  In the present note, we show that the uni-directional flows in a rectangular
channel and in a circular pipe are exact spatio-temporal solutions of the
Navier-Stokes equations over a short time interval. We assert that the
classical plane Poiseuille-Couette flow and Hagen-Poiseuille flow are
time-independent approximations of the exact solutions if an appropriate
initial velocity distribution at starting location is specified. Conceptually,
there do not exist absolute steady flows starting from unspecified initial
data. The classic experimental measurements by Poiseuille can be explained in
terms of the evolutional solutions. In particular, the pipe flow does not have
a time-independent characteristic velocity. The orthodox notion that the
parabolic profile exists for arbitrary Reynolds numbers is unwarranted.
