## Abstract

We have formulated and solved the boundary-value problem of steady, symmetric and one-dimensional electro-osmotic flow of a micropolar fluid in a uniform rectangular microchannel, under the action of a uniform applied electric field. The Helmholtz–Smoluchowski equation and velocity for micropolar fluids have also been formulated. Numerical solutions turn out to be virtually identical to the analytic solutions obtained after using the Debye–Hückel approximation, when the microchannel height exceeds the Debye length, provided that the zeta potential is sufficiently small in magnitude. For a fixed Debye length, the mid-channel fluid speed is linearly proportional to the microchannel height when the fluid is micropolar, but not when the fluid is simple Newtonian. The stress and the microrotation are dominant at and in the vicinity of the microchannel walls, regardless of the microchannel height. The mid-channel couple stress decreases, but the couple stress at the walls intensifies, as the microchannel height increases and the flow tends towards turbulence.

## 1. Introduction

When an electrolytic liquid comes into contact with a solid, an electric double layer is formed in the interfacial region. The electric double layer comprises a layer of charges of one polarity on the solid side and a layer of charges of the opposite polarity on the liquid side of the solid–liquid interface. At the same time, free ions in the liquid assemble into a diffuse region beyond the interfacial charge layer. Two electric double layers are formed when a liquid is contained in a channel with two parallel solid walls. When an electric field is applied, the free ions in the liquid experience a force that causes bulk motion of the liquid. This type of flow is classified as electro-osmotic.

Electro-osmosis in microchannels has attracted great interest recently, owing to pharmaceutical, chemical, environmental and defence applications, among others. An industrial example is furnished by reverse-osmosis membranes for desalination of water (Murugan *et al*. 2006). Two other industrial examples are the injection of detoxifying agents and the control of leakage at toxic-waste sites (Probstein 1989, p. 191; Keane 2003). Moreover, the study of electro-osmotic flows is useful to: (i) design channels for fluid transport in biological and chemical instruments (Fluri *et al*. 1996), (ii) design micropumps, microturbines and micromachines (Harrison *et al*. 1991; Fan & Harrison 1994; DeCourtye *et al*. 1998), and (iii) design microchannels in micro/nanoscale chips for the analysis of DNA sequences and for drug delivery (Arangoa *et al*. 1999).

Reuss introduced the concept of electro-osmosis in 1809, after performing an experiment that proved that water can be forced to move through wet clayey soil if an external electric field is applied (Probstein 1989, p. 191). About half a century later, Wiedemann deduced from experiments that electro-osmotic flow is proportional to the applied current. Helmholtz developed the electric-double-layer theory in 1879 to analytically relate electrical and flow parameters. In 1903, Von Smoluchowski measured the electric-double-layer thickness to be much smaller than the height of the channel and derived a slip boundary condition at the walls of the channel. Twenty years later, Debye and Hückel calculated the distribution of ions in a low-ionic-energy solution by using the Boltzmann distribution for the ionic energy (Burgreen & Nakache 1964).

Thereafter, although outstanding contributions have been made to the field of electro-osmosis, theoretically (e.g. Burgreen & Nakache 1964; Hu *et al*. 1999; Yang *et al*. 2001; Chai & Shi 2007) as well as experimentally (e.g. Herr *et al*. 2000; Pikal 2001; Horiuchi *et al*. 2007; Kim *et al*. 2008), attention has been chiefly confined to simple Newtonian fluids. However, many technoscientifically significant liquids are not simple Newtonian fluids; instead, they are generalized Newtonian fluids called *micropolar* fluids (Ariman *et al*. 1973; Eringen 2001). Not only does a micropolar fluid sustain body forces and the usual (Cauchy) stress tensor as simple Newtonian fluids do, but it also sustains body couples and the couple stress tensor; furthermore, the stress tensor is non-symmetric in a micropolar fluid. The additional effects arise from the presence of microscopic aciculate elements in a micropolar fluid—whereby micropolar-fluid motion has six degrees of freedom, three more than of a simple Newtonian fluid—because the length scale of motion is comparable to the length scale of the aciculate elements.

