Abstract:
The Navier-Stokes equations are the fundamental equations governing the dynamics of Newtonian
uids, if one neglects theirs derivation from the Boltzmann equation of molecular dynamics.
The inviscid Navier-Stokes equations are known as the Euler equations. The Reynolds number
of the
ow, which is the ratio of inertial to frictional forces in the
uid, is in nite for an inviscid
uid. For very low Reynolds number
ows it is possible to obtain some exact and approximate
solutions of the Navier-Stokes equations.
Modelling turbulent
ows is a di cult problem in
uid dynamics. The Navier-Stokes equations
are not amenable to mathematical analysis, and they cannot be used to predict detailed
consequences or the emergence of randomness at high Reynolds number. The theory of Richardson
and Kolmogorov views turbulence as a cascade of eddies or coherent structures. However,
most of these studies are statistical and are not likely to answer the lack of universality. The
theory of chaos in
uid
ows, originated by Lorenz, views turbulence as a sensitive dependence
on initial conditions. Simple nonlinear equations with analytical solutions and prescribed initial
conditions were found to exhibit chaotic and apparently random behaviour. Turbulent scales
have a fractal like distribution. However, the randomness generated by the Navier-Stokes equations
may be due to (hidden) intrinsic reasons. This framework cannot be used to predict global
variables in a turbulent
ow, for example the Reynolds number dependence of the resistance
coe cient, in a pipe of given radius and pressure gradient.
