Turbulent Brownian motion

Ladhyx, Ecole Polytechnique - Palaiseau, France

^(a) yvesh.pomeau@gmail.com (corresponding author)

Received: 4 November 2020
Accepted: 10 February 2021

Abstract

An outstanding result of classical physics is (Einstein A., Ann. Phys. (Berlin), 17 (1905) 549; Smoluchowski M., Ann. Phys. (Berlin), 21 (1996) 756; Pomeau Y. and Piasecki J., C. R. Phys., 18 (2017) 570 and references therein) Stokes-Einstein's formula for the diffusion coefficient of tiny spheres driven by pressure fluctuations of a fluid at equilibrium. This formula includes both the macroscopic Stokes formula for the viscous drag and the fluctuations described by Gibbs-Boltzmann statistics. In the present work we consider a somewhat related problem, namely the diffusion (in space) induced by the turbulent fluctuations of the wake behind a body settling at constant speed under the effect of gravity —because of a difference between its mass density and the one of the fluid. We assume that the body is so big and heavy that the Reynolds number is large enough to make valid scaling relations independent of the fluid viscosity in a regime of fully developed turbulence of the wake.