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November  2021, 14(11): 4069-4092. doi: 10.3934/dcdss.2020414

From quasi-incompressible to semi-compressible fluids

Tomáš Roubíček 1,2,

1. 

Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic
2. 

Institute of Thermomechanics, Czech Acad. Sci., Dolejškova 5, CZ-18200 Praha 8, Czech Republic

Received  December 2019 Revised  April 2020 Published  November 2021 Early access  July 2020

Fund Project: This research has been partially supported from the grants 19-04956S of Czech Science Foundation, and the FWF/CSF project I 4052 N32 with BMBWF through the OeAD-WTZ project CZ04/2019, and also from the institutional support RVO: 61388998 (ČR)

A new concept of semi-compressible fluids is introduced for slightly compressible visco-elastic fluids (typically rather liquids than gasses) where mass density variations are negligible in some sense, while being directly controlled by pressure which is very small in comparison with the elastic bulk modulus. The physically consistent fully Eulerian models with specific dispersion of pressure-wave speed are devised. This contrasts to the so-called quasi-incompressible fluids which are described not physically consistently and, in fact, only approximate ideally incompressible ones in the limit. After surveying and modifying models for the quasi-incompressible fluids, we eventually devise some fully convective models complying with energy conservation and capturing phenomena as pressure-wave propagation with wave-length (and possibly also pressure) dependent velocity.

Keywords: Viscoelastic fluids, slightly compressible liquids, pressure waves, dispersion, Bernoulli principle, existence of weak solutions, uniqueness.
Mathematics Subject Classification: 35K55, 76A10, 76N10, 76R50.
Citation: Tomáš Roubíček. From quasi-incompressible to semi-compressible fluids. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 4069-4092. doi: 10.3934/dcdss.2020414
References:
[1]

E. C. Aifantis, On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci., 30 (1992), 1279-1299.  doi: 10.1016/0020-7225(92)90141-3.

[2]

H. Askes and E. C. Aifantis, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Phys. Rev. B, 80 (2009), 195412. doi: 10.1103/PhysRevB.80.195412.

[3]

H. Askes and E. C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Intl. J. Solids Structures, 48 (2011), 1962-1990.  doi: 10.1016/j.ijsolstr.2011.03.006.

[4]

C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I: Formal derivations, J. Stat. Phys., 63 (1991), 323-344. 

[5]

A. Bardow and H. C. Öttinger, Consequences of the Brenner modification to the Navier-Stokes equations for dynamic light scattering, Physica A, 373 (2007), 88-96.  doi: 10.1016/j.physa.2006.05.047.

[6]

H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Qarterly Appl. Math., 50 (1992), 559-583.  doi: 10.1090/qam/1178435.

[7]

A. Berezovski and P. Ván, Internal Variables in Thermoelasticity, Springer, Switzerland, 2017.

[8]

D. Bernoulli, Hydrodynamica, Sive De Viribus Et Motibus Fluidorum Cb ommentarii, 1738.

[9]

L. Berselli, Sufficient conditions for the regularity of the solutions of the Navier-Stokes equations, Math. Meth. Appl. Sci., 22 (1999), 1079-1085.  doi: 10.1002/(SICI)1099-1476(19990910)22:13<1079::AID-MMA71>3.0.CO; 2-4.

[10]

L. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585-3595.  doi: 10.1090/S0002-9939-02-06697-2.

[11]

L. C. Berselli and S. Spirito, On the construction of suitable weak solutions to the 3D Navier-Stokes equations in a bounded domain by an artificial compressibility method, Comm. Contemporary Math., 20 (2018), 1650064. doi: 10.1142/S0219199716500644.

[12] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge Univ. Press, 1993.  doi: 10.1017/CBO9780511599965.
[13]

D. V. Boger, A highly elastic constant-viscosity fluid, J. Non-Newtonian Fluid Mechanics, 3 (1977), 87-91.  doi: 10.1016/0377-0257(77)80014-1.

[14]

H. Brenner, Kinematics of volume transport, Physica A, 349 (2005), 11-59.  doi: 10.1016/j.physa.2004.10.033.

[15]

H. Brenner, Fluid mechanics revisited, Physica A, 370 (2006), 190-224.  doi: 10.1016/j.physa.2006.03.066.

[16]

M. Bulíček, E. Feireisl and J. Málek, On a class of compressible viscoelastic rate-type fluids with stress-diffusion, Nonlinearity, 32 (2019), 4665-4681.  doi: 10.1088/1361-6544/ab3614.

