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 ϕ. We also assume that the pressure gradient is parallel to the cylinders. Under such assumptions, for each pair (l,ϕ), one defines the average of the longitudinal component of the flow velocity Σ[l,ϕ]. Here, we prove that the quantity Σ[l,ϕ] depends analytically on the pair (l,ϕ), which we consider as a point in a suitable Banach space.