Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves

Abstract

Superhydrophobic surfaces have been demonstrated to be capable of reducing fluid resistance in microand nanofluidic applications. The objective of this paper is to present analytical solutions for the Stokes flow through microchannels employing superhydrophobic surfaces with alternating micro-grooves and ribs. Results are presented for both cases where the micro-grooves are aligned parallel and perpendicular to the flow direction. The effects of patterning the grooves on one or both channel walls are also analyzed. The reduction in fluid resistance has been quantified in terms of a dimensionless effective slip length, which is found to increase monotonically with the shearfree fraction and the periodic extent of each groove-rib combination normalized by the channel half-height. Asymptotic relationships have been derived for the normalized effective slip length corresponding to large and small limiting values of the shear-free fraction and the normalized groove-rib period. A detailed comparison has been made between transverse and longitudinal grooves, patterned on one or both channel walls, to assess their effectiveness in terms of enhancing the effective slip length. These comparisons have been carried out for small and large limiting values, as well as finite values of the shear-free fraction and normalized groove-rib period. Results for the normalized effective slip length corresponding to transverse and longitudinal grooves are further applied to model the Stokes flow through microchannels employing superhydrophobic surfaces containing a periodic array of micro-grooves inclined at an angle to the direction of the applied pressure gradient. Results are presented for the normalized effective slip lengths parallel to the direction of the applied pressure gradient and the normalized cross flow rate perpendicular to the direction of the applied pressure gradient.

Superhydrophobic drag reduction in laminar flows: a critical review

“…Following early work (Philip 1972a, b), many analytical models about slip lengths on simple SHPo surfaces have been developed (Lauga and Stone 2003;Ybert et al 2007;Sbragaglia and Prosperetti 2007a, b;Davis and Lauga 2009a, b;Feuillebois et al 2009;Belyaev and Vinogradova 2010a, b;Davis and Lauga 2010;Asmolov and Vinogradova 2012;Cottin-Bizonne et al 2012) and corroborated by numerical simulations (Cottin-Bizonne et al 2003Priezjev et al 2005;Hendy and Lund 2007;Biben and Joly 2008;Hyvalouma and Harting 2008;Teo and Khoo 2008;Cheng et al 2009;Wang 2009, 2010;Teo and Khoo 2010). By fabricating near-perfect microstructures in an extremely clean condition, Lee et al (2008) succeeded to measure the slip length as a function of surface features, finally verifying the early theoretical models (Lauga and Stone 2003;Ybert et al 2007) quantitatively and bringing a conclusion to the issue.…”

Longitudinal pressure-driven flows between superhydrophobic grooved surfaces: Large effective slip in the narrow-channel limit

“…With gas bubbles trapped in a Cassie state within the microstructure pockets, the overall resistance to flow may be reduced compared to that about a homogeneous solid boundary. A common measure of drag reduction in channel flows, employed in both experimental [1,2] and theoretical [3][4][5] work, is the effective slip length.…”

“…The corresponding symmetric configuration, where longitudinal no-shear stripes are prescribed at the two channel walls, was analyzed by Teo & Khoo [5]. Similarly to Ref.…”

“…In particular, we focus upon the symmetric configuration, where Teo & Khoo [5] have observed large superhydrophobic effect for small numerical values of κ -a phenomenon which has no counterpart in the asymmetric configuration. Being unable to provide an asymptotic approximation for the large slip length, Teo & Khoo [5] have proposed instead an empirical correlation involving a κ −2 -like divergence of λ.…”

“…Being unable to provide an asymptotic approximation for the large slip length, Teo & Khoo [5] have proposed instead an empirical correlation involving a κ −2 -like divergence of λ. With the goal of illuminating this slip-length singularity and providing a systematic approximation for the sip length, we here address the narrow-channel limit using singular perturbations.…”