Transverse instability of traveling wave created by longwave Marangoni convection in the liquid layer covered by insoluble surfactant

Near the onset of Marangoni convection, the appearance of different kinds of stationary patterns, like hexagons, squares, and rolls, as well as wave patterns, is possible. The simplest type of the latter ones is one-dimensional single traveling wave (1D STW). The nonlinear dynamics of the wave is usually described by the complex Ginzburg-Landau equation (CGLE).

We consider the case where the liquid interface is covered by insoluble surfactant that plays an active role in the pattern selection together with inhomogeneity of temperature along the interface and surface deformability. In the course of propagation of a modulated STW the local thickness of the liquid layer depends on the convection intensity and distribution of the surfactant over the surface. It leads to a modification of CGLE, as well as to the change of longitudinal phase-modulation criterion (Benjamin-Feir instability) and the appearance of an additional, amplitude-mode, type of instability.

Here we investigate the 2D transverse instability of the STW analytically by means of asymptotic technique. Two cases of transverse modulation are considered – with characteristic modulation wavelengths and , where is the width of the wavenumber interval of the primary instability of the equilibrium. In both cases the coupled systems of amplitude equations including a modified CGLE are obtained and linear instability analysis is performed.