Viscoplastic Fluid

Viscoplastic fluids are materials that exhibit a yield stress: below the yield stress, there is no deformation of the fluid and it behaves like a rigid solid. When the yield stress is exceeded, the material flows like a fluid. These materials are also called Bingham plastics after Bingham, who first described these types of materials.

Such flow behavior can be observed for slurries and suspensions, certain polymer solutions, heavy oils, cosmetic creams, liquid chocolate, and some pastes.

where $T_0$ is the yield stress. For $T>T_0$ , the material flows; for $T<T_0$ , the material remains solid.

Simcenter STAR-CCM+ provides the Herschel-Bulkley model for viscoplastic fluids.

Herschel-Bulkley

The Herschel-Bulkley model for Bingham plastics is commonly used to model materials like concrete, mud, and toothpaste.

$$\mu \left(\dot{\gamma }\right)=\{\begin{array}{ccc}{\mu }_0& ,\text{if}& \dot{\gamma }<\frac{{\tau }_0}{{\mu }_0}\\ \frac{{\tau }_0+k{(\dot{\gamma }-\frac{{\tau }_0}{{\mu }_0})}^n}{\dot{\gamma }}& ,\text{if}& \dot{\gamma }>\frac{{\tau }_0}{{\mu }_0}\end{array}$$

(705)

where:

• $k$ is the consistency factor
• $n$ is the power law exponent
• ${\tau }_0$ is the yield stress threshold
• ${\mu }_0$ is the yielding viscosity
• ${\mu }_{min}$ is the minimum viscosity limit
• ${\mu }_{max}$ is the maximum viscosity limit
• $\dot{\gamma }$ is the shear rate

For low shear rates ( $\dot{\gamma }<{\tau }_0/{\mu }_0$ ), the material behaves like a highly viscous fluid with yielding viscosity ${\mu }_0$ . Once the yield stress threshold is passed with increasing shear rate ( $\dot{\gamma }>{\tau }_0/{\mu }_0$ ), the viscosity is given by the power law expression. In both cases, the viscosity is bounded by ${\mu }_{min}$ and ${\mu }_{max}$ .

The method reduces to the Ostwald de Waele law, (or the standard power law), when the yield stress threshold is set to zero: ${\tau }_0=0$ .