Abstract
Mass conservation and linear momentum balance relations for a porous body and any fluid therein, valid at any given length scale in excess of nearest-neighbour molecular separations, are established in terms of local weighted averages of molecular quantities. The mass density field for the porous body at a given scale is used to identify its boundary at this scale, and a porosity field is defined for any pair of distinct length scales. Specific care is paid to the interpretation of the stress tensor associated with each of the body and fluid at macroscopic scales, and of the force per unit volume each exerts on the other. Consequences for the usual microscopic and macroscopic viewpoints are explored.
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Abbreviations
- \(\mathcal{M}\) :
-
material system; Section 2.1.
- \(\mathcal{P}\) :
-
porous body (example of a material system); Sections 2.1, 3.1, 4.1
- \(\mathcal{F}\) :
-
fluid body (example of a material system); Sections 2.1, 3.1, 4.1
- ψ :
-
weighting function; Sections 2.1, 2.3
- ψ ɛ ,h :
-
weighting function corresponding to spherical averaging regions of radiusε and boundary mollifying layer of thicknessh; Section 3.2
- ε :
-
Euclidean space; Section 2.1
- V :
-
space of all displacements between pairs of points inε; Section 2.1
- ρ ψ :
-
mass density field corresponding toψ; (2.3)1
- ρ P ψ ,ψ f ψ :
-
mass density fields for\(\mathcal{P}\),\(\mathcal{F}\); (4.1)
- P ψ :
-
momentum density field corresponding toψ; (2.3)2
- v ψ :
-
velocity field corresponding toψ; (2.4)
- S r (X):
-
interior of sphere of radiusr with centre at pointx; (3.3)
- \(\partial \mathcal{R}\) :
-
boundary ofany region\(\mathcal{R}\)
- \(^ + \mathcal{R}_{\varepsilon , h}^p \) :
-
region in whichρ p ψ > 0 withψ =ψ ɛ,h; (3.1)
- \(\mathcal{R}_{\varepsilon , h}^p \) :
-
subset of\(^ + \mathcal{R}_{\varepsilon , h}^p \) whose points lie at leastε+h from boundary of\(^ + \mathcal{R}_{\varepsilon , h}^p \); (3.4)
- \(^ + \mathcal{R}_{\varepsilon }^p , \mathcal{R}_\varepsilon ^p \) :
-
abbreviated versions of\(^ + \mathcal{R}_{\varepsilon , h }^p , \mathcal{R}_{\varepsilon , h}^p \); Section 3.2, Remark 4
- \(^ - \mathcal{R}_{\varepsilon }^p \) :
-
strict interior of\(\mathcal{R}_{\varepsilon }^p \); (3.7)
- \(^ + \mathcal{R}_{\varepsilon }^f , \mathcal{R}_{\varepsilon }^f , ^ - \mathcal{R}_{\varepsilon }^f \) :
-
analogues of\(^ + \mathcal{R}_{\varepsilon }^p , \mathcal{R}_{\varepsilon }^p , ^ - \mathcal{R}_{\varepsilon }^p \) for fluid system\(\mathcal{F}\); Section 3.2
- \(^ \pm \mathcal{R}_{\psi }^p , ^ \pm \mathcal{R}_{\psi }^f \) :
-
general version of\(^ \pm \mathcal{R}_{\varepsilon }^p , ^ \pm \mathcal{R}_{\varepsilon }^f \) corresponding to any choice of weighting functionψ; (4.6)
- \(\mathcal{I}_{\varepsilon }^p \) :
-
\(\mathcal{P}\) interfacial region at scaleε; (3.8)
- ε 0 :
-
scale of nearest-neighbour separations in\(\mathcal{P}\); Section 3.2. Remark 1
- \(p_{\varepsilon _{1,} \varepsilon _2 } \) :
-
porosity field at scales (ε 1;ε 2); (3.9)
- \(\mathcal{S}_{\varepsilon _{1,} \varepsilon _2 }^{pore} \) :
-
pore space at scales (ε 1;ε 2); (3.12)
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Murdoch, A.I., Kubik, J. On the continuum modelling of porous media containing fluid: a molecular viewpoint with particular attention to scale. Transp Porous Med 19, 157–197 (1995). https://doi.org/10.1007/BF00626664
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DOI: https://doi.org/10.1007/BF00626664