Asymptotic analysis of boundary value problems in continuum mechanics

We propose a research that uses rigorous asymptotic techniques for deriving effective models governing different processes in continuum mechanics. The research is organized in 6 parts defined by following goals:

• Goal 1. Boundary perturbation and the Darcy-Weisbach law. We want to study the effective behavior of the fluid in presence of the rough boundary. Using the perturbation and homogenization methods developed in [24] i [25]., our first goal is to derive the effective law describing the pressure drop in the rough pipe, due to the friction on the boundary. The traditional engineering approach is to use the Darcy-Weisbach law. The law is phenomenological and the idea is to justify the engineering approach and derive a formula for the friction coefficient or to propose some new approach that could improve the engineering practice.
• Goal 2. Junctions of vessels filled with fluid. The usual and efficient way to study processes in thin domains (pipes, blood vessels, fractures, …) is to derive lower-dimensional models (1 ore 2 D models) by averaging the process in direction where the domain is thin. Different processes for such derivation have been proposed by different authors. When it comes to domains that have the structure constructed from two or more thin domains it is, sometimes, possible to consider the lower-dimensional models in each part of the domain and then some junction condition to link them. That subject has been well studied in theory of elasticity but rather poorly in fluid mechanics, except for few papers devoted to the junction of thin pipes ([2], [7]-[10]). According to our knowledge the problem of the junction of two gaps, gap and pipe, pipe and 3D domain, … was not rigorously studied.
• Goal 3. Non-stationary micropolar fluid flow. Our next goal is to propose new effective models of higher order of accuracy describing non-stationary flow of a micropolar fluid in thin pipe-like domains. This part of research is connected with the ongoing CSF project "Young Researchers' Career Development Project" (2016-2020). We will base our approach on the rigorous asymptotic analysis with respect to the thickness of the pipe and investigate the flows through undeformed pipe, curved pipe and multiple pipe systems. We also plan to deliver new existence, uniqueness and exponential decay results.
• Goal 4. Non-isothermal porous medium flow. Motivated by a broad range of engineering applications, we also intend to study heat transfer phenomena through fluid- saturated porous media. Though isothermal porous medium flow has been successfully analyzed in numerous works, rigorous analytical treatments of non-isothermal flows seems to be sparse throughout the literature. Depending on the problem under consideration, we will introduce small parameter in the problem (pipe's thickness; the magnitude of the Darcy number; boundary perturbation parameter) and try to derive new effective models using asymptotic techniques. We believe that the obtained results will have impact on the known engineering practice related to flows in thermoconductive porous medium.
• Goal 5. Interactions of elastic bodies. We will also work on derivation and justification of different models of interaction of elastic bodies of different sizes and properties. For instance three-dimensional and thin plate-like elastic bodies or thin plate-like and thin rod-like elastic bodies. Example and motivation for the research we find for instance in blood vessels which are built from several layers of tissue which are of different thicknesses and different elastic properties. The other motivation can be found in the structure of heart valve leaflets which are made of the tissue which are reinforces by a mesh of thin struts of different material. Very similar situation we find in stents [70] inserted in blood vessels which after some time and reendothelialization form one structure. We plan to extend the results already present in the literature, for both, linear and nonlinear problems. The main techniques which will be used are asymptotic techniques with respect to the small thickness including Gamma convergence.
• Goal 6. Interaction of an elastic plate and a thin layer of compressible fluid. The idea is to develop a simplified model for the interaction of the thin layer of fluid and the plate which is part of the fluid domain boundary. This problem can be found in many engineering models in which the elastic element is of plate shape and is lubricated with a fluid layer (e.g. air). The complexity of compressible fluid equations causes weaker regularity of the solution. Besides that, the deformation of the elastic plate is described by a hyperbolic partial differential equation and therefore for the nonlinear coupling, in order to keep the Lipschitz property of the fluid domain, the solution should be smooth enough. The original problem is too difficult for numerical solving as well. This is the reason to derive and justify a simplified model. The unique solution of the simpler effective model is the approximation obtained by observing the behavior of the solution when the small parameter, i.e. the thickness of the fluid layer, tends to zero.

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