R. Anthore

Drag force on a sphere moving towards a corrugated wall

From the solution of the creeping-flow equations, the drag force on a sphere becomes infinite when the gap between the sphere and a smooth wall vanishes at constant velocity, so that if the sphere is displaced towards the wall with a constant applied force, contact theoretically may not occur. Physically, the drag is finite for various reasons, one being the particle and wall roughness. Then, for vanishing gap, even though some layers of fluid molecules may be left between the particle and wall roughness peaks, conventionally it may be said that contact occurs. In this paper, we consider the example of a smooth sphere moving towards a rough wall. The roughness considered here consists of parallel periodic wedges, the wavelength of which is small compared with the sphere radius. This problem is considered both experimentally and theoretically. The motion of a millimetre size bead settling towards a corrugated horizontal wall in a viscous oil is measured with laser interferometry giving an accuracy on the displacement of 0.1

The preparation of a ferrofluid by decomposition of dicobalt octacarbonyl: I. Experimental Parameters

\mu

The preparation of a ferrofluid by decomposition of dicobalt octacarbonyl: II. Nucleation and growth of particles

m. Several wedge-shaped walls were used, with various wavelengths and wedge angles. From the results, it is observed that the velocity of the sphere is, except for small gaps, similar to that towards a smooth plane that is shifted down from the top of corrugations. Indeed, earlier theories for a shear flow along a corrugated wall found such an equivalent smooth plane. These theories are revisited here. The creeping flow is calculated as a series in the slope of the roughness grooves. The cases of a flow along and across the grooves are considered separately. The shift is larger in the former case. Slightly flattened tops of the wedges used in experiments are also considered in the calculations. It is then demonstrated that the effective shift for the sphere motion is the average of the shifts for shear flows in the two perpendicular directions. A good agreement is found between theory and experiment.

Precise measurement of particle–wall hydrodynamic interactions at low Reynolds number using laser interferometry

Abstract The experimental parameters which lead to the formation of a stable suspension of metallic cobalt particles in an organic solvent (ferrofluid) were evidenced. The cobalt particles originate from the thermolysis of dicobalt octacarbonyl solution in the presence of a chosen surface active agent. The kinetics of the CO evolution depend on the reaction temperature, the nature of the solvent and of the surfactant, the weight ratio of carbonyl and surfactant, and the initial concentration of the cobalt carbonyl solution. Spherical particles, of a remarkable narrow size distribution, are obtained when the decomposition of Co2(CO)8 is carried out, in an aromatic solvent, above 110°C, in the presence of a surfactant possessing a long hydrocarbon chain and a strong ionic group (sulfonate). The decomposition in toluene, in which ethyl (2 hexyl) sodium sulfo-succinate is dissolved, leads to particles of about 70 A in diameter. The kinetics of the CO evolution are rather complex, but when a ferrofluid is being formed, several sequences are always recorded: an initial and rapid evolution of CO corresponding to the formation of Co4(CO)12. Part of this compound is insoluble in the reaction medium and appears to be a regulating intermediate. After this short initial stage the rate of decomposition of Co4(CO)12 slows down and becomes practically constant. Later the CO formation is accelerated again and finally it decreases as the reaction goes to completion. This S-shaped curve which describes the decomposition of Co2(CO)8 is always observed when a ferrofluid is in progress of formation.

Measurement of the average velocity of sedimentation in a dilute polydisperse suspension of spheres

Abstract The formation of a ferrofluid by thermal decomposition of a toluene solution of dicobalt octacarbonyl , in the presence of ethyl (2 hexyl) sodium sulfosuccinate, has been studied. The carbon monoxide evolution, the diameters of the particles, and the number of growing particles have been measured using also small-angle X-ray scattering and magnetic methods. The different sequences of CO evolution are explained. At least two factors are responsible for the formation of particles of very narrow size distribution: the presence of microreactors in the reaction medium and a diffusion controlled growth mechanism.

