Self-driven particles in linear flows and trapped in a harmonic potential

We present analytical expressions for the mean-square displacement of self-driven particles in general linear flows and trapped in a harmonic potential. The general expressions are applied to three types of linear flows, namely, shear flow, solid-body rotation flow, and extensional flow. By using Brownian dynamics simulations, the effect of trapping and external linear flows on the particles' distribution is also elucidated. These simulations also enabled us to validate our theoretical results.

Figure 1

Spherical self-propelled particle (yellow) of radius $a$, swimming in a two-dimensional fluctuating environment at temperature $T$, and subject to a general uniform linear flow. In the figure, a shear flow $[{\mathbf{U}}_{\infty }=\left(G{x}_2,0\right)]$ is represented. The pink area indicates a harmonic trap.

Figure 2

Effect of external linear flows and trapping on the swimmers' distribution. First row: 3000 active non-interacting spherical swimmers ($D_{\mathrm{\Omega }}=0.161/\mathrm{s},D_B=0.22\mu {\mathrm{m}}^2/\mathrm{s}$) with a constant swimming velocity $U=5\mu \mathrm{m}/\mathrm{s}$, immersed in three linear flows and trapped by a harmonic potential, here, $\mathrm{Pe}=3.125$, and $K=0.625$. The swimmers' distribution is plotted for three different times namely 0.1 s (red), 7.3 s (black), and 200 s (blue). (a), Swimmers' distribution in a solid-body rotation flow; (b), Swimmers' distribution in a shear flow; (c), Swimmers' distribution in an extensional flow. The bottom row represents the same conditions as in the first row but under a strong trap, that is, $\mathrm{Pe}=3.125$, and $K=6.25$ for 0.1 s (red), 0.6 s (black), and 10 s (blue).

Figure 3

Brownian dynamics simulations for 3000 realizations of an active spherical swimmer ($D_{\mathrm{\Omega }}=0.161/\mathrm{s},D_B=0.22\mu {\mathrm{m}}^2/\mathrm{s}$) with a constant swimming speed $U=5\mu \mathrm{m}/\mathrm{s}$, immersed in a shear flow and trapped by a harmonic potential ($\mathrm{Pe}=3.125$, and $K=6.25$); (a) Long time theoretical results according to Eqs. (18, 19, 20) are shown as dashed lines, while the numerical results are shown in circles; (b) Short time limit results given by Eqs. (21)–(23) are shown in dashed lines, while the numerics are in circles.

Figure 4

Brownian dynamics simulations for 3000 realizations of an active spherical swimmer ($D_{\mathrm{\Omega }}=0.161/\mathrm{s},D_B=0.22\mu {\mathrm{m}}^2/\mathrm{s}$) with a constant swimming velocity $U=5\mu \mathrm{m}/\mathrm{s}$ immersed in a shear flow and trapped by a harmonic potential ($\mathrm{Pe}=3.125$, and $K=6.25$). In the figure the asymptotic short and long times results given by Eqs. (19) and (22), respectively, are also plotted. The whole expression Eq. (A1) is in good agreement with the simulations.

Figure 5

Effect of temperature on the swimmers' MSD component $\langle x_1^2\rangle$ under a shear flow. For this case, three different bath's temperature were chosen namely $T=\{280\text{K},300\text{K},\text{and}320\text{K}\}$. (a) Equation (18) showing the effect of temperature for an ABP in an external shear flow ($U=5\mu \mathrm{m}/\mathrm{s}, G=0.5$ 1/s). (b) The same as in (a) but for the case of passive particles ($U=0\mu \mathrm{m}/\mathrm{s}, G=0.5$ 1/s).