Analysis of Fluids and Related Topics: Lattice Hydrodynamics, Dennis Sullivan, Stony Brook University

Title:  Lattice Hydrodynamics

Abstract:  

We construct a particular lattice model of 3D incompressible fluid motion with viscosity parameter. The construction follows the momentum derivation of  the continuum model switching to combinatorial topology just before taking the calculus limit. The lattice consists of two interpenetrating face centered cubic lattices. [the structure of NaCl]. The lattice of sites organizes a chain complex L of four vector spaces built from overlapping uniform cubes, faces, edges and sites giving a multilayered covering of periodic three space. There are two nilpotent operators on L, a duality  involution, each of odd degree, and a combinatorial Laplacian on L. The result of the momentum derivation is an ODE on one degree of L which is a combinatorial version of the continuum model. The goal of work in progress is to use the model both to derive theory and to compute at a given scale.