Stagnant cap

Experimental observations (S3) indicate that a stagnant cap is formed over the rear of the droplet as surface-active agents are added, and that this cap tends to enlarge with increasing concentrations until the entire droplet is enveloped. Thus, circulation may occur only in the front portion of the bubble. In contrast to this mechanism, Thorsen and Terjesen (T3) and Gamer (Gil) concluded that most of the mass transfer takes place at the rear of the bubble. 

Fig. 3.5 Internal circulation in a water drop falling through castor oil [from Savic (SI), reproduced by permission of the National Research Council of Canada] (a) d = 1.77 cm, Uj = 1.16 cm/s, exposure 1/2 s, fully circulating (b) d= 1.21 cm, Uj = 0.62 cm/s, exposure 1 s, stagnant cap at top of drop.

Savic s calculated values of Y, along with values obtained subsequently (D5, H8), are plotted in Fig. 3.6. Also shown is an asymptotic solution (H4) for a small stagnant cap (37t/4 < 9 < n) ... 

Fig. 3.6 Effect of stagnant cap on terminal velocity of a bubble or inviscid drop.

Dispersed phase resistances are increased when surface contaminants reduce interfacial mobility. Huang and Kintner (H9) used Savic s stagnant-cap theory in a semiempirical model for this resistance. A simpler quasi-steady model is proposed here, analogous to that for continuous phase resistance. The Sherwood... 

Fig. 3.23 Variation of fractional approach to equilibrium with time for fluid particles with contaminated interface and Pep/(1 + k) qc (9q = angle excluded by stagnant cap.

Stagnant cap, o- Results lie between the solutions of Newman and Kronig and Brink. This model can easily be extended to include changes of cap angle with time. 

Experimenters who have observed asymmetry of internal circulation patterns have generally attributed this to accumulation of surface-active materials at the rear, causing a stagnant cap (see Chapter 3). It seems likely that at least part of the asymmetry results from the forward shift of the internal vortex at nonzero Re, as predicted numerically. 

S. Sadhal and R. E. Johnson, Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film - Exact solution, J. Fluid Mech. 126, 237-50 (1982). 

The intermediate concentration region is characterised by heterogeneities of the dynamic adsorption layer structure (Deijaguin and Dukhin, 1959 -1961). When increasing the siufactant concentration a stagnant cap is formed and expands, while at decreasing concentration the weakly retarded zone expands. Thus, a double-zone dynamic adsorption layer arises in the... 

In the dynamic adsorption layer theory of Deqaguin-Dukhin, the hydrodynamic field of a bubble can be assumed to be known as first approximate, while the more difficult stagnant cap problem has still to be solved. For the solution of this hydrodynamic problem unusual and very difficult boundary conditions exist which are very inconvenient even after essential simplifications. The hydrodynamic field of a bubble is studied imder the assumption that the stagnant cap is completely immobilised and any motion of the surface beyond the stagnant cap is ignored. Since the description of the stagnant cap is to a large extent a hydrodynamic problem, it has received less attention (cf. Section 8.7). 

The described theory of strong retardation of a bubble surface by DAL at small Reynolds numbers was developed by Dukhin Derjaguin (1961) and Dukhin Buikov (1965) and confirmed by Saville (1973). The balance of Marangoni and viscous stresses, given by Eq. (8.38) as the basis for the determination of the surface concentration distribution was used later in the theory of a stagnant cap (Section 8.7). This stress balance is often characterised qualitatively by a dimensionless number, the Marangoni number (cf He et al. 1991),... 

It is of great interest to know the conditions under which it is advisable to use the model of bubble surface motion proposed by Savic (1953). The surfactant adsorption at the main portion of the bubble strongly deviates from its equilibrium value when the condition (8.71) holds. The condition for a small stagnant cap size v(/ 1 with regard to Eq. (8.148) can be written in the following form. 

The Rear Stagnant Cap and Bubble Buoyant Velocity at Small Re... 

As a purely hydrodynamic problem, the velocity field due to a stagnant cap at the rear of a moving drop was solved exactly in terms of an infinite series of Gegenbauer polynomials with constants depending on the cap angle ( ). From this series, an analytical solution for the drag F(( > ) exerted on the drop can be obtained, from which the terminal velocity was computed once the external force on the drop is resolved,... 

The above procedure was first introduced by Griffith (1962) whose study is incomplete since he did not use the proper hydrodynamic solution, and later by Sadhal Johnson (1983) in their exact solution of the problem. Each of these authors assumed that the surface pressure exerted by the compression of the surfactant in the stagnant cap may be represented by a linear isotherm. 

