Swimming fluctuations of micro-organisms due to heterogeneous microstructure

Swimming microorganisms in biological complex fluids may be greatly influenced by heterogeneous media and microstructure with length scales comparable to the organisms. A fundamental effect of swimming in a heterogeneous rather than homogeneous medium is that variations in local environments lead to swimming velocity fluctuations. Here we examine long-range hydrodynamic contributions to these fluctuations using a Najafi-Golestanian swimmer near spherical and filamentous obstacles. We find that forces on microstructures determine changes in swimming speed. For macroscopically isotropic networks, we also show how the variance of the fluctuations in swimming speeds are related to density and orientational correlations in the medium.

Figure 1

Najafi-Golestanian three-sphere swimmer near obstacles and rods. The isolated circles and arrows show cuts of positions for obstacles, or for the center of rods extending perpendicularly out of the plane, investigated in Figs. 2 and 4, respectively.

Figure 2

Change in velocity [$\Delta {\overline{V}}_1/(d^4\omega /L^3)$, thick lines] of swimmer due to single obstacle placed along cuts 1, 2, 3, and 4 [blue (solid), red (dashes), green (dash-dots), magenta (dots)] in Fig. 1. Comparison to $\frac{1}{3}{\sum }_{\alpha }1/\left(8\pi \mu \right){\left\{\overline{[(\mathbb{I}+\mathbf{r}\mathbf{r}/r^2)/r]\mathbf{f}}\right\}}_1/(d^4\omega /L^3)$ (thin lines), and $\frac{1}{3}{\sum }_{\alpha }1/\left(8\pi \mu \right){\left\{\left[\overline{(\mathbb{I}+\mathbf{r}\mathbf{r}/r^2)/r}\right](-6\pi \mu d\overline{{\mathbf{v}}_0})\right\}}_1/(d^4\omega /L^3)$ (circles on thin lines) corroborates Eqs. (3) and (4).

Figure 3

For a swimmer at the origin, contour plots of (a) change in first component of swimmer velocity $\left(\Delta V_1\right)$ due to obstacle at various positions relative to the swimmer, and (b) first component of swimmer's velocity field in the absence of obstacle $\left(v_{0,1}\right)$. Positive (negative) regions of (a) correlate with the negative (positive) regions of (b). In the whited-out regions within $10d$ of the swimmer spheres, we do not plot results since there the far-field approximations are not valid.

Figure 4

Change in velocity [$\Delta {\overline{V}}_1/(d^4\omega /L^3)$, thick lines] of swimmer due to rod placed along cuts 1, 2, 3, and 4 [blue (solid), red (dashes), green (dash-dots), magenta(dots)] in Fig. 1. Comparison to $\frac{1}{3}{\sum }_{\alpha ,a}{\left[\overline{{\mathbf{S}}^{\alpha a}{\mathbf{f}}^a}\right]}_1/(d^4\omega /L^3)$ (thin lines) and $\frac{1}{3}{\sum }_{\alpha ,a}{\left[\overline{{\mathbf{S}}^{\alpha a}}{\mathbf{R}}^{\mathrm{RFT},a}\overline{{\mathbf{v}}_0^a}\right]}_1/(d^4\omega /L^3)$ (symbols) corroborates Eq. (3), interpreted as additive contributions from each segment of the rod, and Eq. (7).

Figure 5

Variance of swimming velocity calculated from Eq. (10) for isotropically averaged structured mesh as a function of mesh size and random obstacles with equivalent density.

Figure 6

Configuration for slender-body theory.

Figure 7

Straight rod with varying background flow.

Figure 8

Comparison of the $z$ component of the velocity at positions along the $z$ axis passing through the middle of the long rod. In (a) the background wavelength $\lambda$ and rod radius $d_f=0.01\lambda$ are fixed and the segment length $2q$ is varied. In (b) the wavelength $\lambda$ and segment length $2q$ are fixed, and the rod radius $d_f$ is varied.