Modification of the fundamental theorem for transport phenomena in porous media

Takatsu, Yasuyuki

doi:10.1016/j.ijheatmasstransfer.2017.08.067

Cited by 9 publications

(5 citation statements)

References 28 publications

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“…Second, following a rich line of research, e.g., [ 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 ], the fluid or material density

at each position

and time t can be rigorously defined by integration over a small fluid volume

—or equivalently a small fluid mass

—for which there are two interpretations. In the common viewpoint,

must be sufficiently large to enable the fluid to be considered a continuum, e.g., [ 44 , 45 , 46 , 47 , 56 , 57 ].…”

Section: A Hierarchy Of Densitiesmentioning

confidence: 99%

“…Consider the generalized volumetric density

[qty m −3 ] in Equation ( 24 ) based on the fluid or material density

. This can be defined by the convolution [ 41 , 44 , 45 , 46 , 47 , 48 , 51 , 56 , 57 , 58 , 60 , 61 , 62 , 64 , 66 , 67 , 68 ]:

where

indicates a volumetric expected value,

is a small fluid domain,

[m 3 ] is the volume of

, and here

is an infinitesimal volume element in

. Usually Equation ( A1 ) has a geometric interpretation, but V can be reinterpreted as the measure of set

[ 52 , 59 , 61 ].…”

mentioning

confidence: 99%

“…In multiphase systems, variants of Equation ( A1 ) can be defined for each phase, in which many authors use a characteristic function

to indicate the presence of phase

[ 47 , 48 , 51 , 52 , 53 , 54 , 56 , 57 , 58 , 59 , 60 , 66 ]. Equation ( A1 ) has also been generalized to give a variety of averages in zero-, one-, two- and three-dimensional spaces and/or time [ 45 , 46 , 48 , 56 , 57 , 60 , 62 , 63 , 65 ], and used to examine the effects of multiple or time-varying averaging scales [ 48 , 54 , 57 , 58 , 59 , 61 , 62 , 63 , 64 , 65 , 66 , 68 ]. Equation ( A1 ) can be further generalized using a weighting function, here labeled w [ 33 , 41 , 42 , 43 , 50 , 53 , 54 , 58 , 59 , 61 ,…”

mentioning

confidence: 99%

“…The definition of a fluid or generalized density based on spatial averaging has received considerable attention in the literature, especially for the analysis of fluid flow through porous media, where many different averages can be defined and a precise terminology is necessary, e.g., [ 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 ].…”

mentioning

confidence: 99%

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A Hierarchy of Probability, Fluid and Generalized Densities for the Eulerian Velocivolumetric Description of Fluid Flow, for New Families of Conservation Laws

Niven

2022

Entropy

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The Reynolds transport theorem occupies a central place in continuum mechanics, providing a generalized integral conservation equation for the transport of any conserved quantity within a fluid or material volume, which can be connected to its corresponding differential equation. Recently, a more generalized framework was presented for this theorem, enabling parametric transformations between positions on a manifold or in any generalized coordinate space, exploiting the underlying continuous multivariate (Lie) symmetries of a vector or tensor field associated with a conserved quantity. We explore the implications of this framework for fluid flow systems, based on an Eulerian velocivolumetric (position-velocity) description of fluid flow. The analysis invokes a hierarchy of five probability density functions, which by convolution are used to define five fluid densities and generalized densities relevant to this description. We derive 11 formulations of the generalized Reynolds transport theorem for different choices of the coordinate space, parameter space and density, only the first of which is commonly known. These are used to generate a table of integral and differential conservation laws applicable to each formulation, for eight important conserved quantities (fluid mass, species mass, linear momentum, angular momentum, energy, charge, entropy and probability). The findings substantially expand the set of conservation laws for the analysis of fluid flow and dynamical systems.

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at each position

and time t can be rigorously defined by integration over a small fluid volume

—or equivalently a small fluid mass

—for which there are two interpretations. In the common viewpoint,

must be sufficiently large to enable the fluid to be considered a continuum, e.g., [ 44 , 45 , 46 , 47 , 56 , 57 ].…”

Section: A Hierarchy Of Densitiesmentioning

confidence: 99%

“…Consider the generalized volumetric density

[qty m −3 ] in Equation ( 24 ) based on the fluid or material density

. This can be defined by the convolution [ 41 , 44 , 45 , 46 , 47 , 48 , 51 , 56 , 57 , 58 , 60 , 61 , 62 , 64 , 66 , 67 , 68 ]:

where

indicates a volumetric expected value,

is a small fluid domain,

[m 3 ] is the volume of

, and here

is an infinitesimal volume element in

. Usually Equation ( A1 ) has a geometric interpretation, but V can be reinterpreted as the measure of set

[ 52 , 59 , 61 ].…”

mentioning

confidence: 99%

“…In multiphase systems, variants of Equation ( A1 ) can be defined for each phase, in which many authors use a characteristic function

to indicate the presence of phase

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A Hierarchy of Probability, Fluid and Generalized Densities for the Eulerian Velocivolumetric Description of Fluid Flow, for New Families of Conservation Laws

Niven

2022

Entropy

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“…The Forchheimer number has been proposed as the critical factor to estimate the conversion from Darcy flow to non-Darcian flow, and it is expressed as follows 28 : …”

Section: Non-darcian Theorymentioning

confidence: 99%

Re-crushing process and non-Darcian seepage characteristics of broken coal medium in coal mine water inrush

Pang

Zhang

Guo

et al. 2021

Sci Rep

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The initiation process of the mine water inrush accident, the essence of this process is the sudden change of the seepage state of the broken coal medium under pressure and the instability of the skeleton. In order to study the re-crushing mechanism and seepage characteristics of the broken coal medium under load, a set of three-axis seepage system was designed independently. Using the steady-state infiltration method, multiple flow factors under different particle size combinations and different stress conditions of the broken coal medium were obtained. The results of the study indicate: in one hand, the reduction of the porosity of the broken coal medium will cause the flow channel to be rebuilt, and the sudden change of flow rate will directly lead to the non-Darcian flow behavior. The early stage of compaction mainly affects the permeability k value, and the later stage of compaction mainly affects the non-Darcian β value; On the other hand, the seepage throat in the broken coal medium may have a sharp increase in its flow rate, leading to a sudden change in the flow pattern. The critical Reynolds number is also used to determine whether non-Darcian flow is formed, and its value in the water inrush system is about 40–133; at the same time, the non-Darcian flow in the broken coal medium conforms to the Forchheimer-type flow law. By analyzing the dependence relationship between factors, a seepage factor representation algebraic relationship suitable for Forchheimer type non-Darcian flow of broken coal medium is given, which can be used as a calculation basis in the prevention and treatment of mine water inrush accidents.

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A Brief Introduction to Convection in Porous Media

Nield

Simmons

2018

Transp Porous Med

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Modification of the fundamental theorem for transport phenomena in porous media

Cited by 9 publications

References 28 publications

A Hierarchy of Probability, Fluid and Generalized Densities for the Eulerian Velocivolumetric Description of Fluid Flow, for New Families of Conservation Laws

A Hierarchy of Probability, Fluid and Generalized Densities for the Eulerian Velocivolumetric Description of Fluid Flow, for New Families of Conservation Laws

Re-crushing process and non-Darcian seepage characteristics of broken coal medium in coal mine water inrush

A Brief Introduction to Convection in Porous Media

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