## Abstract

Shear-driven flows over superhydrophobic surfaces formed of closely spaced circular bubbles are characterized by giant longitudinal slip lengths, viz., large compared with the periodicity (Schnitzer, Phys Rev Fluids 1(5):052101, 2016). This hints towards a strong superhydrophobic effect in the concomitant scenario of pressure-driven flow between two such surfaces, particularly for non-wide channels where bubble-to-bubble pitch and bubble radius are commensurate with channel width. We show here that such pressure-driven flows can be analyzed asymptotically and in closed form based on the smallness of the gaps separating the bubbles relative to the channel width (and bubble radius). We find that the flow adopts an unconventional plug profile away from the inter-bubble gaps, with the uniform velocity being asymptotically larger than the corresponding Poiseuille scale. For a given solid fraction and channel width, the net volumetric flux is maximized when the length of each semi-circular bubble-liquid interface is equal to the channel width. The plug flow identified herein cannot be obtained via a naive implementation of a Navier condition, which is indeed inapplicable for non-wide channels.

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## Acknowledgements

Ehud Yariv was supported by the Israel Science Foundation (Grant No. 1081/16).

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## Appendix: Boundary-integral computation

### Appendix: Boundary-integral computation

With *w* satisfying Poisson’s equation, it is straightforward to solve the preceding problem using a boundary-integral formulation [23]. We define the harmonic function \(\varphi = w + x^2/2\) and employ Green’s third identity,

wherein \(\mathbf {x}_0\) is a point on the boundary \(\partial \mathcal {D}\) and the Green function \(g(\mathbf {x},\mathbf {x}_0)\) satisfies \(\nabla ^2 g = \delta (\mathbf {x}-\mathbf {x}_0)\). For convenience, we employ here the free-space Green function, \(g = (2\pi )^{-1} \ln |\mathbf {x}-\mathbf {x}_0|\). Discretization of (A.1) in conjunction with the boundary conditions governing \(\varphi \) (which readily follow from those governing *w*) yields 2*N* equations governing the values of \(\varphi \) and \(\partial \varphi /\partial n\) at *N* collocation points on \(\partial \mathcal {D}\). Once \(\varphi \) (and whence *w*) is determined on the boundary, \(\bar{w}\) is readily evaluated upon making use of Green’s second identity.

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Yariv, E., Schnitzer, O. Pressure-driven plug flows between superhydrophobic surfaces of closely spaced circular bubbles.
*J Eng Math* **111, **15–22 (2018). https://doi.org/10.1007/s10665-018-9952-z

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### Keywords

- Lubrication approximation
- Singular perturbations
- Stokes flows