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Drag Reduction for Viscous Laminar Flow on Spray-Coated Non- Wetting Surfaces†

Siddarth Srinivasan,a Wonjae Choi,b, Kyoo-Chul Park,c Shreerang S. Chhatre,a Robert E. Cohen∗a

and Gareth H. McKinley∗c

Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX

First published on the web Xth XXXXXXXXXX 200X

DOI: 10.1039/b000000x

We estimate the effective Navier-slip length for flow over a spray-fabricated liquid-repellent surface which supports a composite solid-air-liquid interface or ’Cassie-Baxter’ state. The morphology of the coated substrate consists of randomly distributed corpuscular micro-structures which encapsulate a film of trapped air (or ’plastron’) upon contact with liquid. The reduction in viscous skin friction due to the plastron is evaluated using torque measurements in a parallel plate rheometer resulting in a measured slip length of bslip ≈ 39 µm, comparable to the mean periodicity of the microstructure evaluated from confocal fluorescence microscopy. The introduction of a large primary length-scale using dual-textured spray-coated meshes increases the magnitude of the effective slip length to values in the range 94 µm ≤ bslip ≤ 213 µm depending on the geometric features of the mesh. The wetted solid fractions on each mesh are calculated from simulations on model sinusoidal mesh geometries and the trend in measured values of bslip with the mesh periodicity L and the computed wetted solid-fraction rφs are observed to follow existing analytic predictions.

1 Introduction

The local shearing flow of Newtonian fluids close to a solid surface is described by the Navier-Stokes equation along with an appropriate boundary condition on the velocity field at the liquid/solid interface. In macroscopic flows past smooth sur- faces, where the length scale of the system is much larger than the molecular length scale, the adoption of the no-slip bound- ary condition is widely accepted as valid.1 A more general condition used in various experimental studies investigating slip at molecular length scales is the Navier-slip hypothesis, Vw = βτw , where Vw is the effective tangential surface ve- locity at the wall and τw is the local tangential shear-stress at the interface. For a Newtonian fluid whose viscosity is η, this expression can also be written in the form Vw = bslipγ̇w, where γ̇w = (dV/dz)w is the local shear rate in the vicinity of the wall, and bslip = βη is the local slip length, a material property of the surface.1–3 Physically, the slip length corresponds to the distance below the surface at which a linear extrapolation of

a Department of Chemical Engineering, Massachusetts Institute of Technol- ogy, Cambridge, MA, USA. b Department of Mechanical Engineering, University of Texas at Dallas, USA. c Department of Mechanical Engineering, Massachusetts Institute of Tech- nology, Cambridge, MA, USA. ∗a E-mail: recohen@mit.edu ∗c E-mail: gareth@mit.edu † Electronic Supplementary Information (ESI) available: [details of any supplementary information available should be included here]. See DOI: 10.1039/b000000x/

the velocity profile would satisfy the no-slip boundary con- dition.4,5 Multiple studies have investigated liquid-slip phe- nomena on smooth surfaces coated with low surface energy materials for which the molecular interactions at the solid- liquid interface become weak.6–10 Although these investiga- tions demonstrate that the no-slip boundary condition is not strictly valid, the resulting slip lengths across the solid-liquid interface are generally too small (bslip ≈ 1 − 10 nm) to affect the macroscopic liquid flow significantly.

A growing body of work has attempted to utilize ’super- hydrophobic’ textured surfaces with regular microfabricated patterns or hierarchical textures11–21 to amplify the effective fluid slip at the interface. In such non-wetting surfaces, the liquid layer sits on a composite solid-air interface (or Cassie- Baxter interface22), by entrapping pockets of air between the individual topographical features. The composite interface can robustly resist pressure-induced wetting transitions over a range of liquid surface tensions and externally imposed pres- sure differences by careful design of the fabricated surface morphology.23,24 In facilitating the establishment of an air layer or ’plastron’ that is stable to externally imposed pres- sure differences, such surfaces can reduce the frictional dissi- pation associated with laminar flows in microdevices,17,25 in rheometers,,20,26–28 in pipes,21,29 over coated spheres30 and in turbulent flows in channels.31 The reduction in viscous skin friction due to the composite micro-textured interface can be significant in confined flows; in a seminal study Watanabe et al demonstrated a 14% reduction of drag in a 16 mm diame-

