Giant superhydrophobic slip of shear-thinning liquids

We theoretically illustrate how complex fluids flowing over superhydrophobic surfaces may exhibit giant flow enhancements in the double limit of small solid fractions ($\epsilon\ll1$) and strong shear thinning ($\beta\ll1$, $\beta$ being the ratio of the viscosity at infinite shear rate to that at zero shear rate). Considering a Carreau liquid within the canonical scenario of longitudinal shear-driven flow over a grooved superhydrophobic surface, we show that, as $\beta$ is decreased, the scaling of the effective slip length at small solid fractions is enhanced from the logarithmic scaling $\ln(1/\epsilon)$ for Newtonian fluids to the algebraic scaling $1/\epsilon^{\frac{1-n}{n}}$, attained for $\beta=\mathcal{O}(\epsilon^{\frac{1-n}{n}})$, $n\in(0,1)$ being the exponent in the Carreau model. We illuminate this scaling enhancement and the geometric-rheological mechanism underlying it through asymptotic arguments and numerical simulations.