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

Crowdy, Darren

doi:10.1017/jfm.2016.88

Cited by 47 publications

(65 citation statements)

References 13 publications

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“…16) mixing is expected to enhance surfactant fluxes, potentially inducing a qualitative change in Marangoni stress. To enable general predictions, it would be important to extend existing effective slip models (27,29,34) to include surfactant.…”

Section: Pressure-relaxation Experiments For Surfactant Effectmentioning

confidence: 99%

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Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces

Peaudecerf

Landel

Goldstein

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

162

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Superhydrophobic surfaces (SHSs) have the potential to achieve large drag reduction for internal and external flow applications. However, experiments have shown inconsistent results, with many studies reporting significantly reduced performance. Recently, it has been proposed that surfactants, ubiquitous in flow applications, could be responsible by creating adverse Marangoni stresses. However, testing this hypothesis is challenging. Careful experiments with purified water already show large interfacial stresses and, paradoxically, adding surfactants yields barely measurable drag increases. To test the surfactant hypothesis while controlling surfactant concentrations with precision higher than can be achieved experimentally, we perform simulations inclusive of surfactant kinetics. These reveal that surfactant-induced stresses are significant at extremely low concentrations, potentially yielding a no-slip boundary condition on the air-water interface (the "plastron") for surfactant concentrations below typical environmental values. These stresses decrease as the stream-wise distance between plastron stagnation points increases. We perform microchannel experiments with SHSs consisting of streamwise parallel gratings, which confirm this numerical prediction, while showing near-plastron velocities significantly slower than standard surfactant-free predictions. In addition, we introduce an unsteady test of surfactant effects. When we rapidly remove the driving pressure following a loading phase, a backflow develops at the plastron, which can only be explained by surfactant gradients formed in the loading phase. This demonstrates the significance of surfactants in deteriorating drag reduction and thus the importance of including surfactant stresses in SHS models. Our time-dependent protocol can assess the impact of surfactants in SHS testing and guide future mitigating designs.superhydrophobic surface | drag reduction | surfactant | Marangoni stress | plastron S uperhydrophobic surfaces (SHSs) combine hydrophobic surface chemistry and micro-or nanoscale patterning to retain a network of air pockets when exposed to a liquid (e.g., reviews in refs. 1-3). Because a large portion of the interface between the solid wall and the liquid is replaced by an air-liquid interface, which can be considered almost as a shear-free surface (known as a "plastron"), SHSs could be used to obtain significant drag reduction in fluid flow applications (4, 5). Microchannel tests have recorded drag reductions of over 20% (e.g., refs. 6-11) and rheometer tests reported slip lengths of up to 185 µm (12). Turbulent flow experiments have reduced drag by up to 75% (13-16). However, a wide range of experiments have provided inconsistent results, with several studies reporting little or no drag reduction (16)(17)(18)(19)(20)(21)(22)(23)(24)(25).A key step toward solving this puzzle has come with the realization that surfactants could induce Marangoni stresses that impair drag reduction. This was first hypothesized to account for experiments that rev...

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Section: Pressure-relaxation Experiments For Surfactant Effectmentioning

confidence: 99%

“…This was first hypothesized to account for experiments that revealed little measurable slip (23,24), in contradiction with available theoretical predictions based on the absence of surfactants (26)(27)(28)(29)(30)(31)(32)(33)(34). Following this hypothesis, surfactants naturally present in water would adsorb onto the air-water interface, as sketched in Fig.…”

mentioning

confidence: 98%

Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces

Peaudecerf

Landel

Goldstein

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

162

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“…Those prior results, coupled with the complementary results derived here for the same class of superhydrophobic surface geometries, allow us to test this assumption. For longitudinal flow, Crowdy (2016) has derived a formula for the effective slip length for exactly circular no-shear menisci. The effective slip length in (1.4) of Crowdy (2016) is plotted as a function of φ in figure 4 along with the new slip length result (1.5) for a no-slip surface with the same fixed value of 2c/L = 0.5, corresponding to the no-shear fraction used in the experiments of Bolognesi et al (2014).…”

Section: Comparison Of No-slip and No-shear Assumptionsmentioning

confidence: 99%

Effective slip lengths for immobilized superhydrophobic surfaces

Crowdy

2017

J. Fluid Mech.

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Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around 47 • , for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below 90 • can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.

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“…These surfaces reduce friction due to the presence of free surfaces, or menisci, spanning interstitial grooves between protrusions in the substructure of the surface. Much theoretical [3][4][5][6][7][8][9][10][11] , experimental [12][13][14] and numerical work [15][16][17][18] has been done to understand the friction properties of these surfaces. A paper by Philip 3 , which solves a varierty of pertinent mixed boundary value problems, has become a wellknown reference in this area but it only deals with flat menisci and under the assumption that they are shearfree.…”

Section: Introductionmentioning

confidence: 99%

“…A paper by Philip 3 , which solves a varierty of pertinent mixed boundary value problems, has become a wellknown reference in this area but it only deals with flat menisci and under the assumption that they are shearfree. There has been recent efforts to quantify hydrodynamic slip in more general situations where, for example, the menisci are curved [5][6][7][8]14,18 and where the effect of a second subphase fluid is incorporated 10,11,[19][20][21] . Special superhydrophobic microfluidic devices even exist with the capability of actively controlling the meniscus curvature to "tune" surfaces to have desired friction properties 22 .…”

Section: Introductionmentioning

confidence: 99%

Slip length for transverse shear flow over a periodic array of weakly curved menisci

Crowdy

2017

Physics of Fluids

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By exploiting the reciprocal theorem of Stokes flow we find an explicit expression for the first order slip length correction, for small protrusion angles, for transverse shear over a periodic array of curved menisci. The result is the transverse flow analogue of the longitudinal flow result of Sbragaglia & Prosperetti [Phys. Fluids, 19, 043603, (2007)]. For small protrusion angles, it also generalizes the dilute-limit result of Davis & Lauga [Phys. Fluids, 21, 113101 (2009)] to arbitrary no-shear fractions. While the leading order slip lengths for transverse and longitudinal flow over flat no-shear slots are well-known to differ by a factor of 2, the first order slip length corrections for weakly protruding menisci in each flow are found to be identical.

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

Cited by 47 publications

References 13 publications

Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces

Traces of surfactants can severely limit the drag reduction of superhydrophobic surfaces

Effective slip lengths for immobilized superhydrophobic surfaces

Slip length for transverse shear flow over a periodic array of weakly curved menisci

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