  We study the motion of an incompressible fluid in an $n+1$-dimensional
infinite pipe $\,\La\,$ with an $L$-periodic shape in the $z=x_{n+1}$
direction. We set $\,x=(x_1,x_2,\cdots,x_{n})$, and $z=x_{n+1}$. We denote by
$\Sigma_z$ the cross section of the pipe at the level $z\,,$ and by $v_z$ the
$n+1$ component of the velocity. Fluid motion is described by the evolution
Stokes or Navier-Stokes equations together with the non-slip boundary condition
$\bv=\,0\,$. We look for solutions $\bv(x,z,t)$ with a given, arbitrary,
$T-$time periodic total flux $\,\int_{\Sigma_z} \,v_z(x,z,t)\,dx=g(t)\,,$ which
should be simultaneously $T$-periodic with respect to time and $L$-periodic
with respect to $z\,.$ We prove existence and uniqueness of the solution to the
above problems. The results extend those proved in reference \cite{B-05}, where
the cross sections were independent of $z$. The argument is presented through a
sequence of steps. We start by considering the linear, stationary, $z-$periodic
Stokes problem. Then we study the double periodic evolution Stokes equations,
which is the heart of the matter. Finally, we end with the extension to the
full Navier-Stokes equations.
