For Newtonian fluids, a constant viscosity

(10–6)

can be specified. is referred to as the Newtonian or zero-shear-rate viscosity.

The power law for viscosity is

(10–7)

where is the consistency factor, is the natural time or the reciprocal of a reference shear rate, and is the power-law index, which is a property of a given material.

The power law is commonly used to describe the viscous behavior of polymeric materials, such as polyethylene, with shear rates ranging over 2 to 3 decades. However, it fails to describe the behavior at low shear rates. If the behavior at low shear rates needs to be fitted as well, the Bird-Carreau or Cross law will capture the plateau zone of the viscosity curve for low shear rates better than the power law.

The Bird-Carreau law for viscosity is

(10–8)

where	= infinite-shear-rate viscosity
	= zero-shear-rate viscosity
	= natural time (that is, inverse of the shear rate at which the fluid changes from Newtonian to power-law behavior)
	= power-law index

The Bird-Carreau law is commonly used when it is necessary to describe the low-shear-rate behavior of the viscosity. It differs from the Cross law primarily in the curvature of the viscosity curve in the vicinity of the transition between the plateau zone and the power law behavior.

The Cross law for viscosity is

(10–9)

where	= zero-shear-rate viscosity
	= natural time (that is, inverse of the shear rate at which the fluid changes from Newtonian to power-law behavior)
	= Cross-law index (= for large shear rates)

Like the Bird-Carreau law, the Cross law is commonly used when it is necessary to describe the low-shear-rate behavior of the viscosity. It differs from the Bird-Carreau law primarily in the curvature of the viscosity curve in the vicinity of the transition between the plateau zone and the power law behavior.

A modified Cross law for viscosity is also available:

(10–10)

This law can be considered a special case of the Carreau-Yasuda viscosity law (Equation 10–16), where the exponent has a value of 1.

The Bingham law for viscosity is

(10–11)

where is the yield stress and is the critical shear rate, beyond which Bingham’s constitutive equation is applied. For shear rates less than , the behavior of the fluid is normalized in order to guarantee appropriate continuity properties in the viscosity curve.

The Bingham law is commonly used to describe materials such as concrete, mud, dough, and toothpaste, for which a constant viscosity after a critical shear stress is a reasonable assumption, typically at rather low shear rates.

A modified Bingham law for viscosity is also available:

(10–12)

where .

Compared to the standard Bingham law, the modified Bingham law is an analytic expression, which means that it may be easier for Ansys Polyflow to calculate, leading to a more stable solution. The value has been selected so that the standard and modified Bingham laws exhibit the same behavior above the