Nicolas Lecoq

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

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

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

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.

Lubricating motion of a sphere in a conical vessel

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.

Boundary conditions for creeping flow along periodic or random rough surfaces : experimental and theoretical results

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).

Karstmod - A generic modular reservoir model dedicated to spring discharge modeling and hydrodynamic analysis in karst

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.

Phase-Field Modelling of Spinodal Decomposition during Ageing and Heating

Hydrodynamic interactions between particles and walls are relevant for the open problem of specifying boundary conditions for suspension flows. The Reynolds number around a small particle close to a wall is usually low and creeping flow equations apply. From the solution of these equations, the drag coefficient on a sphere becomes infinite when the gap between the sphere and a smooth wall vanishes, so that contact may not occur. Physically, the drag is finite because of various reasons, one of them 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, it may conventionally be said that contact occurs. In this paper, we are considering the example of a smooth sphere moving towards a rough wall. The roughness considered here consist of random rough planes or parallel periodic wedges, the characteristic length 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 0.2μm. Several random rough planes and 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. For the periodic wedges, the creeping flow is calculated as a series in the slope of the roughness grooves. The convergence of the series for the shift distance in term of the slope is accelerated by use of Euler transformation and of the existence of a limit for large slope. 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. The effective theoretical shift for a sphere approaching a wall is obtained from Lorentz reciprocal theorem with an expansion for small roughness compared with the gap between the sphere and the wall. The effective shift is found to be the average of the shifts for shear flows in the two perpendicular directions. A good agreement is found between theory and experiment. The theoretical description of the flow close to the random rough wall represents a difficult, nearly insurmountable problem except in lattice Boltzmann simulations. Statistical analysis is presented in this paper to deduce the effective shift for sand-blasted rough surfaces. To overcome the difficulties of modelling, a regular perturbation expansion is developed, and from Lorentz reciprocal theorem, the first order correction to the drag force due to random roughness is evaluated.

Lubrication in a flat cone: The transition from a cone to a plane wall

On the basis of the characterization of the different karst subsystems (Soil/Epikarst—Unsaturated Zone—Saturated Zone) and mathematical models developed on specific sites, we propose an adjustable modeling platform of karst for both the simulation of spring discharge at outlets and the analysis of the hydrodynamics of the compartments considered in the model. This platform was developed within the framework of the KARST observatory network initiative from the INSU/CNRS, which aims to strengthen knowledge-sharing and promote cross-disciplinary research on karst systems at the national scale.

Interactions between two close spheres in Stokes flow

Despite the tremendous success of phase-field (PF) modelling in predicting many of the experimentally observed microstructures in solids, additional progress is required in order to apply it to predict microstructure evolution in real alloy systems. One way to achieve this is to couple thermodynamic and kinetic databases with PF model. In this work, we present phase-field simulations of spinodal decomposition in Fe-Cr alloy during thermal ageing and anisothermal heating. In the PF method, the local free energy is directly constructed using the CALPHAD method. During isothermal ageing, the morphology of decomposed phases consisted in an interconnected irregular shape for short ageing times, and a further ageing caused the change to a droplet like shape of the decomposed Cr-rich phase. The influence of heating rate on phase transformations is then simulated and compared with experimental results obtained by differential thermal analysis, carried out with heating rates in the range 0.5 °C.min-1 to 15 °C.min-1. The simulation results show that heating rate strongly influences the microstructure morphology.

HYDRODYNAMIC INTERACTIONS BETWEEN TWO SPHERES AT CONTACT

A lubrication formula is derived for the drag on a sphere settling in a viscous fluid on the axis of a flat cone vessel. This formula provides the link between the classical e−1 result for the plane wall and our earlier e−5∕2 result for the cone [Masmoudi et al., Phys. Fluids 10, 1231 (1998)], where e is the gap between the sphere and wall. An excellent agreement is obtained with experimental results based on the measurement of the displacement of a sphere with laser interferometry. Moreover, this formula practically applies for a large range of values of cone angles; it then proves better than the earlier formula [Masmoudi et al. (1998)] in describing experimental results.