Magnetohydrodynamic flow of generalized Maxwell fluids in a rectangular micropump under an AC electric field

Journal of Magnetism and Magnetic Materials 387 (2015) 111–117

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Magnetohydrodynamic flow of generalized Maxwell fluids in a rectangular micropump under an AC electric field Guangpu Zhao a, Yongjun Jian a,n, Long Chang b, Mandula Buren a a b

School of Mathematical Science, Inner Mongolia University, Hohhot, Inner Mongolia 010021, PR China School of Mathematics and Statistics, Inner Mongolia University of Finance and Economics, Hohhot, Inner Mongolia 010051, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 9 November 2014 Received in revised form 7 February 2015 Accepted 27 March 2015 Available online 30 March 2015

By using the method of separation of variables, an analytical solution for the magnetohydrodynamic (MHD) flow of the generalized Maxwell fluids under AC electric field through a two-dimensional rectangular micropump is reduced. By the numerical computation, the variations of velocity profiles with the electrical oscillating Reynolds number Re, the Hartmann number Ha, the dimensionless relaxation time De are studied graphically. Further, the comparison with available experimental data and relevant researches is presented. & 2015 Elsevier B.V. All rights reserved.

Keywords: Magnetohydrodynamics (MHD) Generalized Maxwell fluid Hartmann number Rectangular micropump

1. Introduction The need to transport fluids through microchannel has dramatically increased in recent years due partly to its application in microfluidic system. Many microfluidic devices based upon suitable pumping systems can be used to transport and separate substances in the fields of chemical, medical and biological applications. With this aim, several options have been proposed in the literature, including some mechanical and non-mechanical micropumps, and their characteristics have been analyzed in papers [1–4]. Due to certain advantages such as simple fabrication process, the absence of moving parts, low voltage operation and the possibility to achieve relatively high flow rates, magnetohydrodynamic (MHD) micropumps have attracted the attention of many researchers [5–9]. In principle, these pumps rely on the interaction between an externally-imposed electrical current and a transverse magnetic field for pushing an electrically-conducting fluid through the channel. So far, there are two different versions of MHD micropumps available in the market, one is direct current (DC)-operated, and the other is alternating current (AC)-operated. The former mainly relies on a permanent magnet to generate the Lorentz force, while in the latter, use is made of an electromagnet for this purpose. The feasibility of MHD micropumps has been demonstrated by using both DC and AC electric and magnetic fields [10]. n

Corresponding author. E-mail address: [email protected] (Y. Jian).

http://dx.doi.org/10.1016/j.jmmm.2015.03.094 0304-8853/& 2015 Elsevier B.V. All rights reserved.

In this paper, AC-operated is taken into account. In fact, it has been recognized that Lorentz forces can be used to control and manipulate the flows in microdevices [11]. Many theoretical and experimental researches have been reported in the literature to study the flow behavior of MHD micropumps. Jang and Lee [12] have experimentally shown that the average flow rates in micropumps can be substantially augmented by employing low-magnitude magnetic fields. Verardi et al. [13] simulated the MHD duct flow by employing the finite element method. The results showed that axial velocity profiles were distorted into M-shape as the magnetic field applied. Generally, MHD flows in micropumps can be described by one-dimensional (1D) models based on the classical Hartmann flow [14], which only consider the flow walls perpendicular to magnetic field, usually known as the Hartmann walls. However, neglecting frictional effects of the walls parallel to magnetic field, usually known as the side walls, may lead to misleading predictions. Taking into account the four walls of the rectangular MHD microduct, a better description can be acquired with two-dimensional (2D) models. Wang et al. [15] conducted the numerical simulations of two-dimensional fully developed laminar flow for a MHD micropump. Ho [16] obtained analytical solution of MHD micropump in rectangular ducts and compared the results with those of related experiments. Most of literatures in MHD micropumps were based upon the assumption of the no slip condition in all walls of duct. However, Rivero and Cuevas [17] investigated the MHD micropumps both in one and in two-dimensional flow models by considering the influence of slip condition. They found the 2D slip

