# On the Flows of Fluids Defined through Implicit Constitutive Relations between the Stress and the Symmetric Part of the Velocity Gradient

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. The Implicit Model Defined through $f(\rho ,T,D)=0$

## 4. A Sub-Class of Fluids in whichViscosity Depends only on Pressure

## 5. Stress Power-Law Fluids

## 6. Relevance of Implicit Models to Turbulence

## 7. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

- Burgers, J.M. Mechanical considerations—Model systems-phenomenological theories of relaxation and viscosity. In First Report on Viscosity and Plasticity; Nordemann Publishing: New York, NY, USA, 1939; pp. 5–67. [Google Scholar]
- Oldroyd, J.G. On the formulation of rheological equations of state. Proc. R. Soc. A Math. Phys. Eng. Sci.
**1950**, 200, 523–541. [Google Scholar] [CrossRef] - Prandtl, L. Spannungaversteilung in plastischenkoerpern. In Proceedings of the 1st International Congress in Applied Mechanics, Delft, The Netherlands, 22–26 April 1924; pp. 43–54.
- Reuss, A. Berucksichtigung der elastischenFormanderung in der Plastizitatstheorie. Z. Angew. Math. Mech.
**1930**, 10, 266–271. [Google Scholar] [CrossRef] - Dugas, R. A History of Mechanics; Dover Publications: New York, NY, USA, 1988. [Google Scholar]
- Truesdell, C. A Program towards Rediscovering the Rational Mechanics of the Age of Reason. Arch. Hist. Exact Sci.
**1960**, 1, 1–34. [Google Scholar] [CrossRef] - Navier, C.L.M.H. Sur les lois du mouvement des fluids, en ayantegard a l’adhesiondes molecules. Ann. Chem.
**1821**, 19, 244–260. (In French) [Google Scholar] - Navier, C.L.M.H. Sur les lois du mouvement des fluids, en ayantepard a l’adhesion de leurs molecules. Bull. Soc. Philoma Thique
**1822**, 298, 75–79. (In French) [Google Scholar] - Navier, C.L.M.H. Me’moiresur les lois du mouvement des fluids. Mem. Acad. Sci.
**1823**, 6, 389–416. (In French) [Google Scholar] - Poisson, S.D. Sur les Equations generales de l’_equilibreet du mouvements des Corps solides, elastiques et des uides. J. EcolePolytech.
**1831**, 13, 174. (In French) [Google Scholar] - Stokes, G.G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Camb. Philos. Soc.
**1845**, 8, 287–305. [Google Scholar] - Stokes, G.G. On the effect of internal friction of fluids on the motion pendulums. Trans. Camb. Philos. Soc.
**1851**, 9, 8–106. [Google Scholar] - Rajagopal, K.R. On implicit constitutive theories. Appl. Math.
**2003**, 28, 279–319. [Google Scholar] [CrossRef] - Rajagopal, K.R. Implicit constitutive relations for fluids. J. Fluid Mech.
**2006**, 550, 243–249. [Google Scholar] [CrossRef] - Rajagopal, K. R. The elasticity of elasticity. Z. Angew. Math. Phys.
**2007**, 58, 309–317. [Google Scholar] [CrossRef] - Bridgman, P.W. Physics of High Pressures; MacMillan: London, UK, 1931. [Google Scholar]
- Bulicek, M.; Gwiazda, P.; Malek, J.; Swierczewska-Gwiazda, A. On unsteady ows of implicitly constituted incompressible fluids. SIAM J. Math. Anal.
**2012**, 44, 2756–2801. [Google Scholar] [CrossRef] - Edgeworth, R.; Dalton, B.J.; Parnell, T. The pitch drop experiment. Eur. J. Phys.
**1984**, 5, 198–200. [Google Scholar] [CrossRef] - Boltenhagen, P.; Hu, Y.; Matthys, E.F.; Pine, D.J. Observation of bulk phase separation and coexistence in a sheared micellar solution. Phys. Rev. Lett.
**1997**, 79, 2359–2362. [Google Scholar] [CrossRef] - Perlacova, T.; Prusa, V. Tensorial implicit constitutive relations in mechanics of incompressible non-Newtonian fluids. J. Non-Newton. Fluid Mech.
**2015**, 216, 13–21. [Google Scholar] [CrossRef] - Truesdell, C. A First Course in Rational Continuum Mechanics; Academic Press: New York, NY, USA, 1977. [Google Scholar]
- Noll, W. A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal.
