Chapter 7 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

Volumn 4, Issue , 2007, Pages 407-444

  Malek J ^a  

^a CHARLES UNIVERSITY   (Czech Republic)

Author keywords

implicit constitutive theory; incompressible fluid; pressure dependent viscosity; shear rate dependent viscosity

Indexed keywords

EID: 61749091323     PISSN: 18745792     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5792(07)80011-5     Document Type: Chapter

References (59)

1. Amrouche Ch., and Girault V. Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44 119 (1994) 109-140 (1)
2. Anand M., and Rajagopal K.R. A note on the flows of inhomogeneous fluids with shear-dependent viscosities. Archives of Mechanics 57 (2005) 417-428
3. Andrade C. Viscosity of liquids. Nature 125 (1930) 309-310
4. Bair S., and Kottke P. Pressure-viscosity relationships for elastohydrodynamics. Tribology transactions 46 (2003) 289-295
5. Bair S., and Winer W.O. The high pressure high shear stress rheology of liquid lubricants. J. Tribology 114 (1992) 1-13
6. Barus C. Isothermals isopiestics and isometrics relative to viscosity. American Jour. Sci. 45 (1893) 87-96
7. Bendler J.T., Fontanella J.J., and Schlessinger M.F. A new Vogel-like law: ionic conductivity dialectric relaxation and viscosity near the glass transition. Phys. Rev. Lett. 87 19 (2001) 1955
8. Bogovskii{ogonek} M.E. Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 248 5 (1979) 1037-1040
9. Boussinesq J. Théorie Analytique de la Chaleur (1903), Gauthier-Villars
10. Bridgman P.W. The Physics of High Pressure (1931), MacMillan, New York
11. Bulíček M. Navier's slip and evolutionary Navier-Stokes systems with pressure, shear-rate and temperature dependent viscosity. PhD thesis, Faculty of Mathematics and Physics (2006), Charles University, Czech Republic
12. Bulíček M., Málek J., and Rajagopal K.R. Navier's slip and evolutionary Navier-Stokes systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J. (2006) (to appear)
13. Cutler W.G., McMickle R.H., Webb W., and Schiessler R.W. Study of the compressions of several high molecular weight hydrocarbons. J. Chem. Phys. 29 (1958) 727-740
14. Dowson D., and Higginson G.R. Elastohydrodynamic Lubrication, The Fundamentals of Roller and Gear Lubrication (1966), Pergamon, Oxford
15. Feireisl E., and Málek J. On the Navier-Stokes equations with temperature-dependent transport coefficients. Differential Equations and Nonlinear Mechanics (2006) Article ID 90616, 14 pages (electronic)
16. Franta M., Málek J., and Rajagopal K.R. On steady flows of fluids with pressure-and shear-dependent viscosities. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 2055 (2005) 651-670
17. Gazzola F. A note on the evolution of Navier-Stokes equations with a pressure-dependent viscosity. Z. Angew. Math. Phys. 48 5 (1997) 760-773
18. Gazzola F., and Secchi P. Some results about stationary Navier-Stokes equations with a pressure-dependent viscosity. In: Salvi R. (Ed). Navier-Stokes Equations: Theory and Numerical Methods. volume 388 of Pitman Res. Notes Math. Ser. (1998), Longman, Harlow 31-37
19. Griest E.M., Webb W., and Schiessler R.W. Effect of pressure on viscosity of higher hydrocarbons and their mixtures. J. Chem. Phys. 29 (1958) 711-720
20. Hron J., Málek J., Nečas J., and Rajagopal K.R. Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure-and shear-dependent viscosities. Mathematics and Computers in Simulation 61 (2003) 297-315
21. Hron J., Málek J., and Rajagopal K.R. Simple flows of fluids with pressure dependent viscosities. Proc. R. Soc. A 457 (2001) 1603-1622
22. Huy N.D., and Stará J. On existence and regularity of solutions to a class of generalized stationary Stokes problem. Comment. Math. Univ. Carolin. 47 (2006) 241-264
23. Irwing J.B., and Barlow A.J. An automatic high pressure viscometer. J. Phys. E 4 (1971) 232-236
24. Johnson K.L., and Cameron R. Shear behaviour of elastohydrodynamic oil films at high rolling contact pressures. Proc. Instn. Mech. Engrs. 182 14 (1967) 307-319
25. Johnson K.L., and Greenwood J.A. Thermal analysis of an Eyring fluid in elastohydrodynamic traction. Wear 61 (1980) 355-374
26. Johnson K.L., and Tevaarwerk J.L. Shear behaviour of elastohydrodynamic oil films. Proc. R. Soc. Lond. A 356 (1977) 215-236
27. Kannan K., and Rajagopal K.R. A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid. Mathematics and Mechanics of Solids 9 (2004) 37-59
28. Málek J., Mingione G., and Stará J. Fluids with pressure dependent viscosity: partial regularity of steady flows. In: Dumortier F., Broer H., Mawhin J., Vanderbauwhede A., and Verduyn Lunel S. (Eds). EQUADIFF 2003 Proceedings of the International Conference on Differential Equations (2005), World Scientific, Singapore 380-386
