# Friction Scaling Laws for Transport in Bacterial Turbulence

## TL;DR

This paper investigates how friction affects transport in bacterial turbulence, revealing scaling laws for key quantities and showing that friction influences the transition from ballistic to diffusive behavior.

## Contribution

It introduces a continuum model analyzing friction's impact on passive tracer transport in bacterial suspensions, deriving new scaling laws for transport properties.

## Key findings

- Crossover time $ au_c$ minimizes at zero friction.
- Scaling laws for $ au_c$, $ ext{ell}^*$, $u_{ ext{rms}}$, and $ ext{D}$.
- Friction monotonically decreases characteristic scales.

## Abstract

Understanding the role of frictional drag in diffusive transport is an important problem in the field of active turbulence. Using a continuum model that applies well to bacterial suspensions, we investigate the role of Ekmann friction on the transport of passive (Lagrangian) tracers that go with the local flow. We find that the crossover from ballistic to diffusive regime happens at a time scale $\tau_c$ that attains a minimum at zero friction, meaning that both injection and dissipation of energy delay the relaxation of tracers. We explain this by proposing that $\tau_c \sim 2 \ell^*/u_{\text{rms}}$, where $\ell^*$ is a dominant length scale extracted from energy spectrum peak, and $u_{\text{rms}}$ is a velocity scale that sets the kinetic energy at steady state, both scales monotonically decrease with friction. Finally, we predict robust scaling laws for $\ell^*$, $u_{\text{rms}}$ and the diffusion coefficient $\mathcal{D} \sim \ell^* u_{\text{rms}} / 2$, that are valid over a wide range of fluid friction. Our findings might be relevant to transport phenomena in a generic active fluid.