Читать книгу The Mathematics of Fluid Flow Through Porous Media - Myron B. Allen III - Страница 22

The mass and momentum balance laws

(2.10)

furnish four scalar PDEs involving the 16 scalar functions required to specify , , , and . The symmetry of the stress tensor, , reduces the number of independent scalar functions to 13. From the mathematical point of view, a well posed problem involving Eqs. (2.10) requires additional equations to close the system. We call these equations constitutive relationships.

From the engineer's point of view, constitutive relationships define the physical system being modeled. Since the mass and momentum balance laws apply to all materials, by themselves they provide no way to distinguish among different types of fluids and solids. If we regard the differential equations (2.10) as governing the mass density and velocity , then we need to specify constitutive relationships for the three scalar functions defining the body force and the six independent scalar functions that define the matrix representation of the stress tensor. This book examines only a small number of constitutive relationships, chosen from the myriad that scientists and engineers have developed to model the remarkable variety of materials found in nature.

Figure 2.8 Coordinate system used to define the depth function .