Criminale-Ericksen-Filbey equation

We have seen that a simple apparent viscosity is used to characterize the stress-shear rate relationship for a non-Newtonian fluid. For a viscoelastic fluid, additional coefficients are required to determine the state of stress in any flow. For steady simple shear flow, the additional coefficients are given by the Criminale-Ericksen-Filbey equation... 

The present analysis was based on the lubrication approximation that is, we neglected changes in the x direction. If this assumption is lifted, we are faced with a flow field in which two nonvanishing velocities exist that are functions of two spatial coordinates, vx(x, y), vy (x, y). This is clearly a nonviscometric flow situation, and the Criminale-Ericksen-Filbey (CEF) equation is not applicable. White (13) made an order of magnitude evaluation of normal stress effects for this more realistic flow situation. In this case, the equation of motion reduces to... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Criminale, Ericksen, and Filbey [C21] (see also White [W5]) have shown that for long duration shearing flows the general constitutive equations of form Eqs. (46) and (47) reduce to the form... 

---