  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.
