American Institute of Mathematical Sciences

September  2016, 5(3): 431-448. doi: 10.3934/eect.2016012

Nonlinear diffusion equations in fluid mixtures

 1 DIBRIS, University of Genoa, Via Opera Pia 11A, 16145 Genoa, Italy

Received  October 2015 Revised  November 2015 Published  August 2016

The whole set of balance equations for chemically-reacting fluid mixtures is established. The diffusion flux relative to the barycentric reference is shown to satisfy a first-order, non-linear differential equation. This in turn means that the diffusion flux is given by a balance equation, not by a constitutive assumption at the outset. Next, by way of application, limiting properties of the differential equation are shown to provide Fick's law and the Nernst-Planck equation. Moreover, known generalized forces of the literature prove to be obtained by appropriate constitutive assumptions on the stresses and the interaction forces. The entropy inequality is exploited by letting the constitutive functions of any constituent depend on temperature, mass density and their gradients thus accounting for nonlocality effects. Among the results, the generalization of the classical law of mass action is provided. The balance equation for the diffusion flux makes the system of equations for diffusion hyperbolic, provided heat conduction and viscosity are disregarded. This is ascertained by the analysis of discontinuity waves of order 2 (acceleration waves). The wave speed is derived explicitly in the case of binary mixtures.
Citation: Angelo Morro. Nonlinear diffusion equations in fluid mixtures. Evolution Equations and Control Theory, 2016, 5 (3) : 431-448. doi: 10.3934/eect.2016012
References:
 [1] W. J. Boettinger, J. E. Guyer, C. E. Campbell and G. B. McFadden, Computation of the Kirkendall velocity and displacement field in a one-dimensional binary diffusion couple with a moving interface, Proc. Royal Soc. A, 463 (2007), 3347-3373. doi: 10.1098/rspa.2007.1904. [2] J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry,, Plenum, (). [3] R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials, Int. J. Engng Sci., 7 (1969), 689-722. doi: 10.1016/0020-7225(69)90048-2. [4] M. F. Mc Carthy, Singular Surfaces and Waves,, in Continuum Physics II (ed. A.C. Eringen), (): 449. [5] J. Crank, The Mathematics of Diffusion, Oxford, at the Clarendon Press, 1956. [6] J. A. Dantzig, W. J. Boettinger, J. A. Warren, G. B. McFadden, S. R. Coriell and R. F. Sekerka, Numerical modeling of diffusion-induced deformation, Metall. Mat. Trans. A, 37 (2006), 2701-2714. doi: 10.1007/BF02586104. [7] L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. AIME, 175 (1948), 184-201. [8] M. Fabrizio, C. Giorgi and A.Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002. [9] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [10] J. B. Haddow and J. L. Wegner, Plane harmonic waves for three thermoelastic theories, Math. Mech. Solids, 1 (1996), 111-127. [11] P. M. Jordan, Second-sound propagation in rigid, nonlinear conductors, Mech. Res. Comm., 68 (2015), 52-59. doi: 10.1016/j.mechrescom.2015.04.005. [12] P. J. A. M. Kerkhof and M. A. M. Geboers, Analysis and extension of the tehory of multicomponent fluid diffusion, Chem. Engng Sci., 60 (2005), 3129-3167. [13] B. J. Kirby, Micro- and Nanoscale Fluid Mechanics, Transport in microfluidic devices. Paperback reprint of the 2010 original. Cambridge University Press, Cambridge, 2013. [14] J. C. Maxwell, On the dynamical theory of gases, The Scientific Papers of J.C. Maxwell, 2 (1965), 26-78. [15] A. Morro, Governing equations in non-isothermal diffusion, Int. J. Non-Linear Mech., 55 (2013), 90-97. doi: 10.1016/j.ijnonlinmec.2013.04.010. [16] A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3. [17] I. Müller, Thermodynamics of mixtures of fluids, J. Mécanique, 14 (1975), 267-303. [18] I. Müller, Thermodynamics, Pitman, London 1985, ξ6.7. [19] I. Müller, Thermodynamics of mixtures and phase field theory, Int. J. Solids Structures, 38 (2001), 1105-1113. [20] I. Müller and T. Ruggeri, Extended Thermodynamics, Springer, New York 1993, ξ 2.5. doi: 10.1007/978-1-4684-0447-0. [21] S. Rehfeldt and J. Stichlmair, Measurement and calculation of multicomponent diffusion coefficients in liquids, Fluid Phase Equilibria, 256 (2007), 99-104. doi: 10.1016/j.fluid.2006.10.008. [22] R. F. Sekerka, Similarity solutions for a binary diffusion couple with diffusivity and density dependent on composition, Prog. Mat. Sci, 49 (2004), 511-536. doi: 10.1016/S0079-6425(03)00033-1. [23] J. Stefan, Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemishen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 63 (1871), 63-124. [24] I. Steinbach and M. Apel, Multi phase field model for solid state transformation with elastic strain, Physica D, 217 (2006), 153-160. doi: 10.1016/j.physd.2006.04.001. [25] B. Straughan, Heat Waves, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4. [26] C. Truesdell, Rational Thermodynamics, Springer, New York, 1984. doi: 10.1007/978-1-4612-5206-1. [27] C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, (1960), 226-793; appendix, 794-858.

