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Flow through porous media due to high pressure gradients. (English) Zbl 1228.76161
Summary: While Darcy’s equations are adequate for studying a large class of flows through porous media, there are several situations wherein it would be inappropriate to use Darcy’s equations. One such example is a flow wherein the range of pressures involved is very large, and high pressures and pressure gradients are at play. Here, after developing an approximation for the flow through a porous solid, that is a generalization of a Brinkman equation, we study a simple boundary value problem that clearly delineates the difference between the solution to these equations and those due to the equations that are referred to as “Darcy law”. We find that the solutions for the equations under consideration exhibit markedly different characteristics from the counterpart for the Brinkman equations or Darcy’s equations (or the Navier-Stokes equation if one neglects the porosity) in that the solutions for velocity as well as for vorticity lack symmetry, and one finds the maximum value of the vorticity occurs at the boundary near which the fluid is less viscous in virtue of the pressure being lower. We also find that for a certain range of values for non-dimensional parameters describing the flow, boundary layers develop and the vorticity is confined next to the boundary adjacent to which the viscosity is lower; such boundary layers being absent in other classical cases.

76S05 Flows in porous media; filtration; seepage
Full Text: DOI
[1] Darcy, H., LES fontaines publiques de la ville de Dijon, (1856), Victor Dalmont Paris
[2] 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) · Zbl 0041.54204
[3] Brinkman, H.C., On the permeability of media consisting of closely packed porous particles, Appl. sci. res. A, 1, 81-86, (1947)
[4] Brinkman, H.C., The viscosity of concentrated suspensions and solutions, J. chem. phys., 20, 571, (1952)
[5] Biot, M.A., Theory of elastic waves in a fluid-saturated porous solid, I. low frequency range, J. acoust. soc. am., 28, 168-178, (1956)
[6] Biot, M.A., Theory of elastic waves in a fluid-saturated porous solid, II. high frequency range, J. acoust. sci. am., 28, 179-191, (1956)
[7] Biot, M.A., Mechanics of deformation and acoustic propagation in porous media, J. appl. phys., 33, 1482-1498, (1962) · Zbl 0104.21401
[8] Muskat, M., The flow of homogeneous fluids through porous media, (1937), Edwards Ann Arbor · JFM 63.1368.03
[9] Polubarinova-Kochina, P.Ya., The theory of ground water movement, (1962), Princeton University Press Princeton · Zbl 0114.42601
[10] Scheidegger, A.E., The physics of flow through porous media, (1974), University of Toronto Press Toronto · Zbl 0084.42704
[11] Rajagopal, K.R., On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. models meth. appl. sci., 17, 215-252, (2007) · Zbl 1123.76066
[12] Truesdell, C., Sulle basi Della thermomeccanica, Rend. lincei, 22, 33-38, (1957) · Zbl 0098.21002
[13] Truesdell, C., Sulle basi Della thermomeccanica, Rend. lincei, 22, 158-166, (1957) · Zbl 0098.21002
[14] Truesdell, C., Mechanical basis of diffusion, J. chem. phys., 37, 2336-2344, (1962)
[15] Fick, A., Uber diffusion, Ann. phys., 94, 59-86, (1855)
[16] Truesdell, C., Rational thermodynamics, (1984), Springer Berlin · Zbl 0598.73002
[17] R.M. Bowen, in: A.C. Eringen, Theory of Mixtures in Continuum Physics, vol. 3, 1976, p. 8.
