## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2014 (2014), Article ID 217174, 8 pages.

### Asymptotic Modeling of the Thin Film Flow with a Pressure-Dependent Viscosity

Eduard Marušić-Paloka and Igor Pažanin

**Full-text: Access denied (no subscription detected) **

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

#### Abstract

We study the lubrication process with incompressible fluid taking into account the dependence of the viscosity on the pressure. Assuming that the viscosity-pressure relation is given by the well-known Barus law, we derive an effective model using asymptotic analysis with respect to the film thickness. The key idea is to conveniently transform the governing system and then apply two-scale expansion technique.

#### Article information

**Source**

J. Appl. Math., Volume 2014 (2014), Article ID 217174, 8 pages.

**Dates**

First available in Project Euclid: 2 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1425305910

**Digital Object Identifier**

doi:10.1155/2014/217174

**Mathematical Reviews number (MathSciNet)**

MR3240611

**Zentralblatt MATH identifier**

1326.35244

#### Citation

Marušić-Paloka, Eduard; Pažanin, Igor. Asymptotic Modeling of the Thin Film Flow with a Pressure-Dependent Viscosity. J. Appl. Math. 2014 (2014), Article ID 217174, 8 pages. doi:10.1155/2014/217174. https://projecteuclid.org/euclid.jam/1425305910

#### References

- A. Z. Szeri,
*Fluid Film Lubrication: Theory and Design*, Cambridge University Press, 1998. - O. Reynolds, “On the theory of lubrication and its applications to Mr. Beauchamp Towers experiments, including an experimental determination of the viscosity of olive oil,”
*Philosophical Transactions of the Royal Society of London*, vol. 177, pp. 157–234, 1886. - W. R. Jones, “Pressure viscosity measurement for several lubricants,”
*ASLE Transactions*, vol. 18, no. 4, pp. 249–262, 1975. - S. C. Jain, R. Sinhasan, and D. V. Singh, “Consideration of viscosity variation in determining the performance characteristics of circular bearings in the laminar and turbulent regimes,”
*Wear*, vol. 86, no. 2, pp. 233–245, 1983. - D. R. Gwynllyw, A. R. Davies, and T. N. Phillips, “On the effects of a piezoviscous lubricant on the dynamics of a journal bearing,”
*Journal of Rheology*, vol. 40, no. 6, pp. 1239–1266, 1996. - A. Goubert, J. Vermant, P. Moldenaers, A. Göttfert, and B. Ernst, “Comparison of measurement techniques for evaluating the pressure dependence of the viscosity,”
*Applied Rheology*, vol. 11, no. 1, pp. 26–37, 2001. - C. Barus, “Isothermals, isopiestics and isometrics relative to viscosity,”
*The American Journal of Science*, vol. 45, pp. 87–96, 1893. - P. A. Kottke,
*Rheological implications of tension in liquids [Ph.D. thesis]*, Georgia Institute of Technology, 2004. - M. Renardy, “Some remarks on the Navier-Stokes equations with a pressure-dependent viscosity,”
*Communications in Partial Differential Equations*, vol. 11, no. 7, pp. 779–793, 1986.Mathematical Reviews (MathSciNet): MR837931

Zentralblatt MATH: 0597.35097

Digital Object Identifier: doi:10.1080/03605308608820445 - F. Gazzola, “A note on the evolution Navier-Stokes equations with a pressure-dependent viscosity,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 48, no. 5, pp. 760–773, 1997. - J. Málek, J. Nečas, and K. R. Rajagopal, “Global analysis of the flows of fluids with pressure-dependent viscosities,”
*Archive for Rational Mechanics and Analysis*, vol. 165, no. 3, pp. 243–269, 2002.Mathematical Reviews (MathSciNet): MR1941479