**1958**, 2, 197–226. [Google Scholar] [CrossRef] - Coleman, B.D.; Noll, W. An approximation theorem for functionals, with applications in continuum mechanics. Arch. Ration. Mech. Anal.
**1960**, 6, 355–370. [Google Scholar] [CrossRef] - Maxwell, J.C. On the dynamical theory of gases. Philos. Trans. R. Soc. Lond.
**1867**, A157, 49–88. [Google Scholar] [CrossRef] - Rivlin, R.S.; Ericksen, J.L. Stress deformation relations for isotropic materials. J. Ration. Mech. Anal.
**1955**, 4, 323–425. [Google Scholar] [CrossRef] - Andrade, E.C. Viscosity of liquids. Nature
**1930**, 125, 309–310. [Google Scholar] [CrossRef] - Rajagopal, K.R. Remarks on the notion of “pressure”. Int. J. Non-linear Mech.
**2015**, 71, 165–172. [Google Scholar] [CrossRef] - Rajagopal, K.R. A new development and interpretation of the Navier-Stokes equation which reveals why the Stokes assumption is inapt. Int. J. Non-linear Mech.
**2013**, 50, 141–151. [Google Scholar] [CrossRef] - Szeri, A.Z. Fluid Film Lubrication: Theory and Design; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Barus, C. Isotherms, isopiestics and isometrics relative to viscosity. Am. J. Sci.
**1893**, 45, 87–96. [Google Scholar] [CrossRef] - Bair, S.; Kottke, P. Pressure-viscosity relationships for elastohydrodynamics. Tribol. Trans.
**2003**, 46, 289–295. [Google Scholar] [CrossRef] - Renardy, M. Some remarks on the Navier-Stokes equations with a pressure dependent viscosity. Commun. Partial Differ Equ.
**1986**, 11, 779–793. [Google Scholar] [CrossRef] - Gazzola, F. A note on the evolution of Navier-Stokes equations with a pressure-dependentviscosity. Z. Angew. Math. Phys.
**2005**, 48, 760–773. [Google Scholar] [CrossRef] - Gazzola, F.; Secchi, P. Some results about stationary Navier-Stokes equations with apressure-dependent viscosity. In Navier-Stokes Equations: Theory and Numerical Methods; Salvi, R., Ed.; Longman: England, UK, 1998; pp. 31–37. [Google Scholar]
- Malek, J.; Necas, J.; Rajagopal, K.R. Global Analysis of Flows of Fluids with Pressure-Dependent Viscosities. Arch. Ration. Mech. Anal.
**2002**, 165, 243–269. [Google Scholar] [CrossRef] - Malek, J.; Necas, J.; Rajagopal, K.R. Global existence of solutions for flows of fluids with pressure and shear rate dependent velocities. Appl. Math. Letters
**2002**, 15, 961–967. [Google Scholar] [CrossRef] - Hron, J.; Malek, J.; Necas, J.; Rajagopal, K.R. Numerical Simulations and Global Existence of Solutions of Two Dimensional Flows of Fluids with Pressure and Shear Dependent Viscosities. Math. Comput. Simul.
**2003**, 61, 297–315. [Google Scholar] [CrossRef] - Franta, M.; Malek, J.; Rajaogopal, K.R. On steady flows of fluids with pressure and shear dependent viscosities. Proc. R. Soc. AMath. Phys. Eng. Sci.
**2005**, 461, 651–670. [Google Scholar] [CrossRef] - Hron, J.; Malek, J.; Rajagopal, K.R. Simple flows of fluids with pressure-dependent viscosities. Proc. R. Soc. Lond.
**2001**, A457, 1603–1622. [Google Scholar] [CrossRef] - Suslov, S.A.; Tran, T.D. Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid. J. Non-Newton. Fluid Mech.
