Abstract
This work explores diffusion with scale-dependent coefficients, starting from a general advection-diffusion framework from a theoretical standpoint, and then focusing numerically on a purely diffusive regime. Advection-diffusion processes are central to modeling transport phenomena in natural and engineered systems. However, classical models often fail to capture the complexities of systems with spatial and temporal variability. In this work, we present a multiscale advection-diffusion model that incorporates time-dependent diffusion coefficients and spatially inhomogeneous, multiscale body forces. Using the asymptotic homogenization technique, we derive a macroscopic equation that reflects the evolution of transport properties across multiple scales, accounting for both spatial and temporal variations. A key contribution of this study is the formulation of new cell problems associated with the dual time-dependence of the diffusion coefficient and the multiscale forces, which lead to the introduction of additional source terms. Furthermore, we incorporate a novel source term arising from the nonzero divergence of the advective velocity field, which modifies the effective macroscopic advection velocity to capture source and sink effects at the microscale. We apply this model to describe water molecule diffusion in packed erythrocytes, a system exhibiting dual time scales, and by showing how our approach captures the temporal evolution of transport under dynamic diffusion.