1–14 | 1

ter pipeline textured with a superhydrophobic surface.21 The vast majority of subsequent investigations on superhydropho- bic surfaces have involved precisely-fabricated and regularly- patterned geometries which help develop a systematic under- standing of the influence of the wetted solid-fraction and sur- face periodicity in promoting large effective slip lengths and associated friction reduction. It is less clear whether sub- strates with randomly deposited micro-structures, which are cost-effective to manufacture and more readily applicable to large coated areas, would exhibit similar dramatic reduction in drag.5,32 Sbragaglia & Prosperetti33 and Feuillebois et. al34

propose theoretical models to investigate how random textures can enhance the effective slip at a fluid-solid interface. In the present work, we use parallel-plate rheometery to determine the effective slip length for flow over spray-fabricated corpus- cular microtextures that are randomly deposited over both flat substrates and on woven wire meshes.

In Figure 1, we illustrate conceptually the effective slip present at the interface for flow over a spherically textured non-wetting substrate in the presence of an unconfined or pres- sure driven flow (Figure 1a) and also in a laminar Couette flow (Figure 1b). In each of these cases, the conventional no-slip condition is valid on the top of the wetted textures, while the local fluid velocity at the liquid-air interface is determined by a tangential stress-balance. The net dissipative interaction of the fluid with this textured surface can be expressed using an area-averaged effective slip velocity 〈Vw〉, or alternatively in terms of an effective slip length 〈bslip〉, again averaged over the periodicity (L) of the textured surface. As indicated in Figure 1, the slip length 〈bslip〉 can be greater than the charac- teristic scale of the texture (2R); a larger value of slip length indicates higher friction-reducing ability of the corresponding textured surface. For the laminar Couette flow shown in Fig- ure 1b, the shear rate γ̇ in the fluid and the resulting shear stress τ vary linearly except in the immediate vicinity of the surface texture. The apparent shear rate in the Couette flow with the assumption of a no-slip boundary γ̇a = Vplate/h can be related to the true shear rate γ̇t that is established in the fluid in the presence of slip as γ̇ah = γ̇t(h + 〈bslip〉), where h is the gap height.35 For a Newtonian liquid with viscosity η and shear stress τ = ηγ̇, measurement of the frictional forces or torques (in a torsional rheometer) due to flow at a fixed height between two flat parallel rigid surfaces (with no slip) and textured non-wetting surfaces (with slip) enables the ef- fective slip length on the latter to be directly related to the measured viscous friction using a rheometer.20,26,27 The vis- cous stress for a linear Couette flow in the proximity of the top plate can be expressed as τslip = ηVplate(h + 〈bslip〉)−1 = ηγ̇a(1 + 〈bslip〉/h)−1, where Vplate is the velocity of the upper plate. In a parallel-plate rheometer with disc radius R, the to- tal torque M =

∫

2πr2τdr measured by the instrument for a Newtonian fluid is36 M = (πηγ̇aR3)/2. Therefore, for a

Fig. 1 (a) A schematic diagram showing a liquid flow on a textured non-wetting surface, possessing an effective velocity V̄w that is averaged over the texture period L. (b) A laminar Couette flow with average shear rate γ̇a = Vplate/h between a non-wetting textured bottom surface exhibiting an average slip length 〈bslip〉 and a conventional flat solid top surface possessing a no-slip boundary condition.

fixed upper plate velocity, the ratio of the apparent viscosities (or measured torques) between (i) two flat surfaces with no slip and, (ii) a flat surface and a textured non-wetting surface with slip can be directly related to the average slip as,

ηflat ηslip

= τflat τslip

= Mflat Mslip

= 1 + 〈bslip〉 h

(1)

Eqn. 1 implies that in order for fluid slip in confined laminar flows to be manifested as a significant effect, the magnitude of the slip length 〈bslip〉 should be comparable to the length scale of the flow h (i.e., b/h ∼ O(1)). This can be readily achieved in a a rheometer because gaps in the range h . O(100µm) can be attained reliably.50 A more universal measure of fluid slip is the amount of drag reduction (DR) associated with the reduction in the measured apparent viscosities and can thus be written for a torsional Couette flow as:

DR = 1− ηslip ηflat

= bslip

h+ bslip (2)

Rheometric torque measurements can therefore be usefully employed as a macroscopic measurement technique that pro- vides a systematic method to probe area-averaged microscopic liquid-slip phenomenon over a large random (and possibly anisotropic) surface morphology. Care must