Micropolar fluids are known to occur in nature and are also of technoscientific importance. Typically, a micropolar fluid is a suspension of rigid or semi-rigid particles that cannot only translate but also rotate about axes passing through their centroids. Blood has often been modelled as a micropolar fluid (Eringen 1973; Turk *et al*. 1973; Misra & Ghosh 2001), because it contains platelets, cells and other particles. Modelling granular flows as micropolar fluids, Hayakawa (2000) showed that the analytical solutions of certain boundary-value problems are topologically very similar to relevant experimental results. We can expect rigid-rod epoxies to be micropolar fluids as well, because their aciculate molecules exhibit rotation about their centroidal axes (Giamberini *et al*. 1997; Su *et al*. 2000). Liquid crystals and colloidal suspensions are also cited as examples of micropolar fluids (Eringen 2001).

Motivated by this understanding, we decided to analyse the electro-osmotic flow of a micropolar fluid in a rectangular microchannel. Our interest lies in the cross-sectional distribution of fluid speed, stress tensor, microrotation, and couple stress tensor in the microchannel, especially at locations close to the walls and mid-channel. We are also interested in a comparison with the flow of a simple Newtonian fluid in a microchannel in order to isolate the effects of micropolarity.

The plan of this paper is as follows: the formulation of a relevant boundary-value problem is presented in §2, while §3 contains detail of analytical solution obtained by using the Debye–Hückel approximation (Li 2004, p. 19). The Debye–Hückel approximation is not always valid. Therefore, we also need a numerical approach for the solution of the boundary-value problem, as described in §4. Section 5 comprises analytical and numerical results obtained and discussions thereon. The main conclusions are summarized in §6.

## 2. Basic analysis and formulation

Adopting the notation for the position vector, where is the triad of Cartesian unit vectors, we are interested in examining the steady flow of a micropolar fluid in the microchannel |*y*′|≤*h* for *x*′[−*w*,*w*], when the length 2*w* is much greater than the height 2*h* of the microchannel and there is no variation along the *z*′-axis. The walls are assumed to be perfectly insulating and impermeable. Furthermore, we assume that: (i) neither a pressure gradient nor a body couple is present, (ii) the effect of gravity is unimportant, (iii) the flow is symmetric as both walls are identical, (iv) a spatially uniform, electrostatic field is applied to the fluid, (v) the Joule heating effects are small enough to be ignored, and (vi) the micropolar fluid is ionized, incompressible and viscous.

Under these conditions, the three applicable equations of micropolar-fluid flow are as follows (Eringen 2001):(2.1)(2.2)and(2.3)Here, ** V**′ and

**′, respectively, are the fluid velocity and the microrotation; is the applied electric field that is spatially uniform within the microchannel;**

*v**ρ*and

*j*

_{o}, respectively, are the mass density and the microinertia; and

*α*,

*β*and

*γ*are the three spin-gradient viscosity coefficients. The Newtonian shear viscosity coefficient and the vortex viscosity coefficient, respectively, are denoted by

*μ*and

*Χ*; these are related by the inequality 2

*μ*+

*Χ*≥0, where

*Χ*≥0 (Eringen 2001, p. 14). Let us note that

**′,**

*v**j*

_{o},

*Χ*,

*α*,

*β*and

*γ*are null-valued in a simple Newtonian fluid.

In the absence of a significant convective or electrophoretic disturbance to the electric double layers, the charge density is described by a Boltzmann distribution, and takes the following form for a symmetric, dilute and univalent electrolyte (Li 2004):(2.4)Here, *z*_{o} is the absolute value of the ionic valence; *ψ*′ is the electric potential; *e* is the charge of an electron; *n*_{o} is the number density of ions in the fluid far away from any charged surface; *k*_{B} is the Boltzmann constant; and *T* is the temperature. With *ϵ* denoting the static permittivity of the fluid, the charge density and the electric potential are also related by the Gauss law(2.5)thus,(2.6)The Debye length is assumed in this paper to be much smaller than *h*, i.e. *λ*_{D}≪*h*.

