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

Haifa El Jarroudi; Mustapha El Jarroudi; Haifa El Jarroudi; Mustapha El Jarroudi

doi:10.3934/cam.2024030

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 655-699. doi: 10.3934/cam.2024030

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Research article

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

Haifa El Jarroudi ,
Mustapha El Jarroudi ^,

Department of Mathematics, Abdelmalek Essaâdi University FST Tanger, Tangier, Morocco

Received: 10 January 2024 Revised: 07 March 2024 Accepted: 01 August 2024 Published: 13 September 2024
35B40, 28A80, 35J75

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.
- viscous incompressible fluid flow,
- fractal branching tubes,
- asymptotic behavior,
- critical parameter,
- effective flow models
Citation: Haifa El Jarroudi, Mustapha El Jarroudi. Asymptotic behavior of a viscous incompressible fluid flow in a fractal network of branching tubes[J]. Communications in Analysis and Mechanics, 2024, 16(3): 655-699. doi: 10.3934/cam.2024030

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Abstract

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.

References

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