On steady flows of fluids with pressure-and shear-dependent viscosities

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 461, Issue 2055, 2005, Pages 651-670

  Franta, M ^a   Malek J ^a   Rajagopal, K R ^b  

^a CHARLES UNIVERSITY   (Czech Republic)

^b Texas AandM University   (United States)

Author keywords

Dirichlet (no slip) boundary condition; Existence; Incompressible fluid; Pressure dependent viscosity; Shear dependent viscosity; Weak solution

Indexed keywords

ARTICLE; COMPRESSIVE STRENGTH; COMPUTATIONAL FLUID DYNAMICS; FORCE; HYDRODYNAMICS; MEMBRANE FLUIDITY; PARTICLE SIZE; PRESSURE; SHEAR STRENGTH; SHEAR STRESS; STEADY STATE; VISCOELASTICITY;

EID: 17144393380     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2004.1360     Document Type: Article

References (36)

1. Amrouche, C. & Girault, V. 1994 Decomposition of vector spaces and application to the Stokes problems in arbitrary dimension. Czech. Math. J. 44, 109-141.
2. Andrade, E. C. 1930 Viscosity of liquids. Nature 125, 309-310.
3. Antman, S. S. 1982 Material constraints in continuum mechanics. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 70, 256-264.
4. Antman, S. S. & Marlow, R. S. 1991 Material constraints, Lagrange multipliers, and compatibility. Applications to rod and shell theories. Arch. Ration. Mech. Analysis 116, 257-299.
5. Bogovskii, M. E. 1979 Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 5, 1037-1040.
6. Bridgman, P. W. 1931 The physics of high pressure. Macmillan.
7. Cutler, W. G., McMickle, R. H., Webb, W. & Schiessler, R. W. 1958 Study of the compressions of several high molecular weight hydrocarbons. J. Chem. Phys. 29, 727-740.
8. Frehse, J., Málek, J. & Steinhauer, M. 1997 An existence result for fluids with shear dependent viscosity - steady flows. Nonlin. Analysis 30, 3041-3049.
9. Frehse, J., Málek, J. & Steinhauer, M. 2003 On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Analysis 34, 1064-1083.
10. Fuchs, M. & Seregin, G. A. 1997 Some remarks on non-Newtonian fluids including nonconvex perturbations of the Bingham and Powel-Eyring model for visco-plastic fluids. Math. Models Meth. Appl. Sci. 7, 405-433.
11. Fuchs, M. & Seregin, G. A. 2000 Variational methods for problems from plasticity theory and for generalized Newtonian fluids. Lecture Notes in Mathematics 1749. Springer.
12. Gauss, C. F. 1829 Über ein neues Grundgesetz der Mechanik. J. Grelle 5, 367-369.
13. Gazzola, F. 1997 A note on the evolution of Navier-Stokes equations with a pressure-dependent viscosity. Z. Angew. Math. Phys. 48, 760-773.
14. Gazzola, F. & Secchi, P. 1998 Some results about stationary Navier-Stokes equations with a pressure-dependent viscosity. In Navier-Stokes equations: theory and numerical methods (ed. R. Salvi), pp. 31-37. Longman.
15. Griest, E. M., Webb, W. & Schiessler, R. W. 1958 Effect of pressure on viscosity of higher hydrocarbons and their mixtures. J. Chem. Phys. 29, 711-720.
16. Hron, J., Málek, J. & Rajagopal, K. R. 2001 Simple flows of fluids with pressure dependent viscosities. Proc. R. Soc. A 457, 1603-1622.
17. Hron, J., Malek, J., Nečas, J. & Rajagopal, K. R. 2003 Numerical simulations and global existence of solutions of two dimensional flows of fluids with pressure and shear dependent viscosities. Math. Comput. Simulat. 61, 297-315.
18. Johnson, K. L. & Cameron, R. 1967 Shear behaviour of elastohydrodynamic oil films at high rolling contact pressures. Proc. Inst. Mech. Engrs 182, 307-319.
19. Johnson, K. L. & Greenwood, J. A. 1980 Thermal analysis of an Eyring fluid in elastohydrodynamic traction. Wear 61, 355-374.
20. Johnson, K. L. & Tevaarwerk, J. L. 1977 Shear behaviour of elastohydrodynamic oil films. Proc. R. Soc. A 356, 215-236.
21. 1,α-regularity of weak solutions to a class of nonlinear fluids in two dimensions - stationary Dirichlet problem. Zap. Nauchn. Semin. POMI 259, 89-121.
22. Kaplický, P., Málek, J. & Stará, J. 1999b On global existence of smooth two-dimensional steady flows for a class of non-Newtonian fluids under various boundary conditions. In Applied non-linear analysis (ed. A. Sequeira, H. Beirao da Veiga & J. H. Videman), pp. 213-229. New York: Kluwer Academic/Plenum.
23. Ladyzhenskaya, O. A. 1967 On some new equations describing dynamics of incompressible fluids and on global solvability of boundary value problems to these equations. Tr. Mat. Inst. Steklova 102, 85-104.
24. Ladyzhenskaya, O. A. 1968 On some modifications of the Navier-Stokes eqautions for large gradients of velocity. Zap. Nauchn. Semin. LOMI 7, 126-154.
25. Ladyzhenskaya, O. A. 1969 The mathematical theory of viscous incompressible flow. New York: Gordon & Breach.
26. Lagrange, J. L. 1787 Mécanique analytique. Paris: Mme Ve Courcier. (Transl. A. Boissonade & V. N. Vagliente, 1997 Kluwer Academic.)
27. Lions, J. L. 1969 Quelques méthodes de résolution des problèmes aux limites non linéares. Dunod.
28. Málek, J., Nečas, J., Rokyta, M. & Růž ička, M. 1996 Weak and measure-valued solutions to evolutionary PDEs. London: CRC Press.
29. Málek, J., Nečas, J. & Rajagopal, K. R. 2002a Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Analysis 165, 243-269.
30. Málek, J., Nečas, J. & Rajagopal, K. R. 2002b Global existence of solutions for flows of fluids with pressure and shear dependent viscosities. Appl. Math. Lett. 15, 961-967.
31. Růžička, M. 1997 A note on steady flow of fluids with shear dependent viscosity. Nonlin. Analysis 30, 3029-3039.
32. Rajagopal, K. R. 2003 On implicit constitutive theories. Applic. Math. 48, 279-319.
33. Reiner, M. 1945 A mathematical theory of dilatancy. Am. J. Math. 67, 350-362.
34. Renardy, M. 1986 Some remarks on the Navier-Stokes equations with a pressure-dependent viscosity. Commun. PDEs 11, 779-793.
35. Reynolds, O. 1886 On the theory of lubrication and its application to Mr Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil. Phil. Trans. R. Soc. Lond. 177, 157-234.
36. Stokes, G. G. 1845 On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Camb. Phil. Soc. 8, 287-305.