RESUMO
Lattice Boltzmann simulations of water-in-oil (W/O) type emulsions of moderate viscosity ratio (≃1/3) and with oil soluble amphiphilic surfactant were used to study the droplet size distribution in forced, steady, homogeneous turbulence, at a water volume fraction of 20%. The viscous stresses internal to the droplets were comparable to the interfacial stress (interfacial tension), and the droplet size distribution (DSD) equilibrated near the Kolmogorov scale with droplet populations in both the viscous and inertial subranges. These results were consistent with known breakup criteria for W/O and oil-in-water emulsions, showing that the maximum stable droplet diameter is proportional to the Kolmogorov scale when viscous stresses are important (in contrast to the inviscid Hinze-limit where energy loss by viscous deformation in the droplet is negligible). The droplet size distribution in the inertial subrange scaled with the known power law ~d(-10/3), as a consequence of breakup by turbulent stress fluctuations external to the droplets. When the turbulent kinetic energy was sufficiently large (with interfacial Péclet numbers above unity), we found that turbulence driven redistribution of surfactant on the interface inhibited the Marangoni effect that is otherwise induced by film draining during coalescence in more quiescent flow. The coalescence rates were therefore not sensitive to varying surfactant activity in the range we considered, and for the given turbulent kinetic energies. Furthermore, internal viscous stresses strongly influenced the breakup rates. These two effects resulted in a DSD that was insensitive to varying surfactant activity.
RESUMO
Lattice Boltzmann simulations were used to study the coalescence kinetics in emulsions with amphiphilic surfactant, under neutrally buoyant conditions, and with a significant kinematic viscosity contrast between the phases (emulating water in oil emulsions). The 3D simulation domain was large enough (256(3) ~ 10(7) grid points) to obtain good statistics with droplet numbers ranging from a few thousand at early times to a few hundred near equilibrium. Increased surfactant contents slowed down the coalescence rate between droplets due to the Gibbs-Marangoni effect, and the coalescence was driven by a quasi-turbulent velocity field. The kinetic energy decayed at a relatively slow rate at early times, due to conversion of interfacial energy to kinetic energy in the flow during coalescence. Phenomenological, coupled differential equations for the mean droplet diameter D(t) and the number density n(d)(t) were obtained from the simulation data and from film draining theories. Local (in time) power law exponents for the growth of the mean diameter (and for the concomitant decrease of n(d)) were established in terms of the instantaneous values of the kinetic energy, coalescence probability, Gibbs elasticity, and interfacial area. The model studies indicated that true power laws for the growth of the droplet size and decrease of the number of droplets with time may not be justified, since the exponents derived using the phenomenological model were time dependent. In contrast to earlier simulation results for symmetric blends with surfactant, we found no evidence for stretched logarithmic scaling of the form D ~ [ln (ct)](α) for the morphology length, or exponential scalings associated with arrested growth, on the basis of the phenomenological model.
RESUMO
The dispersion of passive scalars and inertial particles in a turbulent flow can be described in terms of probability density functions (PDFs) defining the statistical distribution of relevant scalar or particle variables. The construction of transport equations governing the evolution of such PDFs has been the subject of numerous studies, and various authors have presented formulations for this type of equation, usually referred to as a kinetic equation. In the literature it is often stated, and widely assumed, that these PDF kinetic equation formulations are equivalent. In this paper it is shown that this is not the case, and the significance of differences among the various forms is considered. In particular, consideration is given to which form of equation is most appropriate for modeling dispersion in inhomogeneous turbulence and most consistent with the underlying particle equation of motion. In this regard the PDF equations for inertial particles are considered in the limit of zero particle Stokes number and assessed against the fully mixed (zero-drift) condition for fluid points. A long-standing question regarding the validity of kinetic equations in the fluid-point limit is answered; it is demonstrated formally that one version of the kinetic equation (derived using the Furutsu-Novikov method) provides a model that satisfies this zero-drift condition exactly in both homogeneous and inhomogeneous systems. In contrast, other forms of the kinetic equation do not satisfy this limit or apply only in a limited regime.
