Abstract
A general set of boundary conditions at the interface between dissimilar fluid-filled porous matrices is established starting from an extended Hamilton-Rayleigh principle. These conditions do include inertial effects. Once linearized, they encompass boundary conditions relative to volume Darcy-Brinkman and to surface Saffman-Beavers-Joseph-Deresiewicz dissipation effects.
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D Bibliography
Alazmi B., Vafai K.: Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Int. J. Heat Mass Transf. 44, 1735–1749 (2001)
Albers B.: Monochromatic surface waves at the interface between poroelastic and fluid half-spaces. Proc. R. Soc. Lond. A 462, 701723 (2006)
Allaire G.: Homogenization of the Stokes flow in a connected porous medium. Asymptotic Anal. 2,3, 203–222 (1989).
Allaire G.: Homoge ne isation des e quations de Stokes dans un domaine perfore de petits trous re partis pe riodiquement. C. R. Acad. Sci. Paris, Ser. I, 309,11, 741–746 (1989).
Allaire G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. Arch. Rational Mech. Anal., 113, 209298 (1991).
Allaire G.: Homogenization of the Navier-Stokes equations with a slip boundary condition. Comm. Pure Appl. Math., XLIV, 605–641 (1991).
Allaire, G.: Continuity of the Darcys law in the low-volume fraction limit. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18, 475–499 (1991)
Altay G., Dokmeci M.C.: On the equations governing the motion of an anisotropic poroelastic material. Proc. R. Soc. A 462, 2373–2396 (2006)
Baek S., Srinivasa A.R.: Diffusion of a fluid through an elastic solid undergoing large deformation. Int. J. Non-Linear Mech. 39, 201–218 (2004)
Batra G., Bedford A., Drumheller D.S.: Applications of Hamiltons principle to continua with singular surfaces. Arch. Rational Mech. Anal. 93, 223251 (1986)
Bedford A., Drumheller D.S.: A variational theory of immiscible mixtures. Arch. Rational Mech. Anal. 68, 37–51 (1978)
Bedford A., Drumheller D.S.: A variational theory of porous media. Int. J. Solids Struct. 15, 967–980 (1979) 20 CISM Course C-1006 Udine, July 12–16 2010
Bedford A., Drumheller D.S.: Recent advances: theories of immiscible and structured mixtures. Int. J. Eng. Sci. 21, 863–960 (1983)
Biot M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)
Biot M.A.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 23, 91–96 (1956)
Biot M.A.: Theory of propagation of elastic waves in fluid-saturated porous solid. J. Acoust. Soc. Am. 28, 168–191 (1956)
Biot M.A., Willis D G.: The elastic coefficients of the theory of consolidation. J. Appl. Mech. 24 594–601 (1957)
Biot M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 1482–1498 (1962)
Biot M.A.: Variational principles for acoustic gravity waves. Phys. Fluids 6, 772–778 (1963)
Beavers G.S., Joseph D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)
Burridge R., Keller J.B.: Poroelasticity equations derived from microstructure. J. Acoust. Soc. Am. 70:4, 1140–1146 (1981)
Casertano L., Oliveri del Castillo A., Quagliariello M.T.: Hydrodynamics and geodynamics in Campi Flegrei Area of Italy. Nature 264, 161–164 (1976)
Caviglia G., Morro A.:Harmonic waves in thermo-visco-elastic solids.Int. J. Eng. Sci.43, 1323–1336 (2005)
Chandesris M., Jamet D.: Boundary conditions at a planar fluidporous interface for a Poiseuille flow. Int. J. Heat and Mass Transf. 49, 2137–2150 (2006).
