1 Introduction

In recent years,the electroosmotic flow (EOF) in microchannels has been receiving more and more attention of researchers because of its wide applications in many biomedical lab-ona- chip devices,e.g.,sample analyses of DNA,separation of species,and bio-chemical reactions in different physiological processes^{[1, 2, 3, 4, 5, 6]}. In all the above processes,lab-on-a-chip microfluidic devices are very essential to the transport fluid in a desired/controlled fashion. However,when the channel size shrinks,the precise and minute control of the flow by a mechanical pump becomes increasingly difficult to combat with the necessity of microvalves and micropumps. These microcomponents are very prone to the mechanical failures due to the fatigue and fabrication defects. Moreover,any small change in the pressures between the inlet and the outlet may cause an enormous flow change which may not be desired in most of the cases. With this respect,it is found that the electroosmotic flow has several advantages as it does not require any moving part and can act as a noise-free reliable driver in controlling the flow of a fluid in a microchannel.

Burgeen and Nakache^[7] studied the electrokinetic flow in ultrafine capillary slits,assuming that the channel height was of the order of the thickness of the electrical double layer (EDL). Considering the channel to be symmetric and neglecting the convective term in the momentum equation,they analyzed the velocity field and potential for two equally charged ions. Levine et al.^[8] studied a similar problem for both thin and overlapping double layers in the case of a single pair of monovalent ions,assuming an axial symmetry of the flow along a direction normal to the channel walls. Levine et al.^[9] analytically solved the problem of an electrokinetic flow in a fine cylindrical capillary by the Debye-H¨uckel approximation. By making use of a simplified ad hoc model for the distribution of the charge density,they examined the flow for stronger electrolyte solutions. Yang et al.^[10] showed that when an aqueous solution of low ionic concentration come in contact with a solid surface of high zeta potential,significant changes might take place in the flow and heat transfer characteristics of the fluid in rectangular microchannels owing to the presence of the EDL.

Qu and Li^[11] developed a model to determine the electrical potential distribution and ionic concentration distribution in an overlapped EDL field between two flat plates by using the Debye-H¨uckel approximation. They observed significant differences,especially at small separation distances,between their results and those estimated on the basis of the classical theory. Hsu et al.^[12] evaluated the electrokinetic phenomenological coefficients during the flow of an electrolyte solution through an elliptical microchannel. They showed that,for a constant cross-sectional area,both the volumetric flow rate and the streaming potential under the action of electrokinetic forces increased monotonically with a rise in the aspect ratio. Jian et al.^[13] analyzed the flow behavior of time periodic electroosmosis in a cylindrical microannulus, and showed that the wall zeta potential ratio affected the dimension and direction of the EOF velocity profiles within the EDL in the vicinity of the cylindrical walls in microannulus. Sheu et al.^[14] developed a numerical model to study the electroosmotic flow in human meridian. Sadar et al.^[15] carried out an experimental study by using the nano-particles image velocimetry technique in a dilute aqueous solution of sodium tetraborate. On the basis of the experimental observations,they tried to examine the flow characteristics in the EOF case in a rectangular microchannel. Various aspects of electroosmotic flow were studied by different investigators^{[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]} because of the importance of this electrokinetic phenomenon,where ionized liquid flowed with respect to a charged surface in the presence of an external electric field. However,the above studies are limited to simple Newtonian fluids.

The flow behaviors of non-Newtonian fluids are of greater importance since they exist widely in most physiological fluids and fluids used in different industries. The non-Newtonian behaviors of different physiological and industrial fluids have been widely investigated^{[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]}. Some of these investigations explored a variety of important information on the dynamics of blood flow in arteries in normal/pathological states. However,most of these rheological models do not take the microstructural effects into account,and thus are inadequate to simulate the rotatory motion and gyration of fluid microelements,which characterize the suspensions in blood such as erythrocytes and thrombocytes. Among these non-Newtonian fluid models,the theory of micropolar fluids developed by Eringen^{[44, 45, 46, 47]} is a frontier one during the last few decades. The techno-scientific importance of a micropolar fluid lies in the fact that by means of its six degrees of freedom,the behaviors of microscopic articulate elements,platelets,and cells and other microparticles in blood can be explored. Siddiqui and Lakhtakia^{[47, 48]} formulated and solved a boundary value problem concerning the one-dimensional symmetric electroosmotic flow of a micropolar fluid.

