Abstract
We consider the nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of \(\mathbf{R}^{3}\) bounded with two concentric spheres that present the solid thermo-insulated walls. In the thermodynamical sense the fluid is perfect and polytropic. We assume that the initial density and temperature are bounded from below with a positive constant and that the initial data are sufficiently smooth spherically symmetric functions. The starting problem is transformed into the Lagrangian description on the spatial domain \(]0,1[\). In this work we prove that our problem has a generalized solution for any time interval \([0,T ]\), \(T\in\mathbf{R}^{+}\). The proof is based on the local existence theorem and the extension principle.
Highlights
The model of micropolar fluids, introduced by Eringen, has received considerable attention in the last two decades
From a mathematical point of view, the micropolar fluid model is considered in two directions-one explores the incompressible and the other the compressible flows
The compressible flow of the micropolar fluid has begun to be intensively studied in the last few years
Summary
The model of micropolar fluids, introduced by Eringen (e.g. in [ ]), has received considerable attention in the last two decades. In this paper we consider the model for the compressible flow of the isotropic, viscous and heat-conducting micropolar fluid which is in the thermodynamical sense perfect and polytropic. This work is a natural continuation of the research presented in these two papers, where we prove that the problem has a generalized solution globally in time, i.e. on the domain ] , [ × ] , T[, for any finite T >. A define the new constant L by b η(b) = s ρ (s) ds = L a and introduce the new coordinate x = L– η(ξ ) With this new coordinate the spatial domain becomes ] , [ and we get the following initial-boundary problem:. In this work we consider the properties of the so-called generalized solution to the problem ( )-( ) which is introduced in [ ], p. Ρ , θ ∈ H ] , [ , v , ω ∈ H ] , [ , and that there exists a constant m ∈ R+ such that ρ (x) ≥ m, θ (x) ≥ m for x ∈ ] , [
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