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
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
An alternative way of writing this expression proves useful in subsequent sections. Define the depth function
which has dimension
2.3.2 Stress in Fluids
The stress tensor
(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:
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
Exercise 2.5 Integrate the third component of Eq. (2.12) to obtain the hydrostatic equation,
Thus pressure increases linearly with depth in an ideal fluid at rest.
Equation (2.13) closely models the pressure of Earth's atmosphere. At sea level, the pressure of the atmosphere fluctuates around
