Asymptotic behavior of a viscous incompressible fluid flow in a fractal network of branching tubes

Abstract

<p>We considered a viscous incompressible fluid flow in a varying bounded domain consisting of branching thin cylindrical tubes whose axes are line segments that form a network of pre-fractal curves constituting an approximation of the Sierpinski gasket. We supposed that the fluid flow is driven by volumic forces and governed by Stokes equations with boundary conditions for the velocity and the pressure on the wall of the tubes and inner continuity conditions for the normal velocity on the interfaces between the junction zones and the rest of the pipes. We constructed local perturbations, related to boundary layers in the junction zones, from solutions of Leray problems in semi-infinite cylinders representing the rescaled junctions. Using $ \Gamma $-convergence methods, we studied the asymptotic behavior of the fluid as the radius of the tubes tends to zero and the sequence of the pre-fractal curves converges in the Hausdorff metric to the Sierpinski gasket. Based on the constructed local perturbations, we derived, according to a critical parameter related to a typical Reynolds number of the flow in the junction zones, three effective flow models in the Sierpinski gasket, consisting of a singular Brinkman flow, a singular Darcy flow, and a flow with constant velocity.</p>