The Myth of the Continuum

Predicted and observed tracks of Hurricane Matthew (John Droz).

A turbulent region of a fluid is a region which is less ordered than streamline flow, i.e. where the Boltzmann entropy is higher than for streamline flow.  Both turbulence and entropy are properties of stochastic systems and so cannot be properly represented by the Navier–Stokes equations and their finite difference approximations.  Numerical models based on them are unstable and require parameters such as eddy viscosity be set unrealistically high in order to suppress turbulence. In the real world eddy viscosity is only non-zero where turbulence already exists.

Like Maxwell’s equations, the Navier–Stokes equations fail at high energy densities because they are Newtonian – they are partial differential equations that describe a mythical continuum which is deterministic, continuous and differentiable. They do not allow for the granularity and stochastic behaviour of the systems they are supposed to describe. They do not accommodate the atomic theory nor the granularity of action space.

Planck’s formula for  black-body radiation is based on probabilistic assumptions as are  Kolmogorov’s turbulence spectrum and the Law of the Wall in Fluid Dynamics.

Fluid dynamic models fail because no real fluid is a continuum.

Reference:

Chapter 3 of The Fluid Catastrophe by John Reid

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