Fabien Candelier

On the effect of inertia and history forces on the slow motion of a spherical solid or gaseous inclusion in a solid-body rotation flow

The motion of a spherical inclusion released in a vertical solid-body rotation flow is investigated theoretically and experimentally. Solid spheres and bubbles are considered. The particle Reynolds number, the Taylor number, the Weber number and the capillary number are smaller than unity. The motion equations of the inclusion are obtained by revisiting the hydrodynamic equations. The axial (vertical) motion and the horizontal motion are uncoupled, even though they are sensitive to the rotation rate of the flow. Analytical solutions of the particle motion equation are compared to experimental results obtained by releasing a particle in a rotating tank filled with silicone oil. For solid spheres and bubbles, both the terminal velocity and the particle ejection rate (or trapping rate) predicted by the theory agree with experiments, without any empirical adjustment. In particular, the experimental device enables us to check the validity of various theories involving solid or gaseous inclusions with or without inertia or history effects. It is observed that the mobility tensor obtained by writing the fluid motion equations in the rotating frame accurately predicts the horizontal particle trajectory, like the Boussinesq-Basset equation obtained by writing the fluid motion equations in the non-rotating frame and neglecting the horizontal contribution of inertia effects.

Time-dependent force acting on a particle moving arbitrarily in a rotating flow, at small Reynolds and Taylor numbers

The arbitrary motion of a solid sphere released in a solid-body rotating fluid is investigated theoretically in the limit of small Reynolds and Taylor numbers. The angular velocity of the fluid is assumed to be constant and under the premise that T a ½ » R e , the simplicity of the unperturbed flow enables us to calculate analytically the force acting on a particle moving with a harmonic slip velocity (by means of matched asymptotic expansions), when both inertia and unsteady effects are taken into account. Subsequently, these single-frequency results are used in order to determine the temporal expression of the force acting on an arbitrarily moving sphere, since the problem under study is linear. This force is first determined in a co-rotating reference frame and takes the form of two convolution products involving the particle acceleration and the particle velocity. For convenience, the corresponding expression of this force is also derived in the laboratory reference frame, and the particle motion equation obtained is thereafter illustrated by dealing with two practical situations, where unsteady and inertia effects must be taken into account to predict the particle dynamics accurately.

Inertial drag on a sphere settling in a stratified fluid

We compute the drag force on a sphere settling slowly in a quiescent, linearly stratified fluid. Stratification can significantly enhance the drag experienced by the settling particle. The magnitude of this effect depends on whether fluid-density transport around the settling particle is due to diffusion, to advection by the disturbance flow caused by the particle, or due to both. It therefore matters how efficiently the fluid disturbance is convected away from the particle by fluid-inertial terms. When these terms dominate, the Oseen drag force must be recovered. We compute by perturbation theory how the Oseen drag is modified by diffusion and stratification. Our results are in good agreement with recent direct-numerical simulation studies of the problem at small Reynolds numbers and large (but not too large) Froude numbers.