Shape analysis of the longitudinal flow along a periodic array of cylinders

We study the behavior of the longitudinal flow along a periodic array of cylinders upon perturbations of the shape of the cross section of the cylinders and the periodicity structure, when a Newtonian fluid is flowing at low Reynolds numbers around the cylinders. The periodicity cell is a rectangle of sides of length $l$ and $1/l$, where $l$ is a positive parameter, and the shape of the cross section of the cylinders is determined by the image of a fixed domain through a diffeomorphism $phi$. We also assume that the pressure gradient is parallel to the cylinders. Under such assumptions, for each pair $(l,phi)$, one defines the average of the longitudinal component of the flow velocity $Sigma[l,phi]$. Here, we prove that the quantity $Sigma[l,phi]$ depends analytically on the pair $(l,phi)$, which we consider as a point in a suitable Banach space.