The Linear Analysis of Marangoni Instability

The upper integers denoting phase 1 or 2, to which every value belongs, are omitted for simplification. 

One can use the dynamic boundary conditions of Eqs. (12)-(13) without considering the influence of the osmotic pressure and the Gibbs adsorption (the case of significant contribution of these values to the interfacial balance of forces is analyzed in Refs. [4,19])  

If the interphase surface tension is not dependent on the concentration of transporting substances and is a function of temperature only, it is necessary to complete the system of Eqs. (29)-(30) with the system of Eqs. (8) and the appropriate boundary conditions  

The condition for the secondary derivative from in the z-direction follows from 

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