Investigation of electroosmotically induced pressure gradient in rectangular capillaries

International Journal of Engineering Science 145 (2019) 103172

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Investigation of electroosmotically induced pressure gradient in rectangular capillaries Alison Subiantoro a,∗, Kim Tiow Ooi b a b

Department of Mechanical Engineering, The University of Auckland, 20 Symonds Street, Auckland 1010, New Zealand School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore

a r t i c l e

i n f o

Article history: Received 20 June 2017 Revised 1 March 2018 Accepted 18 September 2019

Keywords: Electroosmosis Electroosmotic flow Closed-end square capillary Pressure gradient Theory Experiment

a b s t r a c t This paper investigates the pressure gradient created by electroosmotic effect in closed-end rectangular capillaries with DC electric field. An existing theoretical model was reviewed. Experiments were carried out to validate the model. The capillaries were square in geometry, 50 mm long with internal sizes of 75 μm and 100 μm. The electrolyte was NaCl solution with concentrations of 0.0 0 05 M and 0.01 M. The range of electric field tested was between 10 and 60 kV/m. Comparison of numerical and experimental data showed that they were generally in good agreement in the range of operating conditions tested. The accuracy of the theoretical model can be improved further by using an accurate zeta potential estimation model. The effects of electric field strength, electrolyte concentration and capillary size to the induced pressure gradient were also studied. It was found that induced pressure gradient was linearly proportional to electric field strength, with a rate of 2 kPa/m for every 10 kV/m increment of field strength. Induced pressure gradient was not affected significantly by electrolyte concentration in the range of conditions tested, regardless of capillary size. Reduction of capillary size increased induced pressure gradient. The rate of increase was higher with a smaller capillary size. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction In the last few decades, major advancement in precision technology has given way to the development of microchannels. Liquid flows in these microchannels exhibit unique effects, including electrokinetic effects. These effects occur when an electrolyte is in contact with a solid surface, causing electric charges to be acquired by the surface, creating an electrical layer that may affect the flows. There are a few categories of electrokinetic phenomena. One of them is electroosmosis, which is when liquid moves in response to an applied electric field. The flow caused by electroosmosis is called an electroosmotic flow. Electroosmotic effect can and have been exploited for various applications, including lab-on-a-chip (Lafleur, Jönsson, Senkbeil & Kutter, 2016) and pumps (Zhou, Zhang, Li & Wang, 2016). Kang, Yang and Huang (2002a,b) analyzed the dynamics of electroosmotic flows in a cylindrical capillary. They found that with a DC electric field applied, the electroosmotic flow velocity field generated exhibited a plug flow profile. The factors affecting the flow include the electric field strength, geometrical properties of the channel, such as diameter and length, and properties of the fluid, such as zeta potential, valence, density, viscosity and concentration. ∗

Corresponding author. E-mail addresses: [email protected] (A. Subiantoro), [email protected] (K.T. Ooi).

https://doi.org/10.1016/j.ijengsci.2019.103172 0020-7225/© 2019 Elsevier Ltd. All rights reserved.

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A. Subiantoro and K.T. Ooi / International Journal of Engineering Science 145 (2019) 103172

Nomenclature

μf υ ρ ɛ0 ɛr

ψ ω ζ

Dh e E0 kb L n0 P Re0 t T u U zv

= = = = = = = = = = = = = = = = = = = = =

dynamic fluid viscosity (Pa s) kinematics fluid viscosity (m2 /s) density (kg m−3 ) permittivity of vacuum (C/V.m) dielectric constant electrical double layer potential (V) angular frequency (Hz) zeta potential (V) hydraulic diameter of the channel (m) elementary charge (V) applied electric field strength (V/m) Boltzmann constant (J/mol.K) capillary length (m) bulk concentration of the ion (M) pressure (Pa) reference Reynolds number time (s) temperature (K) velocity of fluid (m/s) reference velocity (m/s) valence

