Comptes Rendus
no. S1
A review of implicit algebraic constitutive relations for describing the response of nonlinear fluids
Kumbakonam Rajagopal1
1 Department of Mechanical Engineering, Texas A& M University, College Station, TX 77845, United States
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 703-720.

We review the class of implicit algebraic constitutive relations for fluids which includes in its ambit those whose material properties depend on the invariants of the stress, the symmetric part of the velocity gradient, as well as their mixed invariants. Such constitutive relations can describe the response of complex fluids whose material properties depend on the mechanical pressure, shear rate, etc. The class of models under consideration can describe the non-monotone relationship between the shear stress and the shear rate observed in experiments on colloids, as well as other novel response characteristics of non-Newtonian fluids. Constitutive relations for power-law fluids, generalized Stokesian fluids and Piezo-viscous fluids are special sub-classes of the class of fluids considered herein.

Reçu le :
Accepté le :
Première publication :
Publié le :
DOI : 10.5802/crmeca.180
Mots clés : implicit constitutive relations, colloids, stress power-law fluids, non-monotone stress-shear rate relations, monotone graphs
Kumbakonam Rajagopal 1

1 Department of Mechanical Engineering, Texas A& M University, College Station, TX 77845, United States
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Kumbakonam Rajagopal. A review of implicit algebraic constitutive relations for describing the response of nonlinear fluids. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 703-720. doi : 10.5802/crmeca.180. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.180/

[1] Leonhard Euler Sur le mouvement de l’eau par des tuyaux de conduite, Mémoires de l’académie des sciences de Berlin (1754), pp. 111-148

[2] Leonhard Euler Principes généraux de l’état d’équilibre des fluides; Principes généraux du mouvement des fluides; Continuation des recherches sur la théorie du mouvement des fluides, Histoire de l’Académie de Berlin (1755)

[3] C. L. M. H. Navier Sur les Lois des Mouvement des Fluides, en Ayant Egard à L’adhesion des Molécules, Ann. chimie, Volume 19 (1821), pp. 244-260

[4] C. L. M. H. Navier Sur les lois du mouvement des fluides, en ayant égard à l’adhésion de leurs molécules, Bull. Soc. philomath. Paris (1822), p. 75-59

[5] C. L. M. H. Navier Mémoire sur les lois du mouvement des fluides, Mémoires de l’Académie Royale des Sciences de l’Institut de France, Volume 6 (1823) no. 1823, pp. 389-440

[6] Siméon Denis Poisson Mémoire sur les équations générales de l’équilibre et du mouvement des corps solides élastiques et de fluides, L’imprimerie Royale, 1831

[7] George G. Stokes On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, Trans. Camb. Philos. Soc., Volume 8 (1845), pp. 287-308

[8] Clifford Truesdell A program toward rediscovering the rational mechanics of the age of reason, Arch. Hist. Exact Sci., Volume 1 (1960) no. 1, pp. 3-36 | MR | Zbl

[9] René Dugas History of Mechanics, Editions du Griffon, Neuchatel, Switzerland, Dover Publications, 1988

[10] Jan Blechta; Josef Málek; Kumbakonam R. Rajagopal On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion, SIAM J. Math. Anal., Volume 52 (2020) no. 2, pp. 1232-1289 | DOI | MR | Zbl

[11] Clifford Truesdell; Walter Noll The non-linear field theories of mechanics, Springer, 1992 | DOI

[12] Doïna Cioranescu; Vivette Girault; Kumbakonam R. Rajagopal Mechanics and mathematics of fluids of the differential type, Advances in Mechanics and Mathematics, 35, Springer, 2016 | DOI

[13] William R. Schowalter; John L. Lumley Mechanics of Non-Newtonian Fluids, Phys. Today, Volume 33 (1980) no. 4, pp. 57-58 | DOI

[14] Raja Huilgol Continuum Mechanics of viscoelastic liquids, Hindustan Publishing Corporation, Delhi, 1975 | MR

[15] James C. Maxwell Illustrations of the dynamical theory of gases, Philos. Mag., Volume 19 (1867), pp. 19-32

[16] J. M. Burgers Mechanical considerations-model systems-phenomenological theories of relaxation and of viscosity, First report on viscosity and plasticity, Volume 1 (1935)

[17] James G. Oldroyd On the formulation of rheological equations of state, Proc. R. Soc. Lond., Ser. A, Volume 200 (1950) no. 1063, pp. 523-541 | DOI | MR | Zbl

