## måndag 15 oktober 2012

### One Paradox Enough to Kill a Theory

A scientific paradox is a contradiction between the prediction of a scientific theory and observation.

The detection of a paradox of a scientific theory means the end of the theory, unless the contradiction between theory and observation can somehow be resolved, either by changing the theory or the observation.

As an illuminating illustration let us consider D'Alembert's paradox in fluid mechanics, which compares the zero drag of potential flow as stationary incompressible inviscid flow around a body of any shape, with the observation of substantial drag in slightly viscous flow of aero/hydro-mechanics. Theory says drag = 0, while observation shows drag = 1, a glaring contradiction.

D'Alembert's paradox was formulated in 1752 but was not solved until 2008 after 256 years of brooding by the most able minds of mathematics and mechanics.

How was the paradox then handled during its 256 years of existence as a potentially lethal poison to the science of fluid mechanics? Since observation of drag =1 could not be argued away, it was necessary to blame one of the assumptions of potential flow for the unrealistic prediction of zero drag, and then the assumption of zero viscosity of inviscid flow was the first choice:

It was argued that even if the viscosity of air or water is very small, it is not zero and drag = 1 thus results from an arbitrarily small positive viscosity. This led from the zero viscosity Euler equations, allowing unphysical potential solutions which could be computed analytically, to the positive viscosity Navier-Stokes equations with solutions which could not be computed at all.

The paradox was thus wiped away by a gesture without any real scientific substance. If solutions to the Navier-Stokes equations could not be computed then no predictions could be made assuming positive viscosity, which thus was a useless theory. Practical engineering was thus thrown back into d'Alembert's paradox of zero drag of inviscid flow incompatible with the observed lift of a wing in many experiments in the late 19th century preparing for the powered sustained flight demonstrated by the Wright brothers in 1903.

The observations required a change of theory and since positive viscosity had led to a dead end,  a new card had to be pulled and this was the circulation theory of Kutta and Zhukovsky changing potential flow by large scale circulation around the wing section. From circulation followed lift (but still no drag) and the problem now was to explain the generation of circulation by a wing.

To this end, viscosity was again brought in, which of course led to the same dead end as before, but with a bit of mathematics of analytical functions, this was covered up and kept the paradox under control for over 100 years, although many scientists had their doubts.

A non-correct resolution of a paradox thus showed to be better than no resolution at all (which as indicated was unacceptable). One way of maintaining an incorrect theory involving a paradox, thus amounts to presenting one incorrect resolution after the other of the paradox as a way to delay the final verdict that the theory is incorrect.

If you follow this blog, you know about the correct resolution of d'Alembert's paradox presented in 2008 and you may well understand the far-reaching consequences of the the resolution.