Abstract
In this paper, we describe the stationary heat transfer problem in non-reacting gas mixtures confined between two coaxial cylinders or two concentric spheres kept at different temperatures. For the description of this phenomenon, we refer to the extended thermodynamics 13-moments single temperature system and we compare its predictions with those obtained by the classical Navier–Stokes–Fourier–Fick approximations. In contrast with the classical theory, we show that extended thermodynamics takes over also thermodiffusion effects, predicts boundary layers for the common temperature and non-vanishing stress tensors. Furthermore, we compare the results with those already obtained in the planar case, in order to point out the effects of the radial dependence.
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Chapman S., Cowling T.G.: The Mathematical Theory of Non-uniform Gases. 3rd edn. Cambridge University Press, Cambridge (1970)
Enskog, D.:Über eineVerall-gemeinerung derzweitenMaxwellschen Theorie derGase. Physik.Zs. Leipzig 12, 56–60 (1911)
Enskog D.: Bemerkungen zu einer Fundamentalgleichung in der kinetischen Gastheorie. Physik. Zs. Leipzig 12, 533–539 (1911)
Chapman S.: On the kinetic theory of gas, Part II: a composite monoatomic gas: diffusion, viscosity and thermal conduction. Philos. Trans. R. Soc. Lond. A 217, 115–197 (1918)
Chapman S., Dootson F.W.: A note on thermal diffusion. Philos. Mag. 33, 248–256 (1917)
Ibbs T.L.: Thermal diffusion measurements. Proc. R. Soc. Lond. A 107, 470–486 (1925)
Clusius K., Dickel G.: Neues Verfahren zur Gasentmischung und Isotopentrennung. Natruwiss. 26, 546 (1938)
Clusius K., Dickel G.: Zur Trennung der Chlorisotope. Natruwiss. 27, 148 (1939)
Clusius, K., Dickel, G.: Isolierung des leichten Chlorisotops mit dem Atomgewicht 34, 979 im Trennrohr, Natruwiss. 27, 487 (1939)
Clusius K., Dickel G.: Grundlagen eines neuen Verfahren zur Gasentmischung und Isotopentrennung durch Thermodiffusion. Z. Phys. Chem. B 44, 397 (1939)
Furry W.H., Jones R.C., Onsager L.: On the theory of isotope separation by thermal diffusion. Phys. Rev. 55, 1083–1095 (1939)
Leaf B., Wall F.T.: Separation of gas mixtures by thermal diffusion. J. Phys. Chem. 46, 820–826 (1942)
Wall F.T., Holley C.E.: Separation by thermal diffusion of mixtures of gases having the same molecular weight. J. Chem. Phys. 8, 949–953 (1940)
Grew K.E.: Diffusion in mixtures of the inert gases. Proc. R. Soc. Lond. A 189, 402–414 (1947)
Furry W.H., Jones R.C.: Isotope separation by thermal diffusion: the cylindricl case. Phys. Rev. 69, 459–471 (1946)
Jones R.C., Furry W.H.: The separation of isotopes by thermal diffusion. Rev. Mod. Phys. 18, 151–224 (1946)
Rutherford W.M.: Separation of light gas mixtures in the thermal diffusion column. J. Chem. Phys. 53, 4319–4324 (1970)
Rutherford W.M.: Measurement of the thermal-diffusion factor of dilute neon-xenon mixtures with the thermal-diffusion column. J. Chem. Phys. 54, 4542–4546 (1971)
Rutherford W.M.: Isotopic thermal diffusion factors of Ne, Ar, Kr, and Xe from column measurements. J. Chem. Phys. 58, 1613–1618 (1973)
Dybbs A., Springer G.S.: Heat conduction experiments in rarefied gases between concentric cylinders. Phys. Fluids 8, 1946–1950 (1965)
Kassoy D.R.: Heat transfer from circular cylinders at low Reynolds numbers. I theory for variable property flow. Phys. Fluids 10, 938–946 (1967)
Aihara Y., Kassoy D.R., Libby P.A.: Heat transfer from circular cylinders at low Reynolds numbers. II Experimental Results and comparison with theory. Phys. Fluids 10, 947–952 (1967)
Yeh B.T., Frohn A.: Heat conduction in binary gas mixtures between concentric cylinders. Phys. Fluids 16, 801–805 (1973)
de Groot S.R., Mazur P.: Non-Equilibrium Thermodynamics. North-Holland Publishing, Amsterdam (1962)
Müller I.: Thermodynamics. Pitman, London, New York (1985)
Müller I., Müller W.H.: Fundamentals of Thermodynamics and Applications: With Historical Annotations and Many Citations from Avogadro to Zermelo. Springer, Berlin (2009)
Müller I., Ruggeri T.: Rational Extended Rational Thermodynamics. Springer, New York (1998)
Truesdell C.: Sulle basi della termodinamica I, Atti Accad. Naz. Lincei: Rend. Sc. fis. mat. nat. 22, 33–38 (1957)
Truesdell C.: Sulle basi della termodinamica II. Atti Accad. Naz. Lincei: Rend. Sc. fis. mat. nat. 22, 158–166 (1957)
Truesdell C., Toupin R.A.: The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik Prinzipen der klassischen Mechanik und Feldtheorie III/1, Springer, Berlin (1960)
