Close-contact melting on hydrophobic textured surfaces: Confinement and meniscus effects

We investigate the dynamics of close-contact melting (CCM) on gas-trapped hydrophobic surfaces, with specific focus on the effects of geometrical confinement and the liquid-air meniscus below the liquid film. By employing dual-series and perturbation methods, we obtain numerical solutions for the effective slip lengths associated with velocity $\lambda$ and temperature $\lambda_t$ fields, across various values of aspect ratio $\Lambda$ (defined as the ratio of the film thickness $h$ to the structure's periodic length $l$) and gas-liquid fraction $\phi$. Asymptotic solutions of $\lambda$ and $\lambda_t$ for $\Lambda\ll 1$ and $\Lambda \gg 1$ are derived and summarized for different surface structures, interface shapes and $\Lambda$, which reveal a different trend for $\lambda$ and $\Lambda \ll 1$ and the presence of a meniscus. In the context of constant-pressure CCM, our results indicate that transverse-grooves surfaces consistently reduced the heat transfer. However, longitudinal grooves can enhance heat transfer under the effects of confinement and meniscus when $\Lambda \lessapprox 0.1$ and $\phi < 1 - 0.5^{2/3} \approx 0.37$. For gravity-driven CCM, the parameters of $l$ and $\phi$ determine whether the melting rate is enhanced, reduced, or nearly unaffected. We construct a phase diagram based on the parameter matrix $(\log_{10} l, \phi)$to delineate these three regimes. Lastly, we derived two asymptotic solutions for predicting the variation in time of the unmelted solid height.