Piezo-viscous flows over an inclined surface
Authors:
G. Saccomandi and L. Vergori
Journal:
Quart. Appl. Math. 68 (2010), 747-763
MSC (2000):
Primary 76A20; Secondary 76A02
DOI:
https://doi.org/10.1090/S0033-569X-2010-01202-2
Published electronically:
October 15, 2010
MathSciNet review:
2761242
Full-text PDF Free Access
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Additional Information
Abstract: Stokes was the first to recognize that the viscosity of many fluids varies significantly with pressure. Later, several experimental studies showed that such a variation may be exponential. Here, by using the lubrication theory as revised by Rajagopal and Szeri, we study the flow of a piezo-viscous fluid down an incline in various flow regimes.
References
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- E.C. Andrade, Viscosity of liquids, Nature, 125 (1930), 309-310.
- S. Bair, P. Kottke, Pressure-viscosity relationship for elastohydrodynamics, Tribology Trans., 46 (2003), 289–295.
- C. Barus, Isotherms, isopiestics and isometrics relative to viscosity, Amer. J. Sci. 45 (1893), 87-96.
- M.P. Brenner, Instability mechanism at driven contact lines, Phys. Rev. E, Vol. 47, No. 6 (1993), 4597–4599.
- J. Hron, J. Malek, K. R. Rajagopal: Simple flows of fluids with pressure-dependent viscosities. Proc. R. Soc. London A, 457 (2001) 1-20.
- H.M. Laun, Pressure dependent viscosity and dissipative heating in capillary rheometry of polymer melts, Rheol. Acta, $\mathbf {42}$, 295-308 (2003).
- Josef Málek, Jindřich Nečas, and K. R. Rajagopal, Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. Ration. Mech. Anal. 165 (2002), no. 3, 243–269. MR 1941479, DOI https://doi.org/10.1007/s00205-002-0219-4
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- J. Murali Krishnan, K.R. Rajagopal, Review of the uses and modeling of bitumen from ancient to modern times, Appl. Mech. Rev., 56 (2003), 149-214.
- A. Oron, S.H. Davis, S.G. Bankoff, Long-scale evolution of thin liquid films, Reviews of Modern Physics, Vol. 69, No. 3 (1997), 931–980.
- C.A. Perazzo, J. Gratton, Thin film of non-Newtonian fluid on an incline, Phys. Rev. E 67, 016307 (2003), 1–6.
- Sharat C. Prasad and K. R. Rajagopal, Flow of a fluid with pressure dependent viscosity due to a boundary that is being stretched, Appl. Math. Comput. 173 (2006), no. 1, 50–68. MR 2203373, DOI https://doi.org/10.1016/j.amc.2005.02.043
- K. R. Rajagopal, On implicit constitutive theories for fluids, J. Fluid Mech. 550 (2006), 243–249. MR 2263984, DOI https://doi.org/10.1017/S0022112005008025
- K. R. Rajagopal, A semi-inverse problem of flows of fluids with pressure-dependent viscosities, Inverse Probl. Sci. Eng. 16 (2008), no. 3, 269–280. MR 2427495, DOI https://doi.org/10.1080/17415970701529205
- K. R. Rajagopal and G. Saccomandi, Unsteady exact solution for flows of fluids with pressure-dependent viscosities, Math. Proc. R. Ir. Acad. 106A (2006), no. 2, 115–130. MR 2266820, DOI https://doi.org/10.3318/PRIA.2006.106.2.115
- K. R. Rajagopal, G. Saccomandi, and L. Vergori, On the Oberbeck-Boussinesq approximation for fluids with pressure dependent viscosities, Nonlinear Anal. Real World Appl. 10 (2009), no. 2, 1139–1150. MR 2474287, DOI https://doi.org/10.1016/j.nonrwa.2007.12.003
- K. R. Rajagopal, G. Saccomandi, and L. Vergori, Stability analysis of the Rayleigh-Bénard convection for a fluid with temperature and pressure dependent viscosity, Z. Angew. Math. Phys. 60 (2009), no. 4, 739–755. MR 2520609, DOI https://doi.org/10.1007/s00033-008-8062-6
- K. R. Rajagopal and A. Z. Szeri, On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2003), no. 2039, 2771–2786. MR 2015989, DOI https://doi.org/10.1098/rspa.2003.1145
- R.N.J. Saal, G. Koens, Investigations into plastic properties of asphaltic bitumen, Physics, 7 (1933), 408-412.
- R.N.J. Saal, J.W.A. Labout, Rheological properties of asphalt, in: F.R. Eirich (Ed.) Rheology: Theory and Applications, Academic Press, New York, 363-400, 1958.
- G.G. Stokes, On the theories of the internal friction of fluids in motion, and motion of elastic solids, Trans. Camb. Phil. Soc. 8 (1845), 287-305.
- Shankar C. Subramanian and K. R. Rajagopal, A note on the flow through porous solids at high pressures, Comput. Math. Appl. 53 (2007), no. 2, 260–275. MR 2325232, DOI https://doi.org/10.1016/j.camwa.2006.02.023
- A.Z. Szeri, Fluid Film Lubrication: Theory and Design, Cambridge University Press, 1998.
- Macherla Vasudevaiah and Kumbakonam R. Rajagopal, On fully developed flows of fluids with a pressure dependent viscosity in a pipe, Appl. Math. 50 (2005), no. 4, 341–353. MR 2151461, DOI https://doi.org/10.1007/s10492-005-0027-x
References
- C. Ancey, Plasticity and geophysical flows: A review, J. Non-Newtonian Fluid Mech. 142 (2007), 4–35.