[18] Kumbakonam R. Rajagopal On implicit constitutive theories, Appl. Math., Praha, Volume 48 (2003) no. 4, pp. 279-319 | DOI | MR | Zbl

[19] Kumbakonam R. Rajagopal On implicit constitutive theories for fluids, J. Fluid Mech., Volume 550 (2006), pp. 243-249 | DOI | MR | Zbl

[20] Robert Boyle New experiments physico-mechanical, touching the spring of the air, and its effect, H. Hall, Oxford, 1662

[21] Joseph Louis Gay-Lussac Recherches sur la dilation des gaz et vapeurs, lues a l’institut national, le 11 pluviose an 10, Annales de chimie ou recueil de mémoires concernant la chimie et les arts qui en dépendent et spécialement la pharmacie, Volume 334 (1802), pp. 137-175

[22] B. D. Coleman; M. Markovitz; W. Noll Viscometric Flows of Non-Newtonian Fluids, Springer Tracts in Natural Philosophy, 5, Springer, 1966 | DOI

[23] J. E. Dunn; Kumbakonam R. Rajagopal Fluids of differential type: critical review and thermodynamic analysis, Int. J. Eng. Sci., Volume 33 (1995) no. 5, pp. 689-729 | DOI | MR | Zbl

[24] Ronald Samuel Rivlin; Jerald Ericksen Stress-Deformation Relations for Isotropic Materials, J. Ration. Mech. Anal., Volume 4 (1955) no. 2, pp. 323-425 | MR

[25] Markus Reiner A mathematical theory of dilatancy, Am. J. Math., Volume 67 (1945) no. 3, pp. 350-362 | DOI | MR | Zbl

[26] Ronald Samuel Rivlin Hydrodynamics of non-Newtonian fluids, Nature, Volume 160 (1947) no. 4070, p. 611-611 | DOI | MR

[27] Carl Barus Isothermals, Isopiestics and Isometrics relative to Viscosity, American Journal of Science (1880-1910), Volume 45 (1893) no. 266, pp. 87-96 | DOI

[28] Raja Huilgol On the definition of pressure in rheology, Rheology Bulletin, Volume 78 (2009) no. 2, pp. 12-15

[29] Kumbakonam R. Rajagopal Remarks on the notion of “pressure”, Int. J. Non-Linear Mech., Volume 71 (2015), pp. 165-172 | DOI

[30] Scott Bair; Peter Kottke Pressure-viscosity relationships for elastohydrodynamics, Tribol. trans., Volume 46 (2003) no. 3, pp. 289-295 | DOI

[31] D. Dowson; G. Higginson Elasto-hydrodynamic lubrication: The fundamentals of roller and gear lubrication, Pergamon Press, 1966

[32] Anthony J. M. Spencer Part III. Theory of invariants, Volume 1 (1971), pp. 239-353

[33] Kumbakonam R. Rajagopal A generalization of the classical Euler and Korteweg Fluids (2023) (https://arxiv.org/abs/2301.03485)

[34] Diederick Johannes Korteweg Sur la forme que prennent les équations du mouvements des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais connues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité, Archives Néerlandaises des Sciences exactes et naturelles, Volume 6 (1901), pp. 1-24 | Zbl

[35] Kumbakonam R. Rajagopal A new development and interpretation of the Navier–Stokes fluid which reveals why the “Stokes assumption” is inapt, Int. J. Non-Linear Mech., Volume 50 (2013), pp. 141-151 | DOI

[36] L. N. Liebermann The origin of sound absorption in water and in sea water, J. Acoust. Soc. Am., Volume 20 (1948) no. 6, pp. 868-873 | DOI

[37] L. N. Liebermann Bulk viscosity of liquids, Phys. Rev., Volume 73 (1948), pp. 537-538

[38] L. N. Liebermann The second viscosity of liquids, Phys. Rev., Volume 75 (1949) no. 9, pp. 1415-1422 | DOI

[39] S. M. Karim; L. Rosenhead The second coefficient of viscosity of liquids and gases, Rev. Mod. Phys., Volume 24 (1952) no. 2, pp. 108-116 | DOI

[40] Clifford Truesdell On the viscosity of fluids according to the kinetic theory, Z. Phys., Volume 131 (1952) no. 3, pp. 273-289 | DOI | MR | Zbl

