Page 364 - Introduction to Continuum Mechanics

6

Newtonian Viscous Fluid

Substances such as water and air are examples of a fluid. Mechanically speaking they are
different from a piece of steel or concrete in that they are unable to sustain shearing stresses
without continuously deforming. For example, if water or air is placed between two parallel
plates with say one of the plates fixed and the other plate applying a shearing stress, it will
deform indefinitely with time if the shearing stress is not removed. Also, in the presence of
gravity, the fact that water at rest always conforms to the shape of its container is a demonstra-
tion of its inability to sustain shearing stress at rest. Based on this notion of fluidity, we define
a fluid to be a class of idealized materials which, when in rigid body motion(inciuding the state
of rest), cannot sustain any shearing stress. Water is also an example of a fluid that is referred
to as a liquid which undergoes negligible density changes under a wide range of loads, whereas
air is a fluid that is referred to as a gas which does otherwise. This aspect of behavior is
generalized into the concept of incompressible and compressible fluids. However, under
certain conditions (low Mach number flow) air can be treated as incompressible and under
other conditions (e.g. the propagation of the acoustic waves) water has to be treated as
compressible.

In this chapter, we study a special model of fluid, which has the property that the stress
associated with the motion depends linearly on the instantaneous value of the rate of defor-
mation. This model of fluid is known as a Newtonian fluid or linearly viscous fluid which has
been found to describe adequately the mechanical behavior of many real fluids under a wide
range of situations. However, some fluids, such as polymeric solutions, require a more general
model (Non-Newtonian Fluids) for an adequate description. Non-Newtonian fluid models
will be discussed in Chapter 8.

6.1 Fluids

Based on the notion of fluidity discussed in the previous paragraphs, we define a fluid to
be a class of idealized materials which when in rigid body motions (including the state of rest)
cannot sustain any shearing stresses. In other words, when a fluid is in a rigid body motion, the
stress vector on any plane at any point is normal to the plane. That is for any n,