Effective slip in pressure-driven Stokes flow

Abstract

Nano-bubbles have recently been observed experimentally on smooth hydrophobic surfaces; cracks on a surface can likewise be the site of bubbles when partially wetting fluids are used. Because these bubbles may provide a zero shear stress boundary condition and modify considerably the friction generated by the solid boundary, it is of interest to quantify their influence on pressure-driven flow, with particular attention given to small geometries. We investigate two simple configurations of steady pressuredriven Stokes flow in a circular pipe whose surface contains periodically distributed regions of zero surface shear stress. In the spirit of experimental studies probing slip at solid surfaces, the effective slip length of the resulting flow is evaluated as a function of the degrees of freedom describing the surface heterogeneities, namely the relative width of the no-slip and no-shear stress regions and their distribution along the pipe. Comparison of the model with experimental studies of pressure-driven flow in capillaries and microchannels reporting slip is made and a possible interpretation of the experimental results is offered which is consistent with a large number of distributed slip domains such as nano-size and micron-size nearly flat bubbles coating the solid surface. Further, the possibility is suggested of a shear-dependent effective slip length, and an explanation is proposed for the seemingly paradoxical behaviour of the measured slip length increasing with system size, which is consistent with experimental results to date.

Superhydrophobic drag reduction in laminar flows: a critical review

“…Slip lengths of 143 and 61 nm were reported when the grates were parallel and transverse, respectively, to the liquid flow. This ratio was consistent with the 2:1 ratio theoretically predicted by Lauga and Stone (2003). Interestingly, their measured slip lengths were somewhat larger than the theoretical values.…”

“…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.…”

“…Analytical models and numerical simulations are useful tools in understanding slip lengths in laminar flows on a few simple surface patterns such as grates (ridges, trenches) (Philip 1972a, b;Lauga and Stone 2003;Belyaev and Vinogradova 2010a;Asmolov and Vinogradova 2012), posts (pillars) (Ybert et al 2007;Davis and Lauga 2010), and holes (Ybert et al 2007;Davis and Lauga 2009a), as shown in Fig. 2.…”

“…On grates parallel to a liquid flow, the slip length is expressed as a function of the gas fraction and the structural pitch (Philip 1972b;Lauga and Stone 2003): where L is the structural pitch of a grate pattern (i.e., center-to-center distance between two adjacent grates). On grates transverse to a liquid flow, a slip length is given as…”

Microfluidics: The No-Slip Boundary Condition

“…Since the viscosity depends on temperature, high shear rates could lead to the possibility of viscous heating [56], a flow-induced reduction in viscosity, which could be interpreted as an apparent slip. Assuming a traditional exponential law for the viscosity µ = µ 0 exp[−β(T − T 0 )/T 0 ] and flow in a circular capillary of radius a, the apparent slip length due to viscous heating would be [101] …”