note on darcy’s law
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Note On Darcy’s LawA. Cemal Eringen Citation: Journal of Applied Physics 94, 1282 (2003); doi: 10.1063/1.1586951 View online: http://dx.doi.org/10.1063/1.1586951 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/94/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Fluid coupling in DEM simulation using Darcy's law: Formulation, and verification AIP Conf. Proc. 1542, 1134 (2013); 10.1063/1.4812136 Miscible viscous fingering with linear adsorption on the porous matrix Phys. Fluids 19, 073101 (2007); 10.1063/1.2743610 Viscous fingering of miscible slices Phys. Fluids 17, 054114 (2005); 10.1063/1.1909188 Kinetic theory and molecular dynamics simulations of microscopic flows Phys. Fluids 9, 3915 (1997); 10.1063/1.869490 Mechanical Modeling of the Shrinking and Swelling of Porous Elastic Solids J. Rheol. 29, 119 (1985); 10.1122/1.549787
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Note On Darcy’s LawA. Cemal Eringena)
15 Red Tail Drive, Littleton, Colorado 80126-5001
~Received 30 January 2003; accepted 6 May 2003!
Classical Darcy’s Law expresses pressure gradient in a porous media as a linear function of thevelocity difference between solid and fluid constituents. As such, this constitutive equation is framedependent. The present article is an attempt to express this law in terms of frame independentquantities. The resulting expression shows that classical Darcy’s Law may be justified on anapproximate basis when the higher order quantities are neglected. ©2003 American Institute ofPhysics. @DOI: 10.1063/1.1586951#
In the porous media theory, the pressure gradient is ex-pressed as a linear function of the velocity difference be-tween solid and fluid constituents
¹p5D~vS2vF!, ~1!
wherevS andvF denote, respectively, the velocities of solidand fluid constituents at pointsxS and xF, in a referenceframe x. In another frame of referencex8, obtained fromx,by time-dependent rotations and translations,xS and xF gointo x8S andx8F, obtained by
xk8S5Qkl~ t !xl
S1bk~ t !, ~2!
xk8F5Qkl~ t !xl
F1bk~ t !, ~3!
whereQkl denotes the full group of orthogonal transforma-tions, at timet, i.e.:
QQT5QTQ51, detQ561. ~4!
We employ the usual summation convention on repeated in-dices.
Subtracing Eq.~3! from Eq. ~2!, we have
xk8S2xk
F5Qkl~xlS2xl
F!. ~5!
This implies that the difference of position vectors is frameindependent.
The material derivative of Eq.~5! gives
vk8S2vk8
F5Qkl~v lS2v l
F!1Q̇kl~xlS2xl
F!. ~6!
Thus, the velocity difference is not frame independent.It is simple to show that, Eringen1
Q̇kl5Qmlwkm8 2Qkmwml , ~7!
wherewkm8 andwml are the spin tensors, defined by
wkm8 51
2 S ]vk8
]xm82
]vm8
]xk8D , wml5
1
2 S ]vm
]xl2
]v l
]xmD . ~8!
We replaceQ̇kl , in Eq.~6! by Eq.~7!, and use Eq.~5!, toorganize to the form
vk8S2vk8
F2wkl8 ~xl8S2xl8
F!5Qkl@v lS2v l
F2wlm~xmS2xm
F !#.~9!
This shows that
Ãk5vkS2vk
F2wkl~xlS2xl
F! ~10!
is frame independent. Consequently, the frame-independentDarcy’s law may be expressed as a function ofÃ, e.g.
¹p5D@vS2vF2w•~xS2xF!#. ~11!
This result is in the spirit of porous media containingviscous fluids. If the fluid is considered inviscid, then thevelocity gradient and consequentlyw is not admitted, and thesecond term in Eq.~11! would be ignored reverting Eq.~11!to the Darcy’s Law@Eq. ~1!#. Alternatively, we may considerw•(xS2xF) as a second-order quantity, as compared to thefirst term in Eq. ~11!. In both cases, we end up with anapproximate constitutive equation of the form of Eq.~1! thatis not frame independent.
Wilmanski2 arrived at the same conclusion, in anotherway, through neglecting the effects of kinematics in nonin-ertial systems, involving accelerations of the referenceframes.
1A. C. Eringen,Mechanics of Continua, 2nd ed.~Robert E. Krieger Mel-bourne, FL, 1980!, p. 94.
2K. Wilmanski, inSome Questions on Material Objectivity Arising in Mod-els of Porous Materials, Rational Continua, Classical and New,edited byP. Podio-Guidugli and M. Brocato~Springer, Vienna, 2003!, p. 194.a!Electronic mail: [email protected]
JOURNAL OF APPLIED PHYSICS VOLUME 94, NUMBER 2 15 JULY 2003
12820021-8979/2003/94(2)/1282/1/$20.00 © 2003 American Institute of Physics
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
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