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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
.
2.3.1 Body Force
For the body force, which is gravity in all of the problems examined here, we adopt the constitutive relationship . Here m denotes the gravitational acceleration, which varies across Earth's surface, and we adopt a Cartesian coordinate system in which points away from Earth's center, as shown in Figure 2.8.
An alternative way of writing this expression proves useful in subsequent sections. Define the depth function as the mapping that assigns to each spatial point its depth below some datum, at which , as drawn in Figure 2.8. We often take the datum to be Earth's surface, but other choices are possible. Observe that
which has dimension . Therefore, we write the constitutive equation for the body force as .
2.3.2 Stress in Fluids
The stress tensor enjoys a richer set of possibilities. The simplest is the constitutive relationship for an ideal fluid, in which . Here, is a scalar function called the mechanical pressure, having dimension (force/area). The SI unit for pressure is 1 pascal, abbreviated as 1 Pa and defined as 1 kg . The symbol denotes the identity tensor. With respect to any orthonormal basis, the stress of an ideal fluid has matrix representation
(2.11)
Thus, in an ideal fluid, there are no shear stresses, and the fluid experiences only compressive and tensile stresses. Also, there are no preferred directions: . We describe this fact by saying that the stress tensor is isotropic. Section 3.7 discusses isotropic tensors in more detail.
For an ideal fluid in the presence of gravity, the momentum balance reduces to the following equation:
In problems for which inertial terms are negligible, for example when the fluid is at rest, this equation reduces to
(2.12)
Exercise
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