Establishing a Quantitative Functional Relationship between Capillary Pressure Saturation and Interfacial Area Page: 84 of 119
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W.G. Gray / International Journal of Multiphase Flow 26 (2000) 467-501
volume at a point contribute to the definition of the thermodynamic state at that point. A
variational analysis of the system thermodynamics has provided macroscale relations among
the pressures, surface tensions, lineal tension, effective contact angle, effective interfacial
curvatures, and effective common line curvature that must be satisfied at the equilibrium state.
Additionally, information about the required relations among variations of the geometric
densities motivates the appropriate rearrangement of the dynamic entropy inequality such that
near-equilibrium, linearized equations for the geometric quantities may be obtained to close the
problem formulation. By basing the analysis of the changes of geometric parameters on a
variational approach rather than on investigation of the averaging theorems as in Gray (1999),
it is possible to reduce the number of approximations that must be made and therefore obtain
more complete dynamic equations for the geometric variables.
Acknowledgements
This work was supported, in part, by the U.S. Department of Energy under grant DE-
FG07-96ER14701 and by a subgrant from Cornell University of U.S. Department of Energy
grant DE-FG07-96ER14703. The author was a Gledden Visiting Fellow at the Centre for
Water Research at the University of Western Australia during the latter stages of this work.
Appendix. Derivation of variational relations
A.]. Derivation of variations for phase properties
Here the property of a phase will be considered. At the microscale, the amount of the
property per unit volume (i.e. the density of the property) will be denoted as b. Since
averaging will be done to the macroscale, ba may be a function of the microscale
thermodynamic parameters of the system and of x + , where x is the location of the centroid
of the averaging volume and is the distance from this centroid to a microscale point of
interest. The macroscale density of the quantity for the a-phase will be denoted as B . Note
that for convenience this quantity is defined per unit volume of the averaging volume, not just
per volume of a phase. The macroscale property B is a function of the macroscale
thermodynamic parameters of the system and of the location of the averaging volume, x. The
total amount of the property of interest in volume Y is denoted as 9" and is related to its
macroscale and microscale counterparts by:
f = f d 1 b[ d V di (A 1)
Y", F V v
where V is the averaging volume and V1 is the volume occupied by the a-phase within the
averaging volume.
Now consider the variation of the first equality in Eq. (Al). This may be expressed as:493
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Montemagno, Carlo. Establishing a Quantitative Functional Relationship between Capillary Pressure Saturation and Interfacial Area, report, April 23, 2002; United States. (https://digital.library.unt.edu/ark:/67531/metadc736299/m1/84/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.