Digital Object Identifier: doi:10.1007/s00205-002-0219-4 - E. Marušić-Paloka and S. Marušić, “Analysis of the Reynolds equation for lubrication in case of pressure-dependent viscosity,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 710214, 10 pages, 2013.Mathematical Reviews (MathSciNet): MR3044275 - E. Marusic-Paloka, “An analysis of the Stokes system with pressure dependent viscosityčommentComment on ref. [13?]: Please update the information of this reference, if possible.,” In press.
- E. Marušić-Paloka and I. Pažanin, “A note on the pipe flow with a pressure-dependent viscosity,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 197, pp. 5–10, 2013. - M. van Dyke,
*Perturbation Methods in Fluid Mechanics*, The Parabolic Press, Stanford, Calif, USA, 1975.Mathematical Reviews (MathSciNet): MR0416240 - G. H. Wannier, “A contribution to the hydrodynamics of lubrication,”
*Quarterly of Applied Mathematics*, vol. 8, pp. 1–32, 1950. - R. K. Zeytouninan,
*Modelisation Asymptotique en Mecanique des Fluides Newtoniens*, Collection SMAI, Springer, Berlin, Germany, 1994.Mathematical Reviews (MathSciNet): MR1618773 - D. Gilbarg and N. S. Trudinger,
*Elliptic Partial Differential Equations of Second Order*, Springer, 1997. - K. R. Rajagopal and A. Z. Szeri, “On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication,”
*Proceedings of the Royal Society of London A*, vol. 459, no. 2039, pp. 2771–2786, 2003. - S. A. Nazarov, “Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid,”
*Siberian Mathematical Journal*, vol. 31, no. 2, pp. 296–307, 1990.Mathematical Reviews (MathSciNet): MR1065588 - A. Duvnjak and E. Marusic-Paloka, “Derivation of the Reynolds equation for lubrication of a rotating shaft,”
*Archivum Mathematicum*, vol. 36, no. 4, pp. 239–253, 2000.Mathematical Reviews (MathSciNet): MR1811168 - J. Wilkening, “Practical error estimates for Reynolds' lubrication approximation and its higher order corrections,”
*SIAM Journal on Mathematical Analysis*, vol. 41, no. 2, pp. 588–630, 2009.Mathematical Reviews (MathSciNet): MR2507463

Zentralblatt MATH: 05696715

Digital Object Identifier: doi:10.1137/070695447 - E. Marušić-Paloka, I. Pažanin, and S. Marušić, “Second order model in fluid film lubrication,”
*Comptes Rendus: Mecanique*, vol. 340, no. 8, pp. 596–601, 2012. - C. Conca, F. Murat, and O. Pironneau, “The Stokes and Navier-Stokes equations with boundary conditions involving the pressure,”
*Japanese Journal of Mathematics*, vol. 20, pp. 263–318, 1994.Mathematical Reviews (MathSciNet): MR1308419 - L. Cattabriga, “Su un problema al contorno relativo al sistema di equazioni di Stokes,”
*Rendiconti del Seminario Matematico della Università di Padova*, vol. 31, pp. 308–340, 1961. \endinputMathematical Reviews (MathSciNet): MR0138894

### More like this

- An unconditional existence result for elastohydrodynamic piezoviscous lubrication
problems with Elrod-Adams model of cavitation

Bayada, Guy, Chupin, Laurent, and Grec, Bérénice, Differential and Integral Equations, 2008 - Mathematical Justification of a Shallow Water Model

Bresch, Didier and Noble, Pascal, Methods and Applications of Analysis, 2007 - MHD Thin Film Flows of a Third Grade Fluid on a Vertical Belt with Slip Boundary Conditions

Gul, Taza, Shah, Rehan Ali, Islam, Saeed, and Arif, Muhammad, Journal of Applied Mathematics, 2013

- An unconditional existence result for elastohydrodynamic piezoviscous lubrication
problems with Elrod-Adams model of cavitation

Bayada, Guy, Chupin, Laurent, and Grec, Bérénice, Differential and Integral Equations, 2008 - Mathematical Justification of a Shallow Water Model

Bresch, Didier and Noble, Pascal, Methods and Applications of Analysis, 2007 - MHD Thin Film Flows of a Third Grade Fluid on a Vertical Belt with Slip Boundary Conditions

Gul, Taza, Shah, Rehan Ali, Islam, Saeed, and Arif, Muhammad, Journal of Applied Mathematics, 2013 - Viscous Dissipation Effects on the Motion of Casson Fluid over an Upper Horizontal Thermally Stratified Melting Surface of a Paraboloid of Revolution: Boundary Layer Analysis

Ajayi, T. M., Omowaye, A. J., and Animasaun, I. L., Journal of Applied Mathematics, 2017 - An Asymptotic Analysis of Linear Interfacial Motion

Wang, Jin, Methods and Applications of Analysis, 2007 - Symmetries and Conservation Laws of Plates and Shells Interacting with Fluid Flow

Djondjorov, Peter A. and Vassilev, Vassil M., , 2005 - Influence of temperature-dependent viscosity on the MHD Couette flow of dusty fluid with heat transfer

Attia, Hazem A., Differential Equations and Nonlinear Mechanics, 2006 - Temperature Dependent Viscosity of a Third Order Thin Film Fluid Layer on a Lubricating Vertical Belt

Gul, T., Islam, S., Shah, R. A., Khan, I., and Dennis, L. C. C., Abstract and Applied Analysis, 2014 - Series Solutions of Lifting and Drainage Problems of a Nonisothermal Modified Second Grade Fluid Using a Vertical Cylinder

Farooq, M., Rahim, M. T., Islam, S., and Arif, M., Journal of Applied Mathematics, 2014 - On Convective Dusty Flow Past a Vertical Stretching Sheet with Internal Heat Absorption

Nandkeolyar, Raj and Sibanda, Precious, Journal of Applied Mathematics, 2013