Phenomenological behavior of multipolar viscous fluids
Authors:
Hamid Bellout, Frederick Bloom and Jindřich Nečas
Journal:
Quart. Appl. Math. 50 (1992), 559-583
MSC:
Primary 76A05; Secondary 35Q30, 76D05
DOI:
https://doi.org/10.1090/qam/1178435
MathSciNet review:
MR1178435
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Abstract: A constitutive theory is formulated to describe the flow of viscous fluids: the theory of multipolar fluids exhibits nonlinear relations among the stress tensors and spatial derivatives of the velocity of order greater than one and is compatible with the basic principles of continuum mechanics and thermodynamics. For an isothermal, incompressible, dipolar fluid the velocity profiles for various steady flows, such as proper Poiseuille flow in a circular pipe, are computed; the results are compared with the velocity profiles obtained from the steady Navier-Stokes equations through an application of the Prandtl boundary-layer theory.
- J. Nečas and M. Šilhavý, Multipolar viscous fluids, Quart. Appl. Math. 49 (1991), no. 2, 247–265. MR 1106391, DOI https://doi.org/10.1090/qam/1106391
J. L. Bleustein and A. G. Green, Dipolar fluids, Internat. J. Engrg. Sci. 5, 323–340 (1967)
- A. E. Green and R. S. Rivlin, Simple force and stress multipoles, Arch. Rational Mech. Anal. 16 (1964), 325–353. MR 182191, DOI https://doi.org/10.1007/BF00281725
- A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Arch. Rational Mech. Anal. 17 (1964), 113–147. MR 182192, DOI https://doi.org/10.1007/BF00253051
- J. Nečas, A. Novotný, and M. Šilhavý, Global solution to the compressible isothermal multipolar fluid, J. Math. Anal. Appl. 162 (1991), no. 1, 223–241. MR 1135273, DOI https://doi.org/10.1016/0022-247X%2891%2990189-7
- Jindřich Nečas, Antonín Novotný, and Miroslav Šilhavý, Global solution to the ideal compressible heat conductive multipolar fluid, Comment. Math. Univ. Carolin. 30 (1989), no. 3, 551–564. MR 1031872
- J. Nečas and A. Novotný, Some qualitative properties of the viscous compressible heat conductive multipolar fluid, Comm. Partial Differential Equations 16 (1991), no. 2-3, 197–220. MR 1104099, DOI https://doi.org/10.1080/03605309108820757
J. Nečas, A. Novotny, and M. šilhavý, Global solutions to the viscous compressible barotropic multipolar gas, preprint
- Hamid Bellout and Frederick Bloom, Steady plane Poiseuille flows of incompressible multipolar fluids, Internat. J. Non-Linear Mech. 28 (1993), no. 5, 503–518. MR 1241108, DOI https://doi.org/10.1016/0020-7462%2893%2990043-K
H. Bellout, F. Bloom, and J. Nečas, Existence, uniqueness, and stability of solutions to the initial and boundary value problem for bipolar viscous fluids, submitted
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
M. Shinbrot, Lectures on Fluid Mechanics, Gordon and Breach, New York, 1973
S. Goldstein, Modern Developments in Fluid Dynamics, vol. I, Oxford University Press, Oxford, 1938
J. Nečas and M. šilhavý, Multipolar viscous fluids, Quart. Appl. Math. XLIX, 247–265 (1991)
J. L. Bleustein and A. G. Green, Dipolar fluids, Internat. J. Engrg. Sci. 5, 323–340 (1967)
A. E. Green and R. S. Rivlin, Simple force and stress multipoles, Arch. Rational Mech. Anal. 16, 325–353 (1964)
---, Multipolar continuum mechanics, Arch. Rational Mech. Anal. 17, 113–147 (1964)
J. Nečas, A. Novotny, and M. šilhavý, Global solution to the compressible isothermal multipolar fluid, J. Math. Anal. Appl. 162, 223–241 (1991)
---, Global solution to the ideal compressible multipolar heat-conductive multipolar fluid, Comment Math. Univ. Carolina 30, 551–564 (1989)
J. Nečas and M. šilhavý, Some qualitative properties of the viscous compressible heat-conductive multipolar fluid, Comm. Partial Differential Equations 16, 197–220 (1991)
J. Nečas, A. Novotny, and M. šilhavý, Global solutions to the viscous compressible barotropic multipolar gas, preprint
H. Bellout and F. Bloom, Steady plane Poiseuille flows of incompressible multipolar fluids, submitted
H. Bellout, F. Bloom, and J. Nečas, Existence, uniqueness, and stability of solutions to the initial and boundary value problem for bipolar viscous fluids, submitted
J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires, Dunrod, Paris, 1969
M. Shinbrot, Lectures on Fluid Mechanics, Gordon and Breach, New York, 1973
S. Goldstein, Modern Developments in Fluid Dynamics, vol. I, Oxford University Press, Oxford, 1938
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Article copyright:
© Copyright 1992
American Mathematical Society