Before proceeding, let us define the non-dimensionalized quantities(2.7)Here, and are, respectively, the components of the couple stress and the stress tensors, where *p*,*q*∈{1,2,3};(2.8)is the characteristic speed to be identified in §3*b*; ; and *ψ*_{o} is called the zeta potential (Li 2004) that is assumed to be temporally constant and spatially uniform on the walls *y*=±1. With these quantities, equations (2.1)–(2.3) and (2.6), respectively, simplify to(2.9)(2.10)(2.11)and(2.12)where(2.13)Here, *k*_{1} couples the two viscosity coefficients; *k*_{2} and *k*_{3} are normalized micropolar parameters; *Re* may be called the Reynolds number; *Ro* may be called the microrotation Reynolds number (Eringen 2001); and *α*_{o} is the ionic-energy parameter (Li 2004). The Gauss law (2.5) can now be written as(2.14)

We have already assumed that ∂*/*∂*z*≡0; now, we ignore the variations along the *x*′-axis and set ∂*/*∂*x*≡0 consistently with the assumption that *w*≫*h*. Furthermore, the one-dimensional flow is supposed to be laminar and symmetric with respect to the *y*′-axis. Let us therefore designate(2.15)for further analysis of steady flow in the microchannel. The applied electric field is oriented parallel to the *x*′-axis, i.e. . In light of the aforementioned assumptions, equations (2.9)–(2.11) yield the following two ordinary differential equations:(2.16)and(2.17)

As the flow is symmetric, i.e. *u*(*y*)=*u*(−*y*) and *ψ*(*y*)=*ψ*(−*y*), the restrictions(2.18)must hold. The boundary conditions on *u*(*y*) and *ψ*(*y*) are(2.19)In addition, the condition(2.20)is engendered by the assumption *h*≫*λ*_{D} (Probstein 1989, p. 187).

Another boundary condition is evident in the literature: ** v**+

*β*

_{o}∇×

**=0 at the walls, where**

*V**β*

_{o}

**[−1,0] is some constant. This boundary condition simplifies in the present case to(2.21)There are two major schools of thought on the boundary condition (2.21). One school ignores microrotation effects near solid walls and sets**

*∈**β*

_{o}=0 (Papautsky

*et al*. 1999; Eringen 2001). But the second school holds that

*β*

_{o}<0 because the shear and couple stresses at the walls must be high in magnitude in comparison with locations elsewhere, as can be reasoned from the existence of boundary layers (Ramachandran

*et al*. 1979; Chiu & Chou 1993; Rees & Bassom 1996; Hegab & Liu 2004). This argument is held valid when thermal transfer and magnetic effects are to be accounted for (Hegab & Liu 2004). In the present case, electric double layers are present; hence,

*β*

_{o}<0 may be reasonable if electro-osmosis occurs. Analytical and numerical results are provided in this paper for zero, as well as non-zero,

*β*

_{o}. In addition,(2.22)

## 3. Solution of boundary-value problem

### (a) Analytical solution of the Poisson–Boltzmann equation

Under the assumptions made in §2, (2.12) reduces to(3.1)The solution of equation (3.1) satisfying boundary conditions (2.18)_{2}, (2.19)_{2} and (2.20) is (Dutta & Beskok 2001)(3.2)An equivalent form derived by us is(3.3)where(3.4)

### (b) Micropolar Helmholtz–Smoluchowski equation and velocity

As the counterparts of the Helmholtz–Smoluchowski equation and the Helmholtz–Smoluchowski velocity for simple Newtonian fluids (Probstein 1989, p. 192) are not available for steady flows of micropolar fluids, let us derive both in this section. We begin by assuming that the gradient of microrotation is much less than the Laplacian of the velocity, i.e.(3.5)Making use of equation (2.14), we see that equation (2.10) reduces to (d^{2}/d*y*^{2})[*u*(*y*)+*ψ*(*y*)]=0, which is merely a relation between viscosity and electromagnetism. After integrating both sides of this equation with respect to *y* and using the conditions (2.18), we get (d*/*d*y*)[*u*(*y*)+*ψ*(*y*)]=0. On integrating again with respect to *y* and using the boundary conditions (2.19), we obtain *u*(*y*)=−*ψ*(*y*)+1, whence(3.6)follows. Thus, *U* can be called the *micropolar Helmholtz*–*Smoluchowski velocity*, while equation (3.6) is the *micropolar Helmholtz*–*Smoluchowski equation*. Setting *Χ*=0 in equations (2.8) and (3.6), we revert to the Helmholtz–Smoluchowski velocity and equation, respectively, for simple Newtonian fluids (Probstein 1989, p. 192).

