A nonstationary problem of complex heat transfer
type: Item
instance of: scholarly article
title: A nonstationary problem of complex heat transfer
zbMATH Open document ID: 1331.80004
DOI: 10.1134/S0965542514110062
published in: Computational Mathematics and Mathematical Physics
publication date: 2015-11-06T00:00:00Z
review text: The following initial value problem for the normalized evolution diffusion model describing radiative, conductive, and convective heat transfer in a bounded region \(\Omega \subset {\mathbb R}^{3} \) is considered: \[ {\partial \theta \mathord{\left/ {\vphantom {\partial \theta \partial t}} \right. \kern-\nulldelimiterspace} \partial t} -a\Delta \theta +\mathbf{v}\cdot \nabla \theta +b\kappa _{a} \left(\left|\theta \right|\theta ^{3} -\varphi \right)=0, \eqno (1) \] \[ {v\partial \varphi \mathord{\left/ {\vphantom {v\partial \varphi \partial t}} \right. \kern-\nulldelimiterspace} \partial t} -\alpha \Delta \varphi +\kappa _{a} \left(\varphi -\left|\theta \right|\theta ^{3} \right)=0,\; x\in \Omega ,\; t\in \left(0, T\right), \eqno (2) \] \[ a\partial _{n} \theta +\beta \left(\theta -\theta _{b} \right)=0,\; a\partial _{n} \varphi +\gamma \left(\varphi -\varphi _{b}^{4} \right)=0, \eqno (3) \] \[ \left. \theta \right|_{t=0} =\theta _{0} ,\; \left. \varphi \right|_{t=0} =\varphi _{0} , \eqno (4) \] where \(\partial _{n} \) denotes the derivative in the direction of the external normal \({\mathbf n}\). The nonnegative function \(\theta _{b} \); the functions \(\beta =\beta (x)\), and \(\gamma =\gamma (x)\), \(x\in \Gamma \), which describe the reflective properties of the boundary and the initial functions \(\theta _{0} \) and \(\varphi _{0} \) are given. It should be noted that conditions of the third kind for the temperature are usually set on a rigid wall, where \(\mathbf v . n = 0\). In the given case, the third-kind conditions impose on the whole boundary and, in particular, on the inflow section simulate heat transfer at low normal velocities, \(\theta \) is the normalized temperature, \(\varphi \) is the normalized radiation intensity averaged over all directions, \(\mathbf v\) is a given velocity field, and \(\kappa _{a} \) is the absorption coefficient. The constants \(a,\, b\), and \(\alpha \), characterize the specifics of the studied process. The main resul…
reviewed by: Mersaid Aripov
Mathematics Subject Classification ID: 80A20
Mathematics Subject Classification ID: 35Q79
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 78A40
zbMATH DE Number: 6505330
zbMATH Keywords: radiative heat transfer equations
zbMATH Keywords: diffusion approximation
zbMATH Keywords: nonlocal solvability
zbMATH Keywords: asymptotic stability
author: G. V. Grenkin
author: A. Yu. Chebotarev
MaRDI profile type: Publication
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full work available at URL: s0965542514110062
OpenAlex ID: W2006982014
description: scientific article; zbMATH DE number 6505330