The Centre of analysis for the interaction of fluids and solids was initiated at the beginning of 2019 and since Fall 2022 is priorly financed through the ERC-CZ Grant LL2105 CONTACT.
Currently the team has 13 members.
The research is focused on the interactions between fluids and solids. Fluid solid interaction happens in many everyday instances. For example blood flow through a vessel or air flow through the trachea, oscillations of suspension bridges, lifting of airplanes, bouncing of elastic balls, or the rotation of wind turbines. The working group aims to systematically develop an analysis for the related theory of partial differential equations. We attack classical questions of existence, uniqueness, regularity and stability, questions about the qualitative behavior of fluids interacting with solids and the quantification of the forces at the free interface between the solid and the fluid-the variable domain. Moreover, we interchange ideas with the field of scientific computing and modeling and progress the related theory of numerical approximation schemes. In case you wish to participate in the program please contact me for more information.

Current main scientific activities

Here you find news and preprints.

• Elastic plates interacting with fluids. Smooth solutions for a beam interacting with the 2D Navier-Stokes equations were constructed, see (Schwarzacher, Su 2023). Weak-strong uniqueness for elastic shells interacting with the 3D Navier-Stokes equation was shown see (Breit, Mensah, Schwarzacher, Su 2023). Weak solutions involving displacements in all coordinate directions are studied for which existence could be shown (Kampschulte, Sch, Sperone, 2023, JMPA).
• A variational approach to fluid-structure interactions. Existence of weak solutions for 3D solids interacting with 3D fluids via DeGiorgi's celebrated minimizing movements method (Benesova, Kampschulte, Sch, accepted at JEMS). The method could recently be advanced to compressible fluids, see (Breit, Kampschulte, Schwarzacher, 2021) For a survey article that introduces the variational methodology at the example of porous media please see (Benesova, Kampschulte, Schwarzacher, Nonlin. Anal. RWA). Current projects are to extend the method to more solids and contact problems. For solids only some recent progress was achieved in (A. Cesik, G. Gravina, M. Kampschulte, 2022). A different line of research introduces numerical approximation schemes along the method, see (Cesik, Schwarzacher 2023)
• Numerical approximation of fluid-structure interaction. The development of (stable) schemes (Schwarzacher She, 2022, Numerische Mathematik) was accepted. The quantification of errors (motivated by the stability result shown in (Schwarzacher, Sroczinski, 2021) below) can be found here (S. Schwarzacher, B. She, K. Tuma, 2023). A long term aim is the study on adaptive methods to obtain fast solvers for ALE based solvers. See Numerics for more information.
• Bouncing of elastic objects in a fluid. We investigate the possibility of bouncing for solid objects hitting a wall in a viscous fluid. It is known that for no-slip boundary condition no contact of the solid with the wall is happening. We aim to find the necessary and sufficient elastic properties of the solid in order to bounce of the ground, even so no contact is happening. First theoretical and numerical investigations can be found here (Gravina, Sch, Soucek, Tuma, JFM). For further numerical strategies and results see (Fara, Schwarzacher, Tuma 2023).
• Time-periodic solutions The appearance of time-periodic motions in fluid-structure interactions is investigated. Numerically by studies on the traction forces (with Cach, Fehling and Tuma). Analytically by providing existence results (with Mindrila and Mosny). Here the estimate of the diffusive impact of the fluid on the (hyperbolic) solid is crucial. The studies are built on (Mindrila, Schwarzacher 2022, 2023), where periodic solutions for elastic plates interacting with fluids are analyzed.
• Further subjects. Free boundary problems in fluid-mechanics (with Niinikoski, Kampschulte, Schwarzacher). Plasticity and fluid-structure interactions (Benesova, Biswas, Kampschulte, Schwarzacher). Temperature effects. See (Breit, Schwarzacher, Ann. Sc. Norm. Sup. Pisa 2023, 24(2), pp. 619-690) and (Almi, Badal, Friedrich, Schwarzacher 2024).

Funding

The Ministry of Education, Youth and Sport of the Czech Republic (MSMT) supports the centre via the Grant LL2105 CONTACT from 09/2021 until 08/2026. Further we thank the support of the University Centre MathMAC (UNCE/SCI/023) and (UNCE/24/SCI/005), and the Swedish research council via the VR Grant 2022-03862.