Matematička analiza problema provođenja topline u cijevi promjenjive duljine ispunjenoj fluidom

doctoral thesis

• Title (english): Mathematical analysis of the heat conduction problem in a dilated pipe filled with a fluid
• Author: Marija Prša
• Mentor: Eduard Marušić-Paloka (mentor)
• Mentor: Igor Pažanin (komentor)
• Committee member: Josip Tambača (predsjednik povjerenstva)
• Committee member: Maja Starčević (član povjerenstva)
• Committee member: Vesna Županović (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-12, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Predmet istraživanja je matematička analiza problema provođenja topline u cijevi ispunjenoj inkompresibilnim viskoznim fluidom, čija se duljina mijenja uslijed zagrijavanja. Pretpostavljamo da je longitudinalno rastezanje cijevi opisano linearnim zakonom (duljina se mijenja proporcionalno sa integralom srednje temperature, uz koeficijent proporcionalnosti \(\varepsilon\)), odnosno da domena problema nije fiksna već se mijenja ovisno o temperaturi fluida. Temperatura fluida nepoznata je te je dana kao rješenje standardne jednadžbe provođenja topline. Brzina fluida u konvektivnom članu je zadana i Poiseuilleovog je oblika. Isto tako, uzimamo u obzir i izmjenu topline s okolinom cijevi i to izražavamo pomoću Newtonovog zakona hlađenja. Dakle, uzimajući u obzir linearnu jednadžbu konvekcije-difuzije, kao i linearni zakon rastezanja cijevi, dobivamo nelinearni vezani sustav u kojem tražimo rješenje zadaće nad domenom koja se istovremeno mijenja upravo ovisno o tom nepoznatom rješenju. Prvi cilj istraživanja bilo je vidjeti pod kojim uvjetima promatrana zadaća ima rješenje i kada je ono jedinstveno. U tu svrhu dokazali smo apriorne ocjene, a potom i odgovarajući rezultat egzistencije i jedinstvenosti rješenja. Uvjeti koji nam se pojave zadovoljavaju realnost fizikalnih situacija budući da se u njima pojavljuje mali parametar \(\varepsilon\), koeficijent termičkog rastezanja cijevi, koji je dovoljno malenog reda veličine jer ovisi o stvarnom materijalu cijevi. Sljedeći cilj bio je konstruirati aproksimaciju rješenja promatrane zadaće koristeći asimptotički razvoj po potencijama spomenutog parametra \(\varepsilon\), a zatim i rigorozno opravdati dobiveni model dokazujući adekvatnu ocjenu pogreške. Za kraj smo promatrali specijalan slučaj tanke cijevi, pritom modelirajući rješenje pomoću asimptotičkog razvoja po radijusu cijevi. Došli smo do aproksimacije rješenja za prva dva člana razvoja i potom smo dobiveni rezultat opravdali dokazima adekvatnih ocjena pogreške.

Abstract (english)

This dissertation studies a heat conduction problem in a pipe filled with incompressible viscous fluid whose length varies depending on the fluid temperature. We assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law (length of the pipe varies proportionally to the integral of the mean temperature, with the coefficient of proportionality \(\varepsilon\), the heat expansion coefficient), i.e. the flow domain is not fixed, but varies depending on the fluid temperature. Fluid temperature is unknown and given as a solution of the standard heat conduction equation. Fluid velocity in the convection term is given and it corresponds to the Poiseuille flow. We also consider heat exchange with the surrounding medium and apply Newton’s law of cooling to describe it. When taking into account linear heat conduction equation and linear heat expansion law, we get an interesting nonlinear effect, where we try to find a solution to the problem posed on a domain that is varying due to that same unknown solution. Our first goal was to see under which conditions there is a solution to the proposed problem and when is the solution unique. For that purpose, we proved a priori estimates, and subsequently the existence and uniqueness result of the solution. Established conditions are realistic, since they impose dependence on the size of a parameter \(\varepsilon\), the heat expansion coefficient, which is small enough since it depends on the actual pipe material. Next goal was construction of an approximation of the solution by means of an asymptotic expansion in powers of the abovementioned small parameter \(\varepsilon\), followed by rigorous justification of the obtained model by proving the appropriate error estimate. Finally, we studied a special case of a thin pipe, where we constructed the approximation of the solution by means of an asymptotic expansion in powers of the pipe radius. We achieved the solution approximation for the first two terms and subsequently we justified the result by proving adequate error estimates.

• Keywords: problem provođenja topline; cijev ispunjena inkompresibilnim viskoznim fluidom; Newtonov zakon hlađenja; Poiseuilleova brzina fluida
• Keywords (english): heat conduction problem; pipe filled with incompressible viscous fluid; Newton’s law of cooling; Poiseuille flow
• Language: croatian
• URN:NBN: urn:nbn:hr:217:250149
• 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: ii, 75 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-21 11:55:14