The segregation of large intruders in an agitated granular system is of high practical relevance, yet the accurate modelling of the segregation (lift) force is challenging as a general formulation of a granular equivalent of a buoyancy force remains elusive. Here, we critically assess the validity of a granular buoyancy model using a generalization of the Archimedean formulation that has been proposed very recently for chute flows. The first model system studied is a convection-free vibrated system, allowing us to calculate the buoyancy force through three different approaches, i.e. a generalization of the Archimedean formulation, the spring force of a virtual spring and through the granular pressure field. The buoyancy force obtained through these three approaches agree very well, providing strong evidence for the validity of the generalization of the Archimedean formulation of the buoyancy force which only requires an expression for the solid fraction of the intruder, hence allowing for a computationally less demanding calculation of the buoyancy force as coarse-graining is avoided. In a second step, convection is introduced as a further complication to the granular system. In such a system, the lift force is composed of granular buoyancy and a drag force. Using a drag model for the slow velocity regime, the lift force, directly measured through a virtual spring, can be predicted accurately by adding a granular drag force to the generalization of the Archimedean formulation of the granular buoyancy. The developed lift force model allowed us to rationalize the dependence of the lift force on the density of the bed particles and the intruder diameter, and the independence of the lift force on the the intruder diameter, and the independence of the lift force on the intruder density and the vibration strength (once a critical value is exceeded).

Bedload transport often exhibits dual-mode behavior due to interactions of spatiotemporal controlling factors with the migrating three-dimensional bedforms (characterized by the fully developed patterns in the bed, such as alternate bars, pools, and clusters). This study explores dual-mode bedload transport based on experimental measurements and develops Einstein's exponential-based model to characterize large fluctuations of bedload sediment discharge. The particle waiting time, particle flux, and bed elevation are measured in a series of well-controlled laboratory experiments. Flume experiments show that the waiting time distribution of sediments gives a bimodal characteristic, two distinct modes can be identified from the measured data. This study encapsulates this dual-mode bedload transport behavior in a hyperexponential distribution of sediment resting times and introduces it into the continuous time random walk (CTRW) framework. Considering the scaling limit of the thin/heavy-tailed CTRW processes, a single-rate mass transfer (SRMT) and fractional-derivative SRMT (F-SRMT) models are obtained, and the model parameters are determined from the hyperexponential distribution. Further analyses reveal that the dual-mode bedload transport behavior is controlled by mass exchange between the mobile and immobile zones, and a dimensionless index η can quantify the intensity of dual-mode behavior. Applications show that the dual-mode bedload transport models are much more accurate in characterizing bedload transport in a mixed-size gravel bed than the traditional exponential-based model, and the nonlocal movement of bedload sediments is significant in the mixed-size gravel bed. Further investigations will focus on the applicability test of the dual-mode models to other flow regimes and conditions.