With the Debye-H¨uckel approximation,Rice and Whitehead^[49] studied the effect of the electrokinetic radius on the behavior of electrokinetic phenomena in a narrow cylindrical capillary. Sorensen and Koefoed^[50] and Mala et al.^[51] conducted various theoretical studies on the electrokinetic phenomena for capillaries by using the Debye-H¨uckel condition. Mala et al.^[52] studied the effects of an EDL on the flow characteristics both experimentally and theoretically. The theoretical analysis was carried out by using the Debye-H¨uckel approximation,and the obtained predicted volumetric flow rates agreed well with their experimental data. Siddiqui and Lakhtakia^[47] put forward the numerical solution of a boundary value problem,concerning the steady,symmetric,and one-dimensional electroosmotic flow of a micropolar fluid in a uniform rectangular microchannel under the action of an external electric field,and solved the problem analytically under the purview of the Debye-H¨uckel approximation. They reported that the fluid speed estimated on the basis of the numerical method was in good agreement with that obtained by using the analytical method along with the Debye-H¨uckel approximation, and the greatest difference was less than 2.685 8 percent. They further observed that near the walls of the microchannel,the results obtained by the two approaches were coincident. Therefore, it turns out that although it is usually held that the Debye-H¨uckel approximation is valid for low surface electric potential,in some situations,the use of this approximation yields fairly satisfactory results for some flow variables,including the fluid speed during electroosmotic flow.

Chen amd Santigo^[53] remarked that among the three different alternatives,i.e.,electrohydrodynamic, magnetohydrodynamic,and electrokinetic pumping mechanisms,only the electrokinetic phenomenon is applicable to the fluids with a wide range of conductivity. Keeping this in view,we study the electrokinetic phenomenon that is primarily motivated to the study of the dynamics of biological molecules in this paper. In different investigations pertaining to the dynamics of microelements contained in physiological fluids in some normal/pathological conditions and also in various medical treatment methods,e.g.,the dialysis of blood,the flows in porous channels are found to have significant applications. Therefore,the present study is devoted to an investigation of the flow behavior of a micropolar fluid through a porous channel under the influence of an external electric field of alternating nature. For a more realistic consideration, the channel flow is considered to take place between two parallel plates,where both the plates are oscillating with a constant velocity in their own plane. The pressure gradient and the velocity are taken to be periodic functions of time. The advantage of considering an alternating current (AC) electric field is that it can combat the changing needs of different voltages during different studies. The instantaneous value and the amplitude of the alternating field can be changed simply by a step down or step up transformer. However,this is not possible in the case of a direct electric field. The frequency of the time-periodic electric field can also be changedby using a frequency converter if it is required. Moreover,AC electroosmosis takes care of the simultaneous changes of polarities in the charge and field directions. This gives rise to steady migratory motion of ions. Secondly,AC electroosmosis possesses the inherent advantage that it can modify the surface flow by varying the amplitude and frequency of the AC signals.

A mathematical model is developed here with the purpose of investigating the effect of different parameters that control the dynamics of electroosmotic flow under the assumptions stated in the sequel. The model is analyzed theoretically by employing suitable analytical techniques. In the present study,the ionized motion of oscillation is specially concerned under the influence of an alternating electric field. The solution is achieved by developing a perturbation scheme. Moreover,the Debye-H¨uckel approximation is taken into consideration to solve the problem. The derived expressions are computed by considering a specific situation. The computational results for different physical quantities of interest are presented graphically. The study clearly shows that the micropolar effect gets intensified with an increase in the zeta potential. It is also revealed from the study that for a given value of dynamic viscosity,an increase in the microrotation viscosity reduces the amplitude of microrotation. The present study along with the results presented here is immensely useful to simulate related experimental results and to validate the results of more complex numerical models that need to be developed to deal with more realistic problems associated with electroosmotic flow. Above all,the study gives some important insight into the characteristic features of electroosmosis that takes place in various microfluidic devices,where the rotation of the microparticles suspended in the fluid is quite significant.

2 Formulation of problem

Let us use the Cartesian coordinate system described in Fig. 1. The unsteady laminar flow of a viscous incompressible micropolar fluid through a microchannel between two oscillatory porous plates located at y = ±h will be studied here under the influence of an alternating electric field. The fluid considered in this study is supposed to be ionized.

At one plate,the fluid is considered to be injected with the constant velocity V ,while at the opposite plate,it is sucked off with the same velocity. Suppose that the fluid flow takes place only along the axis of the channel with a velocity u expressed as u = u(y,t). Then,the continuity equation reduces to the form @u @x = 0. Since the flow is symmetric about the axis y = 0,we may confine our study to the region 0 6 y 6 h only. With these considerations, our specific concern is to explore various important characteristics of the electroosmotic flow, where the system is subject to an alternating electric field.

Under the purview of the boundary layer approximations,the equations governing the timeperiodic electroosmotic flow of a micropolar fluid are given by

The definitions of the symbols appearing in these equations are available in Section Nomenclature. Equations (1) and (2) are derived by following the microcontinuum field theories of Eringen^[46] and using the laws of conservation of linear and angular momentums,respectively.

