Differential constitutive equations for viscoelastic fluids

Depending on the method of analysis, constitutive models of viscoelastic fluids can be formulated as differential or integral equations. 

In the differential models stress components, and their material derivatives, arc related to the rate of strain components and their material derivatives. 

The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950) 

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation 

---