Dynamics of an internally actuated weakly elastic sphere translating parallel to a rigid wall

TL;DR

This paper models the complex dynamics of a weakly elastic sphere translating near a wall in a viscous fluid, revealing how elasticity influences lift, torque, and drag through series solutions in the Stokes flow regime.

Contribution

It introduces a detailed analytical framework combining Navier elasticity and Stokes equations to analyze elastic effects on particle motion near a wall.

Findings

Elastic effects induce hydrodynamic lift at O(α/H²).

Elasticity causes torque and drag modifications depending on force alignment.

Series solutions up to O(1/H³) elucidate particle deformation and motion.

Abstract

We analyse the dynamics of a weakly elastic spherical particle translating parallel to a rigid wall in a quiescent Newtonian fluid in the Stokes limit. The particle motion is constrained parallel to the wall by applying a point force and a point torque at the centre of its undeformed shape. The particle is modelled using the Navier elasticity equations. The series solutions to the Navier and the Stokes equations are utilised to obtain the displacement and velocity fields in the solid and fluid, respectively. The point force and the point torque are calculated as series in small parameters $\alpha$ and $1/H$, using the domain perturbation method and the method of reflections. Here, $\alpha$ is the measure of elastic strain induced in the particle resulting from the fluid's viscous stress, and $H$ is the non-dimensional gap width, defined as the ratio of the distance of the particle centre from the wall to its radius. The results are presented up to $\textit{O}(1/H^3)$ and $\textit{O}(1/H^2)$, assuming $\alpha \sim 1/H$, for cases where gravity is aligned and non-aligned with the particle velocity, respectively. The deformed shape of the particle is determined by the force distribution acting on it. The hydrodynamic lift due to elastic effects (acting away from the wall) appears at $\textit{O}(\alpha/H^2)$, in the former case. In an unbounded domain, the elastic effects in the latter case generate a hydrodynamic torque at \textit{O}($\alpha$) and a drag at \textit{O}($\alpha^2$). Conversely, in the former case, the torque is zero, while the drag still appears at \textit{O}($\alpha^2$).