### (c) Analytical solutions for the Debye–Hückel approximation

Let us now derive an analytical solution of the boundary-value problem for steady flow, subject to the Debye–Hückel approximation (Li 2004), as follows. If the electric potential energy is small compared with the thermal energy of the ions, i.e. |*z*_{o}*eψ*_{o}*ψ*|≪|*k*_{B}*T*|, then |*α*_{o}*ψ*|≪1; accordingly,(3.7)which is called the Debye–Hückel approximation. Making this approximation on the right-hand side of equation (3.1), we get(3.8)whose solution is given by(3.9)The Debye–Hückel solution (3.9) satisfies the symmetry condition as well as the boundary conditions (2.18)_{2} and (2.19)_{2}.

Let us integrate both sides of equation (2.16) with respect to *y* and use the boundary conditions (2.18) and (2.22) to obtain(3.10)Next, on eliminating d*u/*d*y* from equation (2.17), we get(3.11)Using the Debye–Hückel solution (3.9) in equation (3.11), we reduce it to(3.12)where(3.13)As equation (3.12) is a linear, second-order, non-homogeneous, ordinary differential equation, its solution is found as (Kreyszig 2006)(3.14)where and are constants. The boundary condition (2.22) requires that . Furthermore, after using equations (3.10) and (3.14) in the boundary condition (2.21), the solution of equation (3.12) is obtained as(3.15)where(3.16)

Finally, on using equation (3.15) in equation (3.10), integrating with respect to *y*, and using the boundary condition (2.19)_{1}, we obtain the fluid speed as(3.17)

Thus, equations (3.15) and (3.17) constitute the solution of the boundary-value problem for steady flow of a micropolar fluid when the Debye–Hückel approximation (3.7) holds, whether *β*_{o}=0 or *β*_{o}≠0. On setting *Χ*=0, we get *k*_{1}=*k*_{2}=*k*_{4}=0, which implies from equation (3.15) that *N*(*y*)≡0; simultaneously, from equation (3.17), we recover(3.18)for a simple Newtonian fluid (Probstein 1989).

## 4. Numerical approach

The analytical solution for steady flow of a micropolar fluid in a rectangular microchannel with impermeable walls presented in §3*c* is premised on the Debye–Hückel approximation. An analytical solution appears impossible to find, if the solution (3.3) of the Poisson–Boltzmann equation has to be used instead of the Debye–Hückel solution (3.9); therefore, we resorted to the following numerical approach. Furthermore, in the absence of experimental data on micropolar fluids, the numerical approach provides a check against analytical results obtained using the Debye–Hückel approximation.

The interval *y*∈[−1,1] was divided into 2*I* segments of size Δ*y*=1/*I*. We used the fourth-order finite-difference method (Mancera & Hunt 1997), followed by the central difference method (in which the *y*-derivative is approximated by the central difference formula). Accordingly, the differential equations (2.16) and (2.17) transform into difference equations. The difference equations were solved iteratively by the successive overrelaxation (SOR) method (Hoffman 1992, p. 56). For interpolation, the Richardson method was used (Hoffman 1992, p. 195). The iterative procedure was repeated until convergence was obtained according to the following criterions (Hoffman 1992, p. 425):(4.1)Here, *u*_{i}≡*u*(*i*Δ*y*) and *N*_{i}≡*N*(*i*Δ*y*), *i*∈[−*I,I*], and the superscript *m*≥1 represents the iteration number. Different values of *I* were used until convergent solutions were obtained. Solutions obtained in §3*c* after using the Debye–Hückel approximation (3.7) were compared against the numerical results obtained.

## 5. Results and discussion

All calculations were made with the following material properties fixed: *n*_{o}=6.02×10^{22} m^{−3}; *z*_{o}=1; *ϵ*=10*ϵ*_{o}; *ϵ*_{o}=8.854×10^{−12} F m^{−1}; *μ*=3×10^{−2} Pa s; and *γ*=10^{−4} kg m s^{−1}. The temperature was fixed at *T*=290 K. Since the Boltzmann constant *k*_{B}=1.38×10^{−23} J K^{−1} and the electron charge *e*=1.6×10^{−19} C, the Debye length *λ*_{D}=10.72 nm. Consistent with the assumption that *h*≫*λ*_{D}, we chose *h*∈[107.2,5360] nm so that *m*_{o}∈[10,500]. Most results are presented for *ψ*_{0}≈−25×10^{−3} V, which is about the upper limit for the Debye–Hückel approximation to be valid at approximately room temperature (Hunter 1988, p. 25; Li 2004, p. 19), but some results are also presented for higher magnitudes of *ψ*_{o} in order to transcend the limitations of the Debye–Hückel approximation. The parameter *k*_{1}∈[0,0.95] was kept as a variable, after noting that *k*_{1}→1 as *Χ*→∞. The magnitude of the applied electric field was fixed at *E*_{o}=10^{4} V m^{−1}, which is a reasonable practical value.