[18] Atkin, R.J.; Craine, R.E., Continuum theories of mixtures, basic theory and historical development, Quart. J. appl. math., 29, 290, (1976) · Zbl 0339.76003
[19] Bedford, A.; Drumheller, D.S., Theories of immiscible and structured mixtures, Int. J. eng. sci., 21, 863-960, (1983) · Zbl 0534.76105
[20] Rajagopal, K.R.; Wineman, A.S., Developments in the mechanics of interactions between a fluid and a highly elastic solid, (), 236-248
[21] Samohyl, I., Thermodynamics of irreversible processes in fluid mixtures, (1987), Teubner Leipzig · Zbl 0665.76002
[22] Rajagopal, K.R.; Tao, L., Mechanics of mixtures, () · Zbl 0941.74500
[23] Raats, P.A.C., Application of the theory of mixtures in soil physics, appendix 5D, (), 326-343 · Zbl 0234.76068
[24] Crochet, M.J.; Nagdhi, P.M., Small motions superposed on large static deformations, Acta mech., 4, 315-335, (1967) · Zbl 0204.28202
[25] Adkins, J.E., Nonlinear diffusion, I. diffusion flow of mixtures of fluids, Phil. trans. roy. soc. A, 255, 607-633, (1963) · Zbl 0124.41801
[26] Adkins, J.E., Non-linear diffusion, 2. constitutive equations for mixtures of isotropic fluids, Phil. trans. roy. soc. lond. A, 255, 635-648, (1963) · Zbl 0124.41801
[27] Green, A.E.; Adkins, J.E., A contribution to the theory of non-linear diffusion, Arch. ration. mech. anal., 15, 235-246, (1964) · Zbl 0131.40805
[28] Gray, W.G., General conservation equations for multiphase systems, 4. constitutive theory including phase change, Adv. water res., 6, 130-140, (1983)
[29] Hassanizadeh, M.; Gray, W.G., General conservation equations for multiphase systems, 2. mass, momentum, energy and entropy equations, Adv. water res., 2, 191-203, (1979)
[30] Hassanizadeh, M.; Gray, W.G., General conservation equations for multiphase systems, 3. constitutive theory for porous media flow, Adv. water res., 3, 24-40, (1980)
[31] Stokes, G.G., On the theories of internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, Trans. camb. phil. soc., 8, 287-305, (1845)
[32] Barus, C., Isotherms, isopiestics and isometrics relative to viscosity, Am. J. sci., 45, 87-96, (1893)
[33] Bridgman, P.W., The physics of high pressure, (1931), MacMillan New York · Zbl 0049.25606
[34] Andrade, E.C., Viscosity of liquids, Nature, 125, 309-310, (1930) · JFM 56.1264.10
[35] Bridgman, P.W., The effect of pressure on the viscosity of forty three pure fluids, Proc. am. acad. arts sci., 61, 57-99, (1926)
[36] Cutler, W.G.; McMickle, R.H.; Webb, W.; Schiessler, R.W., Study of the compressions of several high molecular weight hydrocarbons, J. chem. phys., 29, 727-740, (1958)
[37] Griest, E.M.; Webb, W.; Schiessler, R.W., Effect of pressure on viscosity of high hydrocarbons and their mixtures, J. chem. phys., 29, 711-720, (1958)
[38] Johnson, K.L.; Cameron, R., Shear behavior of elastohydrodynamic oil films at high rolling contact pressures, Proc. inst. mech. eng., 182, 14, 307-319, (1967)
[39] Johnson, K.L.; Greenwood, J.A., Thermal analysis of an Eyring fluid in elastohydrodynamic traction, Wear, 71, 355-374, (1980)
[40] Johnson, K.L.; Tevaarwerk, J.L., Shear behavior of elastohydrodynamic oil films, Proc. roy. soc. lond. A, 356, 215-236, (1977) · Zbl 0365.76008
[41] Bair, S.; Winer, W.O., The high pressure high shear stress rheology of liquid lubricants, J. tribol., 114, 1-13, (1992)
[42] Bair, S.; Kottke, P., Pressure – viscosity relationships for elastohydrodynamics, Tribol. trans., 46, 289-295, (2003)
[43] Dowson, D.; Higginson, G.R., Elastohydrodynamic lubrication, the fundamentals of roller and gear lubrication, (1966), Pergamon Oxford
[44] Szeri, A.Z., Fluid film lubrication: theory and design, (1998), Cambridge University Press Cambridge · Zbl 1001.76001
[45] Mills, N., Incompressible mixtures of Newtonian fluids, Int. J. eng. sci., 4, 97-112, (1966) · Zbl 0141.42602
[46] G. Johnson, M. Massoudi, K.R. Rajagopal, A Review of Interaction Mechanisms in Fluid-Solid Flows, DOE/PETC/TR-90/9, US Department of Energy, 1990.
[47] Rajagopal, K.R.; Srinivasa, A.R., On the nature of constraints for continua undergoing dissipative processes, Proc. roy. soc. A - math. phys. eng. sci., 461, 2785-2795, (2005) · Zbl 1186.74008
[48] K.R. Rajagopal, G. Saccomandi, On internal constraints in continuum mechanics differential equations and nonlinear mechanics article ID 18572, doi:10.1155/DENM/2006/18572, 1-12, 2005.
[49] Rajagopal, K.R., On implicit constitutive equations, Appl. math., 48, 279-319, (2005) · Zbl 1099.74009
[50] Rajagopal, K.R., On implicit theories for fluids, J. fluid mech., (2006) · Zbl 1097.76009
[51] Munaf, D.; Lee, D.; Wineman, A.S.; Rajagopal, K.R., A boundary value problem in groundwater motion analysis – comparison of predictions based on darcy’s law and the continuum theory of mixtures, Math. models meth. appl. sci., 3, 231-248, (1993) · Zbl 0773.76067
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