Dynamics of a climbing surfactant-laden film II: Stability

• Published in: Journal of colloid and interface science 2012-04, Vol.371 (1), p.121-135
• Main Authors: Mavromoustaki, A., Matar, O.K., Craster, R.V.
• Format: Article
• Language: English
• Subjects: Chemistry; Colloidal state and disperse state; Critical micelle concentration; Diffusion; equations; Exact sciences and technology; Fingering; Flux; General and physical chemistry; Gravitation; Kinetics; Marangoni effect; Mathematical analysis; Mathematical models; Mathematics; Micelles; Micelles. Thin films; Models, Theoretical; Nonlinear dynamics; Perturbation methods; Soluble surfactant; Stability; Stability analysis; Surface Properties; Surface-Active Agents - chemistry; Surfactants; Thin film
• ISSN: 0021-9797; 1095-7103
• DOI: 10.1016/j.jcis.2011.11.033

[Display omitted] ► We examine the spanwise stability of a surfactant-laden film climbing up a plane. ► Small-amplitude disturbances are introduced in the linearised base-state equations. ► In the constant-flux case the thinnest region is unstable to finger formation. ► For finite-volume spreading, the gravity-force-induced capillary ridge is unstable. ► Increasing the initial micellar concentration leads to stabilising effects. The linear and nonlinear stability of a spreading film of constant flux and a drop of constant volume, discussed in [1], are examined here. A linear stability analysis (LSA) is carried out to investigate the stability to spanwise perturbations, by linearisation of the two-dimensional (2-D) evolution equations derived in [1] for the film thickness and surfactant concentration fields. The latter correspond to convective–diffusion equations for the surfactant, existing in the form of monomers (present at the free surface and in the bulk) and micelles (present in the bulk). The results of the LSA indicate that the thinning region, present upstream of the leading front in the constant flux case, and the leading ridge in the constant volume case, are unstable to spanwise perturbations. Numerical simulations of the 2-D system of equations demonstrate that the above-mentioned regions exhibit finger formation; the effect of selected system parameters on the fingering patterns is discussed.