The numerical approach described in §4 was implemented with computations made for *I*=250, 500 and 1000. The dependences of the relevant components of the fluid velocity, microrotation, stress tensor and couple stress tensor on *m*_{o}*, k*_{1} and *β*_{o} were investigated for steady flow.

In addition to the speed *u*′(*y*), the only non-zero components of the stress tensor and the couple stress tensor for the steady flow of an incompressible micropolar fluid in the absence of pressure are (Eringen 2001)(5.1)where . On normalizing the foregoing quantities, equations (5.1) take the following form:(5.2)Whereas *σ*_{12}=*σ*_{21} for a simple Newtonian fluid, let us note that the equality does not hold for a micropolar fluid.

As mentioned in §2, the choice of *β*_{o} affects fluid flow. Graphs from the Debye–Hückel approximation and from the numerical approach show that *u*′(*y*) does not depend on *k*_{1} when *β*_{o}=0, and is the same as for a simple Newtonian fluid. If |*β*_{o}| is increased, the effects of micropolarity grow and become dominant. Therefore, all the numerical results presented in this section are for *β*_{o}<0.

### (a) Electric potential

Let us begin with the electric potential *ψ*′(*y*) when *ψ*_{o}=−25×10^{−3} V. Figure 1 contrasts the solution (3.3) of the Poisson–Boltzmann equation with the Debye–Hückel solution (3.9). Both solutions are virtually identical for all *y* and *m*_{o}, except at and in the vicinity of *y*=0. Whereas (3.3) correctly yields *ψ*(0)*=*0 in conformity with (2.20), the Debye–Hückel solution yields *ψ*(0)=sech(*m*_{o}). The latter agrees with the former increasingly better as *m*_{o}→∞. Moreover, if |*ψ*_{o}| increases, then the difference between the solutions (3.3) and (3.9) increases, as shown in figure 2 for *ψ*_{o}=−100×10^{−3} V and −500×10^{−3} V. Furthermore, these two figures show that *ψ*′(*y*) for all *y*≠0 increases with |*ψ*_{o}|.

### (b) Fluid speed

Next, we examined the fluid speed *u*′(*y*) for different values of *m*_{o}, *k*_{1} and *β*_{o}, using the numerical approach as well as the Debye–Hückel approximation. Some representative results are provided in figure 3 for *ψ*_{o}=−25×10^{−3} V, *m*_{o}∈{50,500}, *k*_{1}∈{0,0.5,0.95} and *β*_{o}∈{−0.1,−0.1,−0.5,−1}. The numerical and the Debye–Hückel solutions in both figures are virtually identical, the greatest difference being less than 2.6858 per cent when *y*=0, *m*_{o}=500, *k*_{1}=0.95 and *β*_{o}=−0.01; moreover, this difference decreases if |*β*_{o}| increases. Near the walls (*y*=±1), the two solutions are indistinguishable from each other.

Figure 3 clearly shows that *u*′(*y*) depends on *k*_{1}, *β*_{o} and *m*_{o}. The dependence on *k*_{1} is almost absent in the vicinity of the walls because the boundary condition (2.19)_{1} was enforced, but that dependence intensifies significantly at locations away from the walls. Furthermore, *u*′(*y*) increases with |*β*_{o}| for all *m*_{o}≫1 and/or for all *k*_{1}>0. As |*β*_{o}| increases, the speed in an increasingly larger central portion of the microchannel exceeds the speed of a simple Newtonian fluid (*k*_{1}=0), and the proclivity of the speed to exceed the speed of the simple Newtonian fluid decreases as *k*_{1} increases.

