  The possibility to derive an equation for the mean velocity field in
turbulent flow by using classical statistical mechanics is investigated. An
application of projection operator technique available in the literature is
used for this purpose. It is argued that the hydrodynamic velocity defined
there, in situations where the fluid is turbulent, is to be interpreted as the
mean velocity field; in that case, the momentum component of the generalized
transport equation derived there is the mean velocity equation. In this paper,
stationary incompressible flow for constant mass density and temperature is
considered. The stress tensor is obtained as a nonlinear functional of the mean
velocity field, the linear part of which is the Stokes tensor. The formula
contains a time correlation function in local equilibrium. Presently, there
exists a microscopic theory for time correlations in total equilibrium only.
For this reason and as a preliminary measure, the formula has been expanded
into a power series in the mean velocity; though this limits the applicability
to low Reynolds number flow. The second order term has been evaluated in a
former paper of the author. For the third order term, the form of the kernel
function is derived. Its calculation with the aid of the mode-coupling theory
is completed; it will be reported in an separate paper. An numerical
application with the data of the circular jet is under way.
