Viscous Settling of Bravais Unit-Cells

TL;DR

This study combines experiments, theory, and simulations to analyze how Bravais lattice unit-cells settle in viscous fluids, revealing a universal power-law relationship between settling speed and porosity.

Contribution

It introduces a comprehensive approach to understanding the sedimentation of porous lattice structures, highlighting the wall effects and deriving a universal scaling law.

Findings

Settling speed follows a power-law with porosity, U ∝ φ^γ, with γ ≈ 0.43 experimentally.

Wall effects significantly influence settling behavior, but can be corrected for using Faxen's boundary correction.

The unbounded domain power-law exponent is approximately 0.30, indicating a universal scaling law.

Abstract

We study experimentally and theoretically the Stokesian settling of a well-known class of porous shapes: Bravais lattice unit-cells, whose porosity we vary controllably by changing their lattice spacing. In our experiments, conducted in a square cuboidal container with its long-axis aligned along gravity, we find that the settling speed U and the solid fraction {\phi} of these lattice units obey a power-law relationship U $\propto$ {\phi}^{\gamma} , with an exponent {\gamma} = 0.43 independent of their shape. To understand the observed scaling exponent, we analytically and numerically investigate the settling of the simple cubic structure under different approximations. We find that the walls of the container, though far from the sinking object, have a defining effect. Our Stokesian boundary integral simulations show that the Faxen's boundary correction captures the wall-effects accurately and enables us to discount the wall-effect from the experimental data, yielding a power-law exponent {\gamma} = 0.30 for settling in an unbounded domain. The power-law relating sinking speed and porosity is a step towards predictively understanding the sedimentation fluxes of complex objects in the clouds and the oceans. However, the applicability of this universal scaling to irregular and biologically richer aggregates found in nature remains an open direction.