POLYFLOW - how to identify the parameters for a Bingham law?
Formally, the Bingham law is given by:
g-dot = 0, for T < Ty
T = visc . g-dot + Ty, for T > Ty
where T and g-dot are respectively the shear stress and shear rate; while Ty and visc are the yield stress and a viscosity factor, respectively. That expression is not analytical. Following a suggestion by Papanastasiou in 1984, the law can be changed into
T = visc . g-dot + Ty . [1 ` exp( -m . g-dot / gcrit ) ]
This expression is analytical and can easily be derived. Simple inspections have allowed us to consider the value 3 for m. Still, the equation involves three parameters, namely visc, Ty and gcrit. How can they be identified?
In POLYDATA, the law is expressed in terms of viscosity instead of shear stress. The equation of the modified Bingham law is simply divided by the shear rate. By altering the name of the parameters (for reasons of software implementation), we obtain for the shear rate dependence f(g) of the viscosity:
f(g) =fac + ystr*(1 - exp(-3*g/gcrit)/g
Figures 1 and 2 show the meaning of fac and ystr, as well as the influence of gcrit. On shear-stress / shear-rate curves, we see that ystr is the yeald stress, while fac is the slope of the curve. We also see that gcrit controls the transition between the yield stress and the affine behaviour.
Figure 1: Bingham_Fig1_param.gif
Figure 2: Bingham_Fig2_gcrit.gif