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

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bounding surface partial-differential script upper V. We account for these forces by introducing tractions, having dimension force per unit area:

dimension left-parenthesis StartFraction Force Over Area EndFraction right-parenthesis equals StartFraction normal upper M normal upper L normal upper T Superscript negative 2 Baseline Over normal upper L squared EndFraction equals normal upper M normal upper L Superscript negative 1 Baseline normal upper T Superscript negative 2 Baseline period integral Underscript partial-differential script upper V Endscripts sans-serif upper T bold n d a comma

      having dimension MLT Superscript negative 2.

      Four additional remarks help clarify the nature of the stress tensor.

      1 With respect to any orthonormal basis , any linear transformation has a matrix representation with entries . For , this representation has the formFigure 2.6 A region in three‐dimensional space with unit outward normal vector field and the traction acting on the boundary .

      2 In accordance with Exercise 2.4, with respect to any orthonormal basis, the diagonal entries represent forces per unit area acting in directions perpendicular to faces that are orthogonal to , , and , respectively. We refer to these entries as tensile stresses when they pull in the same direction as and as compressive stresses when they push in the opposite direction—namely inward—from . The off‐diagonal entries , where , are shear stresses.

      3 A classic theorem in continuum mechanics reduces the angular momentum balance, which we do not discuss here, to the identity with respect to any orthonormal basis. In other words, the stress tensor is symmetric. See [4, Chapter 4] for details.

      4 With respect to an orthonormal basis , the divergence of the tensor‐valued function has the following representation as a vector‐valued function:

 Geometric representation of a cube of material illustrating the interpretations of entries of the stress tensor matrix with respect to an orthonormal basis, from [4, page 109]. StartFraction 1 Over Volume EndFraction left-parenthesis Mass times Acceleration equals sigma-summation Forces right-parenthesis period

      Based on this parallel, fluid mechanicians call

rho StartFraction upper D bold v Over upper D t EndFraction equals rho StartFraction partial-differential bold v Over partial-differential t EndFraction plus rho left-parenthesis bold v dot nabla right-parenthesis bold v

      the inertial terms.

      If we view the momentum balance as an equation for the velocity bold v, the inertial terms make the momentum balance a nonlinear PDE. In many applications to fluid mechanics, this nonlinearity wreaks mathematical havoc. Mercifully, for reasons examined in Chapter 3, the inertial nonlinearity plays a negligible role in the most commonly used models of porous‐media flow. However, this observation furnishes scant grounds for complacency. As subsequent chapters demonstrate, other types of nonlinearity play prominent roles in the fluid mechanics of porous media.

      The mass and momentum balance laws