Abstract
In the paper anomalous diffusion appearing in a porous medium composed of two porous components of considerably different diffusion characteristics is examined. The differences in diffusivities are supposed to result either from two medium types being present or from variations in pore size (double porosity media). The ‘long-tail’ effect is predicted using the homogenization approach based on the application of multiple scale asymptotic developments. It is shown that, if the ratio of effective diffusion coefficients of two porous media is of the order of magnitude smaller or equal O(ε 2), whereε is a homogenization parameter, then the macroscopic behaviour of the composite may be affected by the presence of ‘tail-effect’. The results of the theoretical analysis were applied to a problem of diffusion in a bilaminate composite. Analytical calculations were performed to show the presence of the ‘long-tail’ effect in two particular cases.
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Abbreviations
- c i :
-
the concentration of chemical species in water within the medium ‘i’
- D i :
-
the effective diffusion coefficient for the medium ‘i’
- D eff ij :
-
the macroscopic (or effective) diffusion tensor in the composite
- ERV :
-
the elementary representative volume
- h :
-
the thickness of the period
- l :
-
a chracteristic length of the ERV or the periodic cell
- L :
-
a characteristic macroscopic length
- n :
-
the volumetric fraction of the material 2
- 1−n :
-
the volumetric fraction of the material 1
- N :
-
the unit vector normal to γ
- t :
-
the time variable
- x :
-
the macroscopic (or slow) space variable
- y :
-
the microscopic (or fast) space variable
- c 1c ,C 2c ,D 1c ,D 2c :
-
the characteristic quantities
- T,T 1L ,T 2L ,T 1l ,T 2l :
-
the characteristic times
- c *1 ,c *2 ,D *1 ,D *2 ,t * :
-
the non-dimensional variables
- ε :
-
the homogenization parameter
- Ω1 :
-
the domain occupied by the material 1
- Ω2 :
-
the domain occupied by the material 2
- γ:
-
the interface between the domains 1 and 2
- ∥Ω∥:
-
the total volume of the periodic cell
- ∂/∂xi :
-
the gradient operator
- ∇:
-
the gradient operator
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Auriault, J.L., Lewandowska, J. Non-gaussian diffusion modeling in composite porous media by homogenization: Tail effect. Transp Porous Med 21, 47–70 (1995). https://doi.org/10.1007/BF00615334
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DOI: https://doi.org/10.1007/BF00615334