Determine the convective-layer prefactor α in the virtual-boundary correction

Determine the order-one prefactor α in the effective-radius correction R → R + α δ that accounts for the external convective, solute-depleted boundary layer in the diffusion-limited settling model for porous particles in linear stratification, where the boundary-layer thickness scales as δ ∼ (4 ν^2 R/(g ρ_s))^{1/4} under free-convection conditions (with ν the kinematic viscosity, R the particle radius, g gravitational acceleration, and ρ_s the particle’s excess solid density).

Background

In the diffusion-limited regime, the authors explain that a thin external convective boundary layer of reduced solute concentration forms around a settling porous particle, which slows mass adaptation relative to the simplified model. To incorporate this effect in a reduced model, they propose a ‘virtual’ boundary by extending the particle radius as R → R + α δ, where δ is estimated from classical free-convection boundary-layer scaling.

Although they provide a best-fit example yielding α ≈ 0.60 for one dataset, they explicitly state that the prefactor α is unknown a priori and presumably order one. A general determination or first-principles derivation of α would refine the predictive model for diffusion-limited settling speeds, especially across varying parameter regimes and geometries.