Page 373 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 357

The scalar p in the above equations is called the pressure. It is a somewhat ambiguous
terminology. As is seen from the above equations, when Dy are nonzero, p is only a part of the
total compressive normal stress on a plane. It is in general neither the total compressive normal
stress on a plane (unless the viscous stress components happen to be zero), nor the mean
normal compressive stress, (see next section). As a fluid theory, it is only necessary to
remember that the isotropic tensor -p<5y is that part of Tq which does not depend explicitly
on the rate of deformation.

6.5 Interpretation of A and //

Consider the shear flow given by the velocity field:

For this flow

and

so that

and

Thus, /* is the proportionality constant relating the shearing stress to the rate of decrease of
angle between two mutually perpendicular material lines (see Sect.3.13). It is called the first
coefficient of viscosity or simply viscosity. From Eq. (6.4.3), we have, for a general velocity
field.