Flow past superhydrophobic surfaces containing longitudinal grooves: effects of interface curvature

Abstract

This article considers Couette and Poiseuille flows past superhydrophobic surfaces containing alternating micro-grooves and ribs aligned longitudinally to the flow. The effects of interface curvature on the effective slip length are quantified for different shear-free fractions and groove-rib spatial periods normalized using the channel height. The numerical results obtained demonstrate the importance of considering interface curvature effects in ascertaining the effective slip length. The effective slip length and performance of longitudinal grooves are compared with those corresponding to transverse grooves, for which analytical results are available for small shear-free fractions and normalized groove-rib periodic spacing. For the same shear-free fraction and interface protrusion angle, the effective slip length corresponding to the Poiseuille flow is found to be strongly affected by the normalized groove-rib spacing, in contrast to the Couette flow. For the Poiseuille flow, when the interface deforms by large protrusion angles into the liquid phase, the effective slip length approaches zero or becomes negative for large values of shear-free fraction and normalized groove-rib spacing due to significant flow blockage effects.

Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci

“…We also write down the associated flow field. For no-shear fractions as high as c/l = 0.75 (1.3) gives agreement with maximum relative errors (across protrusion angles) of 6-7 % compared to the numerical results of Teo & Khoo (2010). Clearly (1.3) reduces to (1.2a−c) to leading order in cδ (or, equivalently, to leading order in the no-shear fraction c/l).…”

“…The quantity λ 1 as given by (3.2) is now the required slip length at this order of approximation and produces the result (1.3). Teo & Khoo (2010) report the slip lengths λ TK , say, with the renormalizations…”

“…Figure 2 shows λ TK , as a function of protrusion angle θ, for c/l = 0.1, 0.25, 0.5, 0.75 and 0.9 together with corresponding data points from Teo & Khoo (2010), which serve as our benchmark solution. The maximum relative error of this approximation for c/l = 0.75 is between 6 and 7 % and for c/l = 0.9 is around 25 %.…”

“…Their analysis is valid for any no-shear fraction, but it is limited to small protrusion angles of the menisci (close to flat). Numerical calculations of the slip lengths associated with such bubble mattresses have been performed by Teo & Khoo (2010) and Ng & Wang (2011).…”

Effective slip lengths for immobilized superhydrophobic surfaces

“…Figure 5 shows that for angles well above critical, a no-slip meniscus can be significantly more slippery than a no-shear surface. Indeed, by changing the normalized groove width 2c/L in the interval [0, 0.47] (where Teo & Khoo (2016) have shown the formula of Davis & Lauga (2009) to be 10 % accurate), we find that for downward protrusion angles around 80 • , the surface can be almost twice as slippery if immobilized as if it is free of shear (cf. Kim & Hidrovo 2012).…”