The fluid speed increases in the vicinity of the walls as *m*_{o} increases, whether the fluid is simple Newtonian or micropolar. Mid-channel, the fluid speed increases with *m*_{o} for all *k*_{1} and *β*_{o}, but it decreases as *k*_{1} increases for all *m*_{o} and low values of |*β*_{o}|, as shown in figure 4. Incidentally, *u*′(0) is identical for all values of *k*_{1}>0 and *m*_{o}≫1 when *β*_{o}=0.

In order to further assess the influence of *m*_{o} on *u*′(*y*), we examined the derivative d*u*′(0)/d*m*_{o} when the Debye–Hückel approximation is valid. Equation (3.17) yields(5.3)where ; ; ; ; and . Graphical analysis of this equation indicates that is independent of *m*_{o} for any specific *k*_{1}∈(0,1); hence, , which is in agreement with figure 3. By contrast, for all *m*_{o}≫1 when *k*_{1}=0, which means that *u*′(0) is independent of *m*_{o} for a simple Newtonian fluid. Furthermore, for fixed *k*_{1}, has a maximum with respect to *β*_{o}∈[−1,0], which is in conformity with figure 3. The value of *β*_{o} for maximum lies in the neighbourhood of −0.945 when *ψ*_{o}=−25×10^{−3} V. Finally, using (2.8) and (5.3), we also conclude that , because *u*(*y*) is independent of *ψ*_{o} according to (3.17).

For higher values of |*ψ*_{o}|, the Debye–Hückel solution (3.17) deviates considerably from the numerical solution, as shown in figure 5. This deviation intensifies with increasing |*ψ*_{o}| for all *k*_{1}∈[0,1). Furthermore, *u*′(*y*) ∀ *y*∈[−1,1] increases with |*ψ*_{o}| for all *k*_{1}∈[0,1).

Next, figure 6 shows a comparison of the fluid speed *u*′(*y*) and the micropolar Helmholtz–Smoluchowski velocity *U*, near the walls, for three different values of *k*_{1} and two different values of *ψ*_{o}. Near the walls (regions that include the electric double layers), we see that as *k*_{1}→1, for all *ψ*_{o}. Furthermore, the difference decreases as *y*→0.

### (c) Microrotation

When the Debye–Hückel approximation holds, the microrotation is given by equation (3.15). According to this expression, *N*′(*y*) is an odd function of *y*, in contrast to *u*′(*y*) and *ψ*′(*y*). In a simple Newtonian fluid *k*_{1}=0 and *N*′(*y*)≡0.

Plots of *N*′(*y*) versus *y* when *ψ*_{o}=−25×10^{−3} V, for different values of *m*_{o}, *k*_{1} and *β*_{o} are presented in figure 7. This figure shows that: (i) increases as |*β*_{o}| increases for all *m*_{o}≫1 and *k*_{1}>0 and (ii) decreases as *k*_{1} increases for all *β*_{o}∈(−1,0) and *β*_{o}∈[10,500]. Unlike *u*′(*y*), increases with |*y*| for all *m*_{o}, *k*_{1} and *β*_{o}. Furthermore, the highest value of occurs at the walls (*y*=±1) for all *m*_{o}, *k*_{1} and *β*_{o}, such that(5.4)because tanh(*m*_{o})≈1 for all *m*_{o}∈[10,500]. This equation indicates that the microrotation at the walls is independent of *m*_{o}. Moreover, for *β*_{o}=−1, which is representative of turbulent boundary layers (Rees & Bassom 1996), we get , which is independent of both *m*_{o} and *k*_{1}, in agreement with figure 7*d*,*h*. Consequently, the microscopic aciculate elements near the walls have a greater proclivity to rotate about their centroids, regardless of the height *h* of the microchannel and the coupling parameter *k*_{1}.

In §5*b*, we observed significant differences in the predictions of the fluid speed by the numerical approach and the Debye–Hückel approximation to analyse the effect of higher values of |*ψ*_{o}| on . Figure 8 indicates that, just like and , increases as |*ψ*_{o}| increases for all *k*_{1}∈[0,1). Furthermore, whether |*ψ*_{o}| is low or high, the highest value of occurs at the walls for all *k*_{1}.

### (d) Stress tensor

Analogous to , the components and of the stress tensor are also odd functions of *y*, by virtue of equations (3.17), (3.15) and (5.2).

