Abstract
In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.
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Abbreviations
- b :
-
mobility
- c :
-
concentration of a chemical
- D :
-
rate of deformation tensor
- D :
-
molecular diffusion coefficient
- D effij :
-
macroscopic (or effective) diffusion tensor
- Ē :
-
electric field
- E 0 :
-
initial electric field
- k ij :
-
molecular tensor
- j, j *, Ĵ:
-
current densities
- K ij :
-
macroscopic permeability tensor
- l :
-
characteristic length of the ERV or the periodic cell
- L :
-
characteristic macroscopic length
- L ijkl :
-
coupled flows coefficients
- n i :
-
unit outward vector normal to γ
- p :
-
pressure
- q t ,q +t ,\(\hat q_t \) :
-
heat fluxes
- q c ,q +c ,\(\hat q_c \) :
-
chemical fluxes
- s :
-
specific entropy or the entropy density
- S :
-
entropy per unit volume
- t :
-
time variable
- t ij :
-
local tensor
- T :
-
absolute temperature
- v i :
-
velocity
- V 0 :
-
initial electric potential
- V :
-
electric potential
- x :
-
macroscopic (or slow) space variable
- y :
-
microscopic (or fast) space variable
- α i :
-
local vectorial field
- Β i :
-
local vectorial field
- δ Г :
-
electric charge density on the solid surface γ
- λ, Μ :
-
bulk and shear viscosities of the fluid
- Μ ij :
-
local tensor
- Ω ij :
-
local tensor
- ζ i :
-
local vector
- χ ij :
-
molecular conductivity tensor
- χ effij :
-
effective conductivity tensor
- ε :
-
homogenization parameter
- ρ :
-
fluid density
- σ 0 :
-
ion-conductivity of fluid
- θ ij :
-
dielectric tensor
- ψ 1i , ψ 2i , ψ 3i :
-
local vectors
- ψ4 :
-
local scalar
- Ω S :
-
solid volume in the periodic cell
- Ω L :
-
volume of pores in the periodic cell
- γ:
-
boundary between Ω S and Ω L
- γs :
-
rate of entropy production per unit volume
- ∥Ω∥:
-
total volume of the periodic cell
- ∥Ω l ∥:
-
volume of pores in the cell
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On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdańsk, Poland.
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Auriault, J.L., Lewandowska, J. On the cross-effects of coupled macroscopic transport equations in porous media. Transp Porous Med 16, 31–52 (1994). https://doi.org/10.1007/BF01059775
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DOI: https://doi.org/10.1007/BF01059775