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The distribution of particle sizes along the height of granular flows result from the competing influence of size segregation and diffusion (the random migration of particles within the granular mixture). When segregation driving forces are much greater than diffusion, the separation of particle sizes tend to be very distinct (separated by a sharp jump in the particle size concentration), whereas when diffusion is more relevant, the particles will tend to be mixed. The competing influence of both processes are captured by the segregation-diffusion equation of Gray and Chugunov (2005), which takes the segregation velocity and diffusion coefficient as input variables. This means that if we are able to obtain expressions for these variables that account for the effects of fluids.
In my research I used scaling analysis to derive formulas for the diffusion and segregation velocity. The effects of viscosity is found to be best represented by the Stokes number St for both particle processes. Its effects however is only evident when viscosity is sufficiently large. In the non-viscous limit, segregation and diffusion are controlled by the flow inertia (shear rate, pressure, grain size, etc.) and the buoyancy. The maximum values in this limit are used as the normalizing factors seen on the graphs on the right.
The normalized segregation velocity (left) and the diffusion coefficient (right). Each point represents a simulation wherein either the flow inertia, fluid viscosity, and density are varied.
After inputting the new equations for the segregation velocity and diffusion into the continuum framework of Gray & Chugunov, it is found that theoretical predictions and measurements from simulations agree well.
Using these as inputs to the continuum model, it possible to predict the evolution of vertical size distributions for granular-fluid mixtures measured from the trajectory of the center of mass of large particles with time (graphs on the left). More details on this work can be found in two of my published works: Cui et al. (2022) 'Particle size segregation and diffusion in immersed granular shear flows', Phys. Rev. Fluids, 7.
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