Dynamics of rear stagnant cap formation at the surface of rising bubbles in surfactant solutions at large Reynolds and Marangoni numbers and for slow sorption kinetics

• Based on a quasi-steady approximation, the theory for decelerated rising of bubbles of a radius of 0.4 mm is refined.
• The approach is valid for large Marangoni numbers and slow sorption kinetics.
• A method for determination of adsorption and desorption rate constants is proposed.
• The method can be applied to local velocity profiles of rising bubbles at surfactant concentrations above a critical value.

In spite of the high level in the theory of steady rear stagnant caps (RSC) and its influence on steady rising, its practical application is mostly impossible because the coefficients for the adsorption and desorption rates are separately unknown. The determination of ka and kd separately is an actual task for the adsorption dynamics as whole. While steady RSC and steady rising retardation by surfactants are described in literature in details, only few papers are devoted to the modeling of the decelerated rising. Moreover, steady rising depends on the ratio ka/kd and its investigation is not helpful for the determination of kd. In contrast a possibility to determine kd (or ka independently) from measurements of decelerated rising was shown by Zholkovskij et al. (2000).However, experimental applications of this theory is difficult because of the condition Re < 1, that corresponds to small bubbles which surface is immobilized by impurities even in super clean water. This constraint may be eliminated due to the results presented by Cuenot et al. (1997), where the modeling of decelerated rising is accomplished numerically for Re = 100. However, direct application of this research is possible for a few surfactants, corresponding to the Marangoni number Ma = 61, as assumed in this simulation work.An equation is obtained for the determination of kd in a broad range of large Ma numbers from measurements of decelerated rising at Re = 200 (bubble radius 400 μm) in this work. This equation is obtained on the basis of an equation for slow adsorption kinetics, a quasi-steady approximation and an equation for surfactant accumulation derived by Zholkovskij et al. (2000) as well as due to incorporation in this theory the vorticity distribution, as calculated by Fdhila and Duineveld (1996) for Re = 200. For the determination of kd it is sufficient to measure the time required for the onset of maximal surface retardation for the concentrations above the critical concentration, i.e. the minimum concentration required for the onset of the minimum rising velocity.

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