Page 365 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 349

It is easy to show from Eq. (i), that the magnitude of the stress vector A is the same for every
plane passing through a given point. In fact, let n^ and n 2 be normals to any two such planes,
then we have

and,

Thus,

Since 112 • Tn^ = n t • T n 2 and T is symmetric, therefore, the left side of Eq. (iv) is zero.
Thus,

Since ii} and n 2 are any two vectors, therefore,

In other words, on all planes passing through a point, not only are there no shearing stresses
but also the normal stresses are all the same. We shall denote this normal stress by -p. Thus,
for a fluid in rigid body motion or at rest

Or, in component form

The scalar p is the magnitude of the compressive normal stress and is known as the
hydrostatic pressure.

6.2 Compressible and Incompressible Fluids

What one generally calls a "liquid" such as water or mercury has the property that its density
essentially remains unchanged under a wide range of pressures. Idealizing this property, we
define an incompressible fluid to be one for which the density of every particle remains the
same at all times regardless of the state of stress. That is for an incompressible fluid

It then follows from the equation of conservation of mass, Eq. (3.15.2b)