Frequency-dependent higher-order Stokes singularities near a planar elastic boundary: Implications for the hydrodynamics of an active microswimmer near an elastic interface

The emerging field of self-driven active particles in fluid environments has recently created significant interest in the biophysics and bioengineering communities owing to their promising future for biomedical and technological applications. These microswimmers move autonomously through aqueous media, where under realistic situations they encounter a plethora of external stimuli and confining surfaces with peculiar elastic properties. Based on a far-field hydrodynamic model, we present an analytical theory to describe the physical interaction and hydrodynamic couplings between a self-propelled active microswimmer and an elastic interface that features resistance toward shear and bending. We model the active agent as a superposition of higher-order Stokes singularities and elucidate the associated translational and rotational velocities induced by the nearby elastic boundary. Our results show that the velocities can be decomposed in shear and bending related contributions which approach the velocities of active agents close to a no-slip rigid wall in the steady limit. The transient dynamics predict that contributions to the velocities of the microswimmer due to bending resistance are generally more pronounced than those due to shear resistance. Bending can enhance (suppress) the velocities resulting from higher-order singularities whereas the shear related contribution decreases (increases) the velocities. Most prominently, we find that near an elastic interface of only energetic resistance toward shear deformation, such as that of an elastic capsule designed for drug delivery, a swimming bacterium undergoes rotation of the same sense as observed near a no-slip wall. In contrast to that, near an interface of only energetic resistance toward bending, such as that of a fluid vesicle or liposome, we find a reversed sense of rotation. Our results provide insight into the control and guidance of artificial and synthetic self-propelling active microswimmers near elastic confinements.

Figure 1

Illustration of the system setup. An axisymmetric active microswimmer modeled as a prolate spheroid is trapped at $z=h$ above an elastic interface infinitely extended in the $xy$ plane. The lengths of the short and long semiaxes are denoted by $a$ and $c$, respectively. Setting the orientation of the swimmer, the unit vector $\widehat{\mathit{e}}$ points along the symmetry axis of the swimmer. The pitch angle of the swimmer relative to the horizontal plane is denoted by $\theta \in [-\pi /2,\pi /2]$ (the complement of the polar angle in spherical coordinates). On both sides of the elastic interface, the surrounding fluid is Newtonian and characterized by the same dynamic viscosity $\eta$. The figure shown in the inset is a top view of the local reference frame associated with the microswimmer, where ${\rho }_0$ is the radial distance and $\phi \in [0,2\pi )$ is the azimuthal orientation.

Figure 2

Evolution of the scaled induced translational and rotational swimming velocities associated with a force dipole (a–c), source dipole (d–f), force quadrupole (g–i), and rotlet dipole (j–l), resulting from hydrodynamic interactions with a planar elastic interface of pure shear (green), pure bending (red), or both shear and bending (black) resistances. The swimmer has an aspect ratio $\gamma =4$ and is oriented by a pitch angle $\theta =\pi /6$ relative to the horizontal direction. Here, the velocities are scaled by the corresponding hard wall limits listed in Table 1, except that the $x$ component of the rotlet dipolar contribution shown in panel (j) is scaled by ${\alpha }_{\mathrm{RD}}/\left(8h^4\right)$ (because this component vanishes in the steady limit). The scaled time is $\tau :={\tau }_{\mathrm{S}}=16{\tau }_{\mathrm{B}}$.

Figure 3

Evolution of the scaled swimming velocities associated with a Stokeslet (a–c) and rotlet (d–f) due to hydrodynamic interactions with an elastic interface showing pure shear (green), pure bending (red), or both shear and bending rigidities (black). Here, the swimmer has an aspect ratio $\gamma =4$ and an orientation $\theta =\pi /6$ with respect to the horizontal direction. The velocities are scaled by the corresponding hard wall values except that the $x$ component of the rotlet contribution is scaled by ${\alpha }_{\mathrm{R}}/\left(8h^4\right)$. We set $\tau :={\tau }_{\mathrm{S}}=16{\tau }_{\mathrm{B}}$.