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

Скачать книгу

      Exercise 2.2 Verify the principle used to derive Eq. (2.7) from the integral equation (2.6). An argument by contradiction may be the easiest approach: Assume that the integrand on the left side of Eq. (2.6) is positive at some point bold x at some time t. Since this function is continuous, it must be positive in a neighborhood of bold x at time t. Consider a fixed region contained in this neighborhood. A similar argument dispatches the possibility that the integrand is negative at some point.

      Exercise 2.3 Justify the following equivalent of the mass balance (2.7):

nabla dot bold v equals 0 period

      The mass balance is the simplest of the balance laws of continuum mechanics. Other balance laws include the momentum balance, the angular momentum balance, and the energy balance. A related thermodynamic law known, as the entropy inequality, also plays an important role in many settings. In each of these laws, an integral version is fundamental, and it is possible to derive differential versions under certain continuity conditions. For a detailed review of the integral balance laws and the derivation of their differential versions, see [4]. With the exception of several applications of the mass balance discussed in Chapters 5 and 6, the remainder of this book focuses on differential balance laws.

      2.2.2 Momentum Balance

      The differential momentum balance equation is

      The terms in Eq. (2.9) require explanation. First, with respect to any orthonormal basis StartSet bold e 1 comma bold e 2 comma bold e 3 EndSet,

bold v dot nabla equals sigma-summation Underscript j equals 1 Overscript 3 Endscripts v Subscript j Baseline StartFraction partial-differential Over partial-differential x Subscript j Baseline EndFraction comma

      so applying this operator to bold v yields

StartLayout 1st Row 1st Column StartFraction upper D bold v Over upper D t EndFraction equals left-parenthesis StartFraction partial-differential Over partial-differential t EndFraction plus bold v dot nabla right-parenthesis bold v 2nd Column equals left-parenthesis StartFraction partial-differential Over partial-differential t EndFraction plus sigma-summation Underscript j equals 1 Overscript 3 Endscripts v Subscript j Baseline StartFraction partial-differential Over partial-differential x Subscript j Baseline EndFraction right-parenthesis sigma-summation Underscript i equals 1 Overscript 3 Endscripts v Subscript i Baseline bold e Subscript i Baseline 2nd Row 1st Column Blank 2nd Column equals sigma-summation Underscript i equals 1 Overscript 3 Endscripts left-parenthesis StartFraction partial-differential v Subscript i Baseline Over partial-differential t EndFraction plus sigma-summation Underscript j equals 1 Overscript 3 Endscripts v Subscript j Baseline StartFraction partial-differential v Subscript i Baseline Over partial-differential x Subscript j Baseline EndFraction right-parenthesis bold e Subscript i Baseline comma EndLayout

      which is clearly a vector‐valued function.

integral Underscript script upper V Endscripts rho bold b d v comma

      where script upper V is the region occupied by the part.

      Third, the function sans-serif upper T left-parenthesis bold x comma t right-parenthesis is the stress tensor. This entity deserves more extended discussion, starting with the term tensor. A second‐order tensor is a linear transformation that maps vectors into vectors. Its geometric action remains fixed under changes in coordinate systems, a requirement discussed in more detail in Section 3.7. The stress tensor is a linear transformation that describes a type of force different from the body force.

      More specifically, any part of a body occupying a region script upper V in three‐dimensional space can experience forces acting

Скачать книгу