engleski

Mathematical analysis of the nonsteady flow of micropolar fluid in a thin domain

The main subject of this dissertation consists of studying the micropolar fluid model, which represents a generalization of the well--known classical Navier--Stokes model, mainly in the sense that it takes into account the microstructure of the fluid and effects such as shrinking and rotation of the particles of the fluid. Our first goal is to prove the existence and uniqueness of a generalized nonsteady micropolar Poiseuille solution in an infinite cylinder. This is achieved by decomposing the problem into two parts: the classical two-- dimensional micropolar problem with known existence and uniqueness results and the micropolar inverse problem, where we prove the corresponding results using semidiscretization in time. Next, we derive an asymptotic model for the nonsteady micropolar fluid flow in a thin pipe assuming the solution has the uni-- directional nonsteady micropolar Poiseuille form. The problem is separated by linearity and we treat two problems by constructing two--scale asymptotic approximations in powers of ϵ, representing the pipe's thickness. The model is justified via error estimates in the appropriate norms. In the case of circular cross--section and external force functions depending only on time, naturally appearing in many applications, the asymptotic expansion is explicitly computed up to the second order and numerical illustrations are provided to indicate the effect of the correctors. Finally, a more complex model is proposed describing the nonsteady micropolar fluid flow in a thin curved pipe, and the effects of flexion, torsion, micropolarity and time derivative are studied. The asymptotic expansions are explicitly computed up to the second--order and the model is justified by means of the error estimate in general as well as for special cases.

micropolar fluid, nonsteady flow, thin domain, curved pipe, micropolar Poiseuille solution, asymptotic expansion, rigorous justification

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