Starting from the three-dimensional Newtonian and incompressible Navier-Stokes equations in a compliant straight vessel, we derive a reduced one-dimensional model by an averaging procedure which takes into consideration the elastic properties of the wall structure. In particular, we neglect terms of the first order with respect to the ratio between the vessel radius and length. Furthermore, we consider that the viscous effects are negligible with respect to the propagative phenomena. The result is a one-dimensional nonlinear hyperbolic system of two equations in one space dimension, which describes the mean longitudinal velocity of the flow and the radial wall displacement. The modelling technique here applied to straight cylindrical vessels may be generalized to account for curvature and torsion. An analysis of well posedness is presented which demonstrates, under reasonable hypotheses, the global in time existence of regular solutions.
Citation: Debora Amadori, Stefania Ferrari, Luca Formaggia. Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels[J]. Networks and Heterogeneous Media, 2007, 2(1): 99-125. doi: 10.3934/nhm.2007.2.99
Related Papers:
[1] |
Debora Amadori, Stefania Ferrari, Luca Formaggia .
Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels. Networks and Heterogeneous Media, 2007, 2(1): 99-125.
doi: 10.3934/nhm.2007.2.99
|
[2] |
Andro Mikelić, Giovanna Guidoboni, Sunčica Čanić .
Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem. Networks and Heterogeneous Media, 2007, 2(3): 397-423.
doi: 10.3934/nhm.2007.2.397
|
[3] |
Grigory Panasenko, Ruxandra Stavre .
Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks and Heterogeneous Media, 2008, 3(3): 651-673.
doi: 10.3934/nhm.2008.3.651
|
[4] |
Steinar Evje, Kenneth H. Karlsen .
Hyperbolic-elliptic models for well-reservoir flow. Networks and Heterogeneous Media, 2006, 1(4): 639-673.
doi: 10.3934/nhm.2006.1.639
|
[5] |
Serge Nicaise, Cristina Pignotti .
Asymptotic analysis of a simple model of fluid-structure interaction. Networks and Heterogeneous Media, 2008, 3(4): 787-813.
doi: 10.3934/nhm.2008.3.787
|
[6] |
Jean-Marc Hérard, Olivier Hurisse .
Some attempts to couple distinct fluid models. Networks and Heterogeneous Media, 2010, 5(3): 649-660.
doi: 10.3934/nhm.2010.5.649
|
[7] |
Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn .
Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks and Heterogeneous Media, 2018, 13(3): 479-491.
doi: 10.3934/nhm.2018021
|
[8] |
Tom Freudenberg, Michael Eden .
Homogenization and simulation of heat transfer through a thin grain layer. Networks and Heterogeneous Media, 2024, 19(2): 569-596.
doi: 10.3934/nhm.2024025
|
[9] |
François Baccelli, Augustin Chaintreau, Danny De Vleeschauwer, David R. McDonald .
HTTP turbulence. Networks and Heterogeneous Media, 2006, 1(1): 1-40.
doi: 10.3934/nhm.2006.1.1
|
[10] |
María Anguiano, Renata Bunoiu .
Homogenization of Bingham flow in thin porous media. Networks and Heterogeneous Media, 2020, 15(1): 87-110.
doi: 10.3934/nhm.2020004
|
Abstract
Starting from the three-dimensional Newtonian and incompressible Navier-Stokes equations in a compliant straight vessel, we derive a reduced one-dimensional model by an averaging procedure which takes into consideration the elastic properties of the wall structure. In particular, we neglect terms of the first order with respect to the ratio between the vessel radius and length. Furthermore, we consider that the viscous effects are negligible with respect to the propagative phenomena. The result is a one-dimensional nonlinear hyperbolic system of two equations in one space dimension, which describes the mean longitudinal velocity of the flow and the radial wall displacement. The modelling technique here applied to straight cylindrical vessels may be generalized to account for curvature and torsion. An analysis of well posedness is presented which demonstrates, under reasonable hypotheses, the global in time existence of regular solutions.