Project start: 1.10.2025.

The classical Navier-Stokes equations are the most well-known model of fluid dynamics, but structurally more complex materials and flows in non-standard domains pose challenges to this classical description. Therefore, a more sophisticated modeling approach with additional degrees of freedom is necessary. The micropolar model introduces local micro-motions of the continuum in the form of micro-rotational velocity and conservation of angular momentum, significantly improving the modeling of flows with small characteristic dimensions (microtubes) and fluids with pronounced particle structures, such as biological fluids.

The main focus of this project is compressible thermally conductive micropolar real fluid, which deviates from idealized assumptions through a generalized equation of state for real gases. The model incorporates the law of energy conservation, where temperature is not neglected, resulting in a system of four nonlinear partial differential equations. Even the classical three-dimensional (3D) Navier-Stokes model is extremely complex for analysis, as evidenced by the unsolved Millennium Problem posed by the Clay Mathematics Institute. Consequently, simplifications such as spherical symmetry of solutions are often introduced.

Fundamental research in this area is relevant and ongoing, and the project will expand it in several directions. First, the 3D spherically symmetric micropolar real gas model will be generalized through the introduction of nonhomogeneous boundary conditions, non-constant viscosity coefficients, external forces, and free boundaries, alongside analysis of solution regularity and development of more efficient numerical schemes. Second, the project addresses the open problem of global existence of solutions to reactive micropolar real gas model describing chemical reaction dynamics. The micropolar real model configuration proves particularly suitable for this purpose due to its effective description of extreme pressure/temperature conditions with particles dispersed in the fluid. Finally, a general 3D micropolar model without temperature and symmetry assumptions will be investigated in unbounded domains with non-compact boundary.