Marcos, Yang, Wong and Ooi (2004) and Zhao, Zhang and Yang (2017) studied the dynamics of electroosmotic flows in a rectangular capillary. The velocity profile and the factors affecting the flow were found to be similar to that in a cylindrical capillary. However, there was a “corner effect” due to the overlapping of electrical double layer (EDL) at the edges. Experimental studies of such flows were carried out, among others, by Sinton and Li (2003), Sinton, Escobedo-Canseco, Ren and Li (2002), and Wang, Wong, Yang and Ooi (2007). They investigated electroosmotic flows in circular and rectangular microchannels and found that the observed flow behaviors agreed well with the theoretical predictions. Various factors that may affect electroosmotic flows in microchannels and capillaries have also been investigated. These include the existence of a pressure gradient along a rectangular microchannel (Bera & Bhattacharyya, 2013 and Vakili, Sadeghi & Saidi, 2014), an AC electric field instead of a DC field (Reppert & Morgan, 2002), irregular shape channels (Zhang, Wong, Yang & Ooi, 2005), a rotating channel (Ng & Qi, 2015 and Gheshlaghi, Nazaripoor, Kumar & Sadrzadeh, 2016), surface roughness (Keramati, Sadeghi, Saidi & Chakraborty, 2016) and temperature (Yavari, Sadeghi, Saidi & Chakraborty, 2013). A closed channel electroosmotic flow is a special case of electroosmosis where the net flow rate is zero. Marcos, Kang, Ooi, Yang and Wong (2004) and Marcos, Ooi, Yang and Wong (2005) presented a comprehensive theoretical work on electroosmotic flow in a closed-end cylindrical microchannel. It was found that the zero net flow condition was caused by an induced backpressure that generate a reverse flow to counter the electroosmotic flow effect. The flow velocity profile was parabolic. Visual observation of the electroosmotic flow in a closed-end capillary was carried out by Manz, Stilbs, Joensson, Soederman and Callaghan (1995) and it was found that the velocity profile was indeed as predicted by the theoretical model. Mishchuk and González-Caballero (2006) developed a mathematical model for electroosmotic flows in a closed end cylindrical capillary under a pulsating electric field, while the entropy generation of electroosmotic flows in closed-end microchannels has been studied by Zhao and Liu (2010). Electroosmotic flows in a closed-end rectangular capillaries have also been studied theoretically by Marcos, Yang, Ooi, Wong and Masliyah (2004). Equations to predict velocity, induced pressure gradient and electrical potential distribution were derived and used to analyze the case. However, to the authors’ knowledge, no experimental validation is available yet for this mathematical model. As mentioned, one interesting feature of the electroosmotic flow in a closed-end microchannel is the existence of an induced pressure gradient along the channel. This, among others, can be used to design a linear actuator without any moving driving component (Subiantoro, 2006). It is the objective of this paper to investigate the electroosmotically induced pressure gradient in microchannels. The investigation was focused on rectangular capillaries with DC electric field. The investigation was carried out numerically and experimentally. The model of Marcos et al. (2004) was adopted and validated. Effects of various parameters to the induced pressure gradient in steady-state conditions were investigated. These parameters were electric field strength, electrolyte concentration and capillary size, which are the three most common controllable factors in such a system.

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Fig. 1. Geometry of the capillary.

2. Mathematical model The equations used to formulate the model were based from those developed by Marcos et al. (2004). The formulation was for a rectangular microchannel with schematics as shown in Fig. 1. In the case of square capillaries, the inner width and inner height (or called inner dimensions) are equal. The main assumptions adopted were: 1. 2. 3. 4. 5. 6. 7. 8.

Rectangular cross section Laminar, fully-developed and steady flow Fluid flow is only in z-direction External DC electric field as the only body force on the fluid Incompressible and Newtonian fluid Symmetric electrolyte fluid Pressure gradient along the axial direction only Thin electrical double layer (EDL)

The flow profile was derived from the Navier-Stokes equation. With all the assumptions, the equation can be expressed as Eq. (1).



μf

∂ 2 u(x, y ) ∂ 2 u(x, y ) + ∂ x2 ∂ y2



=





dP0 zv e + 2E0 zv en0 sinh ψ (x, y ) dz kb T

(1)

The boundary conditions were:

 ∂ u  =0 ∂ x x=0

 ∂ u  =0 ∂ y y=0

u|x=b = 0

u|y=a = 0

Introducing the following dimensionless parameters:

ν=

μf ρf

u¯ =

u U

P0 =

Re0 = X=

P0

ρfU2

x Dh

E0 =

Dh U

ν

y Dh E0 Dh Re0 Y =

ζ

4ab a+b z Z= Dh Re0 z eψ = v kb T

Dh =

G=

2zv en0 ζ ρU 2

Substituting the dimensionless parameters to Eq. (1) gave Eq. (2).

