RESEARCH PAPER Effects of interface curvature on Poiseuille ﬂow through microchannels and microtubes containing superhydrophobic surfaces with transverse grooves and ribs C. J. Teo • B. C. Khoo Received: 22 June 2013 / Accepted: 15 February 2014 Ó Springer-Verlag Berlin Heidelberg 2014 Abstract This paper presents numerical results pertaining to the effects of interface curvature on the effective slip behavior of Poiseuille ﬂow through microchannels and microtubes containing superhydrophobic surfaces with transverse ribs and grooves. The effects of interface curvature are systematically investigated for different normalized channel heights or tube diameters, shear-free fractions, and ﬂow Reynolds numbers. The numerical results show that in the low Reynolds number Stokes ﬂow regime, when the channel height or tube diameter (normalized using the groove–rib spacing) is sufﬁciently large, the critical interface protrusion angle at which the effective slip length becomes zero is h c & 62°–65°, which is independent of the shear-free fraction, ﬂow geometry (channel and tube), and ﬂow driving mechanism. As the normalized channel height or tube diameter is reduced, for a given shear-free fraction, the critical interface protrusion angle h c decreases. As inertial effects become increasingly dominant corresponding to an increase in Reynolds number, the effective slip length decreases, with the tube ﬂow exhibiting a more pronounced reduction than the channel ﬂow. In addition, for the same corresponding values of shear-free fraction, normalized groove–rib spacing, and interface protrusion angle, longitu- dinal grooves are found to be consistently superior to trans- verse grooves in terms of effective slip performance. 1 Introduction Due to the rapid advancement in microfabrication tech- niques and processes, numerous microﬂuidic devices for a diverse range of scientiﬁc and engineering applications have been successfully developed. Many of these micro- ﬂuidic devices contain complex networks of microchannels or microtubes. It is well known that the pressure gradient required for maintaining a ﬁxed volumetric ﬂow rate through a device scales inversely as the fourth power of the characteristic cross-sectional length scale (Lauga and Stone 2003; Davis and Lauga 2009). The pressure drop require- ments for maintaining a speciﬁed ﬂow rate thus increase appreciably corresponding to a reduction in the cross-sec- tional dimensions of the microdevices. It is therefore imperative to devise novel techniques for alleviating the excessive pressure drop requirements. One particularly promising technology that has proven to be effective is the use of superhydrophobic surfaces. Such surfaces consist of regular arrangements of microsize features, such as ridges or posts, patterned on a solid substrate, which is subse- quently coated with a thin layer of hydrophobic material. The hydrophobicity of the surface prevents the ﬂowing liquid from fully penetrating the cavities in between the microprotrusions. A Cassie state is thus maintained, and pockets of air or vapor become trapped in the cavities, which reduce the effective contact area between the ﬂow- ing liquid and the solid wall. This results in a concomitant reduction in ﬂow resistance and thus pressure drop requirements. A considerable amount of theoretical (Philip 1972a, b; Lauga and Stone 2003; Maynes et al. 2007; Sbragaglia and Prosperetti 2007; Teo and Khoo 2009; Ng et al. 2010; Ng and Wang 2011) and numerical studies (Cheng et al. 2009; Martell et al. 2009; Maynes et al. 2007; Priezjev et al. 2005) has previously been carried out to investigate the use of superhydrophobic surfaces for reducing the ﬂow resis- tance through microchannels and microtubes. Many researchers have assumed undeformed or ﬂat liquid–gas C. J. Teo (&)  B. C. Khoo Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore e-mail: mpeteocj@nus.edu.sg 123 Microﬂuid Nanoﬂuid DOI 10.1007/s10404-014-1367-1