Faxén's first and second laws for a sphere in activity-driven capillary transport

In this paper, we use Faxén's laws for calculating the drag force and the torque on spheres of different radii in the presence of a background flow of an active liquid inside a circular capillary. The active liquid contains active particles that demonstrate vortex defects: the resulting circular polarization field induces an axial pressure-gradient (proportional to the axial gradient in the activity), thereby triggering a fluid flow inside the capillary. The profile of this activity-gradient-driven fluid flow is different from the externally-imposed pressure-gradient-driven Hagen–Poiseuille flow inside the capillary. This difference leads to a significant variation in the axial drag force and the torque experienced by the sphere. For example, the axial drag force on the sphere in the presence of activity-gradient-driven flow varies non-monotonically with the position of the particle center: it shows a maximum at locations away from the capillary center and with an increase in the sphere radius, this location shifts more and more away from the capillary center. On the contrary, the axial drag force due to the pressure-driven background flow varies monotonically with the position of the particle center. Also, the torque on the sphere due to the activity-gradient-driven flow (pressure-driven flow) varies non-monotonically (monotonically) with the position of the particle center. Finally, the strengths of both this axial drag force as well as this torque due to activity-driven flow are found to be larger (smaller) than that due to the pressure-gradient driven flow for locations of the sphere closer to the capillary center (wall).