Marangoni effect, defined

Now, from a qualitative point of view, the role of surfactant is twofold One is to simply reduce the interfacial tension everywhere by an amount that depends on the mean value of T, and the second is to produce Marangoni effects that are due to flow-induced nonuniformity in T. As a consequence, it is convenient to define T as being... 

In view of the term (jJl in Eq. (9), the Marangoni effects are hidden in the left-hand side of the interfacial momentum balance equation (8) through the surface gradients of (Ja. The thermodynamic surface tension, dJa, depends on the adsorption and temperature. The derivatives of with respect to In and InT define the Gibbs elasticity for the i-th surfactant species, Et, and the thermal analogue of the Gibbs elasticity, Ef. 

In a foam where the films are interconnected, the related time-dependent Marangoni effect is more relevant. A similar restoring force to expansion results because of transient decreases in surface concentration (increases in surface tension) caused by the finite rate of surfactant adsorption at the surface. Such nonequilibrium surface tension effects are best described in terms of dilatational moduli. The complex dilatational modulus e of a single surface is defined in the same way as the Gibbs elasticity as in equation (2) (the factor 2 is halved as only one surface is considered). 

We write = to define e in terms of Ga). Then the effects of energy output (due to heat and viscous dissipation) and input (due to the Marangoni effect) will be of the same order as nonlinearity and dispersion. The latter two are in appropriate (local) balance for the Bounssinesq-Korteweg-de Vries (BKdV) equation for long waves in shallow inviscid liquid layers. 

We will see in the following that the ability of the surface layer to oppose the resistance to smface motion is also a very important factor. Surface flow induces surface concentration gradients leading to surface tension gradients responsible for very large surface forces (Marangoni effect). This can be quantified by the surface elasticity sq defined as... 

To quantify the impact of Marangonl effects, as compared to those of the interfacial tension only, sometimes the Marangoni number Ma is used. It is defined as... 

In (ll.l), the summation is restricted to wavenumbers close to q<3. In the case of the R.B instability, the quadratic term is due to non-Boussinesq effects (e.g. temperature dependent transport coefficients) a similar term always appears in the case of the Marangoni or the Turing instability. When these equations have a gradient structure and this property is sometimes satisfied near the bifurcation point, one can define a Lyapunov functional that decreases in any dynamics. In that case if one describes local fluctuations by a gaussian white noise term then the (T.D.G.L.) equation can be written as ... 

---