On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow

Bellout, Hamid; Bloom, Frederick

doi:10.1090/S0033-569X-2013-01330-9

Authors: Hamid Bellout and Frederick Bloom
Journal: Quart. Appl. Math. 71 (2013), 773-785
MSC (2010): Primary 76A05, 35K52
DOI: https://doi.org/10.1090/S0033-569X-2013-01330-9
Published electronically: September 4, 2013
MathSciNet review: 3136995
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The higher-order boundary conditions associated with the flow of an incompressible, nonlinear, bipolar viscous fluid in a bounded domain are derived; these boundary conditions differ from the various ad hoc sets of higher-order boundary conditions that have been used in work involving fluid dynamics models employing higher-order gradients of the velocity field. The derivation presented is based on a principle of virtual work and some deep results of Heron on higher-order traces of divergence-free vector fields.

References

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

Bleustein, J. L. and A. E. Green, Dipolar Fluids, Int. J. Eng. Sci., vol. 5 (1967), 323–340.

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

Hamid Bellout, Frederick Bloom, and Jindřich Nečas, Phenomenological behavior of multipolar viscous fluids, Quart. Appl. Math. 50 (1992), no. 3, 559–583. MR 1178435, DOI https://doi.org/10.1090/qam/1178435

Shinbrot, M., Lectures on Fluid Mechanics, Gordon and Breach Pub., N.Y. (1973).

David Ruelle, The turbulent fluid as a dynamical system, New perspectives in turbulence (Newport, RI, 1989) Springer, New York, 1991, pp. 123–138. MR 1126934

O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Revised English edition, Gordon and Breach Science Publishers, New York-London, 1963. Translated from the Russian by Richard A. Silverman. MR 0155093

Ladyzhenskaya, O. A., New Equations for the Description of the Viscous Incompressible Fluids and Solvability in the Large of the Boundary Value Problems for Them, in Boundary Value Problems of Mathematical Physics V, AMS, Providence, R.I. (1970).

Shmuel Kaniel, On the initial value problem for an incompressible fluid with nonlinear viscosity, J. Math. Mech. 19 (1969/1970), 681–707. MR 0253629

Qiang Du and Max D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow, J. Math. Anal. Appl. 155 (1991), no. 1, 21–45. MR 1089323, DOI https://doi.org/10.1016/0022-247X%2891%2990024-T

Lions, J. L., Quelques methodes de resolution des Problemes aux Limites Nonlineaires, Dunrod, Paris (1969).

Yuh-Roung Ou and S. S. Sritharan, Analysis of regularized Navier-Stokes equations. I, II, Quart. Appl. Math. 49 (1991), no. 4, 651–685, 687–728. MR 1134747

Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312

B. Héron, Quelques propriétés des applications de trace dans des espaces de champs de vecteurs à divergence nulle, Comm. Partial Differential Equations 6 (1981), no. 12, 1301–1334 (French, with English summary). MR 640159, DOI https://doi.org/10.1080/03605308108820212

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

Hamid Bellout and Frederick Bloom, On the uniqueness of plane Poiseuille solutions of the equations of incompressible dipolar viscous fluids, Internat. J. Engrg. Sci. 31 (1993), no. 11, 1535–1549. MR 1234135, DOI https://doi.org/10.1016/0020-7225%2893%2990030-X

Hamid Bellout and Frederick Bloom, Existence and asymptotic stability of time-dependent Poiseuille flows of isothermal bipolar fluids, Appl. Anal. 50 (1993), no. 1-2, 115–130. MR 1281207, DOI https://doi.org/10.1080/00036819308840188

Frederick Bloom, Bubble stability in a class of non-Newtonian fluids with shear dependent viscosities, Internat. J. Non-Linear Mech. 37 (2002), no. 3, 527–539. MR 1877242, DOI https://doi.org/10.1016/S0020-7462%2801%2900029-4

Frederick Bloom and Wenge Hao, Steady flows of nonlinear bipolar viscous fluids between rotating cylinders, Quart. Appl. Math. 53 (1995), no. 1, 143–171. MR 1315453, DOI https://doi.org/10.1090/qam/1315453

Hamid Bellout and Jindrich Nečas, The exterior problem in the plane for a non-Newtonian incompressible bipolar viscous fluid, Rocky Mountain J. Math. 26 (1996), no. 4, 1245–1260. MR 1447585, DOI https://doi.org/10.1216/rmjm/1181071986

Hamid Bellout and Sheryl L. Wills, Perturbation of the domain and regularity of the solutions of the bipolar fluid flow equations in polygonal domains, Internat. J. Non-Linear Mech. 30 (1995), no. 3, 235–262. MR 1336912, DOI https://doi.org/10.1016/0020-7462%2894%29E0023-3

Hamid Bellout, Frederick Bloom, and Jindřich Nečas, Solutions for incompressible non-Newtonian fluids, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 8, 795–800 (English, with English and French summaries). MR 1244433

Hamid Bellout, Frederick Bloom, and Jindřich Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations 19 (1994), no. 11-12, 1763–1803. MR 1301173, DOI https://doi.org/10.1080/03605309408821073

Hamid Bellout, Frederick Bloom, and Jindřich Nečas, Existence, uniqueness, and stability of solutions to the initial-boundary value problem for bipolar viscous fluids, Differential Integral Equations 8 (1995), no. 2, 453–464. MR 1296135

Frederick Bloom and Wenge Hao, Regularization of a non-Newtonian system in an unbounded channel: existence and uniqueness of solutions, Nonlinear Anal. 44 (2001), no. 3, Ser. A: Theory Methods, 281–309. MR 1817094, DOI https://doi.org/10.1016/S0362-546X%2899%2900264-3

Hamid Bellout, Frederick Bloom, and Jindřich Nečas, Bounds for the dimensions of the attractors of non-linear bipolar viscous fluids, Asymptotic Anal. 11 (1995), no. 2, 131–167. MR 1350404

Frederick Bloom, Lower semicontinuity of the attractors of non-Newtonian fluids, Dynam. Systems Appl. 4 (1995), no. 4, 567–579. MR 1365839

Frederick Bloom, Attractors of non-Newtonian fluids, J. Dynam. Differential Equations 7 (1995), no. 1, 109–140. MR 1321708, DOI https://doi.org/10.1007/BF02218816

Frederick Bloom, Linearized stability of the viscous incompressible bipolar equations, Nonlinear Anal. 27 (1996), no. 9, 1013–1030. MR 1406276, DOI https://doi.org/10.1016/0362-546X%2895%2900115-C

Frederick Bloom, Attractors of bipolar and non-Newtonian viscous fluids, World Congress of Nonlinear Analysts ’92, Vol. I–IV (Tampa, FL, 1992) de Gruyter, Berlin, 1996, pp. 583–596. MR 1389107

Frederick Bloom and Wenge Hao, Regularization of a non-Newtonian system in an unbounded channel: existence of a maximal compact attractor, Nonlinear Anal. 43 (2001), no. 6, Ser. A: Theory Methods, 743–766. MR 1808208, DOI https://doi.org/10.1016/S0362-546X%2899%2900232-1

Frederick Bloom and Wenge Hao, Inertial manifolds of incompressible, nonlinear bipolar viscous fluids, Quart. Appl. Math. 54 (1996), no. 3, 501–539. MR 1402407, DOI https://doi.org/10.1090/qam/1402407

Bellout, H. and F. Bloom, Incompressible Bipolar and non-Newtonian Viscous Fluid Flow, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel-Boston-Berlin, in press.

Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383

R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech. Anal. 17 (1964), 85–112. MR 169425, DOI https://doi.org/10.1007/BF00253050