Piezo-viscous flows over an inclined surface

Saccomandi, G.; Vergori, L.

doi:10.1090/S0033-569X-2010-01202-2

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

Abstract | References | Similar Articles | 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

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).

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

J. Málek, J. Nečas, and K. R. Rajagopal, Global existence of solutions for flows of fluids with pressure and shear dependent viscosities, Appl. Math. Lett. 15 (2002), no. 8, 961–967. MR 1925921, DOI https://doi.org/10.1016/S0893-9659%2802%2900070-8

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.

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