Chandesris M., Jamet D.: Boundary conditions at a fluidporous interface: an a priori estimation of the stress jump coefficients. Int. J. Heat and Mass Transf. 50, 3422–3436 (2007)
Chateau X., Dormieux L.: Homoge ne isation dun milieu poreux non sature: Lemme de Hill et applications. C.R. Acad. Sci. Paris 320, Serie IIb, 627–634 (1995)
Chateau X., Dormieux L.: Approche microme canique du comportement dun milieu poreux non sature. C. R. Acad. Sci. Paris 326, Serie II, 533–538 (1998)
Cieszko M., Kubik J.: Interaction of elastic waves with a fluid-saturated porous solid boundary. J. of Theor. Appl. Mech. 36, 561580 (1998)
Cieszko M. Kubik J.: Derivation of matching conditions at the contact surface between fluid-saturated porous solid and bulk fluid. Transp. Porous Med. 34, 319336 (1999)
Coussy O., Bourbie T.: Propagation des ondes acoustiques dans les milieux poreux sature s. Rev. Inst. Fr. Pet. 39, 47–66 (1984)
Coussy O. Dormieux L. Detournay E.: From mixture theory to Biots approach for porous media. Int. J. Solids Struct. 35, 46194635 (1998)
Coussy O.: Poromechanics, John Wiley Sons, Chichester (2004)
Cowin SC: Bone Mechanics Handbook, Boca Raton, FL: CRC Press (2001)
Debergue P. et al.: Boundary conditions for the weak formulation of the mixed (u,p) poroelasticity problem. J. Acoust. Soc. Am. 106: 5, 2383–2390 (1999)
de Boer R.: Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory. Appl. Mech. Rev. 49: 4, 201–262 (1996)
de Boer R.: Contemporary progress in porous media theory. Appl. Mech. Rev. 53, 323370 (2000)
de Boer R.: Theoretical poroelasticity a new approach. Chaos Solitons Fractals 25, 861–878 (2005)
de Buhan P., Dormieux L., Chateau X.: A micro-macro approach to the constitutive formulation of large strain poroelasticity. Poromechanics a tribute to Maurice A. Biot. Thimus et al., Rotterdam (1998) 21 CISM Course C-1006 Udine, July 12–16 2010
de Buhan P., Chateau X., Dormieux L.: The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach. Eur. J. Mech. A/Solids. 17, 909–921 (1998)
Cryer C.W.: A comparison of the three dimensional consolidation theories of Biot and Terzaghi. Q. J. Mech. Appl. Math. 16: 4, 401–412 (1963)
de la Cruz V., Hube J., Spanos T.J.T.: Reflection and transmission of seismic waves at the boundaries of porous media. Wave Motion 16, 323–338 (1992)
dellIsola F., Rosa L., Wozniak C.: A micro-structured continuum modelling compacting fluid-saturated grounds: the effects of pore-size scale parameter. Acta Mech. 127, 1:4, 165–182 (1998)
dellIsola, F., Guarascio, M., Hutter, K.: A variational approach for the deformation of a saturated porous solid. A second gradient theory extending Terzaghis effective stress principle. Arch. Appl. Mech. 70, 323–337 (2000).
dellIsola, F., Sciarra, G., Batra, R.C.: Static Deformations of a Linear Elastic Porous Body filled with an Inviscid Fluid. J. Elasticity 72, 99–120 (2003).
Deresiewicz H.: The effect of boundaries on wave propagation in a liquidfilled porous solid: I. Reflection of plane waves at a free plane boundary (non-dissipative case). Bull. Seismol. Soc. Am. 50, 599607 (1960)
Deresiewicz H.: The effect of boundaries on wave propagation in a liquid-filled porous solid: III. Reflec-tion of plane waves at a free plane boundary (general case) Bull. Seismol. Soc. Am. 52, 595625 (1962)
Deresiewicz H.: A note on Love waves in a homogeneous crust overlying an inhomogeneous substratum. Bull. Seismol. Soc. Am. 52, 639–645 (1962)
Deresiewicz H.: The effect of boundaries on wave propagation in a liquidfilled porous solid: IV. surface in a half-space. Bull. Seismol. Soc. Am. 50, 627–638 (1962)
Deresiewicz H.: On uniqueness in dynamic poroelasticity. Bull. Seismol. Soc. Am. 53, 783–788 (1963)
Deresiewicz H.: The effect of boundaries on wave propagation in a liquidfilled porous solid: V. Trans-mission across a plane interface. Bull. Seismol. Soc. Am. 54, 409–416 (1964)
Deresiewicz H.: The effect of boundaries on wave propagation in a liquidfilled porous solid: VII. Surface waves in a half-space in the presence of a liquid layer. Bull. Seismol. Soc. Am. 54, 425430 (1964)
Dormieux L., Coussy O., de Buhan P.: Mode lisation me canique dun milieu polyphasique par la me thode des puissances virtuelles. C.R. Acad. Sci. Paris. 313, Serie IIb, 863–868 (1991)
Dormieux L., Stolz C.: Approche variationelle en poroe lasticite. C.R. Acad. Sci. Paris, 315, Serie IIb, 407–412 (1992)
Dormieux L., Kondo D. and Ulm F.-J. Microporomechanics Wiley (2006).
