their position in the arrangement of the multi-layer flow. It is very clear that when the multilayer electroosmotic flow reaches the steady-state regime in
FIGURE 10 Dimensionless velocity profiles of a purely electroosmotic flow with n =3,
In Fig. 11, we show the electroosmotic flow of four layers of immiscible fluids with different thicknesses, and a wide combination of all dimensionless parameters studied here. Therefore, we can found interesting combined behaviors on the transient evolution of the velocity profiles, from the early time
FIGURE 11 Dimensionless velocity profiles of a purely electroosmotic flow with n =4,
4.3. Tracking of the velocity
Figure 13 shows the tracking results of the dimensionless velocity as a function of the dimensionless time, evaluated at the centerline of the capillary. This results are taken from the flows presented in Sec. 4.2. For all cases, we can see a gradual increase in the velocity as the time progresses since the rest to reach the steady-state. It is clear from Figs. 13(a), (b), (d)-(f), that the time to reach the steady-state of the fluid flows is independent of the dimensionless parameters Q
s,n
,
FIGURE 13 Tracking of the velocity in the multi-layer flow as a function of the dimensionless time evaluated at the centerline of the capillary. (a) effect of Q
s,n
and
5. Conclusions
In the present work, we realize a semi-analytical solution of the start-up of combined electroosmotic and pressure driven flow of multi-layer immiscible fluids within a narrow capillary. The parametric study is based on the different fluid properties, geometrical characteristics, and boundary conditions in the solid-liquid and liquid-liquid interfaces. Considering the studied flow conditions was demonstrated that the presence of electric double layers at liquid-liquid interfaces break the continuity of the electric potential distribution and the shear viscous stresses, producing representative changes of the velocity distributions, which could be in favor or against of the flow. In other results, it was determined that the physical and electric properties of the outermost layer of the multi-layer flow, make it govern the global magnitude of the velocity distribution over the cross-section of the capillary. On the other hand, the time to reach the steady-state regime of the fluid flows is strongly controlled by the viscosity ratios, and it is independent of the other dimensionless parameters presented here. Therefore, this investigation is an important theoretical contribution to simulate transient multi-layer fluid flows under electric interfacial effects, covering different implications that emerge in the design of small devices into the chemical, biological and clinical areas.
Finally, several implications can to be analyzed. For example, it is recommended to address the following issues to extend the present work: the analysis can include non-Newtonian fluids, treat the liquid-liquid interfaces as perturbed lines or with shape defects, and also, the interfaces can be treated as transitional layers with non-zero thickness, where two phases partly dissolve in each other, and the properties of the medium gradually change from the properties of one phase to the other.
