Jednodimenzionalni model toka realnog mikropolarnog plina s primjenom na termalnu eksploziju reaktivnog fluida

U ovoj disertaciji razmatramo model jednodimenzionalnog toka viskoznog toplinski provodljivog realnog mikropolarnog plina kojeg karakterizira generalizirana jednadžba stanja. Iz konstitutivnih jednadžbi mehanike fluida i zakona očuvanja izvodimo pripadni početno-rubni problem s homogenim rubnim uvjetima, prvo u Eulerovim, a zatim u masenim Lagrangeovim koordinatama. Izvedeni model zatim primjenjujemo u slučaju toka i termalne eksplozije reaktivnog fluida. Za oba modela konstruiramo niz aproksimativnih rješenja korištenjem Faedo-Galerkinove metode, opisujemo algoritam za numericko rješavanje i diskutiramo rezultate numeričkih testova. Kako bismo potvrdili znacaj novih modela, ispitujemo utjecaj mikropolarnosti i genera- ˇ lizirane jednadžbe stanja na ponašanje fluida. Naposljetku, za svaki od modela pokazujemo da ima jedinstveno rješenje lokalno po vremenu, a zatim i na proizvoljnom vremenskom intervalu. In this dissertation, we consider a model of the one-dimensional flow of a viscous and thermally conductive real micropolar fluid characterized by a generalized equation of state. From the constitutive equations of fluid mechanics and the conservation laws, we derive the corresponding initial-boundary value problem with homogeneous boundary conditions, first in Euler and then in Lagrangian mass coordinates. The derived model is then applied to the case of flow and thermal explosion of a reactive fluid. For each model, we first show that it has a solution locally in time. We prove this result using a constructive technique based on the Faedo-Galerkin projection. In particular, we show that the constructed sequence of approximate solutions is bounded, and then we obtain convergence (on a subsequence) using classical compactness theorems. Next, we show that the governing initial-boundary problem has at most one solution. Based on the theorems on local existence and uniqueness just proved, we then prove that the problem under consideration has a solution in any finite time interval by applying the principle of expansion. In proving this result, we bound the solutions independently of the finite time interval chosen, using the generalized energy method and the Kazhikov representation of the mass density. Finally, using the semi-discretized approximate systems constucted in the proof of the local existence theorem, we develop a fully discretized numerical scheme for the initial-boundary value problems considered. We perform several numerical tests to confirm the validity of the numerical method and the model itself. We do this by discussing the stabilization properties of the solution. To confirm the significance of the new models, we also investigate the influence of micropolarity and the generalized equation of state on the fluid behavior.