Creeping flow past a fluid globule when a trace of surfactant is present

Abstract

The effects of a trace quantity of a surface-active agent on creeping flow past a bubble or droplet are investigated. The equations describing mass and momentum transfer are simultaneously solved by a perturbation technique, consistent with the jump mass and momentum batances a t the phase interface. The stream function for the velocity distribution is evaluated as an infinite series of spherical harmonics. Golerkin's method, which reduces the partial differential equation of continuity to a set of ordinary differential equations, is used to evaluate the concentration distribution of surfactant.A sample calculation is carried out for relative motion between an air bubble and an infinite body of water which contains a trace of isoamyl alcohol. The relative velocity of the water a t an infinite distance from the bubble is found to be highly sensitive to small changes in surfactant concentration from zero, although the bubble varies imperceptibly from a spherical shape.Consider a fluid globule which moves at a constant velocity under the action of gravity through a second immiscible phase. Hadamard and Rybczynski (1, p. 395) independently presented a solution to this problem which neglects interfacial effects, but their results failed to explain available experimental data for settling velocities.This led Boussinesq (2) to hypothesize that a skin which inhibits the internal circulation can form around a moving droplet. He described this quantitatively by a constitutive equation which expresses the stress in the interface as a linear function of the rate of deformation of the interface. This relationship, often called the Newtonian surface fluid model ( 3 to 6 ) , has two parameters in addition to surface tension: surface shear viscosity and surface dilatational viscosity. Acceptable laboratory measurements of these surface viscosities have not been obtained to date ( 7 ) . Boussinesq (2) obtained an exact solution for creeping flow past a droplet under the assumptions that interfacial behavior could be described by the Newtonian surface fluid model and that surface tension and the two surface viscosities were independent of position on the phase interface. The results of Boussinesq's analysis do not appear to be in any better agreement with available experimental data than those of Hadamard andNeither the analysis of Boussinesq nor the theory of Hadamard-Rybczynski considers the possible presence of surfactants (materials which have an affinity for a phase interface and which consequently alter the surface tension of a system). In practice it is very difficult to eliminate trace quantities of surface-active materials from an experiment. The observation of a fairly immobile cap on the trailing surface of a droplet ( 8 ) suggests that any surfaceactive agents present are not uniformly distributed around the surface, but rather that they are continuously swept toward the rear of the droplet by the fluid motion. If we express surface stress in terms of surface tension alone, the resultant accumulation of surfactants ...

Retardation of Droplet Motion by Surfactant. Part 2. Numerical Solutions for Exterior Diffusion, Surface Diffusion, and Adsorption Kinetics

“…Four limiting interfacial velocity profiles have been adopted. First, investigators have assumed that the profile is unretarded and perturbed the Hadamard-Rybczynski or Boussinesq velocity profiles for an infinitesimal quantity of surfactant present in the exterior liquid (Wasserman and Slattery, 1969;Saville, 1973;Harper, 1974;LeVan and Newman, 1976;Agrawal and Wasan, 1979). Second, the interfacial velocity has been assumed to be retarded uniformly over the entire surface (Levich, 1962;Schechter and Farley, 1963;Newman, 1969;Holbrook and LeVan, 1983).…”

The effects of surfactants on the motion and transport mechanisms of a condensing droplet in a high reynolds number flow

“…He obtained the drag coefficient from the value of a completely surfactant-free surface through a perturbation method, concluding that this model is often a good one if the surfactant is highly active and the Peclet number is high. Wasserman and Slattery (1969) also investigated the creeping flow past a fluid globule when a trace of surfactant is present; the result of a perturbation analysis show that the internal circulation and the free moving velocity are highly sensitive to small changes in surfactant concentration, although the bubble varies imperceptibly from a spherical shape. Recently, Sadal and Johnson (1983) obtained exact flow solutions for a fluid sphere with a surfactant cap in low Reynolds number flow.…”

Migration Movements of Accidentally Spilled Oil in Environmental Waters: A Review

“…For this reason, Boussinesq put forward the view that the oil droplets will lack internal circulation because an interfacial monolayer which acts as a viscous membrane, constructed a constitutive equation including surface shear viscosity, surface dilational viscosity and surface tension, and finally obtained an exact solution in the case of creeping flow analogous to the H-R result (Equation ( 3)). But it is very difficult to obtain reliable measurements for parameter C [59,60]. In response to the discrepancy between the theoretical analysis and the experimental results, many researchers tend to believe that it is due to the impurity of the system, the H-R solution is the terminal velocity of the oil droplets in the pure system, and the Stokes formula is the terminal velocity when the surface of the oil droplets is completely covered with contaminants [61].…”