Boundary Effects on Ideal Fluid Forces and Kelvin's Minimum Energy Theorem

TL;DR

This paper explores how boundaries influence ideal fluid forces, revealing conditions under which forces can switch from attractive to repulsive, based on Kelvin's minimum energy theorem, with implications for steady and unsteady flows.

Contribution

It demonstrates boundary effects on ideal fluid forces, including force sign reversal, using Kelvin's minimum energy theorem, extending understanding of fluid-boundary interactions.

Findings

Boundaries always increase added mass in unsteady flows.

Force can become attractive or repulsive depending on boundary geometry.

Force sign reversal occurs near energy minima predicted by Kelvin's theorem.

Abstract

The electrostatic force on a charge above a neutral conductor is generally attractive. Surprisingly, that force becomes repulsive in certain geometries (Levin & Johnson 2011), a result that follows from an energy theorem in electrostatics. Based on the analogous minimum energy theorem of Kelvin (1849), valid in the theory of ideal fluids, we show corresponding effects on steady and unsteady fluid forces in the presence of boundaries. Two main results are presented regarding the unsteady force. First, the added mass is proven to always increase in the presence of boundaries. Second, in a model of a body approaching a boundary, where the unsteady force is typically repulsive (Lamb 1975, {\S}137), we present a geometry where the force can be attractive. As for the steady force, there is one main result: in a model of a Bernoulli suction gripper, for which the steady force is typically attractive, we show that force becomes repulsive in some geometries. Both the unsteady and steady forces are shown to reverse sign when boundaries approximate flow streamlines, at energy minima predicted by Kelvin's theorem.