Page 483 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 467

It is interesting to consider the limiting cases of the Maxwell element. If G = °°, then the
spring element becomes a rigid bar and the element no longer possesses elasticity. That is, it
is a purely viscous element. In creep experiment, there will be no instantaneous elongation,
the element simply creeps linearly with time (see Eq. (8.1.6)) from the unstretched initial
position. In the stress relaxation experiment, an infinitely large force is needed at t =0 to
produce the finite jump in elongation (from 0 to 1). The force however is instantaneously
returned to zero (i.e., the relaxation time A = rj/G -*0). We can write the relaxation function
for the purely viscous element in the following way

where d(t) is known as the Dirac delta function which may be defined to be the derivative of
the unit step function H(t) defined by:

Thus,

and

Example 8.1.2
Consider a linear Maxwell fluid, defined by Eq. (8.1.1), in steady simple shearing flow:

Find the stress components.
Solution. Since the given velocity field is steady, all field variables are independent of time.
*Y -
Thus, — = 0 and we have
dt

Thus, the stress field is exactly the same as that of a Newtonian incompressible fluid and the
viscosity is independent of the rate of shear for this fluid.