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

TL;DR

The study addresses the dynamics of an internally actuated weakly elastic sphere translating in a general unbounded quadratic flow under inertialess conditions. It formulates a coupled Stokes–Navier elasticity problem for a compressible sphere driven by a localized point force and torque, and solves it using a domain-perturbation approach with small elastic strain $α$ up to $O(α^2)$. The general results show that elasticity induces a force along the velocity at $O(α)$ and a more complex $O(α^2)$ force with oblique components, while a hydrodynamic torque is nonzero at leading elastic orders in general flow but vanishes on the centreline of the three Poiseuille centerlines; the leading deformation $f^{(1)}$ and the second-order deformation $f^{(2)}$ are quantified, and the deformation increases with confinement and decreases with the elastic modulus parameter $Γ$. The analysis is specialized to elliptical, plane, and Hagen–Poiseuille centerlines, revealing how geometry and material properties shape disturbance fields, deformation patterns, and the force/torque balance, with a qualitative comparison to an active compound drop showing the role of compressibility in morphology.

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}($α^2$), assuming $α\ll 1$. Here, $α$ 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}($α$) is aligned with the particle velocity, while the force at \textit{O}($α^2$) acts at an angle to the velocity. Furthermore, the torque is non-zero due to elastic effects at \textit{O}($α$) and \textit{O}($α^2$). For all the three Poiseuille flows, the point force until \textit{O}($α^2$) is aligned with the particle velocity, while the torque comes as zero.

Figures (8)

• Figure 1: (a) Schematic of $Fe_3O_4$ nanoparticles embedded in a polymer bead.peng2008magnetically Schematic of an elastic sphere translating with velocity $V_0 \mathbf{\hat{k}}$ along the centerline of a channel: (b) inside a straight tube of elliptic cross-section, with $a$ and $b$ as the semi-minor and semi-major axes, respectively; elliptic Poiseuille flow ($\mathbf{v}_e$, top subfigure) and its quadratic component ($\mathbf{v}_{q_e}$, bottom subfigure), (c) between two infinite parallel plates separated by a distance $2h$; plane Poiseuille flow ($\mathbf{v}_p$, top subfigure) and its quadratic component ($\mathbf{v}_{q_p}$, bottom subfigure), and (d) inside a cylindrical tube of radius $R_t$; Hagen-Poiseuille flow ($\mathbf{v}_h$, top subfigure) and its quadratic component ($\mathbf{v}_{q_h}$, bottom subfigure). The sphere is subjected to an external point force ($\mathbf{F}$) and a point torque ($\mathbf{T}$).
• Figure 2: Streamlines of (a) the quadratic component of elliptical Poiseuille flow and its irreducible components (b) $\boldsymbol{\gamma}_{q_e}$, (c) $\mathbf{Q}_{q_e}$ and (d) $\boldsymbol{\tau}_{q_e}$. (e)-(h) are the streamlines of the total velocity field (ambient plus disturbance field) corresponding to (a)-(d), at $\alpha=0.2$, $\Gamma=2$, $P_0=0.1$, $\psi_x=0.1$, and $\psi_y=0.08$.
• Figure 3: Streamlines of (a) the quadratic component of plane Poiseuille flow, and its irreducible components (b) $\boldsymbol{\gamma}_{q_p}$, (c) $\mathbf{Q}_{q_p}$, and (d) $\boldsymbol{\tau}_{q_p}$. (e)-(h) are the streamlines of the total velocity field (ambient plus disturbance field) corresponding to (a)-(d), at $\alpha=0.2$, $\Gamma=2$, $P_0=0.1$, and $\psi=0.1$.
• Figure 4: Streamlines of (a) the quadratic component of Hagen-Poiseuille flow, and its irreducible components (b) $\boldsymbol{\gamma}_{q_h}$ and (c) $\boldsymbol{\tau}_{q_h}$. (d)-(f) are the streamlines of the total velocity field (ambient plus disturbance field) corresponding to (a)-(d), at $\alpha=0.2$, $\Gamma=2$, $P_0=0.1$, and $\psi=0.1$.
• Figure 5: Deformed shape of the sphere for different confinement ratios, (a)-(c) for elliptical Poiseuille flow, (d)-(f) for plane Poiseuille flow and (g)-(i) for Hagen-Poiseuille flow plotted on the $xz$-, $yz$- and $xy$-planes, respectively, at $\alpha=0.2$, $P_0=0.1$, and $\Gamma=2$.