  Citation: Sznajder, P.; Cichocki, B.; Ekiel-Je ˙ zewska, M. Lack of Plasma-like Screening Mechanism in Sedimentation of a Non-Brownian Suspension. Symmetry 2022, 14, 63. https://doi.org/10.3390/ sym14010063 Academic Editor: Andrzej Herczy ´ nski and Roberto Zenit Received: 19 November 2021 Accepted: 17 December 2021 Published: 3 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article Lack of Plasma-like Screening Mechanism in Sedimentation of a Non-Brownian Suspension Pawel Sznajder 1, * , Bogdan Cichocki 2 and Maria Ekiel-Je ˙ zewska 1 1 Institute of Fundamental Technological Research, Polish Academy of Sciences, 02-106 Warsaw, Poland; mekiel@ippt.pan.pl 2 Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland; bogdan.cichocki@fuw.edu.pl * Correspondence: psznaj@ippt.pan.pl Abstract: We investigate qualitatively a uniform non-Brownian sedimenting suspension in a station- ary state. As a base of our analysis we take the BBGKY hierarchy derived from the Liouville equation. We then show that assumption of the plasma-like screening relations can cancel some long-range terms in the hierarchy but it does not provide integrable solutions for correlation functions. This suggests breaking the translational symmetry of the system. Therefore a non-uniform structure can develop to suppress velocity fluctuations and make the range of correlations finite. Keywords: non-Brownian sedimentation; stability; BBGKY hierarchy; hydrodynamic screening; correlation functions; low-Reynolds-number hydrodynamics 1. Introduction Sedimentation is a process of falling of particles in a fluid due to gravity (the particles are more dense then the fluid). Considerations in this paper are restricted to the limit of vanishing Reynolds number and infinite Peclet number. Reynolds number is given by Re = aη −1 ρ l U S , (1) where a is radius of a particle, η is dynamic viscosity coefficient of the fluid, ρ l is density of the fluid and U S is the Stokes velocity of a single particle falling in unbounded fluid motionless at infinity. When Re → 0, the fluid flow instantly adjusts to the boundary conditions. Peclet number is given by Pe = aD −1 U S , (2) where D is diffusion coefficient of a single particle in a fluid. In case of Pe → ∞, Brow- nian motion is negligible compared to the motion caused by gravity and hydrodynamic interactions. In such a system there are difficulties with divergent expressions due to hydrodynamic field disturbance slowly decaying over distance (inversely proportional to distance—same as electric potential produced by an isolated charge in a vacuum). Chal- lenges of theoretical approach to system with vanishing Reynolds number (and divergent Peclet number) are known for more than hundred years. In 1911, Smoluchowski [1,2] investigated a particle surrounded by other particles suspended in a Newtonian fluid. His observation was that, considering larger and larger systems leads to a divergent expression for the particle falling velocity, caused by the long-range velocity disturbance produced by other particles. Now this is known in literature as the Smoluchowski paradox [3]. Solution of this paradox was given by Batchelor [4] sixty years after the work of Smoluchowski and then reanalyzed by Beenakker and Mazur [5]. Batchelor ’s main idea was to calculate the average velocity of suspended particles 〈U〉 relative to the flow 〈v〉 of the whole suspension which resulted in 〈U〉−〈v〉 where cancel- lation of divergent terms secures that the average relative velocity is finite. Nevertheless Symmetry 2022, 14, 63. https://doi.org/10.3390/sym14010063 https://www.mdpi.com/journal/symmetry