Thermo- and soluto-capillary convection and instability near the air–liquid interface

Steady thermo- and solute-capillary convection, instability, and pattern evolution near the air–liquid interface driven by point heat and mass sources are investigated. First, under the assumption of the conically similar viscous flow, an exact axi-symmetric solution of the steady thermo- and soluto-capillary convection near the air–liquid interface is determined due to the constant heat and mass fluxes. It is shown that the constant heat, mass fluxes, 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 thermo- and soluto-capillary convection in response to the azimuthal disturbance is analyzed. At a given Peclet (or Schmidt), Marangoni (or Prandtl), and elasticity numbers, the steady basic flow loses its stability when Reynolds number is beyond the critical value. It is found that for small Schmidt numbers, a half-loop structure between the critical Reynolds number and the harmonic wave number of the azimuthal disturbance appears. For large Schmidt numbers, the critical Reynolds number monotonously increases as the harmonic wave number increases. The critical patterns of the velocity fields and the isothermal, iso-concentration lines in the three-dimensional flow system in response to the disturbance harmonic wave number are dominated by both the radial and azimuthal surface tensions. The results not only provide a valuable understanding of Marangoni convection and pattern information near the air–liquid interface but also inspire its potential practical applications in the material chemistry and biotechnology.