[17]

M. Bulíček, J. Málek, V. Průša and E. Süli, PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion, Mathematical Analysis in Fluid Mechanics–selected Recent Results, 25–51, Contemp. Math., 710, Amer. Math. Soc., Providence, RI, 2018. doi: 10.1090/conm/710/14362.

[18]

M. Bulíček, J. Málek and K. Rajagopal, Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 56 (2007), 51-85.  doi: 10.1512/iumj.2007.56.2997.

[19]

J. Burczak, J. Málek and P. Minakowski, Stress-diffusive regularization of non-dissipative rate-type materials, Disc. Cont. Dynam. Systems - S, 10 (2017), 1233-1256.  doi: 10.3934/dcdss.2017067.

[20]

C.-T. A. Chen and F. J. Millero, Speed of sound in seawater at high pressures, J. Acoustical Soc. Amer., 62 (1977), 1129-1135.  doi: 10.1121/1.381646.

[21]

R. M. Chen, W. Layton and M. McLaughlin, Analysis of variable-step/non-autonomous artificial compression methods, J. Math. Fluid Mech., 21 (2019), Paper No. 30, 20 pp. doi: 10.1007/s00021-019-0429-2.

[22]

A. Chorin, A numerical method for solving incompressible viscous flow problems, J. Computational Physics, 2 (1967), 12-26.  doi: 10.1016/0021-9991(67)90037-X.

[23]

A. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2.

[24]

M. E. Denson Jr, Longitudinal waves through the Earth's core, Bull. Seismological Soc. Amer., 42 (1952), 119-134. 

[25]

D. Donatelli and P. Marcati, A dispersive approach to the artificial compressibility approximations of the Navier–Stokes equations in 3D, J. Hyperbolic Diff. Eqns., 3 (2006), 575-588.  doi: 10.1142/S0219891606000914.

[26]

D. Donatelli and S. Spirito, Weak solutions of Navier-Stokes equations constructed by artificial compressibility method are suitable, J. Hyperbolic Diff. Eqns., 8 (2011), 101-113.  doi: 10.1142/S0219891611002330.

[27]

J. Engelbrecht, A. Berezovski, F. Pastrone and M. Braun, Waves in microstructured materials and dispersion, Phil. Mag., 85 (2005), 4127-4141.  doi: 10.1080/14786430500362769.

[28]

A. Eringen, On differential equations of nonlinear elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54 (1983), 4703-4710. 

[29]

E. Feireisl, Y. Lu and J. Málek, On PDE analysis of flows of quasi-incompressible fluids, Zeit. angew. Math. Mech., 96 (2016), 491-508.  doi: 10.1002/zamm.201400229.

[30]

E. Feireisl and A. Novotnỳ, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[31]

E. Feireisl and A. Vasseur, New perspectives in fluid dynamics: Mathematical analysis of a model proposed by Howard Brenner, New Directions in Mathematical Fluid Mechanics, 153–179, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, 2010.

[32]

R. Fine and F. Millero, Compressibility of water as a function of temperature and pressure, J. Chem. Phys., 59 (1973), 5529-5536.  doi: 10.1063/1.1679903.

[33]

E. Fried and M. Gurtin, Second-gradient Fluids: A Theory for Incompressible Flows at Small Length Scales, Technical Report TAM Reports 1064, Dept. Theoretical & Appl. Mech., 2005.

[34]

E. Fried and M. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales, Archive Ration. Mech. Anal., 182 (2006), 513-554.  doi: 10.1007/s00205-006-0015-7.

[35]

B. Gutenberg, Wave velocities in the Earth's core, Bull. Seismological Soc. of America, 48 (1958), 301-314. 

[36]

D. James, Boger fluids, Annu. Rev. Fluid Mech., 41 (2009), 129-142.  doi: 10.1146/annurev.fluid.010908.165125.

[37]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Sringer, Switzerland, 2019.

[38]

M. Lazar, G. A. Maugin and E. C. Aifantis, On a theory of nonlocal elasticity of bi-Helmholtz type and some applications, Intl. J. Solids Structures, 43 (2006), 1404-1421.  doi: 10.1016/j.ijsolstr.2005.04.027.

[39]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. A, 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[40]

A. Madeo, P. Neff, E. C. Aifantis, G. Barbagallo and M. V. d'Agostino, On the role of micro-inertia in enriched continuum mechanics, Proc. R. Soc. A, 473 (2017), 17 pp. doi: 10.1098/rspa.2016.0722.

