Dynamics of an internally actuated weakly elastic sphere in a general quadratic flow

Abstract

Internally actuated elastic particles are widely used in biomedical applications. It is imperative to understand the dynamics of such particles in pressure-driven microfluidic devices to manipulate their motion. We analytically examine the dynamics of an internally actuated elastic particle translating in a general unbounded quadratic flow in the inertialess limit. We consider the particle as a compressible weakly elastic sphere, and its motion is controlled by applying an external point force and a point torque at the centre of its undeformed shape. The fluid and the particle are modelled using the Stokes and the Navier elasticity equations, respectively. We use the domain perturbation method to capture the particle deformation. The point force and the point torque are obtained until \textit{O}($\alpha^2$), assuming $\alpha\ll 1$. Here, $\alpha$ is the measure of the particle elastic strain induced due to the fluid viscous stress. We present the results for the particle motion in a general unbounded quadratic flow. The results are simplified further for the motion along the centreline in the quadratic component of three Poiseuille flows: 1) elliptical Poiseuille, 2) plane Poiseuille, and 3) Hagen-Poiseuille flows. In the general quadratic flow, the point force at \textit{O}($\alpha$) is aligned with the particle velocity, while the force at \textit{O}($\alpha^2$) acts at an angle to the velocity. Furthermore, the torque is non-zero due to elastic effects at \textit{O}($\alpha$) and \textit{O}($\alpha^2$). For all the three Poiseuille flows, the point force until \textit{O}($\alpha^2$) is aligned with the particle velocity, while the torque comes as zero.