# 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.

## Papanastasiou

Papanastasiou in 1987, who took into account earlier works in the early 1960’s (Shangraw et al.,[6]) as well as a well-accepted practice in the modelling of soft solids (Gavrus et al.),[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:

${\displaystyle {\vec {\vec {\tau }}}=(\mu +{\tau _{y} \over \mid {\dot {\gamma }}\mid }[1-\exp(-m\mid {\dot {\gamma }}\mid )])({\vec {\vec {\dot {\gamma }}}})}$

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:

• ${\displaystyle \tau =\tau _{y}[1-\exp(-m{\dot {\gamma }})]+\mu {\dot {\gamma }}}$
• ${\displaystyle \eta =\mu +{\tau _{y} \over \mid {\dot {\gamma }}\mid }[1-\exp(-m\mid {\dot {\gamma }}\mid )]}$

where η is the apparent viscosity.

## References

1. A numerical investigation of inertia flows of Bingham-Papanastasiou fluids by an extra stress-pressure-velocity galerkin least-squares method. http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1678-58782010000500004
2. FLOWS OF VISCOPLASTIC MATERIALS: MODELS AND COMPUTATIONS. E. Mitsoulis, Rheology Reviews 2007, 135 - 178.
3. Bingham, E.C., Fluidity and plasticity, McGraw-Hill, New York (1922).
4. Papanastasiou, T.C., Flow of materials with yield, J. Rheol., 31 (1987) 385-404.
5. Ellwood, K.R.J., Georgiou, G.C., Papanastasiou, T.C., Wilkes, J O, Laminar jets of Bingham-plastic liquids, J. Rheol., 34 (1990) 787-812.
6. Shangraw, R., Grim, W., Mattocks, A.M., An equation for non-Newtonian flow, Trans. Soc. Rheol., 5 (1961) 247-260.
7. Gavrus, A., Ragneau, E., Caestecker, P., A rheological behaviour formulation of solid metallic materials for dynamic forming processes simulation, Proc. 4th Intern. ESAFORM Conf. Mat. Form., Ed. A.M. Habraken, Universite de Liege, Vol. 1, pp. 403-406 (2001).
8. Hirt, C.W., Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39 (1981) 201-225.
9. Beris, A.N., Tsamopoulos, J.A., Armstrong, R.C., Brown, R.A., Creeping motion of a sphere through a Bingham plastic, J. Fluid Mech., 158 (1985) 219-244.