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modeling gives a much better prediction than that of 1D modeling by comparing their results with those of previous experiments. In virtually all works cited above, the working fluid has been assumed to be Newtonian. However, numerous operations in modern mechanical, chemical and biomedical engineering involve non-Newtonian fluid. In the electroosmotic flow (EOF) aspect, Liu et al. [18] first studied the time periodic EOF of generalized Maxwell fluids between two micro-parallel plates. Jian et al. [19] derived an analytical solution for periodical electroosmotic flow of viscoelastic fluid in a complex two-dimensional rectangular microchannel. In the MHD micropump aspect, Shahidian et al. [20] studied MHD micropump of power-law fluids in a circular microchannel. Based upon finite difference method, Moghaddam [21] numerically investigated MHD micropumps of power-law fluids in a 2D rectangular microchannel. The purpose of the present article is to investigate analytically MHD flow of generalized Maxwell fluids in a rectangular micropump under AC electric field. The dependence of MHD micropump velocity on non-dimensional parameters such as oscillating Reynolds number Re, Hartmann number Ha and the relaxation time De will be discussed. Also, a comparison with available experimental data and corresponding studies is carried out.

ρ

→ → → → J = σ (E + u × B )

(2)

→ where E ¼(Ex, 0, 0), Ex is the intensity of the AC electric current in → x direction, B the magnetic field in y direction with strength B, s the electrical conductivity of fluid. There is the only axial velocity component w(x, y, t) and other velocity components in the x–y plane disappear. In addition, an open-end horizontally place channel is assumed and hence the pressure gradient term in the Cauchy momentum equation disappears. Substituting Eq. (2) into Eq. (1) yields two-dimensional Cauchy momentum equation in the z direction.

2. Formulation of the problem

The MHD of the incompressible generalized Maxwell fluids through a two-dimensional rectangular micropump with width a and height b is sketched in Fig. 1. The origin of coordinates is placed at the lower left corner of the pump. The x-axis points to the lateral wall from left to right, y-axis direction is vertically upward and the z-axis is the axial velocity direction. The Cauchy momentum can be expressed as

(1)

→ where ρ is the fluid density, u is the velocity field, t is the time, τ is → → the stress tensors, and J × B is the net body force contributed by the Lorentz force acting on the fluid. In this paper, we suppose that the magnetic Reynolds number is so small that the induced magnetic field is negligible relative to the imposed one. In this case, the magnetic field is independent of the flow velocities. In → general, the electric current density J through the fluid is given by Ohm's law

ρ

2.1. Cauchy momentum equation and constitutive relation

→ → → Du = − ∇⋅τ + J × B Dt

∂w (x, y , t) ∂ ∂ (τxz ) − (τyz ) + σBEx (t) − σB2w (x, y , t) =− ∂t ∂x ∂y

(3)

For generalized Maxwell fluids, the constitutive equation satisfies t

⎧η

⎡

⎤⎫

τxz = −

∫−∞ ⎨⎩ λ01 exp ⎢⎣ − (t −λ1t′) ⎥⎦ ⎬⎭ ∂w (x∂,xy, t′) dt′

τyz = −

∫−∞ ⎨⎩ λ01 exp ⎢⎣ − (t −λ1t′) ⎥⎦ ⎬⎭ ∂w (x∂,yy, t′) dt′

t

⎧η

⎡

(4a)

⎤⎫

(4b)

where λ1 is the relaxation time, η0 is the zero shear rate viscosity. Substituting Eq. (4) into Eq. (3), the Cauchy momentum equation becomes

ρ

∂w (x, y , t) = ∂t ⎡ (t − t′) ⎤ ⎫ ⎡ ∂ 2w (x, y , t′) t ⎧η ∂ 2w (x, y , t′) ⎤ ⎨ 0 exp ⎢ − ⎥ dt ′ + ⎥⎬ ⎢ 2 −∞ ⎩ λ1 λ1 ⎦ ⎭ ⎣ ⎣ ∂x ∂y2 ⎦

∫

+ σBEx (t) − σB2w (x, y , t)

(5)

