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ABSTRACT
Understanding the dynamics of wet granular materials is important for a range of applications, including levee safety, beach erosion, and scour around hydraulic structures. Several continuum models have been proposed recently to model the relevant processes for these applications at computationally tractable resolutions, which were derived using mixture theory approaches for granular and aerated flows. The thermodynamically constrained averaging theory is used to formulate a framework for three-phase flow models involving mechanistic conservation and balance principles for a set of entities consisting of phases, interfaces, and a common curve. The derived entropy inequality provides a basis for formulating permissibility conditions on closure relations. An example is provided showing how the entropy inequality derived in this work can be used to examine the closure relations in existing models. Results from this example show that this existing model violates entropy permissibility conditions. Several possible extensions and uses of this foundational work are summarized.
NotationFootnote1
| b | = | entropy body source density (JK−1 s−1 m−3) |
| = | Green's deformation tensor (−) | |
| = | the set of domains occupied by entities within the averaging volume (−) | |
| = | rate of strain tensor, | |
| E | = | internal energy density (Jm−3) |
| = | partial mass internal energy (m2 s−2) | |
| = | partial mass energy of an entity averaged over the boundary of the entity (m2 s−2) | |
| = | the set of entities (−) | |
| = | macroscale entity total energy conservation equation, Eq. (Equation4 | |
| = | subscript denoting an equilibrium state (−) | |
| F | = | scalar thermodynamic force |
| = | vector thermodynamic force | |
| = | tensor thermodynamic force | |
| f | = | arbitrary continuous function (−) |
| = | body force vector in addition to gravitational acceleration (kg m−2s−2) | |
| = | indicator of an averaging theorem involving the gradient (−) | |
| = | macroscale transfer rate of potential energy due to variability of mass exchange per volume (Jm−3s−1) | |
| = | macroscale entity body force potential balance equation, Eq. (Equation7 | |
| = | body force per unit mass, gravity (m s−2) | |
| g | = | gas phase entity index (−) |
| gs | = | index of interface between g and s phases (−) |
| h | = | energy source density (Jm−3s−1) |
| I | = | unit tensor (−) |
| I′ | = | unit tensor in a surface (−) |
| I′′ | = | unit tensor in a common curve (−) |
| = | unit tensor associated with 3−n-dimensional entity α, where | |
| = | index set of entities (−) | |
| = | index set of common curve entities (−) | |
| = | index set of all entities connected to entity α, | |
| = | index set of entities connected to entity α that are of one dimension lower than entity α (−) | |
| = | index set of entities connected to entity α that are of two dimensions lower than entity α (−) | |
| = | index set of connected entities of lower dimension than the dimension of entity α, | |
| = | index set of entities connected to entity α that are of one dimension higher than entity α (−) | |
| = | index set of entities connected to entity α that are of two dimensions higher than entity α (−) | |
| = | index set of connected entities of higher dimension than the dimension of entity α(−), | |
| = | index set of fluid-phase entities (−) | |
| = | index set of interface entities (−) | |
| = | index set of phase entities (−) | |
| = | set of species indices (−) | |
| = | set of entity indices except the solid phase (−) | |
| i | = | species index (−) |
| J | = | scalar thermodynamic flux |
| = | vector thermodynamic flux | |
| = | tensor thermodynamic flux | |
| = | set of thermodynamic fluxes (−) | |
| = | macroscale surface curvature, | |
| j | = | Jacobian (−) |
| = | kinetic energy per mass due to velocity fluctuations (J kg−1) | |
| = | macroscale deviation kinetic energy equation for entity α (J m−3 s−1) | |
| = | interfacial area relaxation rate coefficient (s−1) | |
| = | unit vector tangent to a common curve (−) | |
| = | macroscale transfer rate of mass of entity κ to entity α per entity extent (kg m−3 s−1) | |
| = | microscale transfer rate of mass of entity κ to entity α (kg m−3 s−1) | |
| = | macroscale entity mass conservation equation, Eq. (Equation2 | |
| = | microscale entity mass conservation equation (kg m−3 s−1) | |
| = | unit normal vector on the boundary of entity α oriented to be positive outward (−) | |
| = | macroscale entity momentum conservation equation, Eq. (Equation3 | |
| = | microscale entity momentum conservation equation (kg m−2s−2) | |
| p | = | pressure (kg m−1 s−2) |
| = | macroscale transfer of energy from entity κ to entity α resulting from heat transfer and deviation from mean processes (Jm−3s−1) | |
| = | non-advective energy flux (J m−2s−1) | |
