×

Balance and constitutive equations for diffusion in mixtures of fluids. (English) Zbl 1299.76245

Summary: The paper deals with the balance equations and constitutive models for mixtures of reacting fluids and addresses particular attention to the modelling of diffusion. Once the balance equations are established, both in the Eulerian and in the Lagrangian description, the second law of thermodynamics is exploited by letting the constitutive functions of the single constituents depend on temperature, mass density, their gradients, and the stretching tensor pertaining to the constituents. The results provide some generalizations of known relations about pressure and the law of mass action. Next the balance equations for the whole mixture are considered and the exploitation of the corresponding entropy inequality is provided thus showing that the modelling with single constituents is more appropriate. The balance equations for the diffusion fluxes are determined also in the case that a single constituent is considered to select the reference field as is natural in the case of a solute-solvent mixture. The balance equation so established allows the Maxwell-Stefan model of molecular friction to be incorporated as the interaction force between constituents. Next, the restriction to binary mixtures, in stationary conditions, provides a simple model for the diffusion flux that, in appropriate limit approximations, leads to the classical Fick’s law and the well-known Nernst-Planck law.

MSC:

76R50 Diffusion
76V05 Reaction effects in flows
80A17 Thermodynamics of continua
Full Text: DOI

References:

[1] J. Crank J (1964) The mathematics of diffusion, Chap 1. Clarendon, Oxford. · Zbl 0157.56703
[2] Gurtin ME (1992) Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92:178-192 · Zbl 0885.35121 · doi:10.1016/0167-2789(95)00173-5
[3] Bokris JO’M, Reddy AKN (1997) Modern electrochemistry, Chap 4. Plenum, New York
[4] Kirby BJ (2010) Micro- and nanoscale fluid mechanics, Chap 11. University Press, Cambridge · Zbl 1283.76002
[5] Brady JF (2011) Particle motion driven by solute gradients with application to autonomous motion: continuum and colloidal perspectives. J Fluid Mech 667:216-259 · Zbl 1225.76288 · doi:10.1017/S0022112010004404
[6] Truesdell C (1984) Rational thermodynamics, Chap 5. Springer, New York · Zbl 0598.73002
[7] Bowen RM, Wiese JC (1969) Diffusion in mixtures of elastic materials. Int J Eng Sci 7:689-722 · Zbl 0188.59301 · doi:10.1016/0020-7225(69)90048-2
[8] Gurtin ME (1971) On the thermodynamics of chemically reacting fluid mixtures. Arch Ration Mech Anal 43:198-212 · Zbl 0227.76012
[9] Müller I (1975) Thermodynamics of mixtures of fluids. J Mécanique 14:267-303 · Zbl 0321.76041
[10] Whitaker S (2012) Mechanics and thermodynamics of diffusion. Chem Eng Sci 68: 362-375 · doi:10.1016/j.ces.2011.09.050
[11] Passman SL, Nunziato JW (1984) A theory of multiphase mixtures. In Truesdell (ed) Rational thermodynamics. Springer, New York
[12] Müller I (1968) A thermodynamic theory of mixtures of fluids. Arch Ration Mech Anal 28:1-39 · Zbl 0157.56703 · doi:10.1007/BF00281561
[13] Sekerka RF (2004) Similarity solutions for a binary diffusion couple with diffusivity and density dependent on composition. Prog Mater Sci 49:511-536 · doi:10.1016/S0079-6425(03)00033-1
[14] Müller I (2001) Thermodynamics of mixtures and phase field theory. Int J Solids Struct 38:1105-1113 · Zbl 1007.74010 · doi:10.1016/S0020-7683(00)00076-7
[15] Morro A (2013) Governing equations in non-isothermal diffusion. Int J Non-Linear Mech 55:90-97 · doi:10.1016/j.ijnonlinmec.2013.04.010
[16] Maxwell JC (1965) On the dynamical theory of gases. Sci Pap JC Maxwell 2:26-78
[17] Stefan J (1871) Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemishen. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien 63:63-124
[18] Rehfeldt S, Stichlmair J (2007) Measurement and calculation of multicomponent diffusion coefficients in liquids. Fluid Phase Equilibria 256:99-104 · doi:10.1016/j.fluid.2006.10.008
[19] Moran MJ, Shapiro HN (2006) Fundamentals of engineering thermodynamics. Wiley, Chichester
[20] Gouin H, Ruggeri T (2008) Identification of an average temperature and a dynamical pressure in a multitemperature mixture of fluids. Phys Rev E 78:016303 · doi:10.1103/PhysRevE.78.016303
[21] Jou D, Casas-Vázquez J, Lebon G (1999), Extended irreversible thermodynamics revisited (1988-1998). Rep Prog Phys 62:1035-1142 · doi:10.1088/0034-4885/62/7/201
[22] De Groot SR, Mazur P (1984) Non-equilibrium thermodynamics. Dover, New York · Zbl 1375.82004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.