Supposing the electric field is alternating current and the velocity of time periodical MHD of generalized Maxwell fluids can be written in complex forms as

w(x, y , t) = R {w0 (x, y) eiωt }, Ex (t) = R {E0 eiωt }

(6a, b)

where R{⋅} denotes the real part of the function, ω is imposed AC electric field oscillating angular frequency. After substitution of Eq. (6) into Eq. (5), we have

⎛ ∂ 2w 1 + iλ1ω 1 + iλ1ω ∂ 2w0 ⎞ 0 ⎟⎟ − (σB2 + iρω) w0 ⎜⎜ + = − σBE0 η0 η0 ∂y2 ⎠ ⎝ ∂x2

(7)

For simplicity, the following dimensionless groups are introduced

x¯ =

ρw0 ρE 0 b σ x y σ ρωb2 ¯0 = , y¯ = , w , Ha = Bb ,S= , Re = , a b η0/ b η0 η0/ b η0 η0

De = λ1ω Fig. 1. Schematics of the MHD micropump in a frame (a) 3D view of the MHD micropump and (b) Duct's cross-section of the MHD micropump.

(8)

where η0/ρb is the characteristic velocity, Ha is the Hartmann number giving an estimate of the magnetic forces compared to the

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where ε ¼b/a is the aspect ratio (height to width) of the rectangular cross-section.

2.2. Small aspect ratio case Under the situation of small aspect ratio ε { 1, the first term on the left hand side of Eq. (9) can be neglected and the velocity becomes independent of the x-coordinate. Thus the two dimensional spatial dependent velocity field reduces to one dimensional problem and Eq. (9) becomes

¯0 ∂ 2w ¯ 0 = − HaS (1 + iDe) − (Ha2 + i Re)(1 + iDe) w ∂y¯ 2

(10)

Accordingly, the no slip conditions at the wall are given by

¯ 0 (0) = w ¯ 0 (1) = 0 w

Fig. 2. Normalized one-dimensional MHD velocity profiles of generalize Maxwell fluids between infinite plates for different De (ε ¼1.0, Re¼ 0.2, Ha¼0.8, S ¼2).

viscous forces, S is a non-dimensional parameter representing the strength of the z direction electric field, Re is electric oscillating Reynolds number and De represents normalized relaxation time. The dimensionless form of Eq. (7) reads

ε2

¯0 ¯0 ∂ 2w ∂ 2w ¯ 0 = − HaS(1 + iDe) + − (Ha2 + iRe)(1 + iDe)w 2 ∂x¯ ∂y¯ 2

(9)

(11)

For convenient, the coefficient on the left hand side of Eq. (10) can be denoted as

(Ha2 + iRe)(1 + iDe) = (α + iβ)2

(12)

where

α=

2 [ (Ha4 + Re2)(1 + De2) + (Ha2 − De Re)]1/2 2

(13a)

β=

2 [ (Ha4 + Re2)(1 + De2) − (Ha2 − De Re)]1/2 2

(13b)

Fig. 3. Normalized two-dimensional MHD velocity profiles of generalize Maxwell fluids in the rectangular micropump for different Reynolds number Re (ε ¼1.0, Ha ¼0.8, S ¼2, De ¼ 0.5). (a) Re¼ 5.0, (b) Re¼ 20, (c) Re¼ 50, and (d) Re¼ 100.

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Fig. 4. Normalized two-dimensional MHD velocity profiles of generalize Maxwell fluids in the rectangular micropump for different Ha (ε¼ 1.0, Re¼ 20, S¼ 2, De ¼0.5). (a) Ha¼ 0.001, (b) Ha¼ 0.05, (c) Ha¼ 0.1, and (d) Ha¼ 1.0.

Eq. (10) is second-order linear inhomogeneous differential equation, and its solution can be expressed as the sum of the general solution corresponding homogeneous equation and a special solution Ys, etc.