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References:
 [1] W. J. Boettinger, J. E. Guyer, C. E. Campbell and G. B. McFadden, Computation of the Kirkendall velocity and displacement field in a one-dimensional binary diffusion couple with a moving interface, Proc. Royal Soc. A, 463 (2007), 3347-3373. doi: 10.1098/rspa.2007.1904. [2] J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry,, Plenum, (). [3] R. M. Bowen and J. C. Wiese, Diffusion in mixtures of elastic materials, Int. J. Engng Sci., 7 (1969), 689-722. doi: 10.1016/0020-7225(69)90048-2. [4] M. F. Mc Carthy, Singular Surfaces and Waves,, in Continuum Physics II (ed. A.C. Eringen), (): 449. [5] J. Crank, The Mathematics of Diffusion, Oxford, at the Clarendon Press, 1956. [6] J. A. Dantzig, W. J. Boettinger, J. A. Warren, G. B. McFadden, S. R. Coriell and R. F. Sekerka, Numerical modeling of diffusion-induced deformation, Metall. Mat. Trans. A, 37 (2006), 2701-2714. doi: 10.1007/BF02586104. [7] L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. AIME, 175 (1948), 184-201. [8] M. Fabrizio, C. Giorgi and A.Morro, A thermodynamic approach to non-isothermal phase-field evolution in continuum physics, Physica D, 214 (2006), 144-156. doi: 10.1016/j.physd.2006.01.002. [9] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. [10] J. B. Haddow and J. L. Wegner, Plane harmonic waves for three thermoelastic theories, Math. Mech. Solids, 1 (1996), 111-127. [11] P. M. Jordan, Second-sound propagation in rigid, nonlinear conductors, Mech. Res. Comm., 68 (2015), 52-59. doi: 10.1016/j.mechrescom.2015.04.005. [12] P. J. A. M. Kerkhof and M. A. M. Geboers, Analysis and extension of the tehory of multicomponent fluid diffusion, Chem. Engng Sci., 60 (2005), 3129-3167. [13] B. J. Kirby, Micro- and Nanoscale Fluid Mechanics, Transport in microfluidic devices. Paperback reprint of the 2010 original. Cambridge University Press, Cambridge, 2013. [14] J. C. Maxwell, On the dynamical theory of gases, The Scientific Papers of J.C. Maxwell, 2 (1965), 26-78. [15] A. Morro, Governing equations in non-isothermal diffusion, Int. J. Non-Linear Mech., 55 (2013), 90-97. doi: 10.1016/j.ijnonlinmec.2013.04.010. [16] A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3. [17] I. Müller, Thermodynamics of mixtures of fluids, J. Mécanique, 14 (1975), 267-303. [18] I. Müller, Thermodynamics, Pitman, London 1985, ξ6.7. [19] I. Müller, Thermodynamics of mixtures and phase field theory, Int. J. Solids Structures, 38 (2001), 1105-1113. [20] I. Müller and T. Ruggeri, Extended Thermodynamics, Springer, New York 1993, ξ 2.5. doi: 10.1007/978-1-4684-0447-0. [21] S. Rehfeldt and J. Stichlmair, Measurement and calculation of multicomponent diffusion coefficients in liquids, Fluid Phase Equilibria, 256 (2007), 99-104. doi: 10.1016/j.fluid.2006.10.008. [22] R. F. Sekerka, Similarity solutions for a binary diffusion couple with diffusivity and density dependent on composition, Prog. Mat. Sci, 49 (2004), 511-536. doi: 10.1016/S0079-6425(03)00033-1. [23] J. Stefan, Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemishen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 63 (1871), 63-124. [24] I. Steinbach and M. Apel, Multi phase field model for solid state transformation with elastic strain, Physica D, 217 (2006), 153-160. doi: 10.1016/j.physd.2006.04.001. [25] B. Straughan, Heat Waves, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4. [26] C. Truesdell, Rational Thermodynamics, Springer, New York, 1984. doi: 10.1007/978-1-4612-5206-1. [27] C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, (1960), 226-793; appendix, 794-858.
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