Croatian Science Foundation project IP2019041140
Multiscale problems in fluid mechanics (MultiFM)
Principal investigator: Igor Pažanin
Host institution: Faculty of Science, University of Zagreb
From: January 2020; Duration: 48 months
Public presentation of the project. ... to be announced ...

Problems of interest:
 Investigating effective behaviour of the nonnewtonian fluid flows
 Proposing new models for fluid flow through domains with porous/poroelastic walls
 Study of multispecies diffusion systems in porous media
 Existence, uniqueness and regularity results

Physical proceses:
 Convection, diffusion, reactions, conduction; Newtonian or micropolar fluids, transport of chemical mixtures in porous media

Analytical tools:
 Asymptotic analysis, homogenization techniques, entropy methods, FaedoGalerkin approach, fixedpoint theorems, compactness results

Applications:
 Mechanical engineering, hydrogeology and biomedicine
The problems in fluid mechanics are usually described by the systems of partial differential equations resulting from various conservation laws and constitutive relations. Such systems are most often too complex to be directly solved numerically, due to its nonlinearity and/or coupled nature. In some situations, simpler empiric laws proposed in the engineering literature can represent good substitutes for the original physical laws. However, to be sure of that, one needs to justify the usage of such simplified models, i.e. to deduce the information about its order of accuracy. This justification can be done either theoretically by establishing to what extent the empirical model deviates from the original conservation law, or numerically by performing numerical simulations and using experimental data. Both means of justifications, theoretical and numerical, are, in many situations, complementary and we will employ them both in this project.
The goal of this project is to derive and analyse higherorder mathematical models describing various multiscale processes in fluid mechanics. In the framework of this particular project we will tackle the problems being demanding from the mathematical point of view and directly motivated by the reallife applications.
More precisely, we will address the fluid flow through thin domains whose walls are rigid, but also could be porous or even poroelastic. We plan to explore the effects of boundary irregularities on the lubrication problems, and to study the low permeability domains such as porous media. The fluids considered will be classical Newton fluids, but most attention will be devoted to nonnewtonain fluids (powerlaw fluids, micropolar fluids). They can be in liquid or gas phase but we will also investigate multiphase fluids where the two phases exist together. Inspired by the applications, nonstandard but physically relevant boundary conditions will be imposed and different physical processes occurring in the fluids will be considered. In view of that and taking into account the track record of PI and team members, the project should result with substantial number of papers published in highranked journals of applied mathematics.