[41]

Y. Marcus, Internal pressure of liquids and solutions, Chem. Rev., 113 (2013), 6536–6551.

[42]

F. J. Millero, C.-T. Chen, A. Bradshaw and K. Schleicher, A new high pressure equation of state for seawater, Deep-Sea Research, 27 (1980), 255-264. 

[43]

K. G. Nayar, M. H. Sharqawy, L. D. Banchik and J. H. Lienhard V, Thermophysical properties of seawater: A review and new correlations that include pressure dependence, Desalination, 390 (2016), 1-24.  doi: 10.1016/j.desal.2016.02.024.

[44]

J. Nečas, A. Novotnỳ and M. Šilhavỳ, Global solution to the ideal compressible heat conductive multipolar fluid, Comment. Math. Univ. Carolinae, 30 (1989), 551-564. 

[45]

J. Nečas and M. Růžička, Global solution to the incompressible viscous-multipolar material problem, J. Elasticity, 29 (1992), 175-202.  doi: 10.1007/BF00044516.

[46]

A. P. Oskolkov, A small-parameter quasi-linear parabolic system approximating the Navier-Stokes system, J. Math. Sci., 1 (1973), 452-470.  doi: 10.1007/BF01084587.

[47]

H. C. Öttinger, H. Struchtrup and M. Liu, Inconsistency of a dissipative contribution to the mass flux in hydrodynamics, Phys. Rev. E, 80 (2009), 056303.

[48]

P. Podio-Guidugli, Inertia and invariance, Ann. Mat. Pura Appl., 172 (1997), 103-124.  doi: 10.1007/BF01782609.

[49]

A. Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, Springer, Wiesbaden, 1997. doi: 10.1007/978-3-663-11171-9.

[50]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013.

[51]

D. Schnack, Lectures in Magnetohydrodynamics, Springer, Berlin, 2009. doi: 10.1007/978-3-642-00688-3.

[52]

R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I), Archive Ration. Mech. Anal., 32 (1969), 135-153.  doi: 10.1007/BF00247678.

[53]

R. Temam, Navier-Stokes Equations – Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.

[54]

G. Tomassetti, An interpretation of Temam's stabilization term in the quasi-incompressible Navier-Stokes system, 2019., Preprint, arXiv: 1909.11168.

[55]

P. Ván, M. Pavelka and M. Grmela, Extra mass flux in fluid mechanics, J. Non-Equilib. Thermodyn., 42 (2017), 133-152. 

[56]

W. Wagner and A. Pruß, The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, J. Phys. Chem. Reference Data, 31 (2002), 387–535.

show all references

References:
[1]

E. C. Aifantis, On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci., 30 (1992), 1279-1299.  doi: 10.1016/0020-7225(92)90141-3.

[2]

H. Askes and E. C. Aifantis, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Phys. Rev. B, 80 (2009), 195412. doi: 10.1103/PhysRevB.80.195412.

[3]

H. Askes and E. C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Intl. J. Solids Structures, 48 (2011), 1962-1990.  doi: 10.1016/j.ijsolstr.2011.03.006.

[4]

C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I: Formal derivations, J. Stat. Phys., 63 (1991), 323-344. 

[5]

A. Bardow and H. C. Öttinger, Consequences of the Brenner modification to the Navier-Stokes equations for dynamic light scattering, Physica A, 373 (2007), 88-96.  doi: 10.1016/j.physa.2006.05.047.

[6]

H. Bellout, F. Bloom and J. Nečas, Phenomenological behavior of multipolar viscous fluids, Qarterly Appl. Math., 50 (1992), 559-583.  doi: 10.1090/qam/1178435.

[7]

A. Berezovski and P. Ván, Internal Variables in Thermoelasticity, Springer, Switzerland, 2017.

[8]

D. Bernoulli, Hydrodynamica, Sive De Viribus Et Motibus Fluidorum Cb ommentarii, 1738.

[9]

L. Berselli, Sufficient conditions for the regularity of the solutions of the Navier-Stokes equations, Math. Meth. Appl. Sci., 22 (1999), 1079-1085.  doi: 10.1002/(SICI)1099-1476(19990910)22:13<1079::AID-MMA71>3.0.CO; 2-4.

[10]

L. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585-3595.  doi: 10.1090/S0002-9939-02-06697-2.