[41] Winslow H. Herschel; Ronald Bulkley Konsistenzmessungen von gummi-benzollösungen, Kolloid-Zeitschrift, Volume 39 (1926) no. 4, pp. 291-300 | DOI

[42] Malcolm M. Cross Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems, J. Colloid Interface Sci., Volume 20 (1965) no. 5, pp. 417-437 | DOI

[43] Pierre J. Carreau Rheological equations from molecular network theories, Transactions of the Society of Rheology, Volume 16 (1972) no. 1, pp. 99-127 | DOI

[44] Tasos C. Papanastasiou Flows of materials with yield, J. Rheol., Volume 31 (1987) no. 5, pp. 385-404 | DOI | Zbl

[45] Josef Málek; Vít Průša; Kumbakonam R. Rajagopal Generalizations of the Navier–Stokes fluid from a new perspective, Int. J. Eng. Sci., Volume 48 (2010) no. 12, pp. 1907-1924 | DOI | MR | Zbl

[46] Christiaan Le Roux; Kumbakonam R. Rajagopal Shear flows of a new class of power-law fluids, Appl. Math., Praha, Volume 58 (2013) no. 2, pp. 153-177 | DOI | MR | Zbl

[47] Sai Manikiran Garimella; Mohan Anand; Kumbakonam R. Rajagopal A new model to describe the response of a class of seemingly viscoplastic materials, Appl. Math., Volume 67 (2022) no. 2, pp. 153-165 | DOI | MR | Zbl

[48] Sai M. Garimella; Mohan Anand; Kumbakonam R. Rajagopal Start-up shear flow of a shear-thinning fluid that approximates the response of viscoplastic fluids, Applied Mathematics and Computation, Volume 412 (2022), 126571 | MR | Zbl

[49] Sai M. Garimella; Mohan Anand; Kumbakonam R. Rajagopal Jeffery–Hamel flow of a shear-thinning fluid that mimics the response of viscoplastic materials, International Journal of Non-Linear Mechanics, Volume 144 (2023), 104084

[50] Shriram Srinivasan; Satish Karra Flow of “stress power-law” fluids between parallel rotating discs with distinct axes, Int. J. Non-Linear Mech., Volume 74 (2015), pp. 73-83 | DOI

[51] Lorenzo Fusi; Angiolo Farina Flow of a class of fluids defined via implicit constitutive equation down an inclined plane: Analysis of the quasi-steady regime, Eur. J. Mech. B Fluids, Volume 61 (2017), pp. 200-208 | DOI | MR | Zbl

[52] S. P. Atul Narayan; Kumbakonam R. Rajagopal Unsteady flows of a class of novel generalizations of the Navier–Stokes fluid, Appl. Math. Comput., Volume 219 (2013) no. 19, pp. 9935-9946 | DOI | MR | Zbl

[53] Lorenzo Fusi; Andrea Ballotti Squeeze Flow of Stress Power Law Fluids, Fluids, Volume 6 (2021) no. 6, 194 | DOI

[54] Yuntao T. Hu; Philippe Boltenhagen; David J. Pine Shear thickening in low-concentration solutions of wormlike micelles. I. Direct visualization of transient behavior and phase transitions, J. Rheol., Volume 42 (1998) no. 5, pp. 1185-1208 | DOI

[55] Yuntao T. Hu; Philippe Boltenhagen; Eric F. Matthys; David J. Pine Shear thickening in low-concentration solutions of wormlike micelles. II. Slip, fracture, and stability of the shear-induced phase, J. Rheol., Volume 42 (1998) no. 5, pp. 1209-1226 | DOI

[56] Philippe Boltenhagen; Yuntao T. Hu; Eric F. Matthys; D. J. Pine Observation of bulk phase separation and coexistence in a sheared micellar solution, Phys. Rev. Lett., Volume 79 (1997) no. 12, pp. 2359-2362 | DOI

[57] E. R. Macias; A. Gonzalez; O. Manero; R. Gonzales-Nunez; J . F. A. Soltero; P. Attané Flow regimes of dilute surfactant solutions, J. Non-Newton. Fluid Mech., Volume 101 (2001) no. 1-3, pp. 149-172 | DOI

[58] Georgina M. H. Wilkins; Peter D. Olmsted Vorticity banding during the lamellar-to-onion transition in a lyotropic surfactant solution in shear flow, Eur. Phys. J. E, Volume 21 (2006) no. 2, pp. 133-143 | DOI