The profiles of with *y*, at and in the vicinity of the wall *y*=−1, are presented in figure 9, when *ψ*_{o}=−25×10^{−3} V, for different values of *m*_{o}, *k*_{1} and *β*_{o}. This figure shows that decreases as *k*_{1} increases for all *m*_{o}∈[10,500], *k*_{1}∈(0, 1) and *β*_{o}∈(−0.5, 0); however, increases with *k*_{1} when *β*_{o}∈[−1,−0.5]. Furthermore, increases with |*β*_{o}| for all *m*_{o} and *k*_{1}. The maximum value of in the boundary layer always occurs at the walls for all *m*_{o}, *k*_{1} and *β*_{o}. These observations can be confirmed as follows.

Using equations (2.21) and (5.2)_{1}, we obtain(5.5)which shows that is directly proportional to the velocity gradient at the wall. As the magnitude of the velocity gradient must be large near the walls owing to the no-slip boundary condition (2.19)_{1} (Currie 1974, p. 276), and because figure 3 indicates that the velocity gradient does have maximum magnitude at the walls, it is not surprising that the maximum value of occurs at the walls *y*=±1.

Next, on using equations (3.9), (3.10), (5.4) and (5.5), with the argument that for all *m*_{o}∈[10,500], we get(5.6)accordingly, is independent of *m*_{o} for all *k*_{1} and *β*_{o}, which is in agreement with figure 9. In addition, for simple Newtonian fluids (*k*_{1}=0), equation (5.6) yields(5.7)which also agrees with figure 9.

Furthermore, if we set *β*_{o}=−1/2 in equation (5.6), we get , which means that is independent of *k*_{1}, in conformity with figure 9*d–f*. This explains the divergent dependences of on *k*_{1} in the regimes *β*_{o}∈(−0.5, 0) and *β*_{o}∈[−1,−0.5). In addition, when *β*_{o}=−1, equation (5.6) reduces to , which shows that is inversely proportional to (1−*k*_{1}) for *β*_{o}=−1, in agreement with figure 9*g–i*.

In contrast to , , which can be proved by using the boundary conditions (2.18)_{1} and (2.22) in equation (5.1)_{1}. Accordingly, in the micropolar fluid is significant only at locations close to the walls of the microchannel, just as in a simple Newtonian fluid.

For higher values of |*ψ*_{o}| (i.e. |*ψ*_{o}|>25×10^{−3} V), the maximum value of occurs at the walls *y*±1, similar to the case when |*ψ*_{o}|=25×10^{−3} V, for all *k*_{1}∈[0,1), as shown in figure 10. This figure also indicates that increases as |*ψ*_{o}| increases for all *k*_{1}.

For the other non-null component of the stress tensor, on using equations (2.7) and (3.10) in equation (5.2)_{2}, we obtain(5.8)Hence, does not depend on *k*_{1} and *β*_{o}, and is the same as that in a simple Newtonian fluid, regardless of *ψ*_{o}. However, the graphs in figure 11, plotted for *ψ*_{o}=−25×10^{−3} V, show that does depend on *m*_{o}. This figure also shows that is maximum at the walls *y*±1 and rapidly drops to a null value as *y*→0 in accordance with equation (2.18)_{2}.

On using equations (2.8) and (3.9) in equation (5.8), consistently with the argument that for all , we obtain , which is independent of *m*_{o}, *k*_{1} and *β*_{o}. This expression agrees with figure 11 and it also coincides with equation (5.7).

Figure 12 shows the profiles of with *y* for . This figure indicates that increases with for all *m*_{o}. In addition, the maximum value of occurs at the walls. These characteristics hold whether or not the Debye–Hückel approximation is valid.

### (e) Couple stress tensor

A high magnitude of a component of the couple stress tensor indicates a greater tendency of the microscopic aciculate elements in a micropolar fluid to rotate about their respective centroids. Under the validity of the Debye–Hückel approximation, we find that the sole non-null component of the couple stress tensor is an even function of *y*, by virtue of equations (3.15) and (5.2)_{3}. This conclusion turned out to be valid even when the Debye–Hückel approximation is not.

