Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid

In this paper we examine some general features of the time-dependent dynamics of drop deformation and breakup at low Reynolds numbers. The first aspect of our study is a detailed numerical investigation of the ‘end-pinching’ behaviour reported in a previous experimental study. The numerics illustrate the effects of viscosity ratio and initial drop shape on the relaxation and/or breakup of highly elongated droplets in an otherwise quiescent fluid. In addition, the numerical procedure is used to study the simultaneous development of capillary-wave instabilities at the fluid-fluid interface of a very long, cylindrically shaped droplet with bulbous ends. Initially small disturbances evolve to finite amplitude and produce very regular drop breakup. The formation of satellite droplets, a nonlinear phenomenon, is also observed

Motion of a sphere in the presence of a plane interface. Part 2. An exact solution in bipolar co-ordinates

A general solution for Stokes’ equation in bipolar co-ordinates is derived, and then applied to the arbitrary motion of a sphere in the presence of a plane fluid/fluid interface. The drag force and hydrodynamic torque on the sphere are then calculated for four specific motions of the sphere; namely, translation perpendicular and parallel to the interface and rotation about an axis which is perpendicular and parallel, respectively, to the interface. The most significant result of the present work is the comparison between these numerically exact solutions and the approximate solutions from part 1. The latter can be generalized to a variety of particle shapes, and it is thus important to assess their accuracy for this case of spherical particles where an exact solution can be obtained. In addition to comparisons with the approximate solutions, we also examine the predicted changes in the velocity, pressure and vorticity fields due to the presence of the plane interface. One particularly interesting feature of the solutions is the fact that the direction of rotation of a freely suspended sphere moving parallel to the interface can either be the same as for a sphere rolling along the interface (as might be intuitively expected), or opposite depending upon the location of the sphere centre and the ratio of viscosities for the two fluids

The motion of a deformable drop in a second-order fluid

The cross-stream migration of a deformable drop in a unidirectional shear flow of a second-order fluid is considered. Expressions for the particle velocity due to the separate effects of deformation and viscoelastic rheology are obtained. The direction and magnitude of migration are calculated for the particular cases of Poiseuille flow and simple shear flow and compared with experimental data

Natural convection in a shallow cavity with differentially heated end walls. Part 2. Numerical solutions

Numerical solutions of the full Navier-Stokes equations are obtained for the problem of natural convection in closed cavities of small aspect ratio with differentially heated end walls. These solutions cover the parameter range Pr = 6.983, 10 ≤ Gr ≤ 2x10^4 and 0.05 ≤ A ≤ 1. A comparison with the asymptotic theory of part 1 shows excellent agreement between the analytical and numerical solutions provided that A ≾ 0.1 and Gr^2A^3Pr^2 ≾ l0^5. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit of part 1 and the boundary-layer limit; A fixed, Gr → ∞

The creeping motion of a spherical particle normal to a deformable interface

Numerical results are presented for the approach of a rigid sphere normal to a deformable fluid-fluid interface in the velocity range for which inertial effects may be neglected. Both the case of a sphere moving with constant velocity, and that of a sphere moving under the action of a constant non-hydrodynamic body force are considered for several values of the viscosity ratio, density difference and interfacial tension between the two fluids. Two distinct modes of interface deformation are demonstrated: a film drainage mode in which fluid drains away in front of the sphere leaving an ever-thinning film, and a tailing mode where the sphere passes several radii beyond the plane of the initially undeformed interface, while remaining encapsulated by the original surrounding fluid which is connected with its main body by a thin thread-like tail behind the sphere. We consider the influence of the viscosity ratio, density difference, interfacial tension and starting position of the sphere in deter-mining which of these two modes of deformation will occur

Wakes in stratified flow past a hot or cold two-dimensional body

This paper considers the general problem of laminar, steady, horizontal, Oseen flow at large distances upstream and downstream of a two-dimensional body which is represented as a line source of horizontal or vertical momentum, or as a line heat source or heat dipole. The fluid is assumed to be incompressible, diffusive, viscous and stably stratified. The analysis is focused on the general properties of the horizontal velocity component, as well as on explicit calculation of the horizontal velocity profiles and disturbance stream-function fields for varying degrees of stratification. For stable stratifications, the flow fields for all four types of singularities exhibit the common feature of multiple recirculating rotors of finite thicknesses, which leads to an alternating jet structure both upstream and downstream for the horizontal velocity component and to leewaves downstream in the overall flow. The self-similar formulae for the velocity, temperature and pressure at very large distances upstream and downstream are also derived and compared with the Oseen solutions

Oxidation of Dibenzothiophene to Dibenzothiophene Sulfone Using TiO2

The oxidation of dibenzothiophene (DBT) to dibenzothiophene-sulfone has been considered as an alternate method to remove sulfur from crude oil. This study demonstrates that the oxidative desulfurization (ODS) of DBT to DBT-sulfone occurs using rutile TiO2 as a catalyst with the aid of O2. In addition, anatase and brookite were synthesized and tested in the reaction. Particle size of the catalysts was calculated and morphology was explored using SEM. A kinetic study was performed to determine the catalytic effect and activation energy of TiO2. The oxidation reaction was carried out by refluxing TiO2 and DBT in decahydronapthalene while flowing O2 through the mixture. Oxygen ratios and catalyst ratios were investigated. The disappearance of DBT was followed with gas chromatography and the product was confirmed with RAMAN and XRD. The reaction demonstrated first order kinetics with the Arrhenius activation energy to be 47.4KJ/mol