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Longstanding problems regarding the causality of the diffusion equation areresolved through a class of exact solutions. A universal differential solutionfor diffusive processes is derived that is causal and exact at any analyticpoint in the data, albeit infinite order in spatial derivatives. This is truefor systems both relativistic and nonrelativistic and shows that the hyperbolicand other relativistic extensions of the heat equation are not valid. A similareffect is demonstrated for flow enhanced mixing of solutions but with a newnonanalytic feature. Viscous hydrodynamics of liquids have both features. BothNewtonian and non-Newtonian viscous liquids give a more important andconfounding alteration of the N-S equations for nonstationary flows. A carefulanalysis of liquids in terms of microscopic constituents and Lagrangian pathsshow there is a well-defined unique microscopic decomposition of fluiddeformations into rotation and two modes corresponding to organized layeredflow that do not enhance mixing and a mode that induces mixing and preferredstatic orientations. The resulting equations are both infinite order in spatialderivatives almost everywhere and are divided into two disjoint classes by anessential nonanalytic hypersuface. These give important rheological effectscoupling hydrodynamic flow to diffusion and reaction kinetics. Practicalconsequences include catalysis and reaction yield control by rheological means.Implications of this work should percolate through almost all of continuummechanics.

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