Solutocapillary instability in slipping falling films

Abstract

We present a comprehensive framework for gravity-driven, surfactant-laden thin films flowing over slippery substrates, elucidating how wall slip modifies the coupled hydrodynamics and interfacial transport. A long-wave model is formulated with a conservative bulk-surface mass balance and a Navier slip condition. The Orr-Sommerfeld eigenvalue problem governs the linear regime, while a weighted-residual model captures the nonlinear evolution over a range of equilibrium surfactant coverages, Marangoni strengths, and adsorption kinetics. The analysis predicts a non-monotonic variation of the critical Reynolds number with equilibrium coverage, exhibiting a maximum at intermediate $\Gamma_e$, and a slip-induced transition from single- to double-hump solitary structures with increasing Marangoni number, accompanied by attenuated capillary ripples. Under fast adsorption kinetics, the surface field homogenizes, preserving the mean film shape and flux while flattening both the surface concentration $\Gamma$ and the bulk inventory $\chi + h\phi$. A spurious interfacial mass growth reported by Pascal et al.(PRF, 2019) and D'Alessio et al.(JFM, 2020) is resolved through a revised surface balance ensuring strict conservation. Wall slip thus emerges as a key control parameter, reducing viscous resistance and mitigating Marangoni back-stress. The slip parameter $\beta$ is a useful control knob for surfactant-laden films. Slip prevents fragile multi-hump bound states, promoting a single broad crest or an almost flat, uniform sheet by carefully bonding $\beta$ to wave selection, ripple damping, and the bulk-surface surfactant balance.