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Motivated by the technology of magnetically targeted drug and gene delivery, in which a magnetic field is used to direct magnetic carrier particles from the circulation to a target site, we develop a continuum model for the motion of particles (magnetic carriers) subject to an external body force (magnetic field) in a flow of a concentrated suspension of a species of neutrally buoyant particles (blood). An advection–diffusion equation describes the evolution of the carrier particles as they advect in the flow under the action of an external body force, and diffuse as a result of random interactions with the suspension of neutrally buoyant particles (shear-induced diffusion). The model is analysed for the case in which there is steady Poiseuille flow in a cylindrical vessel, the diffusive effects are weak and there is weak carrier uptake along the walls of the vessel. The method of matched asymptotic expansions is used to show that carriers are concentrated in a boundary layer along the vessel wall and, further, that there is a carrier flux along this layer which results in a sub-layer, along one side of the vessel, in which carriers are even more highly concentrated. Three distinguished limits are identified: they correspond to cases for which (i) the force is sufficiently weak that most particles move through the vessel without entering the boundary layers along the walls of the vessel and (ii) and (iii) to a force which is sufficiently strong that a significant fraction of the particles enter the boundary layers and, depending upon the carrier absorption from the vessel walls, there is insignificant/significant axial carrier flux in these layers.

Original publication




Journal article


European Journal of Applied Mathematics


Cambridge University Press (CUP)

Publication Date





77 - 107