Arbitrary Lagrangian - Eulerian formulation of the equations system which modeling the interaction of airflow with the pulmonary alveolus

Authors

  • Raúl E. Reupo Vallejos Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas-Universidad Nacional Pedro Ruiz Gallo,Perú
  • Obidio Rubio . Universidad Nacional de Trujillo
  • Alexis Rodriguez . Universidad Nacional de Trujillo
  • Luis J. Caucha . Universidad Nacional de Tumbes, Tumbes-Perú

DOI:

https://doi.org/10.17268/sel.mat.2017.02.06

Keywords:

Fluid-Estructure interaction, airflow, pulmonary alveolus, Arbitrary Lagrangian-Eulerian (ALE) fra- mework, Navier-Stokes equations, equilibrium equation

Abstract

In this article, we formulate a physiological process through a two-dimensional fluid-structure interaction problem between the airflow and the pulmonary alveolus in the Eulerian-Lagrangian Arbitrary (ALE) frame.
This problem arises by coupling the equations of the fluid and the structure, described by the Navier-Stokes equations of evolution for incompressible flows, and an equilibrium equation, respectively.

References

Barker, A., Cai, X. Scalable parallel methods for monolithic coupling in fluid-structure interaction with application to blood flow modeling, Journal of Computational Physics. 229, (2010), 642-659.

Coutand, D., Shkoller, S. On the motion of an elastic solid inside of an incompressible viscous fluid, Archieve for Rational Mechanics and Analysis, 176, N◦01 (2005), 25-102.

Coutand, D., Shkoller, S. On the interaction between quasilinear elastodynamics and the Navier-Stokes equations, Archieve for Rational Mechanics and Analysis, 179, N◦03 (2006), 303-352.

Dailey, H., Ghadiali, S. Fluid-structure analysis of microparticle transport in deformable pulmonary alveoli, Journal of AerosolScience , 38 (2007), 269-288.

Dailey, H., Yalcin, H., Ghadiali, S. Fluid-structure modeling of flow-induced alveolar epithelial cell deformation, Computers and Structures, 85 (2007), 1066-1071.

Du, Q., Gunzburger, M., Hou, L., Lee, J. Analysis of a Linear Fluid-Structure Interaction Problem, Discrete and Continuous Dynamical Systems , 9, N◦03 (2003), 633-650.

Dunne, T., Rannacher, R., Richter, T. Numerical Simulation of Fluid-Structure Interaction based on Monolithic VariationalFormulations, Institute of Applied Mathematics University of Heidelberg, Germany.

Eken, A., Sahin, M. A parallel monolithic approach for fluid-structure interaction in a cerebral aneurysm, Computers and Fluids, 153, (2017), 61-75.

Formaggia, L., Quarteroni, A., Veneziani, A. Cardiovascular Mathematics: Modeling and simulation of the circulatory system, Springer-Verlag. Milano(2009).

Frei, S. Eulerian finite element methods for interface problems and fluid-structure interactions, PhD thesisUniversitat Heidelberg.

Hughes, T., Liu ,W., Zimmermann, T. Lagrange-Eulerian finite element formulation for incompressible viscous flows, Comput.Meths. Appl. Mech. Engrg. 29, (1981), 329-349.

Ignatova, M., Kukavica, I., Lasiecka I., Tuffaha, A. On well-posedness and small data global existence for an interface dampedfree boundary fluid-structure model, Nonlinearity. 27, (2014), 467-499.

Kukavica, I., Tuffaha, A. Solutions to a fluid-structure interaction free boundary problem, Discrete and Continuous Dynamical Systems. 32, N◦04 (2012), 1355-1389.

Meyers, M., Chawla, K. Mechanical Behavior of Materials, Cambridge University Press, New York 2009.

Richter, T. Fluid Structure Interactions. Modeling, Mathematical Analysis and Finite Elements, Special, preliminar edition exclusively for the participants of theWinter School on Modeling, Adaptative Discretizations and Solvers for Fluid-Structure Interactions, Linz, 2016.

Richter, T. A Monilithic Geometric Multigrid Solver for Fluid-Structure Interactions in ALE formulation, International journal for Numerical Methods in Engineering. 104, N◦05 (2015), 372-390.

Wick, T. Solving Monolithic Fluid-Structure Interaction Problems in Arbitrary Lagrangian Eulerian Coordinates with the deal.II Library, Archive of Numerical Software. 1, N◦01 (2013), 1-19.

Wick, T.,Wollner,W. On the differentiability of stationary fluid-structure interaction problems with respect to the problem data, Hamburger Beitrage zur Angewandten Mathematik. (2013).

Wick, T. Adaptive Finite Element Simulation of Fluid-Structure Interaction with Application to Heart-Valve Dynamics. PhD thesis, Universitat Heidelberg, 2012. urn:nbn:de:bsz:16-opus-129926.

Wick, T. Modeling, Discretization, Optimization, and Simulation of Fluid-Structure Interaction, Technische Univesritat Munchen, 2015.

Wick, T. Fluid-structure interactions using different mesh techniques, Computers and Structures. 89, (2011), 1456-1467.

Wu, Y., Chuan Cai, X. A parallel two-level method for simulating blood flows in branching arteries with the resistive boundary ndition, Computers & Fluids. 45, (2011), 92-102.

Wu, Y., Cai, X. A fully implicit domain decomposition based ALE framework for three-dimensional fluid-structure interaction with application in blood flow computation, Journal of Computational Physics. 258, (2014), 524-537.

Yang,Y. Mathematical Modeling and Simulation of the Evolution of Plaques in Blood Vessels. PhD thesis,Universitat Heidelberg, 2014.

Yu, Y., Baek, H., Karniadakis, G. Generalized fictitious methods for fluid-structure interactions: Analysis and simulations, Journal of Computational Physics. 245, (2013), 317-346.

Published

2017-12-15

How to Cite

Vallejos, R. E. R., ., O. R., ., A. R., & ., L. J. C. (2017). Arbitrary Lagrangian - Eulerian formulation of the equations system which modeling the interaction of airflow with the pulmonary alveolus. Selecciones Matemáticas, 4(02), 192-201. https://doi.org/10.17268/sel.mat.2017.02.06