∂ 2 u¯ ∂ 2 u¯ dP0 + = + GE0 sinh (X, Y ) dZ ∂ X 2 ∂Y 2 The boundary conditions became:

 ∂ u¯  =0 ∂ X X=0

u¯ |X=b/Dh = 0

 ∂ u¯  =0 ∂ Y Y =0

u¯ |Y =a/Dh = 0

(2)

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Eq. (2) and its boundary conditions can be solved using the Green’s function method, resulting in Eq. (3).

u¯ (X, Y ) = −



b/Dh



X  =0

a/Dh

Y  =0



G(X, Y |X  , Y  )



dP0 + GE0 sinh (X  , Y  ) dX  dY  dZ

(3)

The term G(X, Y|X , Y ) in Eq. (3) was the Green’s function as expressed in Eq. (4), which was obtained by solving the corresponding homogeneous equation of Eq. (2).











∞ ∞ 4Dh 2   cos 2m2−1 πbDh X  cos 2n2−1 πaDh Y 



ab × cos 2m2−1 πbDh X cos 2n2−1 πaDh Y m=1 n=1

G(X, Y |X  , Y  ) =

(4)

The EDL potential distribution was described by the Poisson-Boltzmann equation, as expressed in Eq. (5).

  ∂ 2 ψ ∂ 2 ψ 2zv en0 z v eψ + = sinh εr ε0 kb T ∂ x2 ∂ y2

(5)

The boundary conditions were

 ∂ψ  =0 ∂ x x=0

 ∂ψ  =0 ∂ y y=0

ψ|x=b = ζ

ψ|y=a = ζ

Substituting the dimensionless parameters into Eq. (5) and assuming that zeta potential was small, so the Debye-Huckel linear approximation (sinh ≈ ) can be adopted, gave Eq. (6).

∂ 2 ∂ 2 2zv 2 e2 n0 2 + = D εr ε0 kb T h ∂X2 ∂Y 2

(6)

The boundary conditions became:

 ∂  =0 ∂ X X=0

 ∂  =0 ∂ Y Y =0

|X=b/Dh = ζ where ζ =

z v eζ kb T

|Y =a/Dh = ζ

.

The solution of Eq. (6) with its boundary conditions was found to be as expressed in Eq. (7).

=

4ζ

where: K =



π

∞ ∞  (−1 )m+1 cos (αm X ) cosh (δmY )  (−1 )n+1 cos (βnY ) cosh (εn X ) δm a



+ ( 2m − 1 ) ( 2n − 1 ) cosh cosh εn b

m=1

n=1

Dh

e2 2zv 2 n0 Dh εr ε0 kb T ,

αm =

(2m−1)π Dh 2b

, βn =

(2n−1)π Dh 2a



(7)

Dh

, δm = (K 2 + αm 2 )1/2 , εn = (K 2 + βn )1/2 . 2

Substituting all the relevant expressions into Eq. (3) and assuming that the EDL was thin, a closed form expression of the flow velocity was obtained and is shown in Eq. (8).

u¯ (X, Y ) = −

64

π 2 Dh 2

∞ ∞ dP0   (−1 )m+n cos (αm X ) cos (βnY ) dZ 2m − 1 ) ( 2n − 1 ) Rmn ( m=1 n=1

⎛

∞ ∞ 32ζ GE0   ⎝ − π abDh m=1 n=1

2

(8)

⎞ (−1 )m+n b (−1 )m+n a cos (αm X ) cos (βnY )  2 + εn 2

⎠ Rmn δm (2n − 1 ) αm + αm (2m − 1 ) βn + βn

(8)

2

) ) where: Rmn = π 2 ( (2mb−1 + (2na−1 ). 2 2 When the ends of the capillaries are closed, a backflow is created by an induced pressure gradient to counter-balance the forward flow near the walls (see Fig. 2 for illustration). The corresponding net flow was zero, as expressed in Eq. (9).



b/Dh



X=−b/Dh

a/Dh

Y =−a/Dh

u¯ (X, Y )dX dY = 0

(9)

Finally, pressure gradient along the capillary length can be expressed by Eq. (10). It can be seen that the factors influencing the pressure gradient include electrolyte properties, capillary geometry, zeta potential value and electric field strength. Time is not a factor because of the steady assumption.