**2008**, 154, 170–178. [Google Scholar] [CrossRef] [Green Version] - Hron, J.; Malek, J.; Prusa, V.; Rajagopal, K.R. Further remarks on simple flows of fluids with pressure dependent viscosities. Nonlinear Anal. Real World Appl.
**2011**, 12, 394–402. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Saccomandi, G.; Vergori, L. Unsteady flows of fluids with pressure dependent viscosities. J. Math. Anal. Appl.
**2013**, 404, 362–372. [Google Scholar] [CrossRef] - Reynolds, O. On the theory of lubrication and its application to Mr Tower’s experiments. Philos. Trans. Roy. Soc. Lond.
**1886**, 177, 159–209. [Google Scholar] - Schafer, C.T.; Giese, P.; Rowe, W.B.; Woolley, N.H. Elastohydrodynamically lubricated line contact based on the Navier-Stokes equations. Tribology
**2000**, 38, 57–69. [Google Scholar] - Greenwood, J.A. Thinning Films and Tribological Interfaces. In Proceedings of the 26th Leeds-Lyon Symposium on Tribology, Leeds, UK, 14–17 September 1999; Tribology Series. Dowson, D., Ed.; Elsevier: Amsterdam, The Netherlands, 2000; Volume 28, pp. 793–794. [Google Scholar]
- Rajagopal, K.R.; Szeri, A.Z. On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication. Proc. R. Soc. Lond. A
**2003**, 459, 2771–2786. [Google Scholar] [CrossRef] - Saccomandi, G.; Vergori, L. The flow of piezo-viscous fluids over an inclined surface. Q. Appl. Math.
**2010**, 68, 747–763. [Google Scholar] [CrossRef] - Gustafsson, T.; Rajagopal, K.R.; Stenberg, R.; Videman, J. Nonlinear Reynolds’ equation for hydrodynamic lubrication. Appl. Math. Model.
**2015**, 39, 5299–5309. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Saccomandi, G.; Vergori, L. Flows of fluids with pressure and shear dependent viscosity down an inclined plane. J. Fluid Mech.
**2012**, 706, 173–189. [Google Scholar] [CrossRef] - Málek, J.; Pruša, V.; Rajagopal, K.R. Generalizations of the Navier-Stokes fluid from a new perspective. Int. J. Eng. Sci.
**2010**, 48, 1907–1924. [Google Scholar] [CrossRef] - Narayanan, S.P.A.; Rajagopal, K.R. Unsteady flows of a class of novel generalizations of a Navier-Stokes fluid. Appl. Math. Comput.
**2013**, 219, 9935–9946. [Google Scholar] - Rayleigh, L. On the motion of solid bodies through viscous liquid. Philos. Mag.
**1911**, 6, 697–711. [Google Scholar] [CrossRef] - Mohankumar, K.V.; Kannan, K.; Rajagopal, K.R. Exact, approximate and numerical solutions for a variant of Stokes’ first problem for a new class of fluids. Int. J. Non-linear Mech.
**2015**, 77, 41–50. [Google Scholar] [CrossRef] - Le Roux, C.; Rajagopal, K.R. Shear flows of a new class of power-law fluids. Appl. Math.
**2013**, 58, 153–177. [Google Scholar] [CrossRef]

**Figure 3.**Relationship between the norm of the extra stress and the norm of the symmetric part of the velocity gradient for stress power-law models.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rajagopal, K.R.
On the Flows of Fluids Defined through Implicit Constitutive Relations between the Stress and the Symmetric Part of the Velocity Gradient. *Fluids* **2016**, *1*, 5.
https://doi.org/10.3390/fluids1020005

**AMA Style**

Rajagopal KR.
On the Flows of Fluids Defined through Implicit Constitutive Relations between the Stress and the Symmetric Part of the Velocity Gradient. *Fluids*. 2016; 1(2):5.
https://doi.org/10.3390/fluids1020005

**Chicago/Turabian Style**

Rajagopal, Kumbakonam R.
2016. "On the Flows of Fluids Defined through Implicit Constitutive Relations between the Stress and the Symmetric Part of the Velocity Gradient" *Fluids* 1, no. 2: 5.
https://doi.org/10.3390/fluids1020005