In this paper, we analyse a special case of the �Sixth Problem of the Millenium: Navier-Stokes equations, existence and smoothness�, whose method of solution was suggested by Ladyzhenskaya in [7]. Whereas, the latter proposed analysis of the problem for a homogeneous boundary condition, we analyse the problem, at hand, with a non-homogeneous boundary condition. Our situation of a non-homogeneous boundary condition arises out of the application of a boundary permeation model proposed by Sauer [9] for the second grade fluids. The model imposes a zero initial velocity that will be explained later. We have already applied the model and the trace-related canonical operators to confirm existence and uniqueness of the �weak� solution to the stationary version of the current problem [3]. Our solutions are referred to as a �weak� as they possess weak derivatives in the sense of distributions or test functions. Also, in [2] we set the necessary and sufficient conditions for the existence of weak solutions to the problem at hand. In this paper, we wish to confirm existence and uniqueness using trace-related canonical operators; thus under different conditions to those proposed by Ladyzhenskaya in [7].

nonlinear, non-stationary, permeable boundary, flows.