⎡

∞  ∞ 



dP0 =− dZ

μfU Dh 2



π D h ζ GE 0 2ab

 m=1 n=1

⎛

m+n

b(−1 )

δm 2 βn + βn

⎢ ⎜ (2m−1) (−1 )m+n m+n ⎣ (2m−1)(2n−1)Rmn ⎝ a(−1 ) +

∞  ∞  m=1 n=1

(2n−1 )

1 (2m−1) (2n−1)2 Rmn 2

⎞⎤



εn 2 αm +αm



⎟⎥ ⎠⎦ (10)

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Fig. 2. Flow velocity profile and pressure profile of electroosmotic flow in a closed-end channel (adapted from Marcos et al. (2004b)).

Fig. 3. Picture and schematic diagram of the experimental set-up.

For a more comprehensive formulation of the equations, readers are directed to the original papers (Marcos et al., 2004). It is useful to note that the forms of the equations have been rearranged for convenience here. 3. Experimental setup An experimental setup was built to validate the model. It consisted of a bundle of square capillaries, two glass reservoirs with a hollow vertical column on each, an anode and a cathode, as illustrated in Fig. 3. The capillaries were 50(±1) mm long. They were of the fused silica square type from Polymicro Technologies. Two different sizes were used, i.e. inner dimensions (i.d.) of 75(±3) μm and 100(±4) μm flat-to-flat. The electrolyte employed in the experiment was a mixture of DI water with NaCl. Two concentrations were used, i.e. 0.0 0 05 M and 0.01 M. The reservoirs are shown in Fig. 4. They were designed to keep bubbles, which were formed during the process, away from the capillaries so as not to disturb the electroosmotic flow. The design also prevented bubbles from completely covering the electrodes. The internal diameters of the vertical columns were 1 mm each. The top ends of the columns were exposed to the atmosphere. Steady-state pressure gradient along the capillaries were observed by measuring the height difference of the fluid across the two columns. The accuracy was within 1 mm or around 0.2 kPa/m. The electrodes were platinum wires. To provide the required DC electric field, a Spellman CZE10 0 0R high DC voltage power source was used. The potential difference values tested were between 50 0–30 0 0 V. Monitoring and control of the power source were carried out with a computer, a BNC-2110 data acquisition card from National Instrument and the LabView software. 4. Results and discussion Experimental data (represented by dots) and theoretical predictions (represented by lines) of induced pressure gradient along the capillaries at various DC electric field strengths are shown in Fig. 5. At least three readings were taken at every data point. The parameters used for the theoretical predictions are tabulated in Table 1. The values of zeta potential were taken by data-fitting experimental results at the strongest electric field tested. They were then assumed constant throughout the range of electric field tested. In general, Fig. 5 shows that the theoretical predictions were in fairly good agreement with the experimental data, with an average deviation of 18%. Assumptions adopted for zeta potential are thought to be the main causes of discrepancies.

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Fig. 4. Detailed drawings of the reservoir.

Table 1 Parameters for theoretical predictions. Item

Value

Reference velocity Valence of fluid Dielectric constant of fluid Density of fluid Dynamic viscosity of fluid Temperature of fluid Length of capillary

0.001 m/s 1 80 1000 kg/m3 0.0009 Pa.s 293 K 0.05 m

Firstly, the model assumed that zeta potential is very small (Debye-Huckel approximation), while the estimated values used in the study are relatively high. Secondly, the zeta potential values were assumed unaffected by electric field strength and were taken at the highest electric fields. This resulted in the high discrepancies observed in low electric field regions. Moreover, the effects of Joule heating and electrolysis processes may affect the zeta potential values (Venditti, Xuan & Li, 2006)). The accuracy of the theoretical model may, therefore, be improved by coupling it with a good zeta potential estimation model. Fig. 5 also shows the effect of electric field strength variation to the steady-state induced pressure gradient. Induced pressure gradient was found to be linearly proportional to electric field strength. The experimental data showed that induced pressure gradient increased at a rate of 2 kPa/m for every 10 kV/m increment of field strength, almost independent of the electrolyte concentration and capillary size. In practice, electric field strength can be increased either by increasing the electrical potential difference along the capillary or by shortening the capillary. The effect of electrolyte concentration to the steady-state induced pressure gradient is shown in Fig. 6. Fig. 6(a) and (b) show the experimental data. The error bars include data spread and experimental inaccuracies as described in Section 3, while the dotted lines are trend-lines. The experimental data showed that changing electrolyte concentration from 0.0 0 05 M to 0.01 M did not significantly affect the pressure gradient. The changes were still within the error bars. This finding was in agreement with the numerical data in Fig. 6(c), which shows numerical data at electric field strength of 30 kV/m. The zeta potential values were −80 mV and −120 mV for capillary sizes of 75 μm and 100 μm, respectively, following the values in Fig. 5. Numerical data also suggested that varying electrolyte concentration from 0.0 0 05 M to 0.03 M did not affect the induced pressure gradient significantly, regardless of the capillary size. The effect of capillary size variation to the steady-state induced pressure gradient is shown in Fig. 7. Fig. 7(a) and (b) show experimental data, while Fig. 7(c) shows numerical data at electric field strength of 30 kV/m and a constant zeta potential of −100 mV, which was the average value observed in the experiment. It can be seen that, in general, reducing

