Dynamics of a climbing surfactant-laden film--I: base-state flow.

The dynamics of a surfactant-laden film climbing up an inclined plane is investigated through a two-dimensional (2-D), nonlinear evolution equation for the interface coupled to convective-diffusion equations for the surfactant, derived using lubrication theory. One-dimensional (1-D) solutions, representing the base-state flow, are investigated for constant flux and constant volume configurations; these flows are parameterised by capillarity, gravity, convection-diffusion ratios (represented by Péclét numbers at the surface and bulk), a solubility parameter, sorption kinetics constants, the number of surfactant monomers in a micelle, and the nonlinearity of the surfactant equation of state. In both configurations studied, a front develops spreading up the substrate against the direction of gravity whereby the leading edge of the front follows a power-law as a function of time. The effect of system parameters on the base-state flow is explored through an extensive parametric study, while the stability of the above-mentioned system to spanwise perturbations is the focus of Part II.

Dynamics of a climbing surfactant-laden film II: stability.

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.