Mathematical models of flows in porous and mixed media

doctoral thesis

• Title (croatian): Matematički modeli tokova u poroznim i miješanim sredinama
• Author: Andrej Novak
• Mentor: Darko Mitrović (mentor)
• Mentor: Mladen Jurak (komentor)
• Committee member: Nenad Antonić (predsjednik povjerenstva)
• Committee member: Henrik Kalisch (član povjerenstva); strani drzavljanin: Nije dostupno
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-01-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

This thesis consists of two parts. In the first part, nontrivial generalizations of the transport-collapse scheme are presented. The initial value problem for heterogeneous scalar conservation laws is considered and as a consequence of the analysis of the scheme a new solution concept of the initial-boundary value problem for scalar conservation laws arose. This is the most important contribution of this part of the thesis. In the both situations under the consideration the rigorous proofs of the convergence of the numerical scheme are given. In the last two chapters two models of fluid flows in different environments are presented. The first one models slow flows of two immiscible fluids with the source term put in the context of water pollution. Using the second model, the analysis of the evolution of the interface between immiscible fluids of different densities in porous media, is performed. The fluids can be compressible (\(CO_2\) or natural gases) or incompressible (oil, water). It is proven that, if the heavier fluid is on top and there is no sink or source term, a tip of the interface will move in the direction of the gravity or the buoyancy. In both situations, the appropriate numerical simulations are given.

Abstract (croatian)

Ovaj rad se sastoji od dva dijela. U prvom dijelu predstavljamo netrivijalnu generalizaciju sheme brzinskog usrednjenja. Promatramo početni problem za heterogeni zakon sačuvanja, a analiza spomenute sheme poslužit će nam kao motivacija za uvođenje novog koncepta rješenja za početno-rubnu zadaću za skalarne zakone sačuvanja. To je i jedan od najznačajnijih znanstvenih doprinosa ove radnje. U obje situacije sve tvrdnje su popraćene rigoroznim dokazima, a radi zornije ilustracije poglavlje završavamo s numeričkim simulacijama. U posljednja dva poglavlja razmatramo dva modela za dinamiku fluida. Prvi opisuje spori protok dva skoro nepomiješiva fluida s izvorom stavljen u kontekst onečišćenja vode. Koristeći drugi model, analiziramo granicu između dva nepomješiva fluida različitih gustoća u poroznoj sredini. Fluidi mogu biti stlačivi (kao prirodni plinovi) ili nestlačivi (kao ulje ili voda). Dokazat ćemo da ako je teži fluid na vrhu i ako nema izvora ili ponora da se granica počinje kretati u smjeru gravitacije ili uzgona. U obje situacije dajemo i numeričke simulacije složenijih situacija.

• Keywords: Scalar conservation laws; transport-collapse scheme; kinetic formulation; immiscible fluids; porous media
• Keywords (croatian): skalarni zakoni sačuvanja; shema usrednjenja prijenosa; kinetička formulacija; nepomiješivi fluidi; porozna sredina
• Language: english
• URN:NBN: urn:nbn:hr:217:872665
• 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: vi, 106 str.
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-03-20 10:44:38