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