Theoretical calculation of the buoyancy force on a particle in flowing granular mixtures

Simulations and experiments of granular mixtures comprising two different sizes flowing over an inclined plane under the influence of gravity show segregation, where large particles rise to the free surface. The segregation results from differential forces acting on the particles. The buoyancy force experienced by the particles is an important component of the total force acting on the particles and, in this work, we theoretically calculate the buoyant force on intruder particles of different sizes in a flowing granular medium. The effective particle volume for buoyancy force calculation is obtained as the partial molar volume of the large particle from the equation of state for a binary mixture of hard spheres. The theoretical results are in good agreement with results obtained from discrete element method (DEM) simulations. Our calculations show that the buoyancy force on a particle is always smaller than the weight of the particle for larger particles. The ratio of the buoyancy force to the weight of the large particles decreases with increasing size ratio and increases with increasing concentration of the particles. This ratio approaches unity in the limiting cases of size ratio and large particle concentration approaching unity. These theoretical calculations establish a method for obtaining the buoyancy force experienced by particles in flowing mixtures.

Figure 1

Schematic showing the system volume change in a granular medium when a large size particle is added.

Figure 2

Comparison of the buoyant force on a single intruder particle (scaled with the particle weight) obtained from Eq. (15) and simulation results of van der Vaart et al. [10] .

Figure 3

Variation of the scaled buoyant force with size ratio of the large size intruder particle for different volume concentration of the large particles.

Figure 4

Variation of the scaled buoyant force with concentration of the large particles for different size ratios $d$.