Derivation of the Reynolds equation and corrections to Navier-Stokes equations using projection techniques
It is shown that it is possible to derive corrections to Navier-Stokes equation from the first hierarchy of equations using projection techniques and a perturbation analysis. In the first order we obtain Reynolds equation and in the second order we derive corrections to the Navier-Stokes equation that raises the possibility of periodic, quasiperiodic and irregular solutions.
The solutions of the Navier-Stokes equations were never found to be irregular or chaotic, thus they can describe only the laminar, periodic flows. We therefore start directly from the Liouville equation. Using a projection formalism applied to the Liouville equation, we arrive at a hierarchy of equations for the reduced distribution function for the N particle system (integro-differential equations). If we consider the integral term in the hierarchy of equations as a perturbation, we are able to develop a perturbative formalism by expanding the reduced distribution function and propagator G, appearing in the hierarchy of equations, in perturbation series. Extracting the terms of the same order and integrating over momentum space, we can obtain kinetic equations for one-particle distribution function for 0th, 1st, 2nd, ..., ...order.
For example, 0th order kinetic equation is nothing but the Boltzmann equation, hence the resulting 0th order equation of motion for the fluid is Navier-Stokes equation. 1st order kinetic equation leads to Reynolds equation, which was derived by Reynolds from Navier-Stokes equation. 2nd order kinetic equation gives us Navier-Stokes equation with 2 correction terms in integral form. We use a renormalization procedure.
Since the renormalized 2nd order equation of motion for the fluid can be transformed into a 4th order (with respect to the time t), non-linear differential equation, there is a possibility of irregular or turbulent-like solutions.