Generalized Newtonian Fluid

The generalized Newtonian fluid model is an explicit constitutive equation that relates the stress tensor to the velocity field through a variable viscosity. This model does not account for elastic effects.

where $\dot{\gamma }$ is the shear rate.

In Simcenter STAR-CCM+, the following generalized Newtonian fluid models are implemented, that is, shear-rate dependent viscosity functions:

Power Law

The simplest model to describe a shear-rate dependent viscosity behavior is the power law.

The value of the power law exponent $n$ determines the class of the fluid:

$$\mu \left(\dot{\gamma }\right)=k{\dot{\gamma }}^{n-1}$$

(701)

• $n=1\Rightarrow$ Newtonian fluid
• $n>1\Rightarrow$ Shear-thickening (dilatant) fluid
• $n<1\Rightarrow$ Shear-thinning (pseudo-plastic) fluid

Cross Fluid

The Cross Fluid is defined from [161] as:

$$\mu \left(\dot{\gamma }\right)={\mu }_{\infty }+\frac{{\mu }_0-{\mu }_{\infty }}{1+{\left(\frac{\dot{\gamma }}{{\dot{\gamma }}_c}\right)}^m}$$

(702)

where:

• $m$ is the Cross rate constant
• ${\mu }_0$ is the zero-shear viscosity, the viscosity at the Newtonian plateau
• ${\mu }_{\infty }$ is the infinite-shear viscosity
• $\dot{{\gamma }_c}$ is the critical shear strain-rate at which shear-thinning starts

Newtonian behavior is regained for $m=0$. As $m$ increases towards unity, the degree of shear-thinning becomes stronger.

Carreau-Yasuda Fluid

The non-Newtonian Generalized Carreau-Yasuda Fluid for viscosity comes from [158] and [219]. It is defined as:

$$\mu \left(\dot{\gamma }\right)={\mu }_{\infty }+\left({\mu }_0-{\mu }_{\infty }\right){(1+{\left(\lambda \dot{\gamma }\right)}^a)}^{\left(n-1\right)/a}$$

(703)

where:

• $n$ power constant
• $\lambda$ relaxation time constant
• $a$ parameter to control shear-thinning. When $a=2$ , the original Carreau model is regained.