¯ 0 (y¯ ) = C1e (α + iβ) y¯ + C2 e−(α + iβ) y¯ + Ys w

¯ 0 (x¯ , 0) = w ¯ 0 (x¯ , 1) = 0 w

Eq. (9) is second-order linear inhomogeneous equation. Similar to our pervious work [19], the method of separation of variables can be used and its solution has the form

(14)

∞

¯ 0 (x¯ , y¯ ) = w

The special solution is

∑ j=1

HaS (1 + iDe) Ys = (α + iβ)2

C1 =

HaS (1 + iDe) (α + iβ)2

e α + iβ

− 2 sinh(α + iβ) − 1 2 sinh(α + iβ)

C2 =

HaS (1 + iDe) 1 − (α + iβ)2 2 sinh(α + iβ)

∑

⎛ λj ⎞ sin ⎜ x¯ ⎟⎟ ϕ j (y¯ ) = − (1 + iDe) HaS ⎝ε ⎠

(20)

where

(15c) ϕ j (y¯ ) =

2.3. Mediate aspect ratio case For mediate aspect ratio ε EO (1) case, the MHD velocity field cannot be taken as one dimensional and we must resolve the two dimensional problem of Eq. (9). The corresponding dimensionless no slip boundary conditions at the four side walls are

¯ 0 (0, y¯ ) = w ¯ 0 (1, y¯ ) = 0 w

(19)

Inserting Eq. (18) into original inhomogeneous Eq. (9) yields

j=1

e α + iβ

(18)

λj are found as

λ j = εjπ , j = 1, 2, 3, ...

∞

(15b)

⎛ λj ⎞ sin ⎜ x¯ ⎟⎟ Yj (y¯ ) ⎝ε ⎠

where the eigenvalue

(15a)

Employing boundary conditions (11), the two constants C1 and C2 can be written as

(17)

(16)

d2Yj (y¯ ) dy¯ 2

− [λ j2 + (Ha2 + i Re)(1 + iDe)] Yj (y¯ )

(21)

Taking ϕj (y¯ ) as the expanding coefficient of Fourier sine series of Eq. (20), we have

ϕ j (y¯ ) = 2(1 + iDe) εHaS

⎤ ⎛ λj ⎞ 1⎡ ⎢ cos ⎜⎜ ⎟⎟ − 1⎥ λ j ⎢⎣ ⎦⎥ ⎝ε⎠

(22)

with ϕj (y¯ ) having been solved, the second-order inhomogeneous ordinary differential Eq. (21) can be solved. It can be expressed as

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115

Fig. 5. Normalized two-dimensional MHD velocity profiles of generalize Maxwell fluids in the rectangular micropump for different De (ε ¼1.0, Ha ¼0.8, Re¼ 20, S ¼2). (a) De ¼0.05, (b) De ¼0.5, (c) De¼ 2.0, and (d) De ¼ 6.0.

Its solution can be expressed as the sum of a general solution corresponding homogeneous equation and a special solution Yjs, etc.

Yj (y¯ ) = D1e B j y¯ + D2 e−B j y¯ + Yjs

(24)

According to method of variation of constant, the special solution is given by

Y js =

⎛ λ j ⎞⎞ 2(1 + iDe)εHaS ⎛ ⎜1 − cos⎜ ⎟⎟ 2 ⎝ ε ⎠⎠ ⎝ B j λj

(25)

Inserting Eq. (25) into Eq. (24), the solution of Eq. (23) is given by

Yj (y¯ ) = D1e B j y¯ + D2 e−B j y¯ +

λj 2(1 + iDe) εHaS (1 − cos( )) ε B 2j λ j

(26)

Applying the following boundary conditions obtained by the method of separation of variables Fig. 6. Comparison of our present flow velocities with Rivero and Cuevas's 2D model (when the magnitude G ¼0, α ¼0) and the experimental data obtained by Lemoff and Lee. (a ¼4 mm, b ¼38 mm, x¼ 0.5, y¼ 0, L ¼0.8 mm, ρ¼ 1025 kg m  3, η0 ¼ 1.09  10  3 kg/m  1 s  1, s ¼ 4 S/m, ω ¼0 rad/s).