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Fig. 5. Variation of pressure gradient across the ends of the capillaries with electric field strength of 75 μm and 100 μm i.d. square capillaries in 0.0 0 05 M and 0.01 M NaCl solutions.

Fig. 6. Variation of induced pressure gradient with electrolyte concentration.

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Fig. 7. Variation of induced pressure gradient with capillary size.

capillary size increased induced pressure gradient. The rate of increase varied exponentially with smaller capillary sizes. Reducing the capillary size from 100 μm to 75 μm at electric field of 30 kV/m and electrolyte concentration of 0.01 M increased the induced pressure gradient by 1.5 kPa/m experimentally. Numerically, the increase was 1.3 kPa/m, which was in good agreement with the experimental value.

5. Conclusion Pressure gradient induced by electroosmotic effect in rectangular microchannels under DC electric field was investigated numerically and experimentally in this study. A previously developed theoretical model (Marcos et al. 2004) was adopted and validated experimentally. Effects of various parameters, including electric field strength, electrolyte concentration and capillary size, to the induced pressure gradient in steady-state were investigated. An experimental setup was built for the study. It includes 50 mm long square capillaries, two glass reservoirs with a hollow vertical column on each and electrodes. The electrolyte was NaCl solution. Steady-state pressure gradient along the capillaries were observed by measuring the height difference of the fluid across the two columns. The range of electric field tested was between 10–60 kV/m. The capillary sizes tested were 75 μm and 100 μm. The electrolyte concentrations experimented were 0.0 0 05 M and 0.01 M. Validation of the model with experimental data showed that the predictions were fairly accurate in the range of operating conditions tested, with an average deviation of 18%. From the parametric studies, it was found that the steady-state induced pressure gradient was linearly proportional to electric field strength. The experimental data showed that induced pressure gradient increased at a rate of 2 kPa/m for every 10 kV/m increment of field strength, almost independent of the electrolyte concentration and capillary size. Induced pressure gradient was found unaffected significantly by electrolyte concentration in the range of conditions tested. The experimental data showed that changing electrolyte concentration from 0.0 0 05 M to 0.01 M did not significantly affect the pressure gradient. This finding was in agreement with the theoretical predictions, which suggested that varying electrolyte concentration even further from 0.0 0 05 M to 0.03 M still did not affect the induced pressure gradient significantly, regardless of capillary size. Reducing capillary size was found to increase induced pressure gradient. The rate of increase varied exponentially with smaller capillary sizes. Reducing the capillary size from 100 μm to 75 μm at electric field of 30 kV/m and electrolyte concentration of 0.01 M increased the induced pressure gradient by 1.5 kPa/m experimentally and 1.3 kPa/m numerically, which showed that they were in good agreement.