Truesdell C.: Sulle basi della Termodinamica delle Miscele. Atti Accad. Naz. Lincei: Rend. Sc. fis. mat. nat. 44, 381–383 (1968)
Müller I.: A thermodynamic theory of mixture of fluids. Arch. Ration. Mech. Anal. 28, 1–39 (1968)
Müller I.: A new approach to thermodynamics of simple mixtures. Zeit. Naturforschung 28, 1801–1813 (1973)
Ruggeri T., Lou J.: Heat conduction in multi-temperature mixtures of fluids: the role of the average temperature. Phys. Lett. A 373, 3052–3055 (2009)
Barbera E., Brini F.: Heat transfer in a binary gas mixture between two parallel plates: an application of linear extended thermodynamics. Acta Mech. 220, 87–105 (2011)
Barbera E., Brini F.: Heat transfer in gas mixtures: advantages of an extended thermodynamics approach. Phys. Lett. A 375, 827–831 (2011)
Barbera, E., Brini, F.: Heat transfer in multi-component gas mixtures described by extended thermodynamics, Meccanica (2011). doi:10.1007/s11012-011-9474-5
Heckl M., Müller I.: Frame dependence, entropy, entropy flux, and wave speed in mixtures of gases. Acta Mech. 50, 71–95 (1983)
Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)
Kosuge S., Aoki K., Takata S.: Heat transfer in a gas mixture between two parallel plates: finite-difference analysis of the Boltzmann equation. In: Bartel, T.J., Gallis, M.A. (eds) Rarefied Gas Dynamics, pp. 289–296. AIP, Melville (2001)
Müller I., Ruggeri T.: Stationary heat conduction in radially symmetric situations: an application of extended thermodynamics. J. Nonnewtonian Fluid Mech. 119, 139–143 (2004)
Barbera E., Müller I.: Inherent frame dependence of thermodynamic fields in a gas. Acta Mech. 184, 205–216 (2006)
Barbera E., Müller I.: Secondary heat flow between confocal ellipses—an application of extended thermodynamics. J. Non-Newtonian Fluid Mech. 153, 149–156 (2008)
Barbera E., Brini F.: On stationary heat conduction in 3D symmetric domains: an application of extended thermodynamics. Acta Mech. 215, 241–260 (2010)
Barbera E., Brini F.: Linearized extended thermodynamics for stationary heat conduction in a gas at rest confined to the gap between two tori. In: Greco, A.M., Rionero, S., Ruggeri, T. (eds) Proceedings of 15th Conference on Waves and Stability in Continuous Media (WASCOM 09), pp. 15–20. World Scientific, Singapore (2010)
Barbera E., Müller I., Reitebuch D., Zhao N.: Determination of boundary conditions in extended thermodynamics via fluctuation theory. Cont. Mech. Thermodyn. 16, 411–425 (2004)
Truesdell C.: The physical components of vector and tensor. ZAMM 33, 345–356 (1953)
Barbera E.: Boundary conditions in extended thermodynamics and applications. In: Wang, Y., Hutter, K. (eds) Trends in Applications of Mathematics to Mechanics, pp. 31–40. Shaker Verlag, Aachen (2005)
Barbera E.: An application of extended thermodynamics and fluctuation principle to stationary heat conduction in radial symmetry. In: Manganaro, N., Monaco, R., Rionero, S. (eds) Proceedings of 14th Conference on Waves and Stability in Continuous Media (WASCOM 07), pp. 31–37. World Scientific, Singapore (2008)
Reitebuch D., Weiss W.: Application of high moment theory to the plane Couette flow. Cont. Mech. Thermodyn. 11, 217–225 (1999)
Struchtrup H., Torrilhon M.: Regularization of Grads 13-moment-equations: derivation and linear analysis. Phys. Fluids 15/9, 2668–2680 (2003)
Torrilhon M., Struchtrup H.: Boundary conditions for regularized 13-moment-equations for micro-channel-flows. J. Comput. Phys. 227(3), 1982–2011 (2008)
Taheri P., Struchtrup H.: Effects of rarefaction in microflows between coaxial cylinders. Phys. Rev. E 80, 066317 (2009)
Grad H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)
Grad, H.: Principles of the kinetic theory of gases. In: Flügge, S. Handbuch der Physik XII Springer, Heidelberg, pp. 205–294 (1958)
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Communicated by Manuel Torrilhon.
Dedicated to Professor Ingo Müller on the occasion of his 75th birthday.
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Barbera, E., Brini, F. An extended thermodynamics description of stationary heat transfer in binary gas mixtures confined in radial symmetric bounded domains. Continuum Mech. Thermodyn. 24, 313–331 (2012). https://doi.org/10.1007/s00161-011-0200-2
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DOI: https://doi.org/10.1007/s00161-011-0200-2