Fillunger P.: Erbdaumechanik. Selbst Verlag des Verfassers, Wien (1936).
Gavrilyuk S.L., Gouin H., Perepechko Yu.V.: A variational principle for two-fluid models. C.R. Acad. Sci. Paris, 324, Serie IIb, 483–490 (1997)
Gavrilyuk S.L., Gouin H., Perepechko Yu.V.: Hyperbolic Models of Homogeneous Two-Fluid Mixtures, Meccanica 33, 161–175(1998)
Gavrilyuk S., Perepechko Yu. V.: Variational approach to constructing hyperbolic models of two-velocity media. J. Appl. Mech. Technical Phys. 39:5, 684–698 (1998)
Gavrilyuk S.L., Gouin H.: A new form of governing equations of fluids arising from Hamiltons principle. Int. J. Eng. Sci. 37, 1495–1520 (1999)
Gavrilyuk S., Saurel R.: Rankine-Hugoniot relations for shocks in heterogeneous mixtures J. Fluid Mech. 575, 495–507 (2007)
Germain P.: Cours de me canique des milieux continus tome 1 the orie ge ne rale, Masson (1973)
Gouin H.: Variational theory of mixtures in continuum mechanics. Eur. J. Mech., B/Fluids 9, 469–491 (1990)
Gouin H., Gavrilyuk S.L.: Hamiltons principle and Rankine-Hugoniot conditions for general motions of mixtures. Meccanica 34, 39–47 (1998) 22 CISM Course C-1006 Udine, July 12–16 2010 Goyeau B. et al.: Momentum transport at a fluidporous interface. Int. J. Heat Mass Transf. 46, 4071–4081 (2003)
Haber S., Mauri R.: Boundary conditions for Darcys flow through porous media. Int. J. Multiphase Flow 9, 561574 (1983)
Hassanizadeh S. M. Gray W.: Boundary and interface conditions in porous media. Water Resources Res. 25, 17051715 (1989)
Hornung U.: Homogenization and Porous Media. Interdiscip. Appl. Math., vol. 6, Springer, Berlin (1997)
Houlsbya G.T., Puzrin A.M.: Rate-dependent plasticity models derived from potential functions. J. Rheol. 46(1), 113–126 (2002)
Jager W., Mikelic A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)
Kaasschieter E.F., Frijns A.J.H.: Squeezing a sponge: a three-dimensional solution in poroelasticity Computational Geosciences 7: 49–59, (2003).
Kosin ski W.: Field Singularities and Wave Analysis in Continuum Mechanics. PWN-Polish Scientific Publishers, Warszawa (1986)
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. 17:4, 351–359 (2005)
Kuznetsov A.V.: Influence of the stress jump condition at the porous medium/clear fluid interface on a flow at porous wall. Int. Comm. Heat Mass Transf. 24, 401–410. (1997)
Le Bars M., Grae Worster M.: Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173 (2006)
Lee C.K.: Flow and deformation in poroelastic media with moderate load and weak inertia. Proc. R. Soc. Lond. A 460, 20512087 (2004)
Levy T., Sanchez-Palencia E.: On boundary conditions for fluid flow in porous media Int. J. Eng. Sci. 13, 923–940 (1975)
Madeo A., dellIsola F., Ianiro N., Sciarra G.: A variational deduction of second gradient poroelasticity II: an application to the Consolidation Problem. J. Mech. Mater. Struct. 3:4, 607–625 (2008)
Mandel J.: Consolidation des sols (e tude mathe matique). Geotechnique 3, 287–299 (1953)
Marle C.M.: On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media. Int. J. Eng. Sci. 20:5, 643662 (1982)
Mobbs S.D.: Variational principles for perfect and dissipative fluid flows. Proc. R. Soc. Lond. A 381, 457468 (1982)
Neale G., Nader W.: Practical significance of Brinkmans extension of Darcys law: coupled parallel flows within a channel and a bounding porous medium. Can. J. Chem. Eng. 52, 475–478 (1974)
Ochoa-Tapia J.A., Whitaker S.: Momentum transfer at the boundary between a porous; medium and a homogeneous fluid I. Theoretical development. Int. J. Heat Mass Transf. 38, 2635–2646 (1995)
Ochoa-Tapia J. A., Whitaker S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid II. Comparison with experiment. Int. J. Heat Mass Transf. 38, 2647–2655 (1995).
Ochoa-Tapia J.A., Whitaker S.: Momentum jump condition at the boundary between a porous medium and a homogeneous fluid: inertial effects. J. Porous Media 1, 201–217 (1998).
