Pressure-Gradient Identities for Material Deformation in Incompressible Euler Flow: A Staged Derivation of Affinity Defects, Pair Separation, and a Spherical Mean Law

Abstract

This paper studies a specific question in incompressible Euler flow: what happens to material particle motion when the local deformation is not exactly affine? I do not present this as a solution to the Euler regularity problem, and I do not claim to build a new turbulence model. The contribution is narrower: I write down a deterministic identity for a link that is often left to modelling or approximation in cluster-based and multi-particle descriptions. This law shows that the spherical mean depends only on the size of the local strain tensor.