A contemporary look at Hermann Hankel's 1861 pioneering work on Lagrangian fluid dynamics

TL;DR

This paper critically assesses Hermann Hankel's 1861 work on Lagrangian fluid dynamics, highlighting its innovative variational and geometric approaches and placing it within the historical development of fluid mechanics.

Contribution

It provides a detailed analysis of Hankel's pioneering methods in Lagrangian fluid dynamics, connecting them to both past and future theoretical frameworks.

Findings

Hankel's work includes early variational formulations for elastic fluids.

It emphasizes the importance of Lagrangian coordinates and vorticity transport.

Hankel's contributions are recognized as highly innovative in mathematical fluid dynamics.

Abstract

The present paper is a companion to the paper by Villone and Rampf (2017), titled "Hermann Hankel's On the general theory of motion of fluids, an essay including an English translation of the complete Preisschrift from 1861" together with connected documents. Here we give a critical assessment of Hankel's work, which covers many important aspects of fluid dynamics considered from a Lagrangian-coordinates point of view: variational formulation in the spirit of Hamilton for elastic (barotropic) fluids, transport (we would now say Lie transport) of vorticity, the Lagrangian significance of Clebsch variables, etc. Hankel's work is also put in the perspective of previous and future work. Hence, the action spans about two centuries: from Lagrange's 1760-1761 Turin paper on variational approaches to mechanics and fluid mechanics problems to Arnold's 1966 founding paper on the geometrical/variational formulation of incompressible flow. The 22-year old Hankel - who was to die 12 years later - emerges as a highly innovative master of mathematical fluid dynamics, fully deserving Riemann's assessment that his Preisschrift contains "all manner of good things."