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Dopustiva singularna rešenja sistema gasne dinamike sa nepozitivnim pritiskom

Admissible singular solutions to gas dynamics systems with non-positive pressure

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2020
Disertacija.pdf (2.106Mb)
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Author
Ružičić, Sanja
Mentor
Nedeljkov, Marko
Committee members
Seleši, Dora
Nedeljkov, Marko
Pilipović, Stevan
Simić, Srboljub
Jovanović, Božidar
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Abstract
Karakteristika hiperboličnih sistema zakona odrržanja je da čak i u slučaju glatkog po-četnog uslova rešenja uglavnom razvijaju prekide u konačnom vremena. Zbog toga se posmatraju slaba rešenja koja dati sistem zadovoljavaju u distributivnom smislu i mogu biti čak i neograničena što se ispoljava kroz pojavu Dirakove delta funkcije u rešenju. U ovoj disertaciji se akcenat stavlja na analizu protoka stišljivog neviskoznog fluida koji ne menja pravac prilikom kretanja. Protok je opisan Ojlerovim sistemom iz gasne dinamike koji se sastoji iz zakona održanja mase, količine kretanja i energije, dok su karakteristike fluida određene konstitutivnim relacijama. U slučaju izentropskog ili izotermnog protoka sistem se svodi na zakone održanja mase i količine kretanja. Glatka rešenja takvog sistema automatski zadovoljavaju zakon održanja energije, dok prelaskom na slabu formulaciju dolazi do gubitka energije. Za predstavnike sistema gasne dinamike sa nepozitivnim pritiskom su uzeti sistem gasne di...namike bez pritiska i model za  Čapliginov gas i njegova uopštenja. Data su rešenja Rimanovih problema za te sisteme koja se mogu predstaviti kao kombinacija klasičnih elementarnih talasa i senka talasa koji aproksimiraju rešenja u obliku delta udarnih talasa i koji omogućavaju rešavanje početnog problema koji u početnom uslovu sadrži delta funkciju. Na primeru modela za uopšten Čapliginov gas dokazano je da uslov prekompresivnosti nije jači od entropijskog uslova, što je prvi takav rezultat u literaturi. Dalje su korišćena rešenja Rimanovih problema, kao i problema singularne interakcije i dat je algoritam za konstrukciju globalnog dopustivog približnog rešenja početnog problema za sistem gasne dinamike bez pritiska. Algoritam je univerzalan i ideja se može proširiti na veliki broj sistema zakona održanja i veliki broj početnih uslova. Diskutovane su promene energije u približnom rešenju i posle interakcija. Dobijeno približno rešenje slabo konvergira u prostoru Radonovih mera sa predznakom.

  A solutions to hyperbolic conservation laws systems starting out as smooth often develop singularities in a finite time. As a consequence, we are forced to look for weak solutions that satisfy the system in distributional sense. Those solutions are often unbounded, which is expressed through the appearance of Dirac delta function. In this theses we study a one-dimensional, compressible and inviscid flow of a fluid. The process is described by compressible Euler gas dynamics system which consists of conservation laws of mass, linear momentum and energy, while the characteristics of the fluid are described using constitutive relations. In the case of isentropic or isothermal flow the system reduces to conservation laws of mass and linear momentum. The energy is conserved for smooth solutions to such systems, but while passing to the weak formulation the energy is being dissipated. As representatives, we  consider pressureless gas dynamics system, as well as Chaplygin gas model and its ...generalizations. We give the solutions to Riemann problems which can be represented as a combinations of classical elementary waves and shadow waves that approximate the solutions in the form of delta shock and allow as to solve the problems with initial data containing delta function. We use generalized Chaplygin gas model as demonstration of the fact that overcompressibility condition is not stronger that entropy condition, which is the first result of that kind in the literature. Further, we use solutions  to the Riemann problems, as well as singular interaction problems to give the algorithm for construction of global admissible approximate solution to the pressureless gas dynamics initial value problem. The algorithm is universal and idea can be applied to large number of conservation laws systems and large number of initial data. We discuss  energy changes in approximate solution and after the interactions. The constructed approximate solution converges in the space of signed Radon measures.

Faculty:
Универзитет у Новом Саду, Природно-математички факултет
Date:
23-06-2020
Projects:
  • Methods of Functional and Harmonic Analysis and PDE with Singularities (RS-174024)
Keywords:
Sistemi zakona održanja / Conservation laws systems / singularna rešenja / uslovi dopustivosti / gasna dinamika / Čapliginov gas / senka talasi / početni problem / singular solutions. admissibility conditions / gas dynamics / Chaplygin gas / shadow waves / initial value problem
[ Google Scholar ]
Handle
https://hdl.handle.net/21.15107/rcub_nardus_12384
URI
https://www.cris.uns.ac.rs/DownloadFileServlet/Disertacija158228005206890.pdf?controlNumber=(BISIS)114114&fileName=158228005206890.pdf&id=14915&source=NaRDuS&language=sr
https://nardus.mpn.gov.rs/handle/123456789/12384
https://www.cris.uns.ac.rs/record.jsf?recordId=114114&source=NaRDuS&language=sr
https://www.cris.uns.ac.rs/DownloadFileServlet/IzvestajKomisije158228006728428.pdf?controlNumber=(BISIS)114114&fileName=158228006728428.pdf&id=14916&source=NaRDuS&language=sr

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