The charge density ρe and the electric potential ψ are related by Gauss’s law of charge distribution as follows:

where ǫ is the dielectric constant.

3 Analysis

Since the electric field is considered to vary sinusoidally with time as depicted in Eq. (1), all other time-varying quantities must also be sinusoidally dependent because Eqs. (1)−(3) are linear in time. Taking account of the fact that the electroosmotic flows occurring in physiological processes usually exhibit oscillatory characteristics,for further consideration,we write

where N represents the microrotation which is a measure of the spinning speed of the rigid/semirigid particles in the fluid about their respective centroid,ω is the angular velocity,and t denotes the time variable.

In Eq. (2),we take

as the distribution of the net electric charge density in equilibrium near a charged surface in a fully developed flow. The EDLs are considered to be so thin that there is no interference between the two walls.

The spin gradient viscosity γs is considered as the form

where K = k μ denotes the the material/micropolar parameter. The expression of this spin gradient viscosity is quite appropriate for the purpose of predicting the true behavior of the micropolar fluid flow in the particular situation when the microrotation transforms to an angular velocity^[54].

In the sequel,we shall make use of a set of non-dimensional variables defined by

We take j = h2 as a microinertia parameter per unit mass. The quantity S0 is the ratio between the suction velocity V and the Helmholtz-Smoluchowski electroosmotic velocity UHS which is defined by where M gives the measure of the mobility of ions in the ionized state of the fluid,ζ is the zeta potential,and μ = ρν is the dynamic viscosity.

In terms of the above-written dimensionless variables,Eqs. (1)-(3) may be rewritten as

Since the present study is conducted under the purview of the kinetic theory,the electric potential energy is considered to be small compared with the thermal energy of the ions so that^[55]

Then,Eq. (10) reduces to in which h_λ is the Debye-H¨uckel parameter,which is a non-dimensional quantity and is defined by

where λ is the thickness of the Debye layer.

Solving Eq. (11) with the boundary conditions

we obtain The appropriate boundary conditions for the flow problem in the present context may be put mathematically as follows: where β (β 2 [−1,0]) is a constant. For a study on electroosmosis,such as the present one, where EDLs are presented as pointed out in Refs. [56−58],negative β is a viable consideration. Henceforward,we shall drop the superscript “*” in the dimensionless variables. Substituting Eqs. (12) and (5) into Eqs. (8) and (9),we have the following set of equations: which will be used in analyzing the problem further. Since the Reynolds number Re for the flows of physiological fluids in a microchannel is very small,we develop a perturbation method of solution by writing the expressions for us and Ns as follows: The boundary conditions for the velocity and microrotation can then be put as follows: Substituting the expansions of us and Ns given in Eqs. (17) and (18) into Eqs. (15) and (16), equating like powers of Re,and ignoring quadratic and higher powers of Re,we obtain the following system of ordinary differential equations: Integrating Eq. (21) and applying Eqs. (19) and (20),we obtain Substituting Eq. (25) in Eq. (22) and integrating the resulting equation,we have where

Now,substituting the expression of Ns0 given by Eq. (25) and then solving it,we have the following expression for us0: with

Substituting Eqs. (25) and (27) into Eq. (23) and integrating the resulting equation with respect to y,we have In a similar manner,substituting Eqs. (26) and (28) into Eq. (24) and then solving the resulting second-order differential equation,we obtain the solution for Ns1 as follows: The derived expressions for C3,C4,C5,C6,I1,and I2 are presented in Appendix A.

Introducing Eq. (29) into Eq. (28),we find

Substituting the expressions for us0,us1,and Ns0 given,respectively,by Eqs. (27),(30),(26), and (29) into Eqs. (17) and (18),we get the real parts of u and N as follows: The derived expressions of different symbols used in the above equations are all presented in Appendix A.

Equations (31) and (32) constitute the solution of the present problem that is concerned with the time-periodic electroosmotic flow of a micropolar fluid under the purview of the Debye- H¨uckel approximation.

4 Numerical estimates of blood flow in microfluidic devices

Motivated by the growing interest in the phenomenon of electroosmosis as a non-moving component method to control the dynamics of blood flow in microfluidic devices,we carry out extensive computational work based on the analytical expressions derived in Section 3. Duringthe calculation,the following data are used for the values of the parameters involved in the solution of the problem:

which are in agreement with the data available for blood flow dynamics in microchannels.

The concepts of conservation of the angular and linear momentum equations (used in Section 2) help us to provide a better picture for the microrotation of the microparticle and velocity distribution of the fluid. To study the fluid flow characteristics,the non-dimensional velocity u and the microrotation N are computed by Eqs. (13) and (14).