[11]

L. C. Berselli and S. Spirito, On the construction of suitable weak solutions to the 3D Navier-Stokes equations in a bounded domain by an artificial compressibility method, Comm. Contemporary Math., 20 (2018), 1650064. doi: 10.1142/S0219199716500644.

[12] D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge Univ. Press, 1993.  doi: 10.1017/CBO9780511599965.
[13]

D. V. Boger, A highly elastic constant-viscosity fluid, J. Non-Newtonian Fluid Mechanics, 3 (1977), 87-91.  doi: 10.1016/0377-0257(77)80014-1.

[14]

H. Brenner, Kinematics of volume transport, Physica A, 349 (2005), 11-59.  doi: 10.1016/j.physa.2004.10.033.

[15]

H. Brenner, Fluid mechanics revisited, Physica A, 370 (2006), 190-224.  doi: 10.1016/j.physa.2006.03.066.

[16]

M. Bulíček, E. Feireisl and J. Málek, On a class of compressible viscoelastic rate-type fluids with stress-diffusion, Nonlinearity, 32 (2019), 4665-4681.  doi: 10.1088/1361-6544/ab3614.

[17]

M. Bulíček, J. Málek, V. Průša and E. Süli, PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion, Mathematical Analysis in Fluid Mechanics–selected Recent Results, 25–51, Contemp. Math., 710, Amer. Math. Soc., Providence, RI, 2018. doi: 10.1090/conm/710/14362.

[18]

M. Bulíček, J. Málek and K. Rajagopal, Navier's slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 56 (2007), 51-85.  doi: 10.1512/iumj.2007.56.2997.

[19]

J. Burczak, J. Málek and P. Minakowski, Stress-diffusive regularization of non-dissipative rate-type materials, Disc. Cont. Dynam. Systems - S, 10 (2017), 1233-1256.  doi: 10.3934/dcdss.2017067.

[20]

C.-T. A. Chen and F. J. Millero, Speed of sound in seawater at high pressures, J. Acoustical Soc. Amer., 62 (1977), 1129-1135.  doi: 10.1121/1.381646.

[21]

R. M. Chen, W. Layton and M. McLaughlin, Analysis of variable-step/non-autonomous artificial compression methods, J. Math. Fluid Mech., 21 (2019), Paper No. 30, 20 pp. doi: 10.1007/s00021-019-0429-2.

[22]

A. Chorin, A numerical method for solving incompressible viscous flow problems, J. Computational Physics, 2 (1967), 12-26.  doi: 10.1016/0021-9991(67)90037-X.

[23]

A. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2.

[24]

M. E. Denson Jr, Longitudinal waves through the Earth's core, Bull. Seismological Soc. Amer., 42 (1952), 119-134. 

[25]

D. Donatelli and P. Marcati, A dispersive approach to the artificial compressibility approximations of the Navier–Stokes equations in 3D, J. Hyperbolic Diff. Eqns., 3 (2006), 575-588.  doi: 10.1142/S0219891606000914.

[26]

D. Donatelli and S. Spirito, Weak solutions of Navier-Stokes equations constructed by artificial compressibility method are suitable, J. Hyperbolic Diff. Eqns., 8 (2011), 101-113.  doi: 10.1142/S0219891611002330.

[27]

J. Engelbrecht, A. Berezovski, F. Pastrone and M. Braun, Waves in microstructured materials and dispersion, Phil. Mag., 85 (2005), 4127-4141.  doi: 10.1080/14786430500362769.

[28]

A. Eringen, On differential equations of nonlinear elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54 (1983), 4703-4710. 

[29]

E. Feireisl, Y. Lu and J. Málek, On PDE analysis of flows of quasi-incompressible fluids, Zeit. angew. Math. Mech., 96 (2016), 491-508.  doi: 10.1002/zamm.201400229.

[30]

E. Feireisl and A. Novotnỳ, Singular Limits in Thermodynamics of Viscous Fluids, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8843-0.

[31]

E. Feireisl and A. Vasseur, New perspectives in fluid dynamics: Mathematical analysis of a model proposed by Howard Brenner, New Directions in Mathematical Fluid Mechanics, 153–179, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, 2010.

[32]

R. Fine and F. Millero, Compressibility of water as a function of temperature and pressure, J. Chem. Phys., 59 (1973), 5529-5536.  doi: 10.1063/1.1679903.

[33]

E. Fried and M. Gurtin, Second-gradient Fluids: A Theory for Incompressible Flows at Small Length Scales, Technical Report TAM Reports 1064, Dept. Theoretical & Appl. Mech., 2005.

