Kinematics, statistics and dynamics: these are the basic elements of fluid dynamics. The Lagrangian formulation of the conservation laws for mass, momentum and energy are familiar to fluid dynamicists, as it is the natural way to extend Newtonian particle dynamics to fluids. Less familiar are: the conservation law for particle identity, which is effectively a definition of the independent Lagrangian variables; the path integral relationship between the statistics of random dependent Lagrangian variables and their Eulerian counterparts; the first integrals of Cauchy and Weber for the inviscid Lagrangian momentum equations, and the Cauchy vector invariant; the boundary conditions that must be imposed on compressible flow at boundaries defined by fluid particles (comoving boundaries), and the increasingly useful Lagrangian conservation law for momentum when the particle position is expressed in radial distance, longitude and latitude. The complexity of the divergence of the viscous stress tensor expressed in Lagrangian variables is undeniable, but the structure emphasizes the status of the Jacobi matrix as the Galilean invariant state variable that characterizes the flow. The Cauchy invariant is algebraically related to the Jacobi matrix and its Lagrangian time derivative; the conservation law for the Cauchy invariant in viscous flow is almost elegant.

Motivation

Leaves drifting in streams and blowing in the wind belong amongst our root impressions of the natural world. Plumes discharging into streams and pumping from smoke stacks symbolize our impact on that world. Thus it is baffling when as students we discover that fluid dynamics is seemingly exclusively investigated by measuring pressure at fixed points. The manometers in our first fluids laboratories plainly measure total stagnation pressure; the mechanical flow meters less obviously strike a dynamical balance between the torque of the partial stagnation pressure on the turbine blades and the torque of friction in the turbine bearings. Our hands and faces do feel the rush of a stream or the sweep of the wind, but these are brute sensations in comparison to the incisive information processing at work when our eyes follows a flow marker.

This is a book about the role of kinematics in fluid dynamics. The most revealing mathematical framework for developing kinematics is the Lagrangian formulation, long ago discarded for being unwieldy compared to the Eulerian formulation (Tokaty, 1971). Yet the discarded unwieldiness owes precisely to the richness of the kinematical information. This book might have been written any time in the twentieth century; the motivation now is the emergence of Lagrangian observing technology. The emergence is of course a reemergence; meteorologists have been routinely tracking weather balloons with theodolites since the nineteenth century.

A benefit of the great majority of Lagrangian data being so new is that most of them are available via the Internet. Also, Lagrangian time series analysis may be carried out using conventional time series methods, for which well-supported software libraries abound. Modern dynamical systems theory suggests new and intriguing quantities characterizing the behavior of Lagrangian time series.

Combining Lagrangian data with Lagrangian dynamical models can be conceptually as simple as combining Eulerian data with Eulerian dynamical models, and as complex to implement effectively. Mixing the two formulations of fluid dynamics leads to nonlinearities in the measurement functionals, further complicating effective implementation. Particle pairs make incisive tools for investigating the field of flow: dynamically constrained analysis of float pairs, without having to run all the machinery that is a modern ocean circulation model, is theoretically possible and offers the cability for real-time, even onboard analysis.

Few analytical solutions are known for the Lagrangian formulation of fluid dynamics; as few, in fact, as are known for the Eulerian. Almost all these solutions describe flow free of momentum advection. The Gerstner waves, and their generalizations the Ptolemaic vortices, stand out as extraordinary exceptions. The classical investigative techniques of linearized hydrodynamic stability theory are available to both the Lagrangian and the Eulerian formulation, as are the newer techniques of phase plane analysis. The classical Stokes' problems for viscous flow near plates are solvable in both formulations; the Lagrangian self-similar solution for Blasius' approximate boundary layer dynamics is intriguingly more complicated than the Eulerian. The general solvability of the Lagrangian formulation of inviscid incompressible fluid dynamics appears, to an applied mathematician, to depend upon the choice of dependent variables. The classical solvability of the viscous problem comes exasperatingly close to being provable in the large, but in the end remains an open question for pure mathematicians.