Time-averaged dynamics of rigid and elastic particles in oscillatory flow

Oscillatory flows are powerful tools for manipulating suspended particles and biological cells in microfluidic settings. In particular, soft biological cells may deform under the large oscillatory stresses typical of applications. We present a comprehensive theory of the dynamics of spherical particles suspended in oscillatory flow, including the effects of particle deformation, which are modeled through linear elasticity. The particle responds to the flow by producing a primary oscillatory disturbance, which is coupled with elastic deformations. The inertia of this primary flow drives a secondary disturbance that is near impossible to calculate. However, by applying the Lorentz reciprocal theorem, we are able to relate the secondary force on the particle with its time-averaged velocity using information of the primary (oscillatory) flow and elastic deformation only. We discuss how the density contrast between particle and fluid, the frequency of oscillation, and the stiffness of the particle influences its motion. For rigid particles, we show that the direction of particle motion can be reversed for certain combinations of frequency and density ratio. For elastic particles, the dynamics depend additionally on a dimensionless compliance that characterizes the ratio of elastic to viscous stresses. We find that the force-velocity relations are modified considerably from the strictly rigid case, even for modestly compliant particles. Finally, we apply the theory to compute the motion of rigid and compliant particles in some canonical oscillatory flows.