Longitudinal thermocapillary flow over a dense bubble mattress

Ehud Yariv, Darren Crowdy

Research output: Contribution to journalArticlepeer-review

Abstract

A common form of superhydrophobic surface is made out of a periodically grooved solid substrate, wherein cylindrical bubbles are trapped in a Cassie state. When a macroscopic temperature gradient is externally applied, Marangoni forces generate thermocapillary flow of characteristic magnitude μ = −aG\sigma T /\mu , in which 2a is the groove width, G the applied-gradient magnitude, \mu the liquid viscosity, and \sigma T the derivative of interfacial-tension coefficient with respect to the temperature. We consider the case of a gradient which is applied parallel to the grooves. Assuming a highly conducting solid substrate, we seek to calculate the longitudinal velocity component, antiparallel to the applied gradient, and in particular its ``slip"" value, attained at large distances away from the surface. Normalized by μ, this value depends only upon the bubble protrusion angle and the solid fraction \epsilon . In this paper we consider the small solid-fraction limit \epsilon ≪ 1 and focus upon the case of a 90\circ protrusion angle, for which this limit is known to be highly singular in the comparable problem of shear-driven flow [O. Schnitzer, Phys. Rev. Fluids, 1 (2016), 052101]._ Using matched asymptotic expansions, we find that the dimensionless slip velocity is given by \pi 2 /√8\epsilon + ln \epsilon + I − ln(8\pi 2 ) + o(1). The first two terms in this expansion follow from a lubrication-type analysis of the narrow gap region separating two neighboring bubbles. The subsequent O(1) terms follow from asymptotic matching with the bubble-scale region, where the bubbles appear to be touching. The solution in that region is obtained using conformal mapping techniques, with the constant I given as an explicit integral, evaluated numerically to be 2.27.

Original languageEnglish
Pages (from-to)1-19
Number of pages19
JournalSIAM Journal on Applied Mathematics
Volume80
Issue number1
DOIs
StatePublished - 2020

Keywords

  • bubble mattress
  • conformal mapping
  • lubrication approximation
  • matched asymptotic expansions
  • superhydrophobic surfaces
  • thermocapillary flow

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