Dynamics of an internally actuated elastic particle in a plane Poiseuille flow

Abstract

We analytically analyse the dynamics of an internally actuated particle, modelled as a compressible elastic sphere embedded with a magnetic bead at its undeformed centre, translating in a plane Poiseuille flow in the Stokes limit. The particle is constrained to translate with a prescribed velocity while remaining at an arbitrary position within the flow by applying an external point force and external point torque at its undeformed centre. The governing equations for the fluid and particle are the Stokes and Navier elasticity equations, respectively. We use the series solutions to the governing equations and the domain perturbation method to capture the deformed shape of the particle, assuming $\alpha \ll 1$. Here, $\alpha$ quantifies the elastic strain induced in the particle due to the viscous stress from the fluid. The external force and external torque are obtained until O($\alpha^2$). The particle translating along the channel length experiences an elastic-induced hydrodynamic lift as well as hydrodynamic torque both at O($\alpha$) and O($\alpha^2$). The leading-order lift depends linearly on the local shear rate and on the combined effects of slip velocity and flow curvature, where the slip velocity is defined as the particle velocity relative to the local ambient flow. The particle reaches a stable equilibrium position away from the centreline, where the net lift vanishes. We show that the direction of deformation-induced lateral migration of the internally actuated particle is qualitatively distinct from that of drops, capsules, and vesicles in the Stokes limit and from that of rigid spheres undergoing inertial migration.