Konformne preslikave in tok tekočin : delo diplomskega seminarja

Šifrer, Žan (author), Kuzman, Uroš (mentor)

V svoji diplomski nalogi sem se ukvarjal z mehaniko tekočin. V prvem delu sem svojo analizo omejil na idealne tekočine, fluide, katerih tokovnice lahko ponazorimo z dvodimenzionalnimi vektorskimi polji brez izvorov in vrtincev. Dokazal sem, da lahko le-te povežemo s teorijo holomorfnih funkcij, ter nato s pomočjo Riemannovega upodobitvenega izreka tokove okoli zapletenih objektov reduciramo na nekatere elementarne primere. V zadnjem delu naloge sem nato dodatno predstavil tudi Blasiousov izrek in nekaj primerov ne-idealnih tokov, ki pa generirajo vzgon.

• Language: Slovenian
• Keywords: holomorfizem, biholomorfizem, meromorfizem, harmonične funkcije, konformne preslikave, idealni tok tekočin, kompleksni potencial, tokovnice, ekvipotenciali, Joukowskijeva preslikava, Riemannov upodobitveni izrek, Blasiusov izrek
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-103483
• UDC: 517.5
• COBISS.SI-ID: 18440025
• Publication date in RUL: 19.09.2018

Secondary language

• Language: English
• Title: Conformal mappings and fluid flows

Abstract

In my thesis I dealt with fluid mechanics. In first part of my analysis I focused on ideal fluids. These are fluids, whose streamlines can be represented with twodimenzional vector fields without sources and vortices. I proved that such vector fields can be connected to theory of holomorphic functions. Using Riemann mapping theorem, flows around complicated objects can be reduced to elementary examples. In last part I also additionally presented Blasius theorem and some non-ideal flow examples, which do produce lift.

• Keywords: holomophism, biholomorphism, meromorphism, harmonic functions, conformal maps, ideal fluid flow, complex potential, streamlines, equipotentials, Joukowsky map, Riemann mapping theorem, Blasius theorem