[34]

E. Fried and M. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales, Archive Ration. Mech. Anal., 182 (2006), 513-554.  doi: 10.1007/s00205-006-0015-7.

[35]

B. Gutenberg, Wave velocities in the Earth's core, Bull. Seismological Soc. of America, 48 (1958), 301-314. 

[36]

D. James, Boger fluids, Annu. Rev. Fluid Mech., 41 (2009), 129-142.  doi: 10.1146/annurev.fluid.010908.165125.

[37]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Sringer, Switzerland, 2019.

[38]

M. Lazar, G. A. Maugin and E. C. Aifantis, On a theory of nonlocal elasticity of bi-Helmholtz type and some applications, Intl. J. Solids Structures, 43 (2006), 1404-1421.  doi: 10.1016/j.ijsolstr.2005.04.027.

[39]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. A, 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[40]

A. Madeo, P. Neff, E. C. Aifantis, G. Barbagallo and M. V. d'Agostino, On the role of micro-inertia in enriched continuum mechanics, Proc. R. Soc. A, 473 (2017), 17 pp. doi: 10.1098/rspa.2016.0722.

[41]

Y. Marcus, Internal pressure of liquids and solutions, Chem. Rev., 113 (2013), 6536–6551.

[42]

F. J. Millero, C.-T. Chen, A. Bradshaw and K. Schleicher, A new high pressure equation of state for seawater, Deep-Sea Research, 27 (1980), 255-264. 

[43]

K. G. Nayar, M. H. Sharqawy, L. D. Banchik and J. H. Lienhard V, Thermophysical properties of seawater: A review and new correlations that include pressure dependence, Desalination, 390 (2016), 1-24.  doi: 10.1016/j.desal.2016.02.024.

[44]

J. Nečas, A. Novotnỳ and M. Šilhavỳ, Global solution to the ideal compressible heat conductive multipolar fluid, Comment. Math. Univ. Carolinae, 30 (1989), 551-564. 

[45]

J. Nečas and M. Růžička, Global solution to the incompressible viscous-multipolar material problem, J. Elasticity, 29 (1992), 175-202.  doi: 10.1007/BF00044516.

[46]

A. P. Oskolkov, A small-parameter quasi-linear parabolic system approximating the Navier-Stokes system, J. Math. Sci., 1 (1973), 452-470.  doi: 10.1007/BF01084587.

[47]

H. C. Öttinger, H. Struchtrup and M. Liu, Inconsistency of a dissipative contribution to the mass flux in hydrodynamics, Phys. Rev. E, 80 (2009), 056303.

[48]

P. Podio-Guidugli, Inertia and invariance, Ann. Mat. Pura Appl., 172 (1997), 103-124.  doi: 10.1007/BF01782609.

[49]

A. Prohl, Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, Springer, Wiesbaden, 1997. doi: 10.1007/978-3-663-11171-9.

[50]

T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013.

[51]

D. Schnack, Lectures in Magnetohydrodynamics, Springer, Berlin, 2009. doi: 10.1007/978-3-642-00688-3.

[52]

R. Temam, Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I), Archive Ration. Mech. Anal., 32 (1969), 135-153.  doi: 10.1007/BF00247678.

[53]

R. Temam, Navier-Stokes Equations – Theory and Numerical Analysis, North-Holland, Amsterdam, 1977.

[54]

G. Tomassetti, An interpretation of Temam's stabilization term in the quasi-incompressible Navier-Stokes system, 2019., Preprint, arXiv: 1909.11168.

[55]

P. Ván, M. Pavelka and M. Grmela, Extra mass flux in fluid mechanics, J. Non-Equilib. Thermodyn., 42 (2017), 133-152. 

[56]

W. Wagner and A. Pruß, The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, J. Phys. Chem. Reference Data, 31 (2002), 387–535.

Figure 1.  Dependence of the velocity of sinusoidal waves on the angular wavenumber k (left) and on the wave length λ = 2π/k (right); an illustration of the normal dispersion due to (30) and (28) for K=9 and ϱ = 1, and D = 3. Waves with ultra short lengths (or with ultra high wave numbers) have zero velocity, i.e. cannot propagate
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Figure 2.  Dependence of the velocity of sinusoidal waves on the angular wavenumber k (left) and on the wave length λ = 2π/k (right); an illustration of the anormalous dispersion due to (38) and (37) for K=1, ϱ = 1, and D > 0 very small
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