Solute dispersion in a permeable capillary with wall exchange: Darcy flow model

Abstract

Complex chemical reactions and hydrodynamic dispersion feature in many aspects of hemodynamics. Motivated by examining the reactive phase change dispersion in perfusion, a mathematical study is presented for solute dispersion in incompressible laminar blood flow through a straight circular capillary with a permeable wall (enabling lateral movement across the vessel fenestrations in perfusion and associated with the presence of the endothelial layer). The boundary condition at the vessel wall is considered a reversible phase exchange process based on first-order chemical kinetics. Darcy’s law is deployed to feature the permeability nature of the capillary. A multiple-scale asymptotic analysis is developed, and a non-dimensional transverse averaged “macro-transport” equation for convective diffusion–dispersion is derived. Expressions are then presented for the advection coefficient and Taylor dispersion coefficient. Numerical evaluation of the impact of key control parameters i.e., permeability parameter, pressure parameter, retention parameter (α), Damköhler number (Da) on dispersion coefficient, advection coefficient and leading order concentration of the solute is conducted, and solutions are visualized graphically both for small and large times. The novelty of the present work is, therefore, the collective consideration of complex wall permeability and pressure difference in addition to boundary reaction and Darcian body force effects. The analysis shows that the dispersion coefficient is initially enhanced gradually with an increment in the retention parameter with its initial small value and thereafter exhibits a smooth decay. The hydrodynamic dispersion coefficient markedly decreases with higher values of the permeable parameter. A higher magnitude of dispersion coefficient is computed at the vessel inlet and then decreases towards the outlet. A boost in the leading order concentration of the solute is computed at small times but is stabilized and eventually remains invariant with time. The axial velocity is found to depend strongly on the axial position in the capillary. A displacement in concentration peaks is also observed which is attributable to advection along the axial direction, and the decreasing peaks with respect to time are due to the diffusion of the solute from the fluid phase to the vessel wall. Generally, it is also observed that retention enhances the chemical reaction effect, leading to a greater loss of solute over time. The simulations are relevant to chemo-hemodynamics and also may find applications in drug delivery (pharmacodynamics).

Graphical abstract