Figure 2 illustrates the microrotation N for different values of the micropolar parameter K for a phase value of τ = 0 (steady case). The results obtained on the basis of the present study for the steady state are in good agreement with those reported by Siddiqui and Lakhtakia^[47]. It is observed that the microrotation N decreases with an increase in K. Fig. 3 and Fig. 4 show that the velocity u and the microrotation N increase as the values of the Debye-H¨uckel parameter h_λ increase. Since an increase in the height of the channel gives rise to an increase in h_λ,the results imply that the velocity of the fluid increases as the height of the channel increases. In the case of a simple Newtonian fluid,similar observation has also been reported by some previous researchers.

Fig. 5 and Fig. 6 give the variations in the velocity and microrotation with the phase τ = ωt for different values of the micropolar parameter K. It may be noted that in the case of velocity, the amplitude of oscillation increases with an increase in the micropolar effects,while a reverse trend is observed in the case of microrotation. It may be noted from Fig. 7 and Fig. 8 that for both u and N,with an increase in h_λ,the amplitude of oscillation enhances. However,the increase rate in the case of microrotation is much bigger than that in the case of velocity. Fig. 9 and Fig. 10 illustrate the time-periodic variation of the velocity and the microrotation with different β. The parameter β is a constant related to the occurrence of electroosmosis,and it is regarded as the boundary parameter. In the presence of EDLs,as in the present case,β 6 0,which has already been mentioned in Section 3. Since the micropolar parameter K is the ratio between the microrotation viscosity k and the dynamic viscosity μ,for a fixed value of the microrotation viscosity,K increases when the dynamic viscosity drops down. This implies that the velocity of the fluid increases when the viscous force reduces. This is in agreement with the usual notion. In a similar manner,it can be argued that a rise in the microrotation viscosity brings about a reduction in the microrotation of the microparticles,as long as the dynamic viscosity μ does not alter its value.

The variations of the velocity and microrotation with the parameter S0 defined in Section 3 are presented in Fig. 11 and Fig. 12,respectively. An enlarged view of a part of the velocity profile,which is of specific interest,is given in the onset in Fig. 11. It is observed that with an increase in the suction velocity,the amplitude of the velocity decreases. However,in the case of microrotation,as the suction velocity increases,the amplitude initially decreases with an increase in the suction velocity up to a certain value,and thereafter a phase shift of 180◦ occurs,causing thereby an increase in the amplitude.

Fig. 13 and Fig. 14 display the variations in the velocity and microrotation with different Re. It is seen that although the amplitude of the velocity increases with the increase in Re, as Re increases,the amplitude of oscillation of microrotation initially decreases,and then an occurrence of 180◦ phase shift occurs when the value of Re exceeds a certain critical value.

Fig. 15 and Fig. 16 depict the variations of the velocity and microrotation for different values of the micropolar parameter K. It is observed that the velocity increases with an increase in the micropolar effect. However,for the microrotation,a reversal trend is found. When the micropolar effect is small,a reversal phenomenon may be found if the value of the Reynolds number is in a particular range. However,in the case of microrotation,the reversal rotation direction of the microparticles suspended in the fluid may take place when the value of K is in a wide range while the value of Re exceeds a certain critical value. Figure 16 illustrates the reversal points with much accuracy.

5 Concluding remarks

The present paper describes the ionized motion of oscillation of a micropolar fluid between two oscillatory porous plates under the influence of an alternating electric field. The solution holds for low Reynolds numbers. The flow dynamics is determined by the Debye-H¨uckel approximation,which is valid when the non-dimensional Debye-H¨uckel parameter h_λ takes a value between 10 and 1 000. Significant variation is observed in the amplitude of oscillation caused due to the microrotation when the values of the micropolar and boundary parameters are changed. As expected,the study confirms that any change in the values of the associated parameters affects the dynamics of microparticles significantly. In some cases,the changes in the parametric values bring about gross changes in the average flow dynamics of the fluid. Moreover, the change rates of the fluid velocity and microrotation are much larger in the vicinity of the walls than those towards the central region. This is presumably owing to an increase in the zeta potential originating from the formation of the EDL. The reported results are well suited for biological molecules,and can be used for biomedical lab-on-a-chip microfluidic devices.

To sum up,we remark that although the consideration of the Debye-H¨uckel approximation is usually valid when the surface electric (zeta) potential is low,in view of the discussion made in Section 1,even for moderately high electric potential,the results presented here are expected to serve as reasonably good estimates of the flow variables in many situations under the purview of the present study.

Acknowledgements The authors are highly thankful to the esteemed reviewers (anonymous) for their comments. The corresponding author,Professor J. C. MISRA,wishes to express his deep sense of gratitude to Professor (Dr.) M. R. NAYAK,President of Siksha O Anusandhan University for providing a congenial research environment.