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

doctoral thesis

• Title (english): Matematička analiza nestacionarnog toka mikropolarnog fluida u tankoj domeni
• Author: Marko Radulović
• Mentor: Igor Pažanin (mentor)
• Committee member: Eduard Marušić-Paloka (predsjednik povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Vesna Županović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

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 \(\epsilon\), 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.

Abstract (english)

Glavna tema ove disertacije je proučavanje modela mikropolarnog fluida, koji predstavlja generalizaciju dobro poznatog klasičnog Navier--Stokesovog modela, prvenstveno u smislu da uzima u obzir mikrostrukturu fluida i efekte kao što su smanjivanje i rotacija čestica fluida. Naš prvi cilj je dokazati egzistenciju i jedinstvenost generaliziranog nestacionarnog mikropolarnog Poiseuille rješenja u beskonačnom cilindru. Ovo postižemo rastavljanjem problema na dva dijela: klasični dvodimenzionalni mikropolarni problem s poznatim rezultatima egzistencije i jedinstvenosti te mikropolarni inverzni problem, gdje dokazujemo odgovarajuće rezultate semidiskretizacijom u vremenu. Nadalje, izvodimo asimptotički model za nestacionarni tok mikropolarnog fluida u tankoj cijevi pod pretpostavkom da rješenje ima Poiseuilleov jednosmjerni nestacionarni mikropolarni oblik. Problem razdvajamo po linearnosti i tretiramo dva problema konstruirajući dvoskalnu asimptotičku aproksimaciju u potencijama parametra \(\epsilon\), koji predstavlja debljinu cijevi. Model je opravdan koristeći ocjenu pogreške u odgovarajućim normama. U slučaju kružnog poprečnog presjeka i funkcija vanjskih sila koje ovise samo o vremenu, koji se prirodno pojavljuju u raznim primjenama, asimptotički razvoj je izračunat eksplicitno do drugog reda i numeričke ilustracije su prezentirane kako bi se jasnije naznačio utjecaj korektora. Napokon, na kraju predlažemo i složeniji model koji opisuje nestacionarni tok mikropolarnog fluida u tankoj zakrivljenoj cijevi te se proučavaju efekti fleksije, torzije, mikropolarnosti kao i utjecaja vremenske derivacije. Asimptotički razvoji su izračunati eksplicitno do drugog reda i model je opravdan ocjenom pogreške u općenitom kao i u specijalnim slučajevima.

• Keywords: micropolar fluid; nonsteady flow; thin domain; curved pipe; micropolar Poiseuille solution; asymptotic expansion; rigorous justification
• Keywords (english): mikropolarni fluid; nestacionaran tok; tanka domena; zakrivljena cijev; mikropolarno Poiseuille rješenje; asimptotički razvoj; rigorozno opravdanje
• Language: english
• URN:NBN: urn:nbn:hr:217:347626
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: xii, 168 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-05-28 10:58:22