Breakdown of the Bretherton law due to wall slippage

Yen Ching Li, Ying Chih Liao, Ten Chin Wen, Hsien Hung Wei

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

Against the common wisdom that wall slip plays only a minor role in global flow characteristics, here we demonstrate theoretically for the displacement of a long bubble in a slippery channel that the well-known Bretherton 2/3 law can break down due to a fraction of wall slip with the slip length λ much smaller than the channel depth R. This breakdown occurs when the film thickness h∞ is smaller than λ corresponding to the capillary number Ca below the critical value Ca*∼λ/R) 3/2. In this strong slip regime, a new quadratic law h∞/R∼ Ca2 (R/λ2 is derived for a film much thinner than that predicted by the Bretherton law. Moreover, both the 2/3 and the quadratic laws can be unified into the effective 2/3 law, with the viscosity μ replaced by an apparent viscosity μapp= μ h∞/λ+h∞. A similar extension can also be made for coating over textured surfaces where apparent slip lengths are large. Further insights can be gained by making a connection with drop spreading. We find that the new quadratic law can lead to θdpropto Ca1/2 for the apparent dynamic contact angle of a spreading droplet, subsequently making the spreading radius grow with time as r \propto t1/8. In addition, the precursor film is found to possess ellf&propto Ca-1/2 in length and therefore spreads as ℓf&propto t1/3 in an anomalous diffusion manner. All these features are accompanied by no-slip-to-slip transitions sensitive to the amount of slip, markedly different from those on no-slip surfaces. Our findings not only provide plausible accounts for some apparent departures from no-slip predictions seen in experiments, but also offer feasible alternatives for assessing wall slip effects experimentally.

Original languageEnglish
Pages (from-to)200-227
Number of pages28
JournalJournal of Fluid Mechanics
Volume741
DOIs
Publication statusPublished - 2014 Feb 1

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

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