Defect self-propulsion in active nematic films with spatially-varying activity*

We study the active flow caused by an isolated topological defect of charge ± 1/2 in an active nematic film subject to both friction with a substrate (Γ) and viscus dissipation (η).  The two dissipation mechanisms give rise to a length scale l_d= (η/Γ)^1/2, which sets the scale for the self-propulsion velocity of the +1/2 defect and the scale for the decay of the flow velocity.  When confined to a disc, spanning vortices of alternating vorticity are formed around the defects, and the self-propulsion velocity becomes determined by the ratio between the disc's radius R and l_d. For small discs the self-propulsion velocity is proportional to R, while it is set by the dissipation length for large discs. 

The effects of a spatially varying activity on the defect’s kinematics are also studied in the two limits of a constant activity gradient, and a sharp interface between an active and a passive region. For the constant gradient the self-propulsion of the +1/2 defect is given by the local activity, and a non-zero vorticity is induced at the defects core. This tends to rotate the defect so that it aligns with the gradient. The constant activity gradient does not make the -1/2 defect self-propelled. For the sharp interface the -1/2 defect becomes self-propelled near the activity jump. Depending on the defects orientation it is either attracted to or repelled by the interface. The +1/2 defect slows down when it approaches the interface, and an induced vorticity tends to reorient it so that its polarization becomes perpendicular to the interface.