- E.C. Andrade, Viscosity of liquids, Nature, 125 (1930), 309-310.
- S. Bair, P. Kottke, Pressure-viscosity relationship for elastohydrodynamics, Tribology Trans., 46 (2003), 289–295.
- C. Barus, Isotherms, isopiestics and isometrics relative to viscosity, Amer. J. Sci. 45 (1893), 87-96.
- M.P. Brenner, Instability mechanism at driven contact lines, Phys. Rev. E, Vol. 47, No. 6 (1993), 4597–4599.
- J. Hron, J. Malek, K. R. Rajagopal: Simple flows of fluids with pressure-dependent viscosities. Proc. R. Soc. London A, 457 (2001) 1-20.
- H.M. Laun, Pressure dependent viscosity and dissipative heating in capillary rheometry of polymer melts, Rheol. Acta, $\mathbf {42}$, 295-308 (2003).
- J. Málek, J. Necas, K.R. Rajagopal: Global Analysis of the Flows of Fluids with Pressure-Dependent Viscosities, Arch. Rational Mech. Anal. 165 (2002), 243-269. MR 1941479 (2004h:76003)
- J. Málek, J. Necas, K.R. Rajagopal: Global Existence of Solutions for Flows of Fluids with Pressure and Shear Dependent Viscosities. Applied Mathematics Letters 15 (2002) 961-967. MR 1925921 (2003g:76032)
- M.J. Martín-Alfonso, F.J. Martínez-Boza, F.J. Navarro, M. Fernández, C. Gallegos, Pressure-temperature-viscosity relationship for heavy petroleum fractions, Fuel, 86 (2007), 227-233.
- J. Murali Krishnan, K.R. Rajagopal, Review of the uses and modeling of bitumen from ancient to modern times, Appl. Mech. Rev., 56 (2003), 149-214.
- A. Oron, S.H. Davis, S.G. Bankoff, Long-scale evolution of thin liquid films, Reviews of Modern Physics, Vol. 69, No. 3 (1997), 931–980.
- C.A. Perazzo, J. Gratton, Thin film of non-Newtonian fluid on an incline, Phys. Rev. E 67, 016307 (2003), 1–6.
- S.C. Prasad, K.R. Rajagopal, Flow of a fluid with pressure dependent viscosity due to a boundary that is being stretched, Applied Mathematics and Computation 173 (2006), 50–68. MR 2203373 (2006h:76007)
- K.R. Rajagopal, On implicit constitutive theories for fluids, J. Fluid Mech. 550 (2006), 243–249. MR 2263984 (2007e:76004)
- K.R. Rajagopal, A semi-inverse problem of flows of fluids with pressure-dependent viscosities, Inverse Probl. Sci. Eng. 16 No. 3 (2008), 269–280. MR 2427495 (2009h:76062)
- K.R. Rajagopal, G. Saccomandi, Unsteady exact solution for flows of fluids with pressure-dependent viscosities, Mathematical Proceedings of the Royal Irish Academy 106A (2) (2006), 115–130. MR 2266820 (2008c:35255)
- K.R. Rajagopal, G. Saccomandi, L. Vergori,On the Oberbeck-Boussinesq approximation in fluids with pressure-dependent viscosities, Nonlinear Analysis: Real World Applications 10 (2009), No. 2 1139–1150. MR 2474287
- K.R. Rajagopal, G. Saccomandi, L. Vergori, Stability analysis of the Rayleigh-Bénard convection for a fluid with temperature and pressure dependent viscosity, Z. Angew. Math. Phys. 60 (2009), 739–755. MR 2520609 (2010g:76058)
- K.R. Rajagopal, A.Z. Szeri, On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication, Proc. R. Soc. A 459 (2003), 2771–2786. MR 2015989 (2004i:76069)
- R.N.J. Saal, G. Koens, Investigations into plastic properties of asphaltic bitumen, Physics, 7 (1933), 408-412.
- R.N.J. Saal, J.W.A. Labout, Rheological properties of asphalt, in: F.R. Eirich (Ed.) Rheology: Theory and Applications, Academic Press, New York, 363-400, 1958.
- G.G. Stokes, On the theories of the internal friction of fluids in motion, and motion of elastic solids, Trans. Camb. Phil. Soc. 8 (1845), 287-305.
- S.C. Subramanian, K.R. Rajagopal, A note on the flow through porous solids at high pressures, Comput. Math. Appl. 53 (2007), 260–275. doi:10.1016/j.camwa.2006.02.023. MR 2325232
- A.Z. Szeri, Fluid Film Lubrication: Theory and Design, Cambridge University Press, 1998.
- M. Vasudevaiah, K.R. Rajagopal, On fully developed flows of fluids with a pressure dependent viscosity in a pipe, Applications of Mathematics 50, No. 4 (2005), 341–353. MR 2151461 (2006b:76002)
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Additional Information
G. Saccomandi
Affiliation:
Dipartimento di Ingegneria Industriale, Università di Perugia, Via G. Duranti, 06125 Italy
Email:
saccomandi@mec.dii.unipg.it
L. Vergori
Affiliation:
Dipartimento di Matematica, Università del Salento, Strada Prov. Lecce-Arnesano, 73100 Lecce, Italy
Email:
luigi.vergori@unile.it
Keywords:
Lubrication approximation,
pressure-dependent viscosity,
Barus’ law.
Received by editor(s):
April 1, 2009
Published electronically:
October 15, 2010
Article copyright:
© Copyright 2010
Brown University