Bingham-Papanastasiou model

An important class of non-Newtonian fluids presents a yield stress limit which must be exceeded before significant deformation can occur – the so-called viscoplastic fluids or Bingham plastics. In order to model the stress-strain relation in these fluids, some fitting have been proposed such as the linear Bingham equation and the non-linear Herschel-Bulkley and Casson models.[1]

Analytical solutions exist for such models in simple flows. For general flow fields, it is necessary to develop numerical techniques to track down yielded/unyielded regions. This can be avoided by introducing into the models a continuation parameter, which facilitates the solution process and produces virtually the same results as the ideal models by the right choice of its value.[2]

Viscoplastic materials like slurries, pastes, and suspension materials have a yield stress, i.e. a critical value of stress below which they do not flow are also called Bingham plastics, after Bingham.[3]

Viscoplastic materials can be well approximated uniformly at all levels of stress as liquids that exhibit infinitely high viscosity in the limit of low shear rates followed by a continuous transition to a viscous liquid. 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 new model basically rendered the original discontinuous Bingham viscoplastic model as a purely viscous one, which was easy to implement and solve and was valid for all rates of deformation. 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. Since the early 1990s, other workers in the field also used the Papanastasiou model for many different problems.[citation needed]

## Papanastasiou

Papanastasiou in 1987, who took into account earlier works in the early 1960s[6] as well as a well-accepted practice in the modelling of soft solids[7] and the sigmoidal modelling behaviour of density changes across interfaces.[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, he proposed an exponential regularization of eq., by introducing a parameter m, which controls the exponential growth of stress, and which has dimensions of time. The proposed model (usually called Bingham-Papanastasiou model) has the form:

τ→→=(μ+τy∣γ˙∣[1−exp⁡(−m∣γ˙∣)])(γ˙→→)

and is valid for all regions, both yielded and unyielded. 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:

* τ=τy[1−exp⁡(−mγ˙)]+μγ˙
*   * η=μ+τy∣γ˙∣[1−exp⁡(−m∣γ˙∣)]

where η is the apparent viscosity.

## References

1. ↑ Soto, Hilda Pari; Martins-Costa, Maria Laura; Fonseca, Cleiton; Frey, Sérgio (December 2010). "A numerical investigation of inertia flows of Bingham-Papanastasiou fluids by an extra stress-pressure-velocity galerkin least-squares method". Journal of the Brazilian Society of Mechanical Sciences and Engineering 32 (5): 450–460. doi:10.1590/S1678-58782010000500004.
2. ↑ Mitsoulis, Evan; Tsamopoulos, John (March 2017). "Numerical simulations of complex yield-stress fluid flows". Rheologica Acta 56 (3): 231–258. doi:10.1007/s00397-016-0981-0. ProQuest 2261996678.
3. ↑ Bingham, Eugene Cook (1922). Fluidity and plasticity. McGraw-Hill. OCLC 1118524319. [page needed]
4. ↑ Papanastasiou, Tasos C. (July 1987). "Flows of Materials with Yield". Journal of Rheology 31 (5): 385–404. doi:10.1122/1.549926. Bibcode: 1987JRheo..31..385P.
5. ↑ Ellwood, K. R. J.; Georgiou, G. C.; Papanastasiou, T. C.; Wilkes, J. O. (August 1990). "Laminar jets of Bingham‐plastic liquids". Journal of Rheology 34 (6): 787–812. doi:10.1122/1.550144. Bibcode: 1990JRheo..34..787E.
6. ↑ Shangraw, Ralph; Grim, Wayne; Mattocks, Albert M. (March 1961). "An Equation for Non‐Newtonian Flow". Transactions of the Society of Rheology 5 (1): 247–260. doi:10.1122/1.548898. Bibcode: 1961JRheo...5..247S.
7. ↑ Gavrus, A.; Ragneau, E.; Caestecker, P. (2001). "Rheological behaviour formulation of solid metallic materials for dynamic forming processes simulation". Proceedings of the 4th International ESAFORM Conference on Material Forming. pp. 403–406. OCLC 51843097.
8. ↑ Hirt, C.W; Nichols, B.D (January 1981). "Volume of fluid (VOF) method for the dynamics of free boundaries". Journal of Computational Physics 39 (1): 201–225. doi:10.1016/0021-9991(81)90145-5. Bibcode: 1981JCoPh..39..201H.
9. ↑ Beris, A. N.; Tsamopoulos, J. A.; Armstrong, R. C.; Brown, R. A. (September 1985). "Creeping motion of a sphere through a Bingham plastic". Journal of Fluid Mechanics 158: 219–244. doi:10.1017/S0022112085002622. Bibcode: 1985JFM...158..219B.

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