Bingham-Papanastasiou model

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:

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

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.