Unlike the stress tensor, the couple stress tensor is not null-valued in the middle of the microchannel for micropolar-fluid flow. Consistent with the argument that for all *m*_{o}∈[10,500], can be obtained after using equations (3.15) and (5.2)_{3} as(5.9)where , and the Debye–Hückel approximation is valid. Furthermore, the couple stress tensor is not null-valued at the walls either, as shown in figure 13, which contains the profiles of with respect to *y*, for , *k*_{1}∈{0.25,0.5,0.95}, *m*_{o}∈{10,50,500} and . This figure clearly indicates that, for low values of |*β*_{o}| and *m*_{o}, the magnitude of mid-channel is greater than at any other location. As |*β*_{o}|, as well as *m*_{o}, increases, a decrease in and an increase in at and in the vicinity of the walls are observed. Furthermore, the difference between and increases if: (i) *m*_{o} increases for all *k*_{1} and *β*_{o}, (ii) *k*_{1} increases for all *k*_{1} and *β*_{o}, or (iii) |*β*_{o}| increases for all *m*_{o} and *k*_{1}.

Finally, when the Debye–Hückel approximation does not hold, we examined for two different values of *ψ*_{o} in figure 14. This figure shows that does not alter with |*ψ*_{o}|. It is not surprising because, if we use equations (2.7), (3.15), (5.1)_{3} and (5.9) to calculate , we get an expression that is independent of *ψ*_{o}. But, increases ∀ as |*ψ*_{o}| does.

## 6. Concluding remarks

We formulated the boundary-value problem of steady electro-osmotic flow of a micropolar fluid in a rectangular microchannel, and we solved it numerically as well as analytically using the Debye–Hückel approximation, when the Debye length is no more than 5 per cent of the height of the microchannel and the length of the microchannel is much larger than its height. The numerical results turned out to be virtually identical to the Debye–Hückel results for low magnitudes of the zeta potential *ψ*_{o}. By contrast, the results differ significantly for higher magnitudes of *ψ*_{o}, and the differences intensify with increasing |*ψ*_{o}|. We also defined the micropolar Helmholtz–Smoluchowski equation and velocity *U*.

The micropolar-fluid speed depends significantly not only on the height of the microchannel in relation to the Debye length (*m*_{o}=*h*/*λ*_{D}), analogous to the speed of a simple Newtonian fluid, but also on the micropolar nature of the fluid as expressed through: (i) the boundary parameter *β*_{o} that mediates the velocity gradient and the microrotation at the walls of the microchannel and (ii) the viscosity coupling parameter *k*_{1} relating the Newtonian shear viscosity coefficient and the (micropolar) vortex viscosity coefficient. The fluid speed in the central portion of the microchannel exceeds the speed of a simple Newtonian fluid (*k*_{1}=0). For a fixed Debye length, the mid-channel fluid speed depends on the microchannel height when the fluid is micropolar, but not when the fluid is simple Newtonian. Moreover, in the regions near the walls (including the electric double layers), as *k*_{1}→1, regardless of the magnitude of *ψ*_{o}. In contrast to the fluid speed, the microrotation is null-valued mid-channel, but is dominant at the walls. The microrotation at the walls does not depend on the microchannel height. Furthermore, if the boundary layers are turbulent, the microrotation at the walls also becomes independent of the viscosity coupling parameter.

The stress at the walls is dominant at and in the vicinity of the walls, whether the fluid is micropolar or simple Newtonian. However, as the micropolarity parameter |*β*_{o}|, increases, the stress at the walls intensifies. The mid-channel couple stress decreases, but the couple stress at the walls intensifies, as the microchannel height, as well as |*β*_{o}|, increases. Finally, the electric potential, fluid speed, microrotation and stresses are enhanced as the zeta potential increases in magnitude.

We have begun to examine non-steady electro-osmotic flows of micropolar fluids in microchannels, wherein the work reported here will be incorporated into the solution of an initial-boundary-value problem.

## Acknowledgments

We gratefully acknowledge the very useful comments of the anonymous reviewers. We thank Mr Ambuj Sharma and Prof. Charles E. Bakis of the Pennsylvania State University for discussions. A.A.S. is grateful to the Higher Education Commission of Pakistan for a grant to enable him to visit Penn State. A.L. thanks the Charles Godfrey Binder Endowment at Penn State for partial financial support.

## Footnotes

- Received September 2, 2008.
- Accepted September 30, 2008.

- © 2008 The Royal Society