[59] David Lopez-Diaz; Erick Sarmiento-Gomez; Cristina Garza; Rolando Castillo A rheological study in the dilute regime of the worm-micelle fluid made of zwitterionic surfactant (TDPS), anionic surfactant (SDS), and brine, J. Colloid Interface Sci., Volume 348 (2010) no. 1, pp. 152-158 | DOI

[60] Tereza Perlacova; Vít Průša Tensorial implicit constitutive relations in mechanics of incompressible non-Newtonian fluids, J. Non-Newton. Fluid Mech., Volume 216 (2015), pp. 13-21 | DOI | MR

[61] Andrea Bonito; Vivette Girault; Diane Guignard; Kumbakonam R. Rajagopal; Endre Süli Finite element approximation of steady flows of colloidal solutions, ESAIM, Math. Model. Numer. Anal., Volume 55 (2021), pp. 1963-2011 | DOI | MR | Zbl

[62] Adam Janecka; Josef Málek; Vít Průša; Giordano Tierra Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor, Acta Mech., Volume 230 (2019) no. 3, pp. 729-747 | DOI | MR

[63] Lorenzo Fusi; Giuseppe Saccomandi; Kumbakonam R. Rajagopal; Luigi Vergori Flow past a porous plate of non-Newtonian fluids with implicit shear stress shear rate relationships, Eur. J. Mech. B Fluids, Volume 92 (2022), pp. 166-173 | DOI | MR | Zbl

[64] Michael Renardy Some remarks on the Navier–Stokes equations with a pressure-dependent viscosity, Commun. Partial Differ. Equations, Volume 11 (1986) no. 7, pp. 779-793 | DOI | MR | Zbl

[65] Filippo Gazzola; Paolo Secchi Some results about stationary Navier–Stokes equations with a pressure-dependent viscosity, Navier–Stokes equations: theory and numerical methods (Varenna, 1997) (Rodolfo Salvi, ed.) (Pitman Research Notes in Mathematics Series), Volume 388, Addison Wesley Longman, 1998, pp. 31-37 | MR | Zbl

[66] Martin Lanzendörfer Flows of incompressible fluids with pressure-dependent viscosity (and their application to modelling the flow in journal bearing), Ph. D. Thesis, Univerzita Karlova, Matematicko-fyzikální fakulta, Prague, Czech Republic (2011)

[67] Jaroslav Hron; Josef Málek; Kumbakonam R. Rajagopal Simple flows of fluids with pressure–dependent viscosities, Proc. R. Soc. Lond., Ser. A, Volume 457 (2001) no. 2011, pp. 1603-1622 | DOI | Zbl

[68] Sergey A. Suslov; Thien Duc Tran Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid, J. Non-Newton. Fluid Mech., Volume 154 (2008) no. 2-3, pp. 170-178 | DOI

[69] Jaroslav Hron; Josef Málek; Vít Průša; Kumbakonam R. Rajagopal Further remarks on simple flows of fluids with pressure-dependent viscosities, Nonlinear Anal., Real World Appl., Volume 12 (2011) no. 1, pp. 394-402 | DOI | MR | Zbl

[70] Krishna Kannan; Kumbakonam R. Rajagopal Flows of fluids with pressure dependent viscosities between rotating parallel plates, New Trends in Mathematical Physics, World Scientific, 2005, pp. 172-183 | DOI | Zbl

[71] Kumbakonam R. Rajagopal; Andras Z. Szeri On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Volume 459 (2003) no. 2039, pp. 2771-2786 | DOI | MR | Zbl

[72] Martin Lanzendörfer On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate, Nonlinear Anal., Real World Appl., Volume 10 (2009) no. 4, pp. 1943-1954 | DOI | MR | Zbl

[73] Martin Lanzendörfer; Jan Stebel On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities, Appl. Math., Praha, Volume 56 (2011) no. 3, pp. 265-285 | DOI | MR | Zbl

[74] Martin Lanzendörfer; Josef Málek; Kumbakonam R. Rajagopal Numerical simulations of an incompressible piezoviscous fluid flowing in a plane slider bearing, Meccanica, Volume 53 (2018) no. 1, pp. 209-228 | DOI | MR

[75] Adrian Hirn; Martin Lanzendörfer; Jan Stebel Finite element approximation of flow of fluids with shear-rate-and pressure-dependent viscosity, IMA J. Numer. Anal., Volume 32 (2012) no. 4, pp. 1604-1634 | DOI | MR | Zbl

