Solutocapillary convection and instability near the air-liquid interface

Steady solutocapillary convection and instability near the air-liquid interface are studied. First, under the assumption of the conically similar viscous flow, an exact axisymmetric solution of the steady solutocapillary convection near the air-liquid interface is determined due to a constant mass flux. It is shown that the constant mass flux and the radial surface tension cause the divergent motion at the interface and the Marangoni convection beneath the interface. Then, the linear stability of the steady solutocapillary convection in response to the azimuthal disturbance is analyzed. At a given Peclet number (or Schmidt number), the steady basic flow loses its stability when the Reynolds number is beyond its critical value. It is found that for the fixed Schmidt number, the critical Reynolds number increases monotonously as the harmonic wave number of the azimuthal disturbance increases. However, for the fixed Peclet number, a nonlinear relationship between the critical Reynolds number and the harmonic wave number of the azimuthal disturbance is found. The structures of iso-concentration lines and velocity fields in the three-dimensional flow system depend on the disturbance harmonic wave number, which is dominated by both the radial and the azimuthal surface tensions. This study provides a profound understanding of the soluble surfactant-driven instability of a divergent flow near the air-liquid interface, which is of great significance for practical applications in the micro-fluidics related to chemistry and biology.