Concentration-dependent correlations in ferrofluids studied by small-angle scattering of synchrotron X-radiation

The motion of a spherical particle (with radius 1 to 6 mm) in a viscous fluid is measured using laser interferometry. The typical sensitivity on the measured displacement of the sphere is of the order of 50 nm. The particle is moving on the axis of a closed cylinder. The hydrodynamic interactions between the particle and the walls of the cylinder are compared with the theoretical result of Sano [J. Phys. Soc. Jpn. 56, 2713 (1987)] valid for a very small sphere. The agreement is excellent for the smallest sphere used in the experiment. The experiment also agrees with the result from the theory of lubrication when the sphere is close to a plane end wall. The effect of the particle roughness appears at small distances. Laser interferometry appears as a useful tool to study particle–wall hydrodynamic interactions when the geometry is cumbersome.

Creeping motion of a sphere along the axis of a closed axisymmetric container

An X-ray attenuation technique is used to obtain the local concentration of spherical particles in a polydisperse suspension as a function of vertical position and time. From these experimental data, the average velocity of sedimentation in the homogeneous part of the suspension is derived by considering the variation with time of the total volume of particles located above a given fixed horizontal plane. Measurements have been performed in suspensions of particles which differ from each other in size with a total volume concentration in particles between 0.13% and 2.5%, and also in suspensions of particles which differ from each other both in size and in density, the total volume concentration being 2%. For the first kind of suspension, the experimental hindered settling factor is plotted versus the concentration and a linear regression analysis provides the slope with its 90% confidence limits: S e = −5.3 ± 1.1. This experimental average coefficient of sedimentation is in good agreement with the theoretical average coefficient S t = −5.60 obtained from the results of Batchelor & Wen (1982). The second kind of suspension, for which permanent doublets of spheres may theoretically exist, is not in the range of validity of Batchelor & Wens results. The experimental average coefficient of sedimentation for this case is found to be much larger than the prediction obtained by extrapolating Batchelor & Wens results out of their range of validity. This increased velocity may be experimental evidence of the existence of permanent doublets.

Lubricating motion of a sphere in a conical vessel

Small-angle X-ray scattering has been used to study correlations between the magnetic grains of ferrofluids : for dilute colloiids, the grains look isolated or associated in dimers, trimers, etc ... depending on the nature of the ferrofluids. At higher concentration, an order appears in the spatial distribution of the grains even without magnetic field. The application of a magnetic field can generate alongated fibers of grains. An average interfiber spacing has been put in evidence. Small-angle X-ray scattering reveals as a sensitive quality test of the grain agglomeration in ferrofluids.

Interactions between two close spheres in Stokes flow

The creeping flow around a sphere settling along the axis of a closed axisymmetric container is obtained both theoretically and experimentally. The numerical technique for solving the Stokes equations uses the classical Sampson expansion; the boundary conditions on the sphere are satisfied exactly and those on the container walls are applied in the sense of least squares. This is an extension to the axisymmetric case of the technique for solving various two-dimensional flow problems. Two types of axisymmetric container are considered here as examples: circular cylinders closed by planes at both ends, and cones closed by a base plane. Calculated streamlines patterns show various sets of eddies, depending upon the geometry and the sphere position. Results are in agreement with earlier Stokes flow calculations of eddies in corners and in closed containers. Experiments use laser interferometry to measure the vertical displacement of a steel bead a few millimetres in diameter settling in a container filled with a very viscous silicone oil. The Reynolds number based on the sphere radius is typically of the order of 10 -5 . The accuracy on the vertical displacement is 50 nm. Experiments show that the motion towards the apex of a cone is much slower than that towards a plane; the bead takes hours to reach the micrometre size roughness asperities on a conical wall, as compared with minutes to reach those on a plane wall. The numerical results for the drag force are in excellent agreement with experiments both for the cylindrical and the conical containers. With standard computer accuracy, the present numerical technique applies when the gap between the sphere and the nearby wall is larger than about one radius. For a sphere in the vicinity of any plane horizontal wall, these results also match with a previous analytical solution. That solution is in excellent agreement with our experimental results at small distances from the wall (typically less than a diameter, depending on the sphere size).

HYDRODYNAMIC INTERACTIONS BETWEEN TWO SPHERES AT CONTACT

The final stage of sedimentation of a spherical particle moving along the axis of a conical vessel containing a viscous incompressible fluid is studied both theoretically by lubrication analysis and experimentally by laser interferometry. The particle settling velocity varies like d5/2, where d is the gap. There is an excellent agreement between this result from lubrication theory and experiment, the upper bound being for a gap of about 1/30 radius and the lower practical bound being at the size of the particle roughness.