| = | non-advective energy flux associated with mechanical processes in entity α (J m−2s−1) | |
| = | macroscale entity entropy balance equation, Eq. (Equation6 | |
| s | = | solid phase entity index (−) |
| ss | = | solid surface index (−) |
| = | macroscale transfer of momentum from entity κ to entity α due to stress and deviation from mean processes (kg m−2s−2) | |
| = | microscale transfer of momentum from entity κ to entity α (kg m−2s−2) | |
| = | macroscale Euler equation for all entities except the solid phase, Eq. (Equation8 | |
| = | material derivative of the body source potential, Eq. (Equation10 | |
| t | = | time (s) |
| = | stress tensor (kg m−1 s−2) | |
| = | concentrated Eulerian solid-phase stress tensor (kg m−1 s−2) | |
| = | velocity (ms−1) | |
| = | reference velocity (ms−1) | |
| = | velocity of flow in an entity averaged over the boundary of the entity (ms−1) | |
| = | velocity fluctuation (ms−1) | |
| W | = | weighting function used in averaging (−) |
| w | = | wetting phase entity index (−) |
| wg | = | index of interface between w and g phases (−) |
| wgs | = | index of a common curve where wg, ws, and gs interfaces meet (−) |
| ws | = | index of interface between w and s phases (−) |
| α | = | general entity index (subscript is used to qualify microscale quantities, while superscript is used to denote macroscale quantities) (−) |
| β | = | general entity index (−) |
| = | drag parameter (kg m−3 s−1) | |
| Γ | = | boundary of domain Ω (−) |
| = | boundary of domain | |
| = | portion of | |
| = | portion of | |
| γ | = | interfacial (J m−3) or common curve lineal tension (J m−2) |
| ε | = | porosity (−) |
| εα | = | specific entity measure (−) |
| η | = | entropy density (JK−1m−3) |
| = | partial mass entropy (JK−1 kg−1) | |
| = | partial mass entropy of an entity averaged over the boundary of the entity (JK−1 kg−1) | |
| θ | = | temperature (K) |
| θα | = | entropy-weighted macroscale temperature of entity α (K) |
| = | geodesic curvature, | |
| = | normal curvature, | |
| Λ | = | entropy production rate (J K−1 m−3 s−1) |
| μ | = | chemical potential (m2 s−2) |
| = | turbulent viscosity (m2 s−1) | |
| ρ | = | mass density (kg m−3) |
| = | Lagrangian solid-phase stress tensor (kg m−1 s−2) | |
| = | concentrated Lagrangian solid-phase stress tensor (kg m−1 s−2) | |
| = | Schmidt number (−) | |
| = | viscous stress tensor (kg m−1 s−2) | |
| = | macroscale transfer of entropy from entity κ to entity α (J K−1 m−3 s−1) | |
| = | non-advective entropy flux (J K−1 m−2 s−1) | |
| = | microscale contact angle between ws and wg interfaces (rad) | |
| = | macroscale measure of contact angle (rad) | |
| = | fraction of boundary of entity α in contact with entity κ (−) | |
| Ψ | = | body force potential density (kg m−1 s−2) |
| = | body force potential density of an entity averaged over the boundary of the entity (m2 s−2) | |
| ψ | = | body force potential per unit mass (m2 s−2) |
| = | spatial domain of the averaging volume (−) | |
| = | domain occupied by entity α within the averaging volume (−) | |
| = | closed domain consisting of a domain and its boundary such that | |
| = | mass fraction (−) | |
| = | above a superscript refers to a density-weighted macroscale average | |
| = | above a superscript refers to a uniquely defined macroscale average | |
| = |
| |
| = | ||
| = |
| |
| = |
| |
| = |
| |
| = |
| |
| = | macroscale material derivative with macroscale velocity | |
| = | macroscale material derivative with macroscale velocity reference | |
| = | material derivative for an interface with respect to the reference velocity | |
| = | material derivative for a common curve with respect to the reference velocity | |
| = | material derivative on a 3−n-dimensional entity where the macroscale reference velocity is employed, | |
| = |
| |
| ∇ | = | spatial gradient operator |
| ∇· | = | spatial divergence operator |
| ∇′ | = | microscale surficial gradient operator on a microscale interface |
| ∇′· | = | microscale surface divergence vector |
| ∇′′ | = | microscale gradient operator along a curve |
| ∇′′· | = | microscale divergence operator for a curve |
| = | microscale 3−n-dimensional del operator, | |
| = | time derivative with all spatial coordinates fixed | |
| = | partial time derivative fixed to point on a surface of a function dependent on microscale spatial coordinates and time | |
| = | partial time derivative fixed to a point on a curve of a function dependent on microscale spatial coordinates and time | |
| = | partial time derivative at a point fixed on 3−n-dimensional entity, where | |
| = | spatial dimensionality of the entity |
Notes
1 For consistency, the notation adopted here follows closely that used in the large body of TCAT literature. We hope that this choice will make it easier for interested readers to delve deeper into this literature.