Breakdown of the Bretherton law due to wall slippage

Abstract

AbstractAgainst 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 $\lambda $ much smaller than the channel depth $R$. This breakdown occurs when the film thickness $h_{\infty } $ is smaller than $\lambda $, corresponding to the capillary number $Ca$ below the critical value $Ca^{\ast } \sim (\lambda /R)^{3 / 2}$. In this strong slip regime, a new quadratic law $h_{\infty } /R \sim Ca^{2} (R/\lambda )^{2}$ 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 $\mu $ replaced by an apparent viscosity $\mu _{app}= \mu h_{\infty } /({\lambda } + h_{\infty })$. 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 $\theta _{d} \propto 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 $\ell _{f} \propto t^{1 / 3}$ 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.