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Newtonian Viscous Fluid 383

6.17 Dissipation Functions for Newtonian Fluids

The rate of work done P by the stress vectors and the body forces on a material particle of
a continuum was derived in Chapter 4, Section 4.12 to be given by

where dVis the volume of the material particle. In Eq. (6.17.1), the first term in the right side
is the rate of change of the kinetic energy (K.E.) and the second termPy dFis the rate of work
done to change the volume and shape of the "particle" of volume dV. Per unit volume, this
rate is denoted by P s and is known as the stress working or stress power.
In this section, we shall compute the stress power for a Newtonian fluid.
(A) Incompressible Newtonian Fluid.

We have.

thus

Since the fluid is incompressible, dv/dxj = 0, therefore,

i.e.,

This is the work per unit volume per unit time done to change the shape and this part of the
work accumulates with time regardless of how Z),y vary with time (P s is always positive and is
zero only for rigid body motions). Thus, the function

is known as the dissipation function for an incompressible Newtonian fluid. It represents the
rate at which work is converted into heat.
(B) Newtonian Compressible Fluid

dV;
For this case, we have, with A denoting -r—
dXj