Regularized representation of bacterial hydrodynamics

Fluid flows induced by a flagellated bacterial swimmer are often modeled as a simple force dipole, valid in the far field. Such representations neglect the inherent rotation of these bacteria as they swim, driven by a spinning helical flagellum or fascicle. Here, we present a refined swimmer representation that makes use of regularized singularities, retaining simplicity while capturing details of the complex flow field near the swimmer that have previously been absent from basic models. We illustrate the significance of this representation via a study of bacterial predator-prey dynamics, highlighting the importance of detailed hydrodynamics in models of bacterial interactions and bacterial active matter.

Figure 1

Sample flow fields (a, b, c) and streamlines (d, e, f) around a model bacterium positioned at the origin, with combined plots showing swimmer geometry and orientation (g, h, i), where axis scales are in units of flagellum length. Panels (a, d, g) correspond to the original flow field ${\mathit{u}}_a$, while panels (b, e, h) show ${\mathit{u}}_f$ as generated using the fitted representation. Panels (c, f, i) show the results of fitting singular coincident Stokeslet and rotlet dipoles to the original flow field. The simple four-component regularized representation can be seen to qualitatively reproduce both the flow field and complex streamlines around the swimmer, with the rotational streamlines in particular unable to be well-approximated by the simple singular model. The magnitude of the fluid velocity relative to the maximum magnitude of ${\mathit{u}}_a$ is shown in color in panels (a, b, c, g, h, i). Slices in panels (a, b, c) are taken through $z=0$, with streamlines in panels (d, e, f) projected onto this plane. Flow field slices in panels (g, h, i) correspond to $z=0$ and $x=-1$.

Figure 2

Schematic of fitted swimmer representations and their associated geometries. Here, axis scales are in units of the flagellum length corresponding to case (b), with dots denoting points on the bacterium, while arrows correspond to strengths in both magnitude and direction, where the center and radius of the attached circles are the associated locations ${\mathit{x}}_0^k$ and regularization parameters, with values given in Table 1. We note that the regularized Stokeslet-rotlet pairs are positioned approximately at the geometric centers of the swimmer body and helix, with this observation apparently robust to considerable variations in morphological scales. Arrows are shown offset from their true positions for visual clarity, and strengths are uniformly rescaled for visualization. Relative measures of the size of the residual flow, ${\mathcal{E}}_f$ as defined in the main text, are also reported.

Figure 3

Predator-prey collision dynamics, as computed with (c, d) and without (e, f) the effects of regularized rotlets, including and excluding rotational diffusion. (a, b) Definitions of predator-prey configuration space for relatively slow (a) and fast (b) predators, with orientations defined such that collision would occur in the absence of hydrodynamic interactions or noise, and swimmer headings initially coplanar. For each initial configuration we simulate the unconstrained motion of the predator and prey until eventual collision or separation. In panels (c–f) we report the probability of a collision event occurring from a given initial configuration, with dashed red curves showing the boundaries of deterministic collision/separation regions in the absence of Brownian noise. Rotational diffusion can be seen to minimally impact on the qualitative dynamics, with both relative speed and swimmer representation dominating the predatory behaviors. In particular, we see that the inclusion of the regularized rotlet terms greatly reduces the prevalence of collisions in this case, while increased predator speed increases collision likelihood. Configurations in which swimmers overlap initially are shown white.

Figure 4

Predator-prey collision dynamics, as computed with (a, b) and without (c, d) the effects of regularized rotlets, including and excluding rotational diffusion and initially with $z=0.1$ and $z=-0.1$ for panels (a, c) and (b, d), respectively. For each initial configuration we simulate the unconstrained motion of the predator and prey until eventual collision or separation, with $v_2/v_1=3$. The probability of a collision event occurring from a given initial configuration is represented by color, with dashed red curves showing the boundaries of deterministic collision/separation regions in the absence of Brownian noise. As in the case of $z=0$ in Fig. 3, rotational diffusion can be seen to minimally impact on the qualitative dynamics, with swimmer representation dominating the predatory behaviors. Most notably, we see a qualitative difference between panels (a) and (b), with only the rotlet terms successfully incorporating swimmer chirality. Unable to capture this chirality, we see that the Stokeslet-only models of (c) and (d) predict the same dynamics regardless of the sign of $z$. Configurations in which swimmers overlap initially are shown white.