A one-dimensional model evolution equation is used to describe the nonlinear dynamics that can lead to the breakup of a cylindrical thread of Newtonian uid when capillary forces drive the motion. The model is derived from the Stokes equations by use of rational asymptotic expansions and under a slender jet approximation. The equations are solved numerically and the jet radius is found to vanish after a nite time yielding breakup. The slender jet approximation is valid throughout the evolution leading to pinching. The model admits self-similar pinching solutions which yield symmetric shapes at breakup. These solutions are shown to be the ones selected by the initial boundary value problem, for general initial conditions. Further more, the terminal state of the model equation is shown to be identical to that predicted by a theory which looks for singular pinching solutions directly from the Stokes equations without invoking the slender jet approximation throughout the evolution. It is shown quantitatively, therefore, that the one-dimensional model gives a consistent terminal state with the jet shape being locally symmetric at breakup. The asymptotic expansion scheme is also extended to include unsteady and inertial forces in the momentum equations to derive a n e v olution system modelling the breakup of Navier-Stokes jets. The model is employed in extensive simulations to compute breakup times for di erent initial conditions; satellite drop formation is also supported by the model and the dependence of satellite drop volumes on initial conditions is studied.

In this paper the weakly nonlinear stability of two-phase core-annular film flows in the limit of small film thickness and in the presence of both viscosity stratification and interfacial tension is examined. Rational asymptotic expansions are used to derive some novel nonlinear evolution equations for the interface between the phases. The novel feature of the equations is that they include a coupling between core and film dynamics thus enabling a study of its effect on the nonlinear evolution of the interface. The nonlinear interfacial evolution is governed by modified Kuramoto–Sivashinsky equations in the cases of slow and moderate flow [the former also developed by Frenkel, Sixth Symposium on Energy Engineering Sciences (Argonne Lab. Pub. CONF-8805106, 1988), p.100, using different asymptotic methods], which include new nonlocal terms that reflect core dynamics. These equations are solved numerically for given initial conditions and a range of parameters. Some interesting behavior results, such as transition (in parameter space) of chaotic solutions into traveling-wave pulses with more than one characteristic length scale.

The breakup of viscous liquid threads covered with insoluble surfactant is investigated here; partial differential equations governing the spatio-temporal evolution of the interface and surfactant concentrations are derived in the long wavelength approximation. These one-dimensional equations are solved numerically for various values of initial surfactant concentration, surfactant activity and the Schmidt number (a measure of the importance of momentum, i.e., kinematic viscosity, to surfactant diffusion). The presence of surfactant at the air–liquid interface gives rise to surface tension gradients and, in turn, to Marangoni stresses, that drastically affect the transient dynamics leading to jet breakup and satellite formation. Specifically, the size of the satellite formed during breakup decreases with increasing initial surfactant concentration and surfactant activity. The usual self-similar breakup dynamics found in the vicinity of the pinchoff location for jets without surfactant [Eggers, Phys. Rev. Lett. 71, 3458 (1993)], however, are preserved even in the presence of surfactant; this is confirmed via numerical solutions of the initial boundary value problem.

The buoyant motion of a bubble rising through a continuous liquid phase can be retarded by the adsorption onto the bubble surface of surfactant dissolved in the liquid phase. The reason for this retardation is that adsorbed surfactant is swept to the trailing pole of the bubble where it accumulates and lowers the surface tension relative to the front end. The difference in tension creates a Marangoni force which opposes the surface flow, rigidifies the interface and increases the drag coefficient. Surfactant molecules adsorb onto the bubble surface by diffusing from the bulk to the sublayer of liquid adjoining the surface, and kinetically adsorbing from the sublayer onto the surface. The surface surfactant distribution which defines the Marangoni force is determined by the rate of kinetic adsorption and bulk diffusion relative to the rate of surface convection. In the limit in which the rate of either kinetic or diffusive transport of surfactant to the bubble surface is slow relative to surface convection and surface diffusion is also slow, surfactant collects in a stagnant cap at the back end of the bubble while the front end is stress free and mobile. The size of the cap and correspondingly the drag coefficient increases with the bulk concentration of surfactant until the cap covers the entire surface and the drag coefficient is that of a bubble with a completely tangentially immobile surface. Previous theoretical research on the stagnant cap regime has not studied in detail the competing roles of bulk diffusion and kinetic adsorption in determining the size of the stagnant cap angle, and there have been only a few studies which have attempted to quantitatively correlate simulations with measurements.This paper provides a more complete theoretical study of and a validating set of experiments on the stagnant cap regime. We solve numerically for the cap angle and drag coefficient as a function of the bulk concentration of surfactant for a spherical bubble rising steadily with inertia in a Newtonian fluid, including both bulk diffusion and kinetic adsorption. For the case of diffusion-limited transport (infinite adsorption kinetics), we show clearly that very small bulk concentrations can immobilize the entire surface, and we calculate the critical concentrations which immobilize the surface as a function of the surfactant parameters. We demonstrate that the effect of kinetics is to reduce the cap angle (hence reduce the drag coefficient) for a given bulk concentration of surfactant. We also present experimental results on the drag of a bubble rising in a glycerol–water mixture, as a function of the dissolved concentration of a polyethoxylated non-ionic surfactant whose bulk diffusion coefficient and a lower bound on the kinetic rate constants have been obtained separately by measuring the reduction in dynamic tension as surfactant adsorbs onto a clean interface. For low concentrations of surfactant, the experiments measure drag coefficients which are intermediate between the drag coefficient of a bubble whose surf...