Instability Due to the Marangoni Effect

Let us consider a flat y=0) interface gas-liquid and suppose that a gas is fixed and a liquid, including a dissolved surface-active substance with bulk concentration Cq, is moving parallel to the phase boundary (towards the x-axis). Furthermore, assume that the convective diffusion is a decelerated stage coming up to the phase boundary of the surface-active substance. In case of a low Reynolds numbers without any external field, Eq. (13) for the balance of tangential forces at the phase boundary becomes more simple  

Here c = c(x, y, t, Cq) is the solution of Eq. (9) for the hquid phase (the surface diffusion is not considered for simplicity). One may determine the connection of and F values in a parametric form (i.e., in terms of Cq), using the following system of equations  

It is clear that one can neglect the first item on the left hand side of Eq. (20a). Using instead of its ordered value from Eq. (21a) let us introduce this value into the right hand side of Eq. (20a) which leads to  

It follows from Eq. (23) that F c d, Furthermore, the values 5 and I are physically of the same order (with / S, changing the concentration field would not change a, and the alternate version may be realized only in case of very large that is at variance with the accepted above condition of low Reynolds number). Thus the boundary condition of Eq. (20) in view of the orders of its members can be written as  

Here Aa is the increment of surface tension over a characteristic length / along the phase boundary. 

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