The transition in settling velocity of surfactant-covered droplets from the Stokes to the Hadamard–Rybczynski solution

Abstract

The exact solution for a small falling drop is a classical result by Hadamard and Rybczynski. But experiments show that small drops fall slower than predicted, giving closer agreement with Stokes' result for a falling hard sphere. Increasing the drop size, a transition between these two extremes is found. This is due to surfactants present in the system, and previous work has led to the stagnant-cap model. We present here an alternative approach which we call the continuousinterface model. In contrast to the stagnant-cap model, we do not consider a surfactant advection-diffusion equation at the interface. Taking instead the normal and tangential interfacial stresses into account, we solve the Stokes equation analytically for the falling drop with varying interfacial tension. Some of the solutions thus obtained, e.g. the hovering drop, violate conservation of energy unless energy is provided directly to the interface. Considering the energy budget of the drop, we show that the terminal velocity is bounded by the Stokes and the Hadamard-Rybczynski results. The continuous-interface model is then obtained from the force balance for surfactants at the interface. The resulting expressions gives the transition between the two extremes, and also predicts that the critical radius, below which drops fall like hard spheres, is proportional to the interfacial surfactant concentration. By analysing experimental results from the literature, we confirm this prediction, thus providing strong arguments for the validity of the proposed model.

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

“…A most significant theoretical analysis result when studying the terminal velocity of oil droplets is the H-R analytical solution (Equation ( 2)), which is obtained by considering the effect of the oil droplet's own viscosity (i.e., internal circulation is considered to exist in the oil droplet) on the basis of Stokes solution, mainly for the motion of an ideal spherical oil droplet in the case of creeping flow (Re << 1). Although the H-R solution has only been verified in particular systems, the appearance of full circulating falling photographs of water droplets in castor oil does suggest the possibility of internal circulation [24,55,56]. However, researchers such as Bond and Newton have argued that the terminal velocity of small oil droplets should be between the Stokes solution and the H-R solution [57,58].…”

“…Devies and Rideal also defined a new parameter called "a degree of circulation" Z based on the theory of surface-active pollutants, and used this parameter to revise the Stokes solution in order to provide a more reasonable explanation of the experimental results [50]. Ervik and Bjørklund later reinterpreted this problem using a continuous-interface model that takes into account the normal and tangential interfacial stresses, and solved the Stokes equation for oil droplets under different interfacial tensions, thus obtaining the critical radius between the two velocity extremes [55]. Relevant theoretical studies on the low Re range are still continuing, but it is difficult to have a completely uncontaminated pure system in the real environment [62], which leads to the fact that the surface of small oil droplets will inevitably be covered with different degrees of active contaminants; the difference between small oil droplets and rigid spheres is negligible [63].…”

Effect of Bubbles and Liquid Drops Fluid Dynamics on the Lithium Recombination inside a Lithium Electrolytic Cell with Diaphragm

“…15 was used to calculate the slip velocity between the bubble surface and the electrolytic solution. The Hadamard-Rybczynski model 34 was used to evaluate the drag coefficient, an adequate approach for spherical bubbles with a diameter of less than 2 mm:…”

Hydrodynamic behaviors of settled magnetorheological fluid redispersion under active dispersing mechanism: simulation and experiment

“…One set of Navier-Stokes (N-S) equations are solved for the momentum of the mixture and the pressure distribution is calculated from a mixture-averaged continuity equation. The velocity of the dispersed phase is described by Hadamard-Rybczynski [33] slip model. In detail, since there is a significant difference in the velocity fields in most cases due to the buoyancy of the dispersed phase, the volume fraction of the dispersed phase is tracked by solving the transport equation.…”