Effect of Insoluble Surfactants on Drainage and Rupture of a Film between Drops Interacting under a Constant Force

The deformation, drainage, and rupture of an axisymmetrical film between colliding drops in the presence of insoluble surfactants under the influence of van der Waals forces is studied numerically at small capillary and Reynolds numbers and small surfactant concentrations. Constant-force collisions of Newtonian drops in another Newtonian fluid are considered. The mathematical model is based on the lubrication equations in the gap between drops and the creeping flow approximation of Navier–Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. A nonuniform surfactant concentration on the interfaces, governed by a convection–diffusion equation, leads to a gradient of the interfacial tension which in turn leads to additional tangential stress on the interfaces (Marangoni effects). The mathematical problem is solved by a finite-difference method on a nonuniform mesh at the interfaces and a boundary-integral method in the drops. The whole range of the dispersed to continuous-phase viscosity ratios is investigated for a range of values of the dimensionless surfactant concentration, Peclét number, and dimensionless Hamaker constant (covering both “nose” and “rim” rupture). In the limit of the large Peclét number and the small dimensionless Hamaker constant (characteristic of drops in the millimeter size range) a fair approximation to the results is provided by a simple expression for the critical surfactant concentration, drainage being virtually uninfluenced by the surfactant for concentrations below the critical surfactant concentration and corresponding to that for immobile interfaces for concentrations above it.     

Numerical method for unsteady viscous hydrodynamical problem with free boundaries

A finite element method for the transient incompressible Navier–Stokes equations with the ability to handle multiple free boundaries is presented. Problems of liquid–liquid type are treated by solving two coupled Navier–Stokes problems for two separate phases. The possibility to solve problems of liquid–gas, liquid–liquid–gas or liquid–liquid–liquid type is demonstrated too. Surface tension effects are included at deformable interfaces. The method is of Lagrangian type with mesh redefinition. A predictor-corrector scheme is used to compute the position of the deformable interface with automatic control of its accuracy and smoothness. The method is provided with an automatic choice of the time integration step and an optional spline filtration of the truncation error at the free surface. In order to show the accuracy of the method, tests and comparisons are presented. Numerical examples include motion of bubbles and multiple drops.     

The 4 A 2→2 E transitions of 2[Cr(en)3Cl3]KCl. 6H2O in strong magnetic fields

The Zeeman effect of the R absorption lines for the pure compound 2[Cr(en)₃Cl₃]KCl.6H₂O has been measured photographically, using a strong pulsed magnet with fields of up to 200 000 gauss at 77 K. The results indicate g _∥ (⁴ A ₂) = 1·82 , g _⊥(⁴ A ₂) = 1·90 with g _∥(ē) = -1·82 and g _∥ (2ā) = 1·82. The uncertainties are of the order of ± 0·05.     

A MODEL FOR ESTIMATING PREMIUMS TO REDUCE IRRIGATION ON FARMS

Abstract This paper aims to determine the minimum agri‐environmental premium per hectare that farmers should receive to reduce crop irrigation. To this end, the authors develop a mixed‐integer quadratic model using a new methodology based on traditional positive mathematical programming, which allows the inclusion in the model of crop variants that are not present in the baseline situation of the farms. What is shown is that the results obtained with this new methodology, using the cost average approach of calibration, are the same as those obtained with an empirical method, when there is no change of crop distribution after the changeover from irrigated to nonirrigated farms. The theoretical results are compared with those obtained using the calibration with exogenous elasticities and are illustrated numerically for a representative farm of an area around Spain's Tablas de Daimiel in which the use of groundwater for crop irrigation has led to the ongoing deterioration of wetlands that depend on the same source of water, endangering the region's environmental sustainability.     

Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow

The effect of insoluble surfactants on drop deformation and breakup in simple shear flow is studied using a combination of a three-dimensional boundary-integral method and a finite-volume method to solve the coupled fluid dynamics and surfactant transport problem over the evolving interface. The interfacial tension depends nonlinearly on the surfactant concentration, and is described by the equation of state for the Langmuir isotherm. Results are presented over the entire range of the viscosity's ratio lambda and the surface coverage x, as well as the capillary number Ca that spans from that for small deformation to values that are beyond the critical one Ca(cr). The values of the elasticity number E, which reflects the sensitivity of the interfacial tension to the maximum surfactant concentration, are chosen in the interval 0.1 < or = E < or = 0.4 and a convection dominated regime of surfactant transport, where the influence of the surfactant on drop deformation is the most significant, is considered. For a better understanding of the processes involved, the effect of surfactants on the drop dynamics is decoupled into three surfactant related mechanisms (dilution, Marangoni stress and stretching) and their influence is separately investigated. The dependence of the critical capillary number Ca(cr)(lambda) on the surface coverage is obtained and the boundaries between different modes of breakup (tip-streaming and drop fragmentation) in the (lambda; x) plane are searched for. The numerical results indicate that at low capillary number, even with a trace amount of surfactant, the interface is immobilized, which has also been observed by previous studies. In addition, it is shown that for large Péclet numbers the use of the small deformation theory to measure the interfacial tension in the case where surfactants are present can introduce a significant error.