Vorticity Determines the Force on Bodies Immersed in Active Fluids

When immersed into a fluid of active Brownian particles, passive bodies might start to undergo linear or angular directed motion depending on their shape. Here we exploit the divergence theorem to relate the forces responsible for this motion to the density and current induced by—but far away from—the body. In general, the force is composed of two contributions: due to the strength of the dipolar field component and due to particles leaving the boundary, generating a nonvanishing vorticity of the polarization. We derive and numerically corroborate results for periodic systems, which are fundamentally different from unbounded systems with forces that scale with the area of the system. We demonstrate that vorticity is localized close to the body and to points at which the local curvature changes, enabling the rational design of particle shapes with desired propulsion properties.

Figure 1

Line-symmetric boomerang-shaped body. (a) Reduced density $\rho /\overline{\rho }$ (heat map) and polarization $\mathbf{p}$ (arrows) of active particles (with global density $\overline{\rho }=N/L^2=1.2$ and speed $v_0=80$). Shown is a close-up of the body, the total system is larger. The length of arrows is calculated as the logarithm of the local polarization magnitude. (b) Vorticity map $\omega$. Note the antisymmetric $\omega (-x,y)=-\omega (x,y)$ yielding a vanishing monopole moment $Q=0$ and a nonvanishing dipole moment $\mathbf{P}=P{\mathbf{e}}_x$ (arrow).

Figure 2

Finite-size behavior in periodic systems. (a) Density profile $\rho$ for the same parameters as Fig. 1 but showing the full system. Note the strong departure from a dipolar field with “streaks” connecting through the periodic boundaries. (b) Numerical current $-J_{L}L/v_0$, force $F_L/v_0$, and vorticity dipole moment $-P_L$ as a function of box size $L$. All quantities agree and scale as $L^2$ for small system sizes (dashed line with $j_0/v_0\simeq 0.34$) before crossing over to a constant value $-P_{\infty }$ beyond $L^{*}\simeq 50R$. Numerical data is shown for $v_0=160$.

Figure 3

$C_2$-symmetric body. (a) Density profile and (b) vorticity map. Note the two dipoles forming at the shape’s corners, which are responsible for the nonvanishing torque (while $Q=0$).

Figure 4

Vorticity $\omega$ along the perimeter of three shapes: (a) the shape from Fig. 3 a lune (subtracting a smaller disc) and (c) a bud (union with smaller disc). The vorticity is zero almost everywhere and localized to regions close to changes of the local curvature. Arrows indicate the local dipoles ${\mathbf{P}}_n$, which are oriented oppositely to the orientation of the perimeter.