Modulation instability of Marangoni roll convection in a layer with deformable interface covered by insoluble surfactant

Stationary one-dimensional roll patterns are typical in several instability problems. Here we investigate the stability of stationary Marangoni rolls in a liquid layer covered by insoluble surfactant with respect to the longwave modulation. The nonlinear interaction between three longwave modes: surface deformation, convection mode described by roll’s amplitude and surfactant concentration disturbance is considered near the instability threshold, Ma-Ma_c=O(δ²) (δ is a small parameter of supercriticality),  by means of modified Newell-Whitehead amplitude equations.  With respect to transverse modulation, the problem has two different distinguished limits. The first one, with transverse modulation wavenumber proportional to δ^1/2, is prescribed by the linear stability theory and the second one, with wavenumber proportional to δ, comes from the long-wave surface deformation and surfactant distribution. Both cases are studied separately. The stability maps are plotted on the parameter plane (β,G), where β is the Biot number and G is the Galileo number, for different values of surfactant concentration.