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

Nonlinear diffusion equations in fluid mixtures

Angelo Morro 1,

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.
Keywords: fluid mixture, Diffusion, mass fraction, chemical potential..
Mathematics Subject Classification: Primary: 76R50, 74J30; Secondary: 35L6.
Citation: Angelo Morro. Nonlinear diffusion equations in fluid mixtures. Evolution Equations & 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.  Google Scholar

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J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry,, Plenum, ().   Google Scholar

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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.  Google Scholar

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M. F. Mc Carthy, Singular Surfaces and Waves,, in Continuum Physics II (ed. A.C. Eringen), (): 449.   Google Scholar

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J. Crank, The Mathematics of Diffusion, Oxford, at the Clarendon Press, 1956.  Google Scholar

[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.  Google Scholar

[7]

L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. AIME, 175 (1948), 184-201. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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J. B. Haddow and J. L. Wegner, Plane harmonic waves for three thermoelastic theories, Math. Mech. Solids, 1 (1996), 111-127.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

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B. J. Kirby, Micro- and Nanoscale Fluid Mechanics, Transport in microfluidic devices. Paperback reprint of the 2010 original. Cambridge University Press, Cambridge, 2013.  Google Scholar

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J. C. Maxwell, On the dynamical theory of gases, The Scientific Papers of J.C. Maxwell, 2 (1965), 26-78. Google Scholar

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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.  Google Scholar

[16]

A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3.  Google Scholar

[17]

I. Müller, Thermodynamics of mixtures of fluids, J. Mécanique, 14 (1975), 267-303.  Google Scholar

[18]

I. Müller, Thermodynamics, Pitman, London 1985, ξ6.7. Google Scholar

[19]

I. Müller, Thermodynamics of mixtures and phase field theory, Int. J. Solids Structures, 38 (2001), 1105-1113. Google Scholar

[20]

I. Müller and T. Ruggeri, Extended Thermodynamics, Springer, New York 1993, ξ 2.5. doi: 10.1007/978-1-4684-0447-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Stefan, Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemishen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 63 (1871), 63-124. Google Scholar

[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.  Google Scholar

[25]

B. Straughan, Heat Waves, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar

[26]

C. Truesdell, Rational Thermodynamics, Springer, New York, 1984. doi: 10.1007/978-1-4612-5206-1.  Google Scholar

[27]

C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, (1960), 226-793; appendix, 794-858.  Google Scholar

show all references

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.  Google Scholar

[2]

J. O'M. Bokris and A. K. N. Reddy, Modern Electrochemistry,, Plenum, ().   Google Scholar

[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.  Google Scholar

[4]

M. F. Mc Carthy, Singular Surfaces and Waves,, in Continuum Physics II (ed. A.C. Eringen), (): 449.   Google Scholar

[5]

J. Crank, The Mathematics of Diffusion, Oxford, at the Clarendon Press, 1956.  Google Scholar

[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.  Google Scholar

[7]

L. S. Darken, Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. AIME, 175 (1948), 184-201. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. B. Haddow and J. L. Wegner, Plane harmonic waves for three thermoelastic theories, Math. Mech. Solids, 1 (1996), 111-127.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[13]

B. J. Kirby, Micro- and Nanoscale Fluid Mechanics, Transport in microfluidic devices. Paperback reprint of the 2010 original. Cambridge University Press, Cambridge, 2013.  Google Scholar

[14]

J. C. Maxwell, On the dynamical theory of gases, The Scientific Papers of J.C. Maxwell, 2 (1965), 26-78. Google Scholar

[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.  Google Scholar

[16]

A. Morro, Evolution equations for non-simple viscoelastic solids, J. Elasticity, 105 (2011), 93-105. doi: 10.1007/s10659-010-9292-3.  Google Scholar

[17]

I. Müller, Thermodynamics of mixtures of fluids, J. Mécanique, 14 (1975), 267-303.  Google Scholar

[18]

I. Müller, Thermodynamics, Pitman, London 1985, ξ6.7. Google Scholar

[19]

I. Müller, Thermodynamics of mixtures and phase field theory, Int. J. Solids Structures, 38 (2001), 1105-1113. Google Scholar

[20]

I. Müller and T. Ruggeri, Extended Thermodynamics, Springer, New York 1993, ξ 2.5. doi: 10.1007/978-1-4684-0447-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Stefan, Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemishen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 63 (1871), 63-124. Google Scholar

[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.  Google Scholar

[25]

B. Straughan, Heat Waves, Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar

[26]

C. Truesdell, Rational Thermodynamics, Springer, New York, 1984. doi: 10.1007/978-1-4612-5206-1.  Google Scholar

[27]

C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, (1960), 226-793; appendix, 794-858.  Google Scholar

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