torsdag 20 maj 2021

A ToE for Fluid Mechanics

Einsteins ideal as a Theory of Everything ToE is a mathematical model of physics without any parameters. 

The standard model of particle physics contains 18 parameters. It is a very complicated model. To determine the parameters experimentally is impossible.

The standard model of isotropic linear elasticity contains 2 parameters. This is a very simple model but for a non- isotropic body the number of parameters includes 18 parameters. 

To be a useful model the values of its parameters must be supplied as input determined from experiments or more basic model, which in general is very difficult. The 2 parameters of isotropic linear elasticity can be determined from simple tests, but the 18 parameters for non-isotropic linear elasticity are difficult to determine, not to speak of non-linear elasticity and all the parameters of the standard model. 

Are there any parameter-free models of physics? A basic example is a circle described as the set of points in a plane with a certain distance to a given mid-point from which the value of Pi can be computed as the quotient between circumference and diameter. That is a very simple model. Is there any model of more complex physics which is parameter-free? 

Yes, there is one, and maybe this is the only one: Euler's equations for incompressible fluid flow are expressed in terms of velocity and pressure without any parameter: Input is geometry, in/out-flow conditions and external forces, but no parameter, since viscosity is set to zero.   

The remarkable thing is now that the drag and lift of a body moving through a slightly viscous fluid like air and water can accurately be predicted by computing turbulent solutions to the Euler equations with only geometry of the body as input. This is like computing the ratio of circumference/diameter of a circle (that is computing Pi), but just more astounding. Drag and lift coefficients (scaling with $speed^2$) of a body only depend on the geometry of the body! No parameter input needed! See Computational Turbulent Incompressible Flow and Breakthrough of predictive simulation.

The Euler equations for incompressible flow is a ToE for slightly viscous incompressible flow like air (subsonic) and water.  This is remarkable. Is this is the only ToE in physics.

Well, Newton's law of gravitation contains the gravitational constant G connecting gravitational force to mass as parameter, but may be viewed as a ToE in the sense of correctly predicting that all bodies independent of composition move the same way subject to gravitation. 

PS Von Neuman famously claimed that he (in principle) could model an elephant with 4 parameters, and make it wiggle its trunk with a 5th, but in practice how would he determine the parameters?  Elephant experiments are costly and cumbersome.

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