Solutocapillary instability in slipping falling films

TL;DR

This paper develops a comprehensive theoretical framework for gravity-driven, surfactant-laden thin films flowing over slippery substrates, analyzing how wall slip influences stability, nonlinear evolution, and surfactant transport.

Contribution

It introduces a new long-wave model incorporating slip effects, resolves previous mass conservation issues, and highlights the role of slip as a control parameter in film stability and wave structures.

Findings

Critical Reynolds number varies non-monotonically with surfactant coverage.

Wall slip transitions solitary wave structures from single- to double-hump forms.

Slip reduces viscous resistance and stabilizes the film against Marangoni effects.

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.