Solute dispersion in pulsatile Casson fluid flow with exchange between phases using the extended Gill’s approach

We investigate solute dispersion in a two-phase system comprising a Casson fluid flowing in a tube and its surrounding wall phase that allows interphase solute exchange to mimic solute transport in blood and tissue phases. A pulsatile pressure gradient is imposed, and Gill’s classical methodology is extended to two-phase flows to analyse solute transport. The key parameters are the diffusivity ratio between wall and fluid phases ( $\lambda$ ), the partition coefficient ( $\beta _p$ ), the Womersley number ( $\alpha$ ), the yield stress ( $\tau _y$ ), the wall thickness ( $\delta _h$ ) and the initial dimensionless radius of the solute source ( $a$ ). In the long-time limit, increasing $\lambda$ , $\beta _p$ and $\delta _h$ reduces the phase-averaged convection ( $K_1$ ) and dispersion ( $K_2$ ) coefficients, owing to solute accumulation in the wall where convective and shear-induced transport are absent. Short-time behaviour is dictated by the rate of solute transfer to the wall. Larger $\alpha$ enhances both $K_1$ and $K_2$ , while larger $\tau _y$ suppresses them. The presence of a wall phase permits $K_2$ to reach $O(10^{0})$ , compared with $K_2 \sim O(10^{-3})$ without a wall, and can delay the onset of steady state to dimensionless time $t \sim O(10^{2})$ . Strong solute exchange and increasing wall thickness diminish downstream solute penetration, while non-Newtonian effects promote interphase transfer. These results provide mechanistic insight into solute exchange across fluid–wall interfaces, relevant to solute transport in blood flow and engineered permeable systems.