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

TL;DR

This work analyzes the dynamics of a weakly elastic sphere translating parallel to a rigid wall under Stokes flow, constrained by a point force and torque. By modeling the solid with Navier elasticity and the fluid with Stokes equations, and employing domain perturbation plus the method of reflections, the authors obtain a coupled description of deformation, displacement, and hydrodynamic forces up to $O(1/H^3)$ (aligned gravity) or $O(1/H^2)$ (arbitrary gravity) with the elastic parameter $oldsymbol{lpha}$ scaling as $O(1/H)$. A key finding is that wall effects generate a hydrodynamic lift at $O(oldsymbol{lpha}/H^2)$ in the aligned case, while torque remains zero through the considered orders; for oblique gravity, a wall-induced torque appears at $O(oldsymbol{lpha}/H)$, and the force/torque expressions couple horizontal and vertical components due to nonlinearity in boundary conditions. The results highlight how elasticity, force distribution, and wall proximity together shape the deformation and mobility of deformable particles near boundaries, with implications for magnetically actuated beads and microfluidic separation strategies where precise force/torque control is essential.

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 $α$ and $1/H$, using the domain perturbation method and the method of reflections. Here, $α$ 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 $α\sim 1/H$, 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 $\textit{O}(α/H^2)$, in the former case. In an unbounded domain, the elastic effects in the latter case generate a hydrodynamic torque at \textit{O}($α$) and a drag at \textit{O}($α^2$). Conversely, in the former case, the torque is zero, while the drag still appears at \textit{O}($α^2$).

Figures (7)

• Figure 1: (a) Schematic of an elastic sphere translating parallel to a rigid wall at a specified velocity ($V_0 \mathbf{\hat{k}}$), subjected to an external point force ($\mathbf{F}$) and a point torque ($\mathbf{T}$). (b) Schematic of the $Fe_3O_4$ nanoparticles encapsulated within the PDMS matrix peng2008magnetically. (c) Unbounded velocity field ($\mathbf{v}_{un}$) undergoing subsequent reflections from wall (w) and sphere (s).
• Figure 2: The shape of the deformed sphere on $xz$ plane at $\alpha=0.2$, $\tilde{\rho}=1.05$, $K_{fz}=0.5$ and $\gamma=2$ in the (a) unbounded domain and (b) in the presence of wall at $H=2.5$.
• Figure 3: Variation in the shape of the deformed sphere in the unbounded domain as a function of $K_{fz}$ at $\gamma=2$, $\alpha=0.2$ when (a) the particle is denser than fluid, $\tilde{\rho}=1.05$ and (b) when the particle is lighter than the fluid, $\tilde{\rho}=0.05$.
• Figure 4: Variation of the maximum principal stress (based on its absolute value) along the particle surface in the unbounded domain (a) for $K_{fz}=0.5$ and (b) for different values of $K_{fz}$. (c) Variation of the principal stresses ($\tau_1$, $\tau_2$, and $\tau_3$) and $\tau_{max}$ for $K_{fz}=0.5$, along the particle surface, plotted at $\gamma=2$, $\alpha=0.2$, and $\tilde{\rho}=1.05$ for $\theta$ varying from $0$ to $\pi$.
• Figure 5: Variation of external point force (parallel to the wall) given in (\ref{['eq:wall force gravity aligned']}) relative to the unbounded given in (\ref{['eq:unbound force gravity aligned']}) for different wall-particle gap widths, H varying from 1.5 to 10 at $\tilde{\rho}=1.05$, $K_{fz}=0.5$ and $\gamma=2$.