# Effect of interface wettability on the flow characteristics of liquid in smooth microchannels

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## Abstract

Rapid advances in microfluidic devices have induced interest in the study of the microscale flow mechanism. However, the experimental results of microscale flow often deviate from the classical theory, and we attribute this deviation to the changing liquid viscosity in the microchannels. Because of the effect of the solid–liquid intermolecular force, the viscosity of the liquid near the walls is different from the bulk viscosity. Based on molecular theory and wetting theory, we propose a modified apparent viscosity model. The apparent viscosity of the liquid in microchannels increases with the increase in wettability and decreases with the increase in distance from the wall and the increase in drive pressure. The apparent viscosity near the hydrophilic wall is higher than the bulk viscosity, which increases the flow friction in the microchannels. To validate this model, we experimentally investigate the frictional characteristic of a deionized water flow in smooth parallel-plate microchannels with different wettabilities and heights of approximately 20 and 50 \(\upmu \hbox {m}\). The results indicate that the friction factor is higher than that predicted by the classical theory. Such a difference increases with increasing wettability and decreases with increasing hydraulic diameter and pressure drop, which is consistent with the results of theoretical analysis. The apparent viscosity calculated by the apparent viscosity model notably fit the experimental results, with a relative difference of less than ± 2.1%.

## List of symbols

- \(A_{\mathrm{ch}}\)
Cross-sectional area of the channel (\(\hbox {m}^{2}\))

- \(A_{\mathrm{p}}\)
Cross-sectional area of the plenum (\(\hbox {m}^{2}\))

- \(D_{\mathrm{h}}\)
Hydraulic diameter (\(\upmu \hbox {m}\))

- \(F_{\mathrm{LL}}\)
Liquid–liquid intermolecular forces (\(\hbox {N}\))

- \(F_{\mathrm{LS}}\)
Solid–liquid intermolecular forces (\(\hbox {N}\))

*f*Darcy’s friction factor

*H*Height of the microchannel (\(\upmu \hbox {m}\))

- \(K_{90}\)
Bend loss coefficient

- \(K_{\mathrm{c}}\)
Contraction loss coefficient

- \(K_{\mathrm{e}}\)
Expansion loss coefficient

*k*Coefficient in Eq. (3)

*L*Length of the microchannel (\(\hbox {mm}\))

*n*Coefficient in Eq. (3)

- \(\Delta P\)
Frictional pressure drop (Pa)

- \(\Delta P_{\mathrm{H1}}\)
Inlet hydrostatic pressure losses (Pa)

- \(\Delta P_{\mathrm{H2}}\)
Outlet hydrostatic pressure losses (Pa)

- \(P_{\mathrm{in}}\)
Inlet pressure (Pa)

- \(P_{\mathrm{out}}\)
Outlet pressure (Pa)

*u*Velocity in the

*x*-direction (\(\hbox {m}/\hbox {s}\))- \(u_{\mathrm{m}}\)
Mean velocity (\(\hbox {m}/\hbox {s}\))

*Re*Reynolds number

*W*width of the microchannel (\(\hbox {mm}\))

*x*,*y*Cartesian coordinates (\(\hbox {m}\))

- \(\delta \)
Mean distance between two adjacent molecules (\(\hbox {nm}\))

- \(\mu _0\)
Bulk viscosity [\({\hbox {kg}}/({{\hbox {m}\,\hbox {s}}})\)]

- \(\mu _{\mathrm{a}}\)
Apparent viscosity [\({\hbox {kg}}/({\hbox {m}\,\hbox {s}})\)]

- \(\bar{{\mu }}_{\mathrm{a}}\)
Average apparent viscosity [\(\hbox {kg}/(\hbox {m}\,\hbox {s})\)]

- \(\theta \)
Contact angle (\(^{\circ }\))

- \(\rho \)
Density (\({\hbox {kg}}/{\hbox {m}^{3}}\))

- \(\sigma _{\mathrm{L}}\)
Surface tension of the liquid (N/m)

- \(\sigma _{\mathrm{L}}^0\)
Internal surface force of the liquid (N/m)

- \(\sigma _{\mathrm{LS}}\)
Interfacial tension of the liquid (N/m)

- \(\sigma _{\mathrm{LS}}^0\)
External surface force of the liquid (N/m)

- \(\xi \)
Parameter in Eq. (2)

## Subscripts

- \(\exp \)
Experimental value

- th
Theoretical value

- 1
Value of the top wall

- 2
Value of the bottom wall

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## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11702320) and the Training Program of the Major Research Plan of the National Natural Science Foundation of China (No. 91741107).

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