Mathematical modeling and numerical simulations of processes in thin or porous domains

We frequently deal with systems described by the differential equations coming from conservation laws that are too complicated to be solved and are of limited use in practice. In many such situations the original physical laws can be replaced by empirical laws or other approximations derived on engineering level of rigor. Mathematical point of view requires justification of those models to see their validity, applicability and accuracy. That can be achieved theoretically by investigating the deviation from the original model, or numerically by comparing simulations with experimental data. Each approach, has its advantages and disadvantages and are, frequently, complementary. In the frame of MaSiMo project we will consider fluid flow through thin domains like pipes and fractures, as well as low permeability domains as porous media.The fluids are either Newtonian or micropolar, monophase or multiphase. Physical processes in those fluids are convective, diffusive, dispersive or conductive, isothermal or nonisothermal. Theoretical analysis is based on a priori estimates for governing differential equations, different notions of convergence and compactness. In particular asymptotic analysis and homogenization will be used. For derivation of a priori estimates a powerful tool is the entropy dissipation method that is considered in this project in the context of nonlinear diffusion and population dynamics problems with a perspective of extension to other considered models. Proposed research is connected to many engineering disciplines, such as mechanical and petroleum engineering, hydrogeology etc. Expected results could improve engineering practice. In Croatian mathematical community there is a significant research in this field with several trained scientists. However, they are dispersed in smaller groups. One of the aims of this project is formation of larger research group with focused scientific interests more competitive in a search for European grants.

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