<p>Rivers carry sediments having a wide grain size distribution, ranging from a few hundreds microns to meters. This leads to grain size segregation mechanism that can have huge consequences on morphological evolution.&#160; Accurate comprehension and modeling of this mechanism with continuous equations is a key step to upscale segregation in sediment transport models.<br><em>Thornton et al. (2006) </em>developed continuous equations for bidisperse segregation in the context of the mixture theory. Based on the momentum balance of small particles, a simple advection-diffusion equation for the volumetric concentration of small particles was derived. This equation enables to explicit the advection term, that tends to segregate the different particle sizes and the diffusive term, that tends to remix the particles. However, this approach does not immediately provide the physical characteristics of the granular flow in the advection and diffusion terms.</p><p>Recently, <em>Guillard et al. (2016)</em> showed, using a Discrete Element Method (DEM), that the segregation force on a large intruder in a bath of small particles, can be seen as a buoyancy force proportional to the pressure. In addition, <em>Tripathi and Khakhar (2011)</em> showed that a large particle rising in a pool of small grains experiences a Stokesian drag force proportional to the granular viscosity.<br>These new results enable to infer a force balance for a single coarse particle in bedload transport. Solving this force balance showed that the large particle rises with the accurate dynamics, meaning that this force balance is relevant to model grain-size segregation.</p><p>Based on these new forces, a continuous multi-class model has been developed to generalize to the segregation of a collection of large particles. The concentration and the segregation velocity of the small particles have been compared with coupled-fluid DEM bedload transport simulations from <em>Chassagne et al. (2020)</em> and show that the accurate dynamics of segregation can be modeled using this continuous model.<br>Based on this continuum multi-class model, a similar advection-diffusion equation as <em>Thornton et al. (2006)&#160;</em> has been obtained. The latter appears to provide the physical origin of the advection and diffusion terms by linking them to the parameters of the flow.</p><p>&#160;</p><p>Chassagne R., Maurin R., Chauchat J., and Frey P. Discrete and continuum modeling of grain-size segregation during bedload transport. J. Fluid Mech. 2020 (in revision).</p><p>Gray J. M. N. T., and &#160;Chugunov V. A. Particle-size segregation and diffusive remixing in shallow granular avalanches. J. Fluid Mech. 569: 365-398, 2006.</p><p>Guillard F. Forterre Y., and Pouliquen O. Scaling laws for segregation forces in dense sheared granular flows. J. Fluid Mech. 807, R1, 2016.</p><p>Thornton A. R., Gray J. M. N. T., and Hogg A. J. A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid Mech. 550: 125, 2006.</p><p>Tripathi A., and Khakhar D. V. Numerical simulation of the sedimentation of a sphere in a sheared granular fluid: a granular stokes experiment. Phys. Rev. Lett. 107, 108,001, 2011.</p>
On FTCS Approach for Box Model of Three-Dimension Advection-Diffusion Equation
