  We study the motion of an incompressible fluid in an $n+1$-dimensional
infinite pipe $\,\Lambda\,$ with an $L$-periodic shape in the $z$-axial
direction. Below $\,x=(x_1,x_2,\cdots,x_{n})$, and $z=x_{n+1}$. Fluid motion is
described here by the evolution Stokes or Navier-Stokes equations together with
the non-slip boundary condition $\mathbf{v}=\,0\,$. Let $g(t)$ be a given real
$T$-periodic function. We look for solutions $\bv(x,z,t)$ with periodic total
flux $\,\int_{\Sigma_z}v_z\,d\Sigma_z=g(t)\,,$ where $\Sigma_z$ denotes the
cross section of the pipe at the level $z\,.$ We look for solutions which are
simultaneously $T$-periodic with respect to time and $L$-periodic with respect
to $z\,,$ for $\,z\in \mathbb{R}\,$ and $\,t\in \mathbb{R}\,.$ We prove
existence and uniqueness of the solution to the above problem, see Theorem
\ref{thm:ex} below. Furthermore, we show that in the significant case of a
$\,3-D\,$ symmetrical rotation pipe, $\,\mathbf{v}\,$ is just the
\emph{full-developed} solution.\par% To avoid hiding our main ideas under
technical calculations, the argument is presented through a sequence of steps.
We start by considering the linear, stationary, Stokes problem. Then, by taking
this case as a reference, we study the evolution Stokes equations. Finally, we
end with the extension to the full Navier-Stokes equations.
