Page 372 - Introduction to Continuum Mechanics

356 Newtonian Fluid

shear stress for a fluid in shearing motion is independent of shear deformation, but is
dependent on the rate of shear. For such fluids, no shear stress is needed to maintain a given
amount of shear deformation, but a definite amount of shear stress is needed to maintain a
constant rate of shear of deformation.

Since the state of stress for a fluid under rigid body motion (including rest) is given by an
isotropic tensor, therefore in dealing with a fluid in general motion, it is natural to decompose
the stress tensor into two parts:

where the components of T depend only on the rate of deformation (i.e., not on deformation)
in such a way that they are zero when the fluid is under rigid body motion (i.e., zero rate of
deformation) and/? is a scalar whose value is not to depend explicitly on the rate of deforma-
tion.
We now define a class of idealized materials called Newtonian fluids as follows:

I. For every material point, the values of T)y' at any time t depend linearly on the components
of the rate of deformation tensor

at that time and not on any other kinematic quantities (such as higher rates of deformation)
II. The fluid is isotropic with respect to any configuration.
Following the same arguments made in connection with the isotropic linear elastic material,
we obtain that for a Newtonian fluid, (also known as linearly viscous fluid, the most general
=
form of TJJ is, with A = />ii+D22+^33 Awt»

where A and /* are material constants (different from those of an elastic body) having the
*J
dimension of (Force)(Time)/(Length) . The stress tensor TJy is known as the viscous stress
tensor. Thus, the total stress tensor is

i.e.,