Peter Mucha [email] (Mathematics, Georgia Institute of Technology)
A unifying theory for velocity fluctuations in sedimentation
Abstract: Model differential equations for the concentration of particles sedimenting in a fluid often assume a diffusion coefficient that is a function solely of the particle concentration (expressed in terms of the solid volume fraction). This diffusive behavior is the result of velocity fluctuations of the particles as they fall, which are in turn due to particle density differences that originate in randomness. The size and scaling of these velocity fluctuations at small Reynolds numbers has been the subject of significant study and controversy. Recently, a vertical stratification of the particle concentration has been identified as a parameter controlling these velocity fluctuations. This stratification control is particularly significant near the falling sediment front between particle-laden fluid below and clear fluid above. We propose a simple model for the diffusion coefficient, based on the recent advances in understanding velocity fluctuations, as a function of both particle concentration and its vertical gradient. This model is tested against particle simulations in the limits of low concentration and small Reynolds number. Steadily-falling concentration profiles with these diffusivities are then examined, and an extension of the model to higher solid volume fractions is discussed. |
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