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

where denotes the divergence operator. With respect to an orthonormal basis ,

Applying the identities (2.4)and (2.5)to the integral mass balance (2.3)yields the equivalent equation

(2.6)

valid for any time‐independent region .

If the integrand in Eq. (2.6)is continuous, then the integrand must vanish:

(2.7)

Equation (2.7)is the differential mass balance.

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 at some time . Since this function is continuous, it must be positive in a neighborhood of at time . 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):

(2.8)

The differential mass balance in the form (2.8)facilitates another observation. In certain motions, the density following any particle is constant. In this case, , so the mass balance implies that

In this case, we say that the motion is incompressible. This concept does not imply anything about the material being modeled; it merely describes the motion based on properties of the velocity field. A compressible material can undergo incompressible motion.

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 5and 6, the remainder of this book focuses on differential balance laws.

The differential momentum balance equation is

(2.9)

often called Cauchy's first law. (For its derivation from an integral form, see [4, Chapter 4]. Strictly speaking, the momentum balance states that there exists a frame of reference in which Cauchy's first law holds.) Each term in Eq. (2.9)is a vector‐valued function having dimension . Thus, Cauchy's first law comprises three scalar PDEs.

The terms in Eq. (2.9)require explanation. First, with respect to any orthonormal basis ,

so applying this operator to yields

which is clearly a vector‐valued function.

Second, the function represents the body force per unit mass, having dimension . In this book, the only body force of interest is gravity, and reduces to the gravitational acceleration near Earth's surface. The total body force acting on a part of the body is

where is the region occupied by the part.

Third, the function 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.