Orsi G., Petrazzuoli S.M., Wohletz K.: Mechanical and thermo-fluid behaviour during unrest at the Campi Flegrei caldera (Italy) J. Vulcanol. Geotherm. Res. 91, 453–470 (1999)
Pan Y., Horne R.N.: Generalized Macroscopic Models for Fluid Flow in Deformable Porous Media. Theories Transport in Porous Media 45: 1–27, 2001.
Prat M.: On the boundary conditions at the macroscopic level’ Transp. Porous Med. 4, 259–280 (1988) 23 CISM Course C-1006 Udine, July 12–16 2010
Poulikakos D., Kazmierczak M.: Forced convection in a duct partially filled with a porous material. J. Heat Transf. 109, 653–662 (1987)
Quiroga-Goode G., Carcione J.M.: Heterogeneous modelling behaviour at an interface in porous media. Comput. Geosciences 1, 109125 (1997)
Rajagopal K.R. et al.: On boundary conditions for a certain class of problems in mixture theory. Int. J. Eng. Sci. 24, 14531463 (1986)
Rajagopal K.R., Tao L.: Mechanics of mixtures. Ser. Adva. Math. Appl. Sci. Vol. 35 (1995)
Rasolofosaon N.J.P., Coussy O.: Propagation des ondes acoustiques dans les milieux poreux sature s: effets dinterface I. Rev. Inst. Fr. Pet. 40, 581–594 (1985)
Rasolofosaon N.J.P., Coussy O.: Propagation des ondes acoustiques dans les milieux poreux sature s: effets dinterface II. Rev. Inst. Fr. Pet. 40, 785802 (1985)
Rasolofosaon N.J.P., Coussy O.: Propagation des ondes acoustiques dans les milieux poreux sature s: effets dinterface III. Rev. Inst. Fr. Pet. 41, 91103 (1986)
Saffman P. G.: On the boundary conditions at the surface of a porous medium. Stud. Appl. Math. L, 93–101 (1971)
Sciarra G., dellIsola F., Ianiro N., Madeo A.: A variational deduction of second gradient poroelasticity I: general theory. J. Mech. Mater. Struct. 3:3, 507–526 (2008)
Seliger R.L., Whitham G.B.: Variational principles in continuum mechanics. Proc. R. Soc. Lond. A 305, 1–25 (1968)
Sharma M.D.: Wave propagation across the boundary between two dissimilar poroelastic solids. J. Sound Vib. 314, 657–671 (2008)
Sonnet A.M., Maffettone P.L. and Virga E.G.: Continuum theory for nematic liquid crystals with tensorial order. J. Non-Newtonian Fluid Mech. 119 51–59 (2004)
V. Terzaghi K.: Theoretical soil mechanics. John Wiley Sons, Chichester (1943)
Savare G., Tomarelli F.: Superposition and chain rule for bounded Hessian functions. Adv. Math. 140, 237–281 (1998)
Vafai K., Thiyagaraja R.: Analysis of flow and heat transfer at the interface region of a porous medium. Int. J. Heat Mass Transf. 30, 1391–1405 (1987)
Vafai K., Kim S.J.: Analysis of surface enhancement by a porous substrate. J. Heat Transf. 112, 700–706 (1990)
Valde s-Paradaa F.J., Goyeaub B., Ochoa-Tapiaa J.A.: Diffusive mass transfer between a microporous medium and a homogeneous fluid: Jump boundary conditions. Chem. Eng. Sci. 61 1692–1704 (2006)
Wilmanski K.: Waves in porous and granular materials. In: Kinetic and Continuum Theories of Granular and Porous Media, 131–185, K. Hutter and K. Wilmanski (eds.), Springer Wien New York (1999)
Wilmanski K.: A few remarks on Biots model and linear acoustics of poroelastic saturated materials. Soil Dyn. Earthq. Eng. 26, 509–536 (2006)
Yang J.: Importance of flow condition on seismic waves at a saturated porous solid boundary. J. Sound Vib. 221:3, 391413 (1999)
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dell’Isola, F., Seppecher, P., Madeo, A. (2011). Fluid Shock Wave Generation at Solid-Material Discontinuity Surfaces in Porous Media. In: dell’Isola, F., Gavrilyuk, S. (eds) Variational Models and Methods in Solid and Fluid Mechanics. CISM Courses and Lectures, vol 535. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0983-0_7
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