Now, accepted simplifications of the stagnant cap theory are used. At the adsorbing surface, c can be neglected as compared with c and at the desorbing surface c can be neglected as compared with c... 

The stagnant cap theory permits a quantitative evaluation of the variation of the total amount of surfactant adsorbed at the bubble as a function of its buoyant velocity. The discussion of these results are avoided since they are restricted to Re 1 and experimental data about the mobility of the bubble surface at small Reynolds numbers do not exist. 

With increasing surfactant concentration, the dynamic adsorption layer changes from the state of weak retardation to the transient state, characterised by the appearance of a stagnant cap and its growth with further surfactant concentration increase. In the process of the growth of the stagnant cap, weakly and strongly retarded parts of the bubble surface coexist. 

The theory of weak retardation of surface motion allows relations to be obtained to give an estimate of the minimum surfactant concentration for the appearance of a stagnant cap which exerts and effects the buoyant velocity. The theory of strong retardation yields a maximum surfactant concentration which separates the transient state from a complete retardation of the bubble surface. Thereby, the transition between the theory of limiting states of the dynamic adsorption layer and the theory of the transient state is obtained, which is important for two reasons. First of all, the theories were developed by different teams of scientists independently. Secondly, it allows to conclude the appropriateness of the approximate methods employed which gives a complete picture of different states of the dynamic adsorption layer. This is possible without huge efforts necessary for numerical solutions. 

Although considerable success in the development of the dynamic adsorption layer theory has been reached, there has been less progress experimentally. This is not marked for the transient state, where the theoretical advances are most impressive. It turns out that experimental works devoted to the stagnant cap theory are more or less of empirical interest as they are restricted to small Reynolds numbers. At small, and even intermediate, Reynolds numbers the bubble surface can initially behave immobile and the formation of a stagnant cap is almost impossible. 

The essence of the problem is that on one hand the phenomenon of a stagnant cap appears only at large Reynolds numbers and on the other, the theory of a stagnant cap was developed only for small Re. The development of a stagnant cap theory for large Re should be possible nowadays because the solution of the very complex hydrodynamic part has been presented already in other studies (Rivkind et al. 1971,1976). 

The calculation of the total desorption flow in the case of electrostatic retardation of desorption kinetics follows from Eq. (7.36). The density of the surfactant anions flux within the stagnant cap can be estimated by... 

The Size of the Stagnant Cap of the Bubble (Droplet) Using Surfactants With a Slow Rate Desorption... 

When the rear stagnant cap and the angle T increase, the collision efficiency decreases. Thus, at uniform surface retardation, i.e. under condition (8.71) and during the rear stagnant cap formation, the mechanisms of DAL effect on the transport stage differ qualitatively. In the former case, with increasing surfactant concentration, the normal component of velocity and, respectively, the flow of particles uniformly decrease over the leading surface. In the latter case, the area admissible for sedimentation of particles decreases. 

Using the formulas for the hydrodynamic field of a bubble carrying a rear stagnant cap (cf. Section 8.7), we can calculate the effect of the cap on collision efficiency. It is unlikely that such work is of interest since the theory described in Section 8.7 is restricted to small Reynolds numbers. Thus, we caimot expect agreement between this theory and reality since the leading surface must be either completely or strongly retarded, according to experimental data by Okazaki (1964). 

At first the surface tension gradient is estimated by the ratio of the surface tension difference between y and y (r ) over a characteristic length, equal to the bubble radius. This uncertainty can be bypassed after the elaboration of a stagnant cap theory for high Reynolds number. In the... 

In transient state the DAL has a slight effect on the transport stage if the rear stagnant cap covers a smaller part of the surface. If the rear stagnant cap is not too small and characterised by the angle 9 (cf Section 8.6) essentially less than 7t/2, the possibility of its effect depends substantially on the mechanism of fixation of particles on a bubble surface (see Appendix lOD). 

Reasonable investigations under these conditions are restricted by the state of the DAL theory which has been developed so far only for conditions of very strong and weak surface retardation (cf Section 8.6). Collision efficiency has been derived only for potential flow conditions (Sutherland, 1948). With increasing surfactant concentration up to c[ (Eqs 8.135 -8.136), a beginning decrease of bubble velocity may be expected. A respective rear stagnant cap results in a decrease of collision efficiency only when attachment of the particle is accomplished not due to the of instability of the water interlayer at some thickness h but under the effect of attraction forces (Appendix lOB). 

---