We study the motion of a viscous incompressible fluid in an $n+1$-dimensional double-infinite pipe $\Lambda$ with an $L$-periodic shape in the $z=x_{n+1}$ direction. Denote by $\Sigma_z$ the cross-section of the pipe at the level $z$ and by $v_z$ the $(n+1)-$th component of the velocity. We look for fully developed solutions $\mathbf{v}(x,z,t)$ with a given $T-$time periodic total flux $g(t)=\int_{\Sigma_z} v_z(x,z,t)\,dx$ which should be simultaneously $T$-periodic with respect to time and $L$-space-periodic with respect to $z$. We prove existence and uniqueness for the above problem. The problem of determining $\mathbf{v}$ and $\Gamma$ requires to solve a non-standard parabolic equation involving a non-local term of the solution.

The new results (which extend those proved by the author in a 2005 paper published in Arch. Ration. Mech. Anal.) were obtained in collaboration with Jiaqi Yang, from the Northwestern Polytechnical University, Xi'an, China, and will appear in the J. of Math. Physics.