Experimental investigations of liquid flow in rib-patterned microchannels with different surface wettability

Abstract

The effects of rib-patterned surfaces and surface wettability on liquid flow in microchannels were experimentally investigated in this study. Microchannels were fabricated on single-crystal silicon wafers by photolithographic and wet-etching techniques. Rib structures were patterned in the silicon microchannel, and the surface was chemically treated by trichlorosilane to create hydrophobic condition. Experiments with water as the working fluid were performed with these microchannels over a wide range of Reynolds numbers between 110 and 1914. The results for the rib-patterned microchannels showed that the friction factor with the hydraulic diameter based on the rib-to-upper-wall height was lower than that predicted from incompressible theory with the same height. The friction factor-Reynolds number products for the hydrophobic condition increased as Reynolds number increased in the laminar flow regime. The experimental results were also compared with the predictive expressions from the literature, and it was found that the experimental data for the small rib/cavity geometry was in good agreement with those in the literature.

Appendix: analytical solution for the fluid flow through microchannels with one and two wall slip conditions

The goal is to determine the values of fRe for the fluid flow in microchannels with one and two wall slip conditions. The one-wall slip problem will be solved is assumed to be a two-dimensional, incompressible flow through a channel with a height of 2b (=H) characterized by an unknown slip length, λ, at the lower wall. By solving the Navier–Stokes equations, the streamwise velocity profile is obtained as followed (Teo and Khoo 2009).

$$ u = {\frac{G b^2}{\mu}} \left[ {\frac{1}{2}} \left(1 - {\frac{y^2}{b^2}} \right) + {\frac{\lambda}{2b}} \left( 1 + {\frac{y}{b}} \right) \right] $$

(12)

where G is the pressure gradient driving the flow through the channel. Also, the mean velocity of the flow, u _m, is given by

$$ u_{\rm m} = {\frac{G b^2}{\mu}} \left[ {\frac{1}{3}} + {\frac{\lambda}{2b}} \right] $$

(13)

Therefore, the fRe is obtained as followed

$$ fRe_{I} = {\frac{24}{1+ {\frac{3 \lambda}{2b}} }} $$

(14)

Also, by substituting Eqs. 6 and 7 into Eq. 14, the fRe for the one-wall slip condition is described as,

$$ fRe_{{\rm hydrophilic}, I} ={\frac{fRe _{\rm th}}{1+ {\frac{3w}{2 \pi D_{\rm h}}} \ln \left[ \sec (F_{\rm c} \pi / 2.53) \right] }} $$

(15)

$$ fRe_{{\rm hydrophobic}, I} ={\frac{fRe _{\rm th}}{1+ {\frac{3w}{\pi D_{\rm h}}} \ln \left[ \sec (F_{\rm c} \pi / 2) \right] \left[ c_1 + {\frac{c_2}{(Re_w + c_3)^2 +c_4 }} \right] }} $$

(16)

Following the same approach above, the fRe for two-wall slip condition is described as

$$ fRe_{II} ={\frac{24}{1+ {\frac{3 \lambda}{b}} }} $$

(17)

Also, the predictive expressions for the wetting and superhydrophobic states with transverse rib geometry can be derived as

$$ fRe_{{\rm hydrophilic}, II} = {\frac{fRe _{\rm th}}{1+ {\frac{3w}{\pi D_{\rm h}}} \ln \left[ \sec (F_{\rm c} \pi / 2.53) \right] }} $$

(18)

$$ fRe_{{\rm hydrophobic}, II} = {\frac{fRe _{\rm th}}{1+ {\frac{6w}{\pi D_{\rm h}}} \ln \left[ \sec (F_{\rm c} \pi / 2) \right] \left[ c_1 + {\frac{c_2} {(Re_w + c_3)^2 +c_4 }} \right] }} $$

(19)

Comparing the values of fRe between Eqs. 15–18 and 16–19, it is clear that the second term of the denominator decreases by half for the one-wall slip condition for both hydrophilic and hydrophobic states.