Abstract
In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid. Our optimization problem consists in finding the shape of the tree that minimizes the viscous energy dissipated by the fluid with a constrained volume, under the assumption that the total flow of the fluid is conserved throughout the structure. These hypotheses model situations where a fluid is transported from a source towards a 3D domain into which the transport network also spans. Such situations could be encountered in organs like for instance the lungs and the vascular networks. Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with an arbitrary number of generations using a matricial approach and (ii) non linear flow regime (Navier-Stokes, moderate regime with a Reynolds number 100) in trees of two generations using shape derivatives in an augmented Lagrangian algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes equations. It relies on the study of a finite dimensional optimization problem in the case (i) and on a standard shape optimization problem in the case (ii). We show that the behaviours of both regimes are very similar and that the optimal shape is highly dependent on the boundary conditions of the fluid applied at the leaves of the tree.
Highlights
In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid
As stated in the introduction, we study an infinite dimensional optimization problem
Left: initial geometry Ω0, the inlet is the blue branch while the outlets are the red branches
Summary
Tree structures are very common means to transport a product between two regions of different scales. The circulation of the fluid in such geometries can dissipate a lot of energy by viscous effects and the question of the optimization of the tree geometry arises Important applications of this problem exist, like in industry, in the study of river basins, in water treatment, in medicine. Previous studies have been made on this topic in the past like in [16, 2, 33, 20, 3] each on particular situations and applications In this frame, the goal of this work is to determine the shapes of dyadic trees that would minimize the viscous energy of a Newtonian fluid under a volume constraint on the tree. We focus here on the 3D case, it is easy to see, with very few changes in the reasoning, that our results would hold for the 2D case
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