[76] Andreas Almqvist; Evgeniya Burtseva; Kumbakonam R. Rajagopal; Peter Wall On lower-dimensional models in lubrication, Part A: Common misinterpretations and incorrect usage of the Reynolds equation, Proc. IMechE. Part J: J Engineering Tribology, Volume 235 (2021) no. 8, pp. 1692-1702 | DOI

[77] Andreas Almqvist; Evgeniya Burtseva; Kumbakonam R. Rajagopal; Peter Wall On lower-dimensional models in lubrication, Part B: Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids, Proc. IMechE. Part J: J Engineering Tribology, Volume 235 (2021) no. 8, pp. 1703-1718 | DOI

[78] Martin Řehoř; Vít Průša Squeeze flow of a piezoviscous fluid, Appl. Math. Comput., Volume 274 (2016), pp. 414-429 | DOI | MR | Zbl

[79] Josef Málek; Kumbakonam R. Rajagopal Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities, Handbook of mathematical fluid dynamics (S. Friedlander; D. Serre, eds.), Volume 4, Elsevier, 2007, pp. 407-444

[80] Josef Málek; Jindřich Nečas; Kumbakonam R. Rajagopal Global existence of solutions for flows of fluids with pressure and shear dependent viscosities, Appl. Math. Lett., Volume 15 (2002) no. 8, pp. 961-967 | DOI | MR

[81] Josef Málek; Jindřich Nečas; Kumbakonam R. Rajagopal Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. Ration. Mech. Anal., Volume 165 (2002) no. 3, pp. 243-269 | DOI | MR | Zbl

[82] Jaroslav Hron; Josef Málek; Jindřich Nečas; Kumbakonam R. Rajagopal Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure-and shear-dependent viscosities, Math. Comput. Simul., Volume 61 (2003) no. 3-6, pp. 297-315 | DOI | MR | Zbl

[83] M. Franta; Josef Málek; Kumbakonam R. Rajagopal On steady flows of fluids with pressure–and shear–dependent viscosities, Proc. R. Soc. A: Math. Phys. Eng. Sci., Volume 461 (2005) no. 2055, pp. 651-670 | DOI | MR | Zbl

[84] Miroslav Bulıček; Josef Málek; Kumbakonam R. Rajagopal Navier’s slip and evolutionary Navier–Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J. (2007), pp. 51-85 | MR | Zbl

[85] Miroslav Bulıček; Josef Málek; Kumbakonam R. Rajagopal Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries, SIAM J. Math. Anal., Volume 41 (2009) no. 2, pp. 665-707 | DOI | MR | Zbl

[86] Macherla Vasudevaiah; Kumbakonam R. Rajagopal On fully developed flows of fluids with a pressure dependent viscosity in a pipe, Appl. Math., Praha, Volume 50 (2005) no. 4, pp. 341-353 | DOI | MR | Zbl

[87] Miroslav Bulıček; Piotr Gwiazda; Josef Málek; Agnieszka Świerczewska-Gwiazda On steady flows of incompressible fluids with implicit power-law-like rheology, Adv. Calc. Var., Volume 2 (2009) no. 2, pp. 109-136 | DOI | MR | Zbl

[88] Miroslav Bulıček; Piotr Gwiazda; Josef Málek; Agnieszka Swierczewska-Gwiazda On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., Volume 44 (2012) no. 4, pp. 2756-2801 | DOI | MR | Zbl

[89] Miroslav Bulıček; Piotr Gwiazda; Josef Málek; Kumbakonam R. Rajagopal; Agnieszka Swierczewska-Gwiazda On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, Mathematical Aspects of Fluid Mechanics (James C. Robinson; José L. Rodrigo; Witold Sadowski, eds.) (London Mathematical Society Lecture Note Series), Volume 402, Cambridge University Press, 2012, pp. 23-51 | DOI | MR | Zbl

[90] Miroslav Bulıček; Josef Málek; Endre Süli Existence of global weak solutions to implicitly constituted kinetic models of incompressible homogeneous dilute polymers, Commun. Partial Differ. Equations, Volume 38 (2013) no. 5, pp. 882-924 | DOI | MR | Zbl

[91] Joseph Boussinesq Théorie analytique de la chaleur mise en harmonie avec la thermodynamique et avec la Théorie mécanique de la Lumière: Refroidissement et chauffement par rayonnement, conductibilité des tiges, lames et masses cristallines, courants de convection, Théorie mécanique de la Lumière. xxxii, 625,[1] p, 2, Gauthier-Villars, 1903

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