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A mathematical model and a numerical model for hyperbolic mass transport in compressible flows

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Abstract

A number of contributions have been made during the last decades to model pure-diffusive transport problems by using the so-called hyperbolic diffusion equations. These equations are used for both mass and heat transport. The hyperbolic diffusion equations are obtained by substituting the classic constitutive equation (Fick’s and Fourier’s law, respectively), by a more general differential equation, due to Cattaneo (C R Acad Sci Ser I Math 247:431–433, 1958). In some applications the use of a parabolic model for diffusive processes is assumed to be accurate enough in spite of predicting an infinite speed of propagation (Cattaneo, C R Acad Sci Ser I Math 247:431–433, 1958). However, the use of a wave-like equation that predicts a finite velocity of propagation is necessary in many other calculations. The studies of heat or mass transport with finite velocity of propagation have been traditionally limited to pure-diffusive situations. However, the authors have recently proposed a generalization of Cattaneo’s law that can also be used in convective-diffusive problems (Gómez, Technical Report (in Spanish), University of A Coruña, 2003; Gómez et al., in An alternative formulation for the advective-diffusive transport problem. 7th Congress on computational methods in engineering. Lisbon, Portugal, 2004a; Gómez et al., in On the intrinsic instability of the advection–diffusion equation. Proc. of the 4th European congress on computational methods in applied sciences and engineering (CDROM). Jyväskylä, Finland, 2004b) (see also Christov and Jordan, Phys Rev Lett 94:4301–4304, 2005). This constitutive equation has been applied to engineering problems in the context of mass transport within an incompressible fluid (Gómez et al., Comput Methods Appl Mech Eng, doi:10.1016/j.cma.2006.09.016, 2006). In this paper we extend the model to compressible flow problems. A discontinuous Galerkin method is also proposed to numerically solve the equations. Finally, we present some examples to test out the performance of the numerical and the mathematical model.

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Acknowledgments

H. Gómez gratefully acknowledges the support provided by Ministerio de Educación y Ciencia through the postdoctoral fellowships program. The authors were partially supported by Xunta de Galicia (grants # PGIDIT05PXIC118002PN and # PGDIT06TAM11801PR), Ministerio de Educación y Ciencia (grants # DPI2004-05156, # DPI2006-15275 and # DPI2007-61214) cofinanced with FEDER funds, Universidad de A Coruña and Fundación de la Ingeniería Civil de Galicia. This funding is gratefully acknowledged.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station, C0200, 201 E. 24th Street, Austin, TX, 78712, USA

    Héctor Gómez

  2. Civil Engineering School, Department of Mathematical Methods, University of A Coruña, Campus de Elviña, 15192, A Coruña, Spain

    Héctor Gómez, Ignasi Colominas, Fermín Navarrina & Manuel Casteleiro

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  1. Héctor Gómez
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  2. Ignasi Colominas
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  3. Fermín Navarrina
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  4. Manuel Casteleiro
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Correspondence to Ignasi Colominas.

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Gómez, H., Colominas, I., Navarrina, F. et al. A mathematical model and a numerical model for hyperbolic mass transport in compressible flows. Heat Mass Transfer 45, 219–226 (2008). https://doi.org/10.1007/s00231-008-0418-0

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  • DOI: https://doi.org/10.1007/s00231-008-0418-0

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