For general flow fields, it is necessary to develop numerical techniques to track down yielded/unyielded regions.
This approximation could be made more and more accurate at even vanishingly small shear rates by means of a material parameter that controls the exponential growth of stress.
Thus, a new impetus was given in 1987 with the publication by Papanastasiou[4] of such a modification of the Bingham model with an exponential stress-growth term.
The early efforts by Papanastasiou and his co-workers were taken up by the author and his coworkers,[5] who in a series of papers solved many benchmark problems and presented useful solutions always providing the yielded/unyielded regions in flow fields of interest.
[8] He introduced a continuous regularization for the viscosity function which has been largely used in numerical simulations of viscoplastic fluid flows, thanks to its easy computational implementation.
As a weakness, its dependence on a non-rheological (numerical) parameter, which controls the exponential growth of the yield-stress term of the classical Bingham model in regions subjected to very small strain-rates, may be pointed.
Thus it avoids solving explicitly for the location of the yield surface, as was done by Beris et al.[9] Papanastasiou's modification, when applied to the Bingham model, becomes in simple shear flow (1-D flow): Bingham-Papanastasiou model: where η is the apparent viscosity.