Drift-diffusive liquid migration in partly saturated sheared granular media

We study liquid migration in partly saturated shear bands of granular media where liquid is transported away from the shear-band centre. Earlier studies show that the liquid migration can be modelled as a diffusive process with a shear-rate-dependent diffusion coefficient. Here, we apply this model in a two-dimensional Cartesian split-bottom shear cell with one wide, steady shear band. Initially, a high liquid concentration peak develops at the edges of the shear band, which propagates away from the shear band, splitting the shear cell into a liquid-depleted shear-band region and an outer region not yet affected by the liquid migration. Assuming the liquid transport in the vertical direction is negligible, we simplify the liquid migration model to a one-dimensional form evolving over time. By coordinate transformation, an analytically solvable drift-diffusion model is obtained for liquid migration from the simplified model. From here, we obtain analytical solutions for the liquid concentration as a function of space and time. The significance of the mechanisms is studied in terms of the local Péclet number. While drift enhances drying of the shear band and accumulates the liquid in the peak, diffusion shifts the liquid further away from the shear band. To validate the model, we predict numerically the trajectory of the liquid concentration peak from the continuum model and compare with discrete particle method (DPM) simulations. Our continuum model results give a perfect qualitative and an approximate quantitative agreement with the overall results predicted from the DPM model.