Peter E. Hydon

Chaotic mixing deep in the lung

Our current understanding of the transport and deposition of aerosols (viruses, bacteria, air pollutants, aerosolized drugs) deep in the lung has been grounded in dispersive theories based on untested assumptions about the nature of acinar airflow fields. Traditionally, these have been taken to be simple and kinematically reversible. In this article, we apply the recently discovered fluid mechanical phenomenon of irreversible low-Reynolds number flow to the lung. We demonstrate, through flow visualization studies in rhythmically ventilated rat lungs, that such a foundation is false, and that chaotic mixing may be key to aerosol transport. We found substantial alveolar flow irreversibility with stretched and folded fractal patterns, which lead to a sudden increase in mixing. These findings support our theory that chaotic alveolar flow—characterized by stagnation saddle points associated with alveolar vortices—governs gas kinematics in the lung periphery, and hence the transport, mixing, and ultimately the deposition of fine aerosols. This mechanism calls for a rethinking of the relationship of exposure and deposition of fine inhaled particles.

A Variational Complex for Difference Equations

Abstract An analogue of the Poincaré lemma for exact forms on a lattice is stated and proved. Using this result as a starting-point, a variational complex for difference equations is constructed and is proved to be locally exact. The proof uses homotopy maps, which enable one to calculate Lagrangians for discrete Euler–Lagrange systems. Furthermore, such maps lead to a systematic technique for deriving conservation laws of a given system of difference equations (whether or not it is an Euler–Lagrange system).

Symmetries and first integrals of ordinary difference equations

This paper describes a new symmetry-based approach to solving a given ordinary difference equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the difference equation can be completely solved. Several examples are used to illustrate the technique for transitive and intransitive symmetry groups. It is also shown that every linear second-order ordinary difference equation has a Lie algebra of symmetry generators that is isomorphic to sl(3). The paper concludes with a systematic method for constructing first integrals directly, which can be used even if no symmetries are known.

Symmetries of Integrable Difference Equations on the Quad‐Graph

This paper describes symmetries of all integrable difference equations that belong to the famous Adler Bobenko Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include master symmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.

Conservation laws of partial difference equations with two independent variables

This paper introduces a technique for obtaining the conservation laws of a given scalar partial difference equation with two independent variables. Unlike methods that are based on Nothers theorem, this new technique does not use symmetries. Neither does it require the difference equation to have any special structure, such as a Lagrangian, Hamiltonian or multisymplectic formulation. Instead, it uses a discrete analogue of the variational complex.

Multisymplectic conservation laws for differential and differential-difference equations

Many well-known partial differential equations can be written as multisymplectic systems. Such systems have a structural conservation law from which scalar conservation laws can be derived. These conservation laws arise as differential consequences of a 1-form ‘quasi-conservation law’, which is related to Noethers theorem. This paper develops the above framework and uses it to introduce a multisymplectic structure for differential-difference equations. The shallow water equations and the Ablowitz–Ladik equations are used to illustrate the general theory. It is found that conservation of potential vorticity is a differential consequence of two conservation laws; this surprising result and its implications are discussed.

Conservation laws for integrable difference equations

This paper deals with conservation laws for all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. All inequivalent three-point conservation laws are found, as are three five-point conservation laws for each equation. We also describe a method of generating conservation laws from known ones; this method can be used to generate higher order conservation laws from those that are listed here.

Multisymplectic formulation of fluid dynamics using the inverse map

We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamiltons principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvins circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.

Flow in pipes with non-uniform curvature and torsion

This paper describes steady and unsteady flows in pipes with small, slowly varying curvature and torsion. Four new pipe shapes are studied, using Germano’s extension of the Dean equations. Analytic and numerical solutions are obtained for flows driven by a steady pressure gradient. Oscillatory flows in pipes with non-uniform curvature are obtained by numerical methods. The eects of the non-uniformities in curvature and torsion are discussed, with particular reference to wall shear stress.

Multisymplectic structures and the variational bicomplex

Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold. ∗t.bridges@surrey.ac.uk †p.hydon@surrey.ac.uk ‡jlawson@wcu.edu