Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T02:37:09.664Z Has data issue: false hasContentIssue false
  • English
  • Français

A scaling approach to the derivation of hydrodynamic boundary conditions

Published online by Cambridge University Press:  25 September 2008

TIEZHENG QIAN ,
CHUNYIN QIU  and
PING SHENG
TIEZHENG QIAN
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
CHUNYIN QIU
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
PING SHENG
Affiliation:
Department of Physics and Institute of Nano Science and Technology, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
Get access Rights & Permissions [Opens in a new window]

Abstract

We show hydrodynamic boundary conditions to be the inherent consequence of the Onsager principle of minimum energy dissipation, provided the relevant effects of the wall potential appear within a thin fluid layer next to the solid wall, denoted the surface layer. The condition that the effect of the surface layer on the bulk hydrodynamics must be independent of its thickness h is shown to imply a set of consistent ‘scaling relationships’ between h and the surface-layer variables/parameters. The use of the scaling relations, in conjunction with the surface-layer equations of motion derived from the Onsager principle, directly leads to the hydrodynamic boundary conditions. We demonstrate the surface-layer scaling process both physically and mathematically, and relate the parameters of the boundary conditions to those in the surface-layer equations of motion. In spatial regions outside the surface layer, equivalence between the use of surface-layer dynamics and boundary conditions is numerically demonstrated for Couette flows. As an application of the present approach, we derive the liquid-crystal hydrodynamic boundary conditions in which the rotational and translational dynamics are coupled.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 611 , 25 September 2008 , pp. 333 - 364
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barrat, J.-L. & Bocquet, L. 1999 Influence of wetting properties on hydrodynamic boundary conditions at a fluid/solid interface. Faraday Discuss. 112, 119–128.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bocquet, L. & Barrat, J.-L. 2007 Flow boundary conditions from nano- to micro-scales. Soft Matter 3, 685–693.CrossRefGoogle ScholarPubMed
Briant, A. J. & Yeomans, J. M. 2004 Lattice Boltzmann simulations of contact line motion. II. Binary fluids. Phys. Rev. E 69, 031603.Google ScholarPubMed
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–34.Google Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
Chen, H. Y., Jasnow, D. & Vinals, J. 2000 Interface and contact line motion in a two phase fluid under shear flow. Phys. Rev. Lett. 85, 1686–1689.Google Scholar
Dussan, V., , E. B. 1976 The moving contact line: the slip boundary condition. J. Fluid Mech. 77, 665–684.CrossRefGoogle Scholar
Dussan, V., , E. B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.CrossRefGoogle Scholar
Dussan, V., , E. B. & Davis, S. H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65, 71–95.Google Scholar
Ericksen, J. L. 1960 Anisotropic fluids. Arch. Rat. Mech. Anal. 4, 231–237.Google Scholar
Frank, F. C. 1958 On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28.CrossRefGoogle Scholar
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.Google Scholar
de Gennes, P. G. & Prost, J. 1993 The Physics of Liquid Crystals. Clarendon.Google Scholar
de Groot, S. R. & Mazur, P. 1984 Non-Equilibrium Thermodynamics. Dover.Google Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101.Google Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.Google Scholar
Kirkwood, J. G. & Buff, F. P. 1949 The statistical mechanical theory of surface tension. J. Chem. Phys. 17, 338–343.CrossRefGoogle Scholar
Koplik, J., Banavar, J. R. & Willemsen, J. F. 1988 Molecular dynamics of Poiseuille flow and moving contact lines. Phys. Rev. Lett. 60, 1282–1285.Google Scholar
Leslie, F. M. 1968 Some constitutive equations for liquid crystals. Arch Rat. Mech. Anal. 28, 265–283.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Google Scholar
Navier, C. L. M. H. 1823 Memoire sur les lois du movement des fluides. Mem. l'Acad. R. Sci. l'Inst. France 6, 389–440.Google Scholar
Onsager, L. 1931 a Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426.Google Scholar
Onsager, L. 1931 b Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265–2279.Google Scholar
Oseen, C. W. 1933 The theory of liquid crystals. Trans. Faraday Soc. 29, 883–889.Google Scholar
Parodi, O. J. 1970 Stress tensor for a nematic liquid crystal. J. Phys. (Paris) 31, 581–584.CrossRefGoogle Scholar
Pismen, L. M. & Pomeau, Y. 2000 Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics. Phys. Rev. E 62, 2480–2492.Google Scholar
Qian, T. Z., Wang, X. P. & Sheng, P. 2003 Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E 68, 016306.Google Scholar
Qian, T. Z., Wang, X. P. & Sheng, P. 2004 Power-law slip profile of the moving contact line in two-phase immiscible flows. Phys. Rev. Lett. 93, 094501.CrossRefGoogle ScholarPubMed
Qian, T. Z., Wang, X. P. & Sheng, P. 2006 A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360.CrossRefGoogle Scholar
Rapini, A. & Papoular, M. 1969 Distorsion d'une lamelle nematique sous champ magnetique, conditions d'ancrage aux parois. J. Phys. (Paris) Colloq. 30, C4-54–C4-56.CrossRefGoogle Scholar
Rayleigh, Lord 1873 Some general theorems relating to vibrations. Proc. Math. Soc. Lond. 4, 357–368.Google Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34, 977–992.Google Scholar
Thompson, P. A. & Robbins, M. O. 1989 Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63, 766–769.CrossRefGoogle ScholarPubMed
Yue, P. T., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317.CrossRefGoogle Scholar
Zocher, H. 1933 The effect of a magnetic field on the nematic state. Trans. Faraday Soc. 29, 945–957.Google Scholar