The numerical simulation of systems involving fluid-structure-contact interaction raises many modeling, mathematical and numerical difficulties. It is in particular crucial for numerous biomedical applications such as the simulation of cardiac valve dynamics (native or artificial) for instance. Fluid-structure interaction without contact is already challenging due to the moving geometries and the potential strong coupling between the solid and the fluid subsystems. If contact between solids is to be modeled as well, the complexity increases with additional difficulties among which:
- In some configurations and with no-slip boundary conditions, fluid-structure interaction models are unable to predict contact (see, e.g., [1, 4, 5]), this is the so called no collision paradox;
- The simple addition of a contact constraint (variational inequality) to a fluid-structure interaction model which allows for contact yields a mechanically inconsistent fluid-structure-contact interaction (see [6, 7]).
The first difficulty can be solved by modifying the boundary and interface conditions on the contact walls (see, e.g., [8, 9, 10]). However this is not necessarily enough to obtain a mechanically consistent model. An alternate approach considered in [6, 7] to circumvent these two issues is to consider a poroelastic modeling of the fluid seepage induced by the roughness of the contacting solid. Yet, very little is known on the mathematical foundations of this approache. In this work, we analyze the ability of such a model to encompass contact. We consider a 2D fluid-ball interaction problem and we build on the approach proposed in [4, 11] to evaluate the behaviour of the drag force with respect to the gap between the solid and the wall. The asymptotics of the model with respect to the porous layer parameters are also investigated. Finally, numerical evidence of these theoretical results is provided.
REFERENCES
[1] Burman, Erik, et al. A mechanically consistent model for fluid-structure interactions with contact including seepage.Computer Methods in Applied Mechanics and Engineering 392 (2022): 114637.
[2] Gérard-Varet, D., and Hillairet, M. Computation of the Drag Force on a Sphere Close to a Wall: The Roughness Issue ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1201-224.
[3] T. Hesla, Collisions of smooth bodies in viscous fluids: A mathematical investigation, PhD thesis: University Minnesota, 2004.
[4] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Commun. Part. Diff. Eq., vol. 32, no. 9, pp. 1345-1371, 2007.
[5] M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems,SIAM J. Math. Anal., vol. 40, pp. 2451-2477, 2009.
[6] C. Ager, B. Schott, A.-T. Vuong, A. Popp and W. Wall, A consistent approach for fluid-structure-contact interaction based on a porous flow model for rough surface contact, Internat. J. Numer. Methods Engrg., vol. 119, no. 13, pp. 1345-1378, 2019.
[7] S. Frei, F. Gerosa, E. Burman and M. Fernàndez, A mechanically consistent model for fluid-structure interactions with contact including seepage, HAL preprint, 2021. [Online]. Available: https://hal.archives-0uvertes.fr/hal-03174087.
[8] L. Balilescu, J. S. Martin and T. Takahashi, Fluid-rigid structure interaction system with Coulomb’s law, SIAM J. Math. Anal, vol. 49, no. 6, pp. 4625-4657, 2017.
[9] D. Gerard-Varet, M. Hillairet and C. Wang, The influence of boundary conditions on the contact problem in a 3d Navier-Stokes flow, J. Math. Pures Appl., vol. 103, p. 1- 38, 2015.
[10] D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., vol. 195, p. 375-407, 2010.
[11] D. Gérard-Varet, D., M Hillairet, Computation of the Drag Force on a Sphere Close to a Wall: The Roughness Issue. ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1201-224