Determine the size-ratio dependence of the drag-force evaluation offset

Determine the exact formulation of the dimensionless offset parameter \(\alpha\), including its dependence on the intruder-to-particle size ratio \(S=R/r_p\), in the two-region drag-force model for an intruder in a dense granular flow.

Background

The force model evaluates the velocity and viscosity in the upper and lower regions surrounding an intruder at a distance $\alpha R$ from the intruder centre. This parameter controls the magnitude of the gradient-induced drag contributions, including terms scaling with the shear-rate and viscosity gradients.

The paper states that $\alpha$ is likely governed by the size ratio between the intruder and bulk particles, but does not derive its value or functional dependence. Resolving this dependence is necessary to make the proposed force model predictive rather than dependent on fitted or assumed geometric parameters.