Eur. Phys. J. B 18, 343–352 (2000) T HE EUROPEAN P HYSICAL JOURNAL B c  EDP Sciences Societ` a Italiana di Fisica Springer-Verlag 2000 Velocity measurement of a settling sphere N. Mordant and J.-F. Pinton a ´ Ecole Normale Sup´ erieure de Lyon, Laboratoire de Physique b , 46 All´ ee d’Italie, 69007 Lyon, France Received 12 April 2000 and Received in final form 13 July 2000 Abstract. We study experimentally the motion of a solid sphere settling under gravity in a fluid at rest. The particle velocity is measured with a new acoustic method. Variations of the sphere size and density allow measurements at Reynolds numbers, based on limit velocity, between 40 and 7 000. At all Reynolds numbers, our observations are consistent with the presence of a memory-dependent force acting on the particle. At short times it has a t -1/2 behaviour as predicted by the unsteady Stokes equations and as observed in numerical simulations. At long times, the decay of the memory (Basset) force is better fitted by an exponential behaviour. Comparison of the dynamics of spheres of different densities for the same Reynolds number show that the density is an important control parameter. Light spheres show transitory oscillations at Re ∼ 400, but reach a constant limit speed. PACS. 06.30.Gv Velocity, acceleration, and rotation – 43.60.+d Acoustic signal processing – 47.27.Vf Wakes 1 Introduction The problem of evaluating the hydrodynamic forces on a rigid body in a moving fluid is a long standing issue. It arises in several engineering domains which involve mul- tiphase flows, e.g., in sedimentation, or improvement of combustion or in the minimization of erosion by droplets in large turbines. All these problems are concerned with the dispersion of particles, whose modeling requires some understanding of the particle dynamics. Another question is the ability of dispersed solid particles to follow the fluid motion when their density or initial velocity do not quite match the fluid properties, that is the ability of solid par- ticles to behave as Lagrangian tracers of fluid motion. This issue is of importance for the prediction of disper- sion of pollutants in the atmosphere or in measurement techniques such as particle image velocimetry (PIV). Our own motivation comes from work aiming at developing a Lagrangian tracking technique for the motion of a few solid particles during large intervals of times, in a turbu- lent flow. It raises the question of the response of a particle to rapid changes in the velocity of the fluid, or to a sudden acceleration. Analytical approaches to the time-dependent motion of a solid particle in a given fluid flow have been restricted to zero or small Reynolds numbers. However, they provide a general frame of description of the forces acting on the particle. We briefly recall their main results, as a basis for the analysis of our experiments. Consider the motion of a solid body in a general time-dependent flow: external and a e-mail: pinton@ens-lyon.fr b CNRS UMR 5672 hydrodynamic forces set the body into motion, and this motion in turn modifies the fluid flow since it introduces a moving boundary condition. The equation of motion for the viscous fluid are the Navier-Stokes equations supple- mented by boundary conditions at infinity (in free space) and on the solid body. In order to solve this problem, the general approach is to obtain expressions for the forces on the particle, in terms of the properties of the fluid flow in its absence. Perturbative developments are made, in which the stress tensor is split into contributions from the undis- turbed flow and corrections due to fluid motion induced by the particle displacement. The complexity of the prob- lem stems from the calculation of the latter term which contains the history of the particle motion. In the limit of vanishing Reynolds numbers, Maxey and Riley [1] have proposed for the equation of motion of the particle : m p dv p dt =(m p − m f )g + m f Du Dt +6πaμ f (u − v p ) + 1 2 m f d(u − v p ) dt +6a 2 (πμ f ρ f ) 1 2  t 0 d(u−vp) dτ (t − τ ) 1 2 dτ, (1) where ρ f is the density of the fluid, μ f its viscosity, u the undisturbed flow field, v p is the sphere velocity (u − v p is the slip velocity), a its radius, m p its mass and m f is the mass of the fluid displaced by the sphere (m f = (4/3)πa 3 ρ f ). As customary, we note dv p /dt the acceleration of the particle and Du/Dt ≡ (∂ t + u.∇)u that of the fluid. This expression is established for non- uniform non stationary creeping flow, and providing that ρ f aW 0 /μ f ≪ 1, ρ f a 2 U/(μ f L) ≪ 1 and a/L ≪ 1(U is an