The Mathematics of Fluid Flow Through Porous Media. Myron B. Allen, III

nabla dot left-parenthesis rho omega Subscript alpha Baseline bold v Subscript alpha Baseline right-parenthesis 2nd Column equals r Subscript alpha Baseline comma 3rd Column alpha 4th Column equals 1 comma 2 comma ellipsis comma upper N semicolon 3rd Row 1st Column ModifyingBelow StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis rho omega Subscript alpha Baseline right-parenthesis With presentation form for vertical right-brace Underscript left-parenthesis upper I right-parenthesis Endscripts plus ModifyingBelow nabla dot left-parenthesis rho omega Subscript alpha Baseline bold v right-parenthesis With presentation form for vertical right-brace Underscript left-parenthesis II right-parenthesis Endscripts plus ModifyingBelow nabla dot bold j Subscript alpha With presentation form for vertical right-brace Underscript left-parenthesis III right-parenthesis Endscripts 2nd Column equals ModifyingBelow r Subscript alpha Baseline With presentation form for vertical right-brace Underscript left-parenthesis IV right-parenthesis Endscripts comma 3rd Column alpha 4th Column equals 1 comma 2 comma ellipsis comma upper N semicolon EndLayout"/>

where is the diffusive flux of constituent . In the last form, we refer to the terms labeled (I), (II), (III), and (IV) as the accumulation, advection, diffusion, and reaction terms, respectively.

      The following exercise reassuringly shows that the multiconstituent mass balance reduces to the single‐constituent mass balance if we use the definitions of the mixture density and the barycentric velocity and ignore the distinctions among constituents.

Exercise 2.15 Use the definitions of the multiconstituent density and the barycentric velocity to show that Eq. (2.28) is equivalent to

      2.5.4 Multiconstituent Momentum Balance

      The differential momentum balance for multicomponent continua, in a form paralleling Eqs. (2.29) and (2.30), is

(2.31) (2.32)

      Here, represents the rate of momentum exchange into from other constituents, excluding momentum exchanges associated purely with the transfer of mass into from other constituents. The term gives the rate of momentum exchange into attributable to mass exchange from other constituents. Equation (2.31) plays a central role in modeling fluid velocities in porous media, as discussed in Sections 3.1 and 3.2.

      As with the multiconstituent mass balance equation, one can retrieve the momentum balance for a simple continuum by summing over all constituents and ignoring the distinction among them. This derivation requires a bit of tensor notation encountered again in Section 5.1.

      Exercise 2.16 For any two vectors , the dyadic product is a tensor having the following action on any vector :

      (2.33)

      Verify that the mapping is linear.

      Exercise 2.17 Recall from Section 2.2 that the matrix representation of any tensor with respect to an orthonormal basis has entries . Compute the matrix representation of .

      Exercise 2.18 Sum Eq. (2.31) and use Eq. (2.32), together with the definitions of multiconstituent density and barycentric velocity , to get

       where

       gives the total body force per unit mass and

(2.34)

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