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Global solution to a one-dimensional model of viscous and heat-conducting micropolar real gas flow

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Abstract

We study the global solvability of the initial-boundary value problem which describes a one-dimensional flow of viscous, heat-conducting, and thermodynamically polytropic micropolar real gas through the channel with solid and thermally insulated walls. We first obtain a series of time-independent a priori estimates for the generalized solution of the described problem. Using the extension principle and already obtained local existence theorem, we show that this problem has a solution globally in time, i.e., that for every , there exists a solution to the problem in the time domain .

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... Only recently has the micropolarity been introduced into a classical reactive model in Bašić-Šiško and Dražić [11,12] where the governing system is derived, analyzed experimentally using a numerical scheme, and the existence of the solution is proved locally in time. Research in the field of compressible micropolar fluid models has flourished since the end of the last millennium and is to the credit of Nermina Mujaković and her collaborators [13][14][15][16][17][18] and continues to yield important results [19][20][21][22]. ...
... In our proofs in this section, we use some ideas from earlier studies [10,22,33]. We denote by C, C 1 , C 2 , … constants dependent only on initial data and problem parameters but independent of t, which can in different places assume different values. ...
... The proof of the following two fundamental lemmas can be found in previous research [22,34,35] so we omit the proofs for brevity. ...
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We study the long time behavior of the generalized solution of the flow and thermal explosion model of the reactive real micropolar gas. The dynamics of the chemical reaction involved and the usual laws of conservation of mass, momentum, angular momentum, and energy generate a complex governing system of partial differential equations. The fluid is nonideal and non‐Newtonian. In this work, we prove that the problem can be solved in an infinite time domain and establish the asymptotic properties of the solution. Namely, we conclude that for certain parameter values, the solution stabilizes exponentially to a steady‐state solution, while for others the stabilization occurs but at power decay rate. At the end, we conducted a few numerical tests whereby we experimentally confirmed theoretical findings about long‐term behavior of the solution.
... In this paper, we move away from the ideal fluid model and study a real gas in one dimension. For the micropolar case of a real fluid in one dimension, the local and global existence and the uniqueness of the generalized solution have been carried out so far [6][7][8]. ...
... Using the Faedo-Galerkin method in [14,15], it is proved that the corresponding problem with homogeneous boundary conditions has a unique generalized solution locally in time, that is, on the domain ]0, L[×]0, T 0 [, where T 0 > 0 is sufficiently small. Here, we prove that the problem has a generalized solution globally in time, that is, on the domain ]0, 1[×]0, T[, for any finite T > 0. The proof is based on the local existence theorem, on the series of a priori estimates and the extension principle, using the approach applied in [8]. ...
... For more details on the derivation and physical interpretation of the model, see [17], but for readers' convenience, we will briefly discuss (8), as this equation has the greatest influence on the adjustments in the proof of the main theorem. As mentioned in [11], Equation (8) resulted from the interpolation of the equations of state for ideal and barotropic fluids, taking into account the generalization of the equation of state for gases introduced in [16]. ...
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We consider 1‐D thermal explosion of a compressible micropolar real gas, assuming that the initial density and temperature are bounded from below with a positive constant and that the initial data are sufficiently smooth. The starting problem is transformed into the Lagrangian description on the spatial domain ]0,1[]0,1[ \left]0,1\right[ and contains homogeneous boundary conditions. In this work, we prove that our problem has a generalized solution for any time interval [0,T],T∈R+[0,T],TR+ \left[0,T\right],T\in {\mathbf{R}}^{+} . The proof is based on the local existence theorem and the extension principle.
... The proof is divided into a sequence of lemmas and is based on the a priori estimates of the approximate solutions. The authors of this paper have already published some important results on this model (see Bašić-Šiško and Dražić 22,23 ). ...
... We would like to point out that in Bašić-Šiško and Dražić, 23 it was proved that, assuming that Theorem 1 holds, the generalized solution exists globally in time, that is, in any time interval of finite length. Therefore, the proof of this theorem is essential for the formal completion of a global existence theorem. ...
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We consider the model for one‐dimensional micropolar, viscous, polytropic, and thermally conductive real gas flow with homogeneous boundary conditions, using the generalized equation of state for pressure. The generalization is shown by the fact that the pressure depends on the mass density as a power function. The governing system of partial differential equations is given in the Lagrangian description. Using the Faedo–Galerkin method and a priori estimates, we prove that the generalized solution exists locally in time.
... For the one-dimensional case, the compressible micropolar ideal gas model was first described by Mujaković (see [31]), and then the local existence, global existence, regularity, large time behavior, stity of the solution, and the existence of global attractors for the nonisentropic compressible micropolar ideal gas model were established in [6,13,25,28,32,33,34,35,36,37,42]. Recently, there have been several investigations on the global existence, uniqueness, regularity, and large time behavior of solutions to problems associated with the compressible micropolar real gas model in [8,2,3,4,23]. In addition, for the isentropic micropolar fluid model, we would like to refer to the studies in [5]. ...
... This type of fluid has been considered in the classical case in the context of several mathematical problems, considering in particular the problem of the existence of a solution, the problem of regularity, and the problem of stabilization of the solution [29][30][31][32][33][34]. For the micropolar case of a real fluid in one dimension, the local and global existence and the uniqueness of the generalized solution have been proved so far [35][36][37]. ...
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In this paper, we analyze a quasi-linear parabolic initial-boundary problem describing the thermal explosion of a compressible micropolar real gas in one spatial dimension. The model contains five variables, mass density, velocity, microrotation, temperature, and the mass fraction of unburned fuel, while the associated problem contains homogeneous boundary conditions. The aim of this work is to prove the uniqueness theorem of the generalized solution for the mentioned initial-boundary problem. The uniqueness of the solution, together with the proven existence of the solution, makes the described initial-boundary problem theoretically consistent, which provides a basis for the development of numerical methods and the engineering application of the model.
Chapter
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Preface. Organization of the Book. Notations. I. Convex Functions and Jensen's Inequality. II. Some Recent Results Involving Means. III. Bernoulli's Inequality. IV. Cauchy's and Related Inequalities. V. Holder and Minkowski Inequalities. VI. Generalized Holder and Minkowski Inequalities. VII. Connections Between General Inequalities. VIII. Some Determinantal and Matrix Inequalities. IX. Cebysev's Inequality. X. Gruss' Inequality. XI. Steffensen's Inequality. XII. Abel's and Related Inequalities. XIII. Some Inequalities for Monotone Functions. XIV. Young's Inequality. XV. Bessel's Inequality. XVI. Cyclic Inequations. XVII. The Centroid Method in Inequalities. XVII. Triangle Inequalities. XVIII. Norm Inequalities. XIX. More on Norm Inequalities. XX. Gram's Inequality. XXI. Frejer-Jackson's Inequalities and Related Results. XXII. Mathieu's Inequality. XXIII. Shannon's Inequality. XXIV. Turan's Inequality from the Power Sum Theory. XXV. Continued Fractions and Pade Approximation Method. XXVI. Quasilinearization Methods for Proving Inequalities. XXVIII. Dynamic Programming and Functional Equation Approaches to Inequalities. XXIX. Interpolation Inequalities. XXX. Minimax Inequalities. Name Index.
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An initial-boundary value problem for one-dimensional flow of a compressible viscous heat-conducting micropolar fluid is considered. It is assumed that the fluid is thermodinamically perfect and politropic. A global-in-time existence theorem is proved. The proof is based on a local existence theorem, obtained in the previous paper. http://web.math.hr/glasnik/vol_33/no2_06.html