[3.2.2.1] Consider stationary flow of a fluid inside the pore space . [3.2.2.2] On the pore scale the viscous forces are given quantitatively by Newton’s law of internal friction

(16) |

where
is the fluid viscosity,
is the phase velocity gradient
, and
is the phase velocity
for stationary flow.

1: In general
is a tensor of rank 2 and is a tensor of rank 4
yielding the fluid stress tensor of rank 2.

[page 4, §0]
[4.1.0.1] The capillary forces are quantified by the Young Laplace law as

(17) |

where is the interfacial tension and the interfacial mean curvature in thermodynamic equilibrium between the two phases. [4.1.0.2] Using the same scale in both laws

(18a) |

approximating by its spatial average as

(18b) |

and using

(18c) |

for both phases one arrives at the microscopic capillary number

(19) |

for phase . [4.1.0.3] Note that is a characteristic flow velocity that depends only on the fluid properties. [4.1.0.4] As a consequence the microscopic capillary number depends only on fluid properties, but is independent of the pore space properties.