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 & 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. 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. 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, (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. 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. 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. 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. 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. Google Scholar

[11]

P. M. Jordan, Second-sound propagation in rigid, nonlinear conductors,, Mech. Res. Comm., 68 (2015), 52. 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. Google Scholar

[13]

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

[14]

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

[15]

A. Morro, Governing equations in non-isothermal diffusion,, Int. J. Non-Linear Mech., 55 (2013), 90. 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. doi: 10.1007/s10659-010-9292-3. Google Scholar

[17]

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

[18]

I. Müller, Thermodynamics,, Pitman, (1985). Google Scholar

[19]

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

[20]

I. Müller and T. Ruggeri, Extended Thermodynamics,, Springer, (1993). 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. 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. 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. Google Scholar

[24]

I. Steinbach and M. Apel, Multi phase field model for solid state transformation with elastic strain,, Physica D, 217 (2006), 153. doi: 10.1016/j.physd.2006.04.001. Google Scholar

[25]

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

[26]

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

[27]

C. Truesdell and R. Toupin, The classical field theories,, Handbuch der Physik, (1960), 226. 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. 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. 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, (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. 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. 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. 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. 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. Google Scholar

[11]

P. M. Jordan, Second-sound propagation in rigid, nonlinear conductors,, Mech. Res. Comm., 68 (2015), 52. 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. Google Scholar

[13]

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

[14]

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

[15]

A. Morro, Governing equations in non-isothermal diffusion,, Int. J. Non-Linear Mech., 55 (2013), 90. 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. doi: 10.1007/s10659-010-9292-3. Google Scholar

[17]

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

[18]

I. Müller, Thermodynamics,, Pitman, (1985). Google Scholar

[19]

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

[20]

I. Müller and T. Ruggeri, Extended Thermodynamics,, Springer, (1993). 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. 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. 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. Google Scholar

[24]

I. Steinbach and M. Apel, Multi phase field model for solid state transformation with elastic strain,, Physica D, 217 (2006), 153. doi: 10.1016/j.physd.2006.04.001. Google Scholar

[25]

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

[26]

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

[27]

C. Truesdell and R. Toupin, The classical field theories,, Handbuch der Physik, (1960), 226. Google Scholar

[1]

Yanheng Ding, Fukun Zhao. On a diffusion system with bounded potential. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1073-1086. doi: 10.3934/dcds.2009.23.1073

[2]

Patrick De Kepper, István Szalai. An effective design method to produce stationary chemical reaction-diffusion patterns. Communications on Pure & Applied Analysis, 2012, 11 (1) : 189-207. doi: 10.3934/cpaa.2012.11.189

[3]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427

[4]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[5]

L. Chupin. Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 45-68. doi: 10.3934/dcdsb.2003.3.45

[6]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

[7]

Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161

[8]

Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767

[9]

Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823

[10]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042

[11]

Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033

[12]

Svetlana Katok, Ilie Ugarcovici. Theory of $(a,b)$-continued fraction transformations and applications. Electronic Research Announcements, 2010, 17: 20-33. doi: 10.3934/era.2010.17.20

[13]

Svetlana Katok, Ilie Ugarcovici. Structure of attractors for $(a,b)$-continued fraction transformations. Journal of Modern Dynamics, 2010, 4 (4) : 637-691. doi: 10.3934/jmd.2010.4.637

[14]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106

[15]

Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627

[16]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47

[17]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019173

[18]

Jesus Ildefonso Díaz, David Gómez-Castro, Jean Michel Rakotoson, Roger Temam. Linear diffusion with singular absorption potential and/or unbounded convective flow: The weighted space approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 509-546. doi: 10.3934/dcds.2018023

[19]

Jean-François Clouet, François Golse, Marjolaine Puel, Rémi Sentis. On the slowing down of charged particles in a binary stochastic mixture. Kinetic & Related Models, 2008, 1 (3) : 387-404. doi: 10.3934/krm.2008.1.387

[20]

Stéphane Brull. Ghost effect for a vapor-vapor mixture. Kinetic & Related Models, 2012, 5 (1) : 21-50. doi: 10.3934/krm.2012.5.21

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]