tisdag 13 maj 2014

Parameter-Free Fluid Models: How to Make Einstein Happy

In recent work (here and here) we have shown that mean-value outputs of computed turbulent solutions of the incompressible Navier-Stokes equations with small viscosity, vary little with the absolute value of the viscosity. This makes this mathematical model to an example of Einstein's ideal as a physics model without parameters or coefficients requiring experimental determination.

For example, we show in New Theory of Flight that the lift and drag of an airplane can be accurately computed using this model, thus ab initio without input of any experimental measurement. This is very remarkable and would have made Einstein very happy.

Augmenting this model to include self-gravitation as in the blog post Equivalence of Inertial and Heavy Mass, gives a parameter-free cosmological model, by choosing the unit of mass so that
  • $\Delta\phi (x,t) = \rho (x,t)$, 
where $\phi (x,t)$ is the gravitational potential and $\rho (x,t)$ mass density for a given unit of length specified by the $x$ coordinate, and the unit of time $t$ so that Newton's 2nd law takes the form
  • $\ddot x = - \nabla\phi (x,t)$,
connecting particle accelleration $\ddot x(t)$ to the gradient $\nabla\phi (x,t)$, where $x(t)$ is the trajectory of a particle of unit mass.

Such a  model can describe galactic scales after galaxy and star formation from interstellar dust under compression, as an incompressible fluid of small viscosity under self-gravitation, without any parameter to determine experimentally. This could have made Einstein even happier.

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