Hydrodynamics of active particles in viscosity gradients

In this work, we analyze the motion of an active particle, modeled as a spherical squirmer, in linearly varying viscosity fields. In general, the presence of a particle will disturb a background viscosity field and the disturbance generated depends on the boundary conditions imposed by the particle on the viscosity field. We find that, irrespective of the details of the disturbance, active squirmer-type particle tend to align down viscosity gradients (negative viscotaxis). However, the rate of rotation and the swimming speed along the gradient do depend on the details of the interaction of the particle and the background viscosity field. In addition, we explore the relative importance on the dynamics of the local viscosity changes on the surface of active particles versus the (nonlocal) changes in the flow field due to spatially varying viscosity (from that of a homogeneous fluid). We show that the relative importance of local versus nonlocal effects depends crucially on the boundary conditions imposed by the particle on the field. This work demonstrates the dangers in neglecting the disturbance of the background viscosity caused by the particle as well as in using the local effects alone to capture the particle motion.

Figure 1

A schematic showing the motion of an active particle in linear viscosity fields and the associated coordinate system. The radius of the particle is $a$. It translates and rotates with the velocities $\mathbf{U}$ and $\mathrm{\Omega }$, respectively. The ambient viscosity ${\eta }_0$ increases linearly with $x$.

Figure 2

(a) The ambient viscosity relative to the reference viscosity and (b) the disturbance viscosity due to the no-flux condition at the instant when the particle's center is at the origin in the laboratory frame, i.e., ${\mathbf{x}}_c=\mathit{0}$. These viscosity distributions do not depend on the orientation of the particle.

Figure 3

(a) The source and (b) the source-dipole part of the disturbance viscosity due to a constant viscosity condition at the instant when the particle center is at the origin in the laboratory frame, i.e., ${\mathbf{x}}_c=\mathit{0}$. These viscosity distributions do not depend on the orientation of the particle.