# Essential Infinite Order Non-PDE Behavior in Continuum Mechanics: Corrections to Hydrodynamics and Diffusion

## TL;DR

This paper introduces an exact, causal, infinite-order differential solution for diffusive processes applicable to both relativistic and nonrelativistic systems, challenging traditional heat equation extensions and revealing new rheological effects in liquids.

## Contribution

It derives a universal infinite-order differential solution for diffusion, demonstrating the invalidity of hyperbolic extensions and uncovering new rheological effects in viscous hydrodynamics.

## Key findings

- Exact causal solution for diffusion at any analytic point.
- Invalidity of hyperbolic and relativistic heat equation extensions.
- Identification of infinite-order and nonanalytic features in fluid dynamics.

## Abstract

Longstanding problems regarding the causality of the diffusion equation are resolved through a class of exact solutions. A universal differential solution for diffusive processes is derived that is causal and exact at any analytic point in the data, albeit infinite order in spatial derivatives. This is true for systems both relativistic and nonrelativistic and shows that the hyperbolic and other relativistic extensions of the heat equation are not valid. A similar effect is demonstrated for flow enhanced mixing of solutions but with a new nonanalytic feature. Viscous hydrodynamics of liquids have both features. Both Newtonian and non-Newtonian viscous liquids give a more important and confounding alteration of the N-S equations for nonstationary flows. A careful analysis of liquids in terms of microscopic constituents and Lagrangian paths show there is a well-defined unique microscopic decomposition of fluid deformations into rotation and two modes corresponding to organized layered flow that do not enhance mixing and a mode that induces mixing and preferred static orientations. The resulting equations are both infinite order in spatial derivatives almost everywhere and are divided into two disjoint classes by an essential nonanalytic hypersuface. These give important rheological effects coupling hydrodynamic flow to diffusion and reaction kinetics. Practical consequences include catalysis and reaction yield control by rheological means. Implications of this work should percolate through almost all of continuum mechanics.