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Analysis of matching conditions at the boundary surface of a fluid-saturated porous solid and a bulk fluid: the use of Lagrange multipliers

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Abstract

The compatibility conditions matching macroscopic mechanical fields at the contact surface between a fluid-saturated porous solid and an adjacent bulk fluid are considered. The general form of balance equations at that discontinuity surface are analyzed to obtain the compatibility conditions for the tangent and normal components of the velocity and the stress vector fields. Considerations are based on the procedure similar to that used in the phenomenological thermodynamics for derivation of constitutive relations, where the entropy inequality and the concept of Lagrange multipliers are applied. This procedure made possible to derive the compatibility conditions for the viscous fluid flowing tangentially and perpendicularly to the boundary surface of the porous solid and to formulate the generalized form of the so called slip condition for the fluid velocity field, postulated earlier by Beavers and Joseph, J. Fluid. Mech. 30, 197–207 (1967).

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References

  1. Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid. Mech. 30, 197–207 (1967)

    Article  Google Scholar 

  2. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. J. Acoust. Soc. Am. 28, 2, 168–191 (1956)

    Article  MathSciNet  Google Scholar 

  3. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34 (1947)

    Article  Google Scholar 

  4. Cieszko, M., Kubik J.: On the compatibility conditions in the fluid-fluid saturated porous solid contact problems. Arch. Mech. 45, 1, 77–91 (1993)

    MATH  Google Scholar 

  5. Cieszko, M., Kubik J.: Constitutive relations and internal equilibrium condition for fluid-saturated porous solid. Arch. Mech. 48, 5, 893–923 (1996)

    MATH  Google Scholar 

  6. Cieszko, M., Kubik J.: Derivation of matching conditions at the contact surface between fluid-saturated porous solid and bulk fluid. J. of Transport in Porous Media 34, 319–336 (1999)

    Article  Google Scholar 

  7. Hou, J.S., Holmes, M.H., Lai, W.H., Mow, V.C.: Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J. of Biomechanical Engineering 111, 78–87 (1989)

    Article  Google Scholar 

  8. Hutter, K., Johnk, K., Svendsen, B.: On interfacial transition conditions in two phase gravity flow. ZAMP 45, 746–762 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Johnson, D.L.: Recent developments in the acoustic properties of porous media. In: Frontiers in Physical Acoustics, XCIII Corso, Soc. Italiana di Fisica Bologna (1986)

  10. Kubik, J.: A macroscopic description of geometrical pore structure of porous solid. Int. J. Engng. Sci. 24, 6, 971–980 (1986)

    Article  MATH  Google Scholar 

  11. Kubik, J.: Pore structure in dynamic behavior of saturated materials. Transport in Porous Media. 9, 15–24 (1992)

    Article  Google Scholar 

  12. Liu, I.S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rat. Mech. and Anal. 46, 131–148 (1972)

    MATH  Google Scholar 

  13. Saffman, P.G.: On the boundary condition at the surface of a porous medium. Studies in Appl. Mech. L 2, 93–101 (1971)

    Google Scholar 

  14. Taylor, G.I.: A Model for the boundary condition of a porous material. Part I. J. Fluid Mech. 49, 319–326 (1971)

    Article  Google Scholar 

  15. Wang, Y., Hutter, K.: A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures. Granular Metter 1, 163–181 (1999)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Environmental Mechanics and Applied Computer Science, Bydgoszcz University, Chodkiewicza 30, 85 064, Bydgoszcz, Poland

    J. Kubik & M. Cieszko

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  1. J. Kubik
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  2. M. Cieszko
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Correspondence to M. Cieszko.

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PACS 47.55.Mh

Communicated by Y.D. Shikhmurzaev

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Kubik, J., Cieszko, M. Analysis of matching conditions at the boundary surface of a fluid-saturated porous solid and a bulk fluid: the use of Lagrange multipliers. Continuum Mech. Thermodyn. 17, 351–359 (2005). https://doi.org/10.1007/s00161-005-0001-6

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  • DOI: https://doi.org/10.1007/s00161-005-0001-6

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