Do hydrodynamic interactions affect the swim pressure?

The dynamic behavior of active matter has been a subject of great interest in recent years, though the role of fluid-mediated hydrodynamic interactions (HI) remains largely unknown. We study the motion of a spherical active Brownian particle (ABP) of size a, moving with a fixed speed U₀, and reorienting on a time scale τ_R in the presence of a confining boundary. The ABP, or swimmer, interacts with the boundary through a combination of hard-core collisions and HI; the strength of HI is characterized by a dimensionless parameter Δ. We compute the average force per unit area exerted on the wall by the swimmer, which is equivalent to the mechanical pressure Π = p_f + n₀k_BT + Π^swim, where p_f is the fluid pressure, n₀ is the far-field number density of swimmers and Π^swim is the swim pressure. In the absence of HI, Π^swim depends only on the mechanical properties of the swimmer. We show that HI quantitatively modify this pressure because the run length λ = U₀τ_R and the translational drag on the swimmer ζ may now depend on Δ, however the swim pressure has the same scaling: Π^swim=n₀ζ(Δ)U₀λ(Δ)/6. Similarly, if the swimmers move by a fixed force F^swim, Π^swim = n₀F^swimλ(Δ)/6.