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We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers–Joseph–Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are incorporated through the introduction of the traces of the structure velocity and the Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error estimates are derived for the semi-discrete continuous-in-time mixed finite element approximation. Numerical experiments are presented to verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters.
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Tangential Navier–Stokes equations on evolving surfaces: Analysis and simulations
The paper considers a system of equations that models a lateral flow of a Boussinesq–Scriven fluid on a passively evolving surface embedded in [Formula: see text]. For the resulting Navier–Stokes type system, posed on a smooth closed time-dependent surface, we introduce a weak formulation in terms of functional spaces on a space-time manifold defined by the surface evolution. The weak formulation is shown to be well-posed for any finite final time and without smallness conditions on data. We further extend an unfitted finite element method, known as TraceFEM, to compute solutions to the fluid system. Convergence of the method is demonstrated numerically. In another series of experiments we visualize lateral flows induced by smooth deformations of a material surface.
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- PAR ID:
- 10418339
- Date Published:
- Journal Name:
- Mathematical Models and Methods in Applied Sciences
- Volume:
- 32
- Issue:
- 14
- ISSN:
- 0218-2025
- Page Range / eLocation ID:
- 2817 to 2852
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract We introduce and analyze a partially augmented fully mixed formulation and a mixed finite element method for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Navier–Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of fluid force, conservation of momentum and the Beavers–Joseph–Saffman condition. We apply dual-mixed formulations in both domains, where the symmetry of the Navier–Stokes and poroelastic stress tensors is imposed in an ultra-weak and weak sense. In turn, since the transmission conditions are essential in the fully mixed formulation, they are imposed weakly by introducing the traces of the structure velocity and the poroelastic medium pressure on the interface as the associated Lagrange multipliers. Furthermore, since the fluid convective term requires the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin-type terms. Existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with nonmatching grids, together with the corresponding stability bounds and error analysis with rates of convergence. Several numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for applications to arterial flow and flow through a filter.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0218202522500658
