Some Linear Effects of Rotation on the Heat Transfer Problem in Rarefied Gases and Rarefied Gas Mixtures
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Some Linear Effects of Rotation on the Heat Transfer Problem in Rarefied Gases and Rarefied Gas Mixtures Elvira Barbera · Francesca Brini
Received: 9 December 2013 / Accepted: 12 February 2014 © Springer Science+Business Media Dordrecht 2014
Abstract The linearized field equations of extended thermodynamics for rarefied monatomic gases and gas mixtures are used to describe two different stationary processes. The aim of the paper is to study the linear effects predicted for such phenomena. Comparison between classical and extended theory and between the solutions for a single gas and a gas mixture will be also presented. Keywords Extended thermodynamics · Rarefied gas mixture · Heat transfer · Non-inertial effects
1 Introduction In this paper the equations of 13-moment extended thermodynamics appropriate to a single rarefied monatomic gas [1] and to rarefied binary gas mixture [2] are used in order to analyze two different stationary processes. In both cases, we consider the single gas or the gas mixture enclosed in the gap between two coaxial cylinders, which are maintained at two different temperatures. For the sake of simplicity, we write the field equations in terms of the cylindrical coordinates (r, θ, z), use the physical components [3, 4] and assume that all field variables depend only on r.
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E. Barbera ( ) Department of Mathematics and Computer Science, University of Messina, V.le F. D’Alcontres 31, 98166 Messina, Italy e-mail: [email protected] F. Brini Department of Mathematics, University of Bologna, via Saragozza 8, 40123 Bologna, Italy e-mail: [email protected]
E. Barbera, F. Brini
Furthermore, we linearize the sets of field equations around an equilibrium state, then we are able1 to determine the solutions in terms of known integral functions. The first problem concerns stationary heat transfer in a non-inertial frame. In fact, the frame containing the two cylinders is supposed to rotate with a constant angular velocity around the cylinder axis. The single gas case is presented in Sect. 2. Such a problem was already studied [5, 6] without linearization through numerical methods; but here, thanks to the linearization, the solutions is easily obtained. We show that also the linearized extended equations are able to predict a non-vanishing tangential component of the heat flux and nonvanishing components of the stress tensor. This simple result is already interesting, since it is in agreement with the kinetic theory predictions, but in contrast with the classical NavierStokes-Fourier assumptions. The same physical problem for a binary mixture is briefly summarized in Sect. 3. The full description of the linearized problem is presented in [7]. For gas mixtures, the linearized equations of extended thermodynamics predict the same results as for a single gas together with other effects: the presence of boundary layers in the temperature field, non-vanishing tangential components of the diffusion flux and a non-vanishing tangential velocity of the fluid in the rotating frame. This last result,
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