Yj (0) = 0, Yj (1) = 0 The constants D1 and D2 can be determined as

D1 =

d2 Yj (y¯ ) dy¯ 2 where

− B 2j Yj (y¯ ) = φj (y¯ ) = const,

B 2j

=

λ j2

+

(Ha2

+ i Re)(1 + iDe)

(23)

(27)

D2 =

⎛ ⎛ λ ⎞ ⎞ e B j − 2 sinh B − 1 j 2(1 + iDe) εHaS ⎜ ⎜ j ⎟⎟ − 1 cos ⎜ ⎜ ⎟ ⎟ 2 ε B 2 sinh B j λj j ⎝ ⎝ ⎠⎠

(28a)

⎛ ⎛ λj ⎞⎞ Bj 2(1 + iDe) εHaS ⎜ ⎜ ⎟⎟ 1 − e 1 cos − ⎜ ⎜ ε ⎟ ⎟ 2 sinh B B 2j λ j j ⎝ ⎝ ⎠⎠

(28b)

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G. Zhao et al. / Journal of Magnetism and Magnetic Materials 387 (2015) 111–117

Fig. 7. Comparison of our present flow velocities with Rivero and Cuevas's 1D model when the magnitude G ¼ 0, α ¼0, β¼ 85. (a) Re¼0.001 and (b) Re¼50.

¯ 0(x¯ , y¯ ) can be obtained by inserting Eq. Finally, the solution of w (26) into Eq. (18).

3. Results and discussion In the previous section, analytical solutions were derived for the periodical MHD flow of generalized Maxwell fluids through a two-dimensional rectangular microchannel. They depend greatly on non-dimensional parameters, such as the electric oscillating Reynolds number Re, Hartmann number Ha and normalized relaxation times De. In practical engineering problems, we also need to mention some typical values of the corresponding dimensional parameters. In the following calculations, typical parameters can be taken as follows: ρ  103 kg m  3, η0  10  3 kg m  1 s  1, b¼200 mm, the order of electrical conductivity of fluids s approximately is 2.2  10  4–106 S/m [22], the lower value 2.2  10  4 S/m of the conductivity is the de-ionized water, while the upper value 106 S/m of the conductivity corresponds to the liquid metal such as mercury. If the range of the imposed magnetic field is B  40 mT–0.44 T [12,17], the order of the Hartmann number Ha is valued from 3.6  10  6 to 3 evaluated from Eq. (8). Making E0 change from 1 V/m to 60 V/m [17], the dimensionless parameters S involved in the present study is approximately changed from 2  10  2 to 7.5  104. The parametric region of the external electric field frequency changes from 0.4 Hz to 400 Hz. By considering the range of frequency is 0–1.6  103 Hz in Ref. [23],

corresponding to angle frequency ω changes from 2.5 rad/s to 2500 rad/s. Therefore, oscillating Reynolds number Re can be evaluated from 0.1 to 100. According to Ref. [24], the parametric value of the relaxation time λ1 is wide. Bandopadhyay and Chakraborty further considered λ1 changed from 10  4 s to 10  2 s [25]. However, the relaxation time should be smaller than the oscillation period, i.e. λ1 o2π/ω or Deo 2π must be satisfied. Therefore, λ1 changes from 10  4 s to 2.5  10  3 s in the paper. The influences of the above mentioned parameters on velocity profiles are investigated. Fig. 2 shows one-dimensional MHD velocity profiles of generalized Maxwell fluids between parallel plates for different normalized relaxation time De (ε ¼1.0, Re ¼0.2, Ha ¼0.8, S ¼2.0). It can be found that the magnitude of velocity increases as De gradually grows. The reason is that longer relaxation time De means larger elastic effect of the Maxwell fluids and leads to larger MHD velocity. Fig. 3 shows normalized MHD velocity profiles of generalized Maxwell fluids across a rectangular micropump for different oscillating Reynolds number Re (ε ¼ 1.0, Ha¼0.8, S ¼2.0, De ¼0.5). It can be noted that increasing oscillating Reynolds number Re leads to notable change of MHD velocity profiles. At the same time, the amplitudes of MHD velocity decrease gradually. The reason is that the diffusion time scale is much greater than the oscillation time period. Therefore, for the larger Re, the velocity only varies in the vicinity of the walls, and there is no sufficient time for flow

Fig. 8. Comparison of our present flow velocities with Rivero and Cuevas's 2D model when the magnitude G ¼ 0, α ¼0, ε ¼1, β ¼85. (a) Re¼ 0.001 and (b) Re¼ 50.

