Page 489 - Introduction to Continuum Mechanics

Non-Newtonian Fluids 473

In the above equation, if a^ = 0, the equation is sometimes called the Jeffrey's model.

8.3 Integral Form of the Linear Maxwell Fluid and of the Generalized Linear
Maxwell Fluid with Discrete Relaxation Spectra

Consider the following integral form of constitutive equation:

where

is the shear relaxation function for the linear Maxwell fluid defined by Eq. (8.Lib). If we
differentiate Eq. (8.3.1) with respect to time t, we obtain (note that / appears in both the
integrand and the integration limit, we need to use the Leibnitz rule of differentiation)

That is,

Thus, the integral form constitutive equation, Eqs. (8.3.1) is the same as the rate form
constitutive equation, Eq. (8.1. Ib). Of course, Eq. (8.3.1) is nothing but the solution of the
linear non-homogeneous ordinary differential equation, Eq. (S.l.lb). [See Prob. 8.6]
It is not difficult to show that the constitutive equation for the generalized linear Maxwell
equation with N discrete relaxation spectra, Eq. (8.2.1) is equivalent to the following integral
form

We may write the above equation in the following form:

where the shear relaxation function <p(t} is given by