Stirring by swimming bodies

We consider the stirring of an inviscid fluid caused by the locomotion of bodies through it. The swimmers are approximated by non-interacting cylinders or spheres moving steadily along straight lines. We find the displacement of fluid particles caused by the nearby passage of a swimmer as a function of an impact parameter. We use this to compute the effective diffusion coefficient from the random walk of a fluid particle under the influence of a distribution of swimming bodies. We compare with the results of simulations. For typical sizes, densities and swimming velocities of schools of krill, the effective diffusivity in this model is five times the thermal diffusivity. However, we estimate that viscosity increases this value by two orders of magnitude.

💡 Research Summary

The paper investigates how the locomotion of swimming bodies (modeled as non‑interacting cylinders in two dimensions or spheres in three dimensions) stirs an inviscid fluid and how this stirring translates into an effective diffusion of passive fluid particles. The authors first derive the potential‑flow velocity field generated by a single swimmer moving at a constant speed (U) along a straight line. Using the impact parameter (b) – the minimum distance between the swimmer’s trajectory and a fluid particle – they calculate the particle’s net displacement after the swimmer has passed. For a cylinder the displacement is (\Delta x = 2\pi a^{2}U/(b^{2}+a^{2})); for a sphere the expression is more complex but decays as (b^{-3}) for large (b). This displacement is termed the “stirring function” (S(b)).

Next, the authors treat a suspension of many swimmers as a random ensemble. Each particle experiences a sequence of independent “kicks” whenever a swimmer passes within a distance set by the impact parameter distribution. The rate of kicks per unit time is (\lambda = nU\sigma), where (n) is the number density of swimmers and (\sigma) is the effective cross‑section ( (2a) for cylinders, (\pi a^{2}) for spheres). By integrating the squared displacement over the impact‑parameter distribution they obtain the mean‑square jump (\langle\Delta r^{2}\rangle). Assuming a Poisson process for the kicks, the particle trajectory becomes a random walk, and the effective diffusion coefficient follows the standard relation

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