29. Málek J., Mingione G., and Stará J. Fluids with pressure dependent viscosity: partial regularity of steady flows (2007) in preparation
30. Málek J., Nečas J., and Rajagopal K.R. Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165 3 (2002) 243-269
31. Málek J., and Rajagopal K.R. Mathematical issues concerning the Navier-Stokes equations and some of their generalizations. In: Dafermos C., and Feireisl E. (Eds). Evolutionary Equations. Handbook of Differential Equations vol. I (2005), Elsevier/North Holland, Amsterdam 371-459
32. Málek J., and Rajagopal K.R. Incompressible rate type fluids with pressure and shear-rate dependent material moduli. Nonlinear Analysis: Real World Applications 8 (2007) 156-157
33. Málek J., and Rajagopal K.R. On the modeling of inhomogeneous incompressible fluid-like bodies. Mechanics of Materials 38 (2006) 233-242
34. Oberbeck A. Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem. VII (1879) 271-292
35. Paluch M., Dendzik Z., and Rzoska S.J. Scaling of high-pressure viscosity data in low molecular-weight glass-forming liquids. Phys. Rev. B 60 (1999) 2979-2982
36. Rajagopal K.R. On implicit constitutive theories. Appl. Math. 48 (2003) 279-319
37. Rajagopal K.R. On implicit constitutive theories for fluids. Journal of Fluid Mechanics 550 (2006) 243-249
38. Rajagopal K.R. Mechanics of non-newtonian fluids. In: Galdi G.P., and Nečas J. (Eds). Recent Developments in Theoretical Fluid Mechanics. volume 291 of Pitman Research Notes in Mathematics Series (1993), Longman Scientific & Technical, Harlow, Essex 129-162
39. Rajagopal K.R. Remarks on the notion of pressure and implicit constitutive theories. Appl. Math. (2007) to appear
40. Rajagopal K.R., Ružička M., and Srinivasa A.R. On the Oberbeck-Boussinesq approximation. Math. Models Methods Appl. Sci. 6 (1996) 1157-1167
41. Rajagopal K.R., and Srinivasa A.R. A thermodynamic framework for rate type fluid models. J. Non-Newtonian Fluid Mech 88 (2000) 207-227
42. Rajagopal K.R., and Srinivasa A.R. On the nature of constraints for continua undergoing dissipative processes. Proc. R. Soc. A 461 (2005) 2785-2795
43. Rajagopal K.R., and Szeri A.Z. On an incosistency in the derivation of the equations of elastohydrodynamic lubrication. Proc. R. Soc. A 459 (2003) 2771-2786
44. Rajagopal K.R., and Tao L. Modeling of microwave drying process of aqueous dielectrics. Z. Angew. Math. Phys. 9 (2004) 305-312
45. Renardy M. Some remarks on the Navier-Stokes equations with a pressure-dependent viscosity. Comm. Partial Differential Equations 11 7 (1986) 779-793
46. Renardy M. Parallel shear flows of fluids with a pressure dependent viscosity. J. Non-Newtonian Fluid Mech. 114 (2003) 229-236
47. Roelands C.J.A. Correlation aspects of the viscosity-temeperature-pressure relationship of lubricating oils. PhD thesis (1966), Technische Hogeschool Delft, The Netherlands
48. Saal R.N.J., and Koens G. Investigation into the plastic properties of asphaltic bitumen. J. Inst. Pet. 19 (1933) 176-212
49. Schaeffer D.G. Instability in the evolution equations describing incompressible granular flow. J. Differential Equations 66 1 (1987) 19-50
50. Serrin J. Mathematical principles of classical fluid mechanics. Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell) (1959), Springer-Verlag, Berlin 125-263
51. Solonnikov V.A. Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations. Trudy Mat. Inst. Steklov. 70 (1964) 213-317
52. Spencer A.J.M. Theory of invariants. In: Eringen A.C. (Ed). Continuum Physics I (1971), Academic Press, New York 292-352
53. Stokes G.G. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc. 8 (1845) 287-305
54. Szeri A.Z. Fluid Film Lubrication: Theory and Design (1998), Cambridge University Press
55. Truesdell C.A. A program towards Rediscovering the Rational Mechanics of the Age of Reason. Archive for the history of the Exact Sciences 1 (1960) 1-30
56. Truesdell C.A. A First Course in Rational Continuum Mechanics. 2nd edition volume 1 (1991), Academic Press, New York
57. Truesdell C.A., and Noll W. The Non-Linear Field Theories of Mechanics, volume III, Handbuch der Physik (1965), Springer-Verlag, Berlin-Heidelberg-New York
58. Vasudevaiah M., and Rajagopal K.R. On fully developed flows of fluids with a pressure dependent viscosity in a pipe. Appl. Math. 50 (2005) 341-353
59. Vogel H. Die Bedeutung der Temperatur Abhangigkeit der Viscosität f.d. Beurteilung von Oelen. Zeit. Angew. Chem. 35 (1922) 561-563