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Acknowledgement The authors would like to acknowledge the support received from the technical team in Thermal Fluids Research Laboratory, Nanyang Technological University, Singapore for the completion of this project. References Bera, S., & Bhattacharyya, S. (2013). On mixed electroosmotic-pressure driven flow and mass transport in microchannels. International Journal of Engineering Science, 62, 165–176. Gheshlaghi, B., Nazaripoor, H., Kumar, A., & Sadrzadeh, M. (2016). Analytical solution for transient electroosmotic flow in a rotating microchannel. Royal Society of Chemistry Advances, 6, 17632–17641. Kang, Y., Yang, C., & Huang, X. (2002a). Dynamic aspects of electroomotic flow in a cylindrical microcapillary. International Journal of Engineering Science, 40(20), 2203–2221. Kang, Y., Yang, C., & Huang, X. (2002b). Electroosmotic flow in a capillary annulus with high zeta potentials. Journal of Colloid and Interface Science, 253(2), 285–294. Keramati, H., Sadeghi, A., Saidi, M. H., & Chakraborty, S. (2016). Analytical solutions for thermo-fluidic transport in electroosmotic flow through rough microtubes. International Journal of Heat and Mass Transfer, 92, 244–251. Lafleur, J. P., Jönsson, A., Senkbeil, S., & Kutter, J. P. (2016). Recent advances in lab-on-a-chip for biosensing applications. Biosensors and Bioelectronics, 76, 213–233. Manz, B., Stilbs, P., Joensson, B., Soederman, O., & Callaghan, P. T. (1995). NMR imaging of the time evolution of electroosmotic flow in a capillary. The Journal of Physical Chemistry, 99(29), 11297–11301. Marcos, Kang, Y. J., Ooi, K. T., Yang, C., & Wong, T. N. (2004). Frequency dependent velocity and vorticity fields of electroosmotic flow in a closed-end cylindrical microchannel. Journal of Micromechanics and Microengineering, 15, 301–312. Marcos, Ooi, K. T., Yang, C., Chai, J. C., & Wong, T. N. (2005). Developing electro-osmotic flow in closed-end micro-channels. International Journal of Engineering Science, 43(17–18), 1349–1362. Marcos, Yang, C., Ooi, K. T., Wong, T. N., & Masliyah, J. H. (2004). Frequency dependent laminar electroosmotic flow in a closed-end rectangular microchannel. Journal of Colloid and Interface Science, 275(2), 679–698. Marcos, Yang, C., Wong, T. N., & Ooi, K. T. (2004). Dynamic aspects of electroosmotic flow in rectangular microchannels. International Journal of Engineering Science, 42(13–14), 1459–1481. Mishchuk, N. A., & González-Caballero, F. (2006). Nonstationary electroosmotic flow in closedcylindrical capillaries. Electrophoresis, 27, 661–671. Ng, C. O., & Qi, C. (2015). Electro-osmotic flow in a rotating rectangular microchannel. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471, 20150200. Reppert, P. M., & Morgan, F. D. (2002). Frequency dependent electroosmosis. Journal of Colloid and Interface Science, 254, 372–383. Sinton, D., Escobedo-Canseco, C., Ren, L., & Li, D. (2002). Direct and indirect electroosmotic flow velocity measurements in microchannels. Journal of Colloid and Interface Science, 254(1), 184–189. Sinton, D., & Li, D. (2003). Electroosmotic velocity profiles in microchannels. Colloids and Surfaces A: Physicochemical Engineering Aspects, 222(1–3), 273–283. Subiantoro, A. (2006). Development of an electrokinetic actuator and its applications in compressors Master of Engineering Thesis. Singapore: Nanyang Technological University. Vakili, M. A., Sadeghi, A., & Saidi, M. H. (2014). Pressure effects on electroosmotic flow of power-law fluids in rectangular microchannels. Theoretical and Computational Fluid, 28, 409–426. Venditti, R., Xuan, X., & Li, D. (2006). Experimental characterization of the temperature dependence of zeta potential and its effect on electroosmotic flow velocity in microchannels. Microfluidics and Nanofluidics, 2(6), 493–499. Wang, C., Wong, T. N., Yang, C., & Ooi, K. T. (2007). Characterization of electroosmotic flow in rectangular microchannels. International Journal of Heat and Mass Transfer, 50(15–16), 3115–3121. Yavari, H., Sadeghi, A., Saidi, M. H., & Chakraborty, S. (2013). Temperature rise in electroosmotic flow of typical non-newtonian biofluids through rectangular microchannels. Journal of Heat Transfer, 136(3) 031702–031702–11. Zhang, Y., Wong, T. N., Yang, C., & Ooi, K. T. (2005). Electroosmotic flow in irregular shape microchannels. International Journal of Engineering Science, 43(19–20), 1450–1463. Zhao, C., Zhang, W., & Yang, C. (2017). Dynamic electroosmotic flows of power-law fluids in rectangular microchannels. Micromachines, 8(2), 34. Zhao, L., & Liu, L. H. (2010). Entropy generation analysis of electro-osmotic flow in open-end and closed-end micro-channels. International Journal of Thermal Sciences, 49(2), 418–427. Zhou, C., Zhang, H., Li, Z., & Wang, W. (2016). Chemistry pumps: A review of chemically powered micropumps. Lab on a Chip, 16, 1791–1811.