Blow-up analysis of hydrodynamic forces exerted on two adjacent $M$-convex particles

TL;DR

This paper analyzes how hydrodynamic forces on two close convex particles in a viscous fluid become unbounded as they approach, providing asymptotic formulas and bounds that depend on particle convexity and motion type.

Contribution

It derives asymptotic formulas and optimal bounds for hydrodynamic forces on convex particles, revealing the influence of convexity and motion on force blow-up rates.

Findings

Force blow-up rate is ^{-3} for flat particles in 2D and 3D.

Convexity affects the singularity strength of forces.

Rotational motion induces the largest force singularities in all directions.

Abstract

In a viscous incompressible fluid, the hydrodynamic forces acting on two close-to-touch rigid particles in relative motion always become arbitrarily large, as the interparticle distance parameter $\varepsilon$ goes to zero. In this paper we obtain asymptotic formulas of the hydrodynamic forces and torque in $2\mathrm{D}$ model and establish the optimal upper and lower bound estimates in $3\mathrm{D}$, which sharply characterizes the singular behavior of hydrodynamic forces. These results reveal the effect of the relative convexity between particles, denoted by index $m$, on the blow-up rates of hydrodynamic forces. Further, when $m$ degenerates to infinity, we consider the particles with partially flat boundary and capture that the largest blow-up rate of the hydrodynamic forces is $\varepsilon^{-3}$ both in 2D and 3D. We also clarify the singularities arising from linear motion and rotational motion, and find that the largest blow-up rate induced by rotation appears in all directions of the forces.