G. Zhao et al. / Journal of Magnetism and Magnetic Materials 387 (2015) 111–117

momentum to diffuse far into the deep of the micropump. Fig. 4 illustrates the variation of normalized two-dimnsional MHD velocity of generalize Maxwell fluids in the rectangular micropump with Hartmann number Ha (ε ¼1.0, Re¼20, S ¼2, De ¼0.5). It can be seen from Fig. 4 that the MHD velocity magnitude increases with Ha due to the enhancing electromagnetic induced effects. Fig. 5 shows the effect of the different relaxation time De on the normalized MHD velocity of generalized Maxwell fluids for a prescribed oscillating Reynolds number Re. Physically, due to the “fading memory” phenomenon of the generalized Maxwell fluids, large relaxation time De means larger elastic effect and weaker recovery ability of the generalized Maxwell fluids. Thus the increasing relaxation time De leads more easily to the variation of the MHD velocity profiles. In order to validate the theoretical result of the models presented in the previous sections, a comparison of the model results with experiment data available and corresponding studies in the literature is presented in Figs. 6–8. In Fig. 6, we compare the result of Maxwell fluid for different relaxation time De under the DC electric field case with the result of Newtonian fluid considered by Rivero and Cuevas [17] and the experimental data obtained by Lemoff and Lee [10]. It can be found that for the case of Newtonian fluid, the present result not only agrees qualitatively well with that obtained in the experiment, but also agrees with the theoretical result of Rivero and Cuevas [17]. With the increase of relaxation time De, the normalized MHD velocity of generalized Maxwell fluids becomes larger than that in the experiment. In other word, the larger De means the larger relaxation time and results in the large deviations from Newtonian behavior. The comparison of our present flow velocities with theoretical results obtained by Rivero and Cuevas's 1D model (G ¼0, α ¼0, β ¼ 85) for different Re and Hartmann number Ha can be found in Fig. 7. The Fig. 7(a) illustrates that for the small electric oscillating Reynolds number Re¼ 0.001 (means DC electrical field) and Newtonian fluid by setting De ¼0, the present model agrees well with the theoretical results in Ref. [17]. Moreover, the MHD velocity of generalized Maxwell fluids becomes larger with relaxation time De. For the large electric oscillating Reynolds number Re¼50, i.e., in the case of AC electric field, the derived result deviates from that of Rivero and Cuevas for DC electric field, as shown in Fig. 7 (b). Similarly, a comparison of our 2D model results with those obtained by Rivero and Cuevas [17] in Fig. 8. The same conclusions can be drawn. Therefore, the present theoretical solutions pertinent to general Maxwell fluids are an expansion for Newtonian fluids. It is necessary to carry out this investigation based upon the practical application for microfluid or nanofluid.

4. Conclusions We have presented a theoretical study of the time periodic MHD flow of the general Maxwell fluids through a rectangular micropump. Employing the method of separation of variables, the velocity profiles are obtained analytically. The computational results show that for given relaxation time, Hartmann number, and aspect ratio ε, increasing oscillating Reynolds number Re leads to rapid changing MHD velocity profiles. At the same time, the amplitudes of MHD velocity decrease gradually. For given oscillating Reynolds number, large Hartmann number leads to large amplitudes of MHD velocity. For given oscillating Reynolds number Re and Hartmann number Ha, large relaxation time De easily leads to the larger variation of the MHD velocity profiles because elasticity is physical property of whole fluid. By comparing the result of the

117

proposed model in this paper with that of the model given in Ref. [17], it shows that the proposed model obtained better identification performance.

Acknowledgments The work was supported by the National Natural Science Foundation of China (Nos. 11472140, 11062005 and 11202092), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (No. NJYT-13A02), the Postgraduate Scientific Research Innovation Foundation of Inner Mongolia (No. 14020202), Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (No. NJZY14109).

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