Breakdown of the Bretherton law due to wall slippage

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 μ_app= 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 θ_dpropto Ca^1/2 for the apparent dynamic contact angle of a spreading droplet, subsequently making the spreading radius grow with time as r \propto t^1/8. In addition, the precursor film is found to possess ell_f&propto Ca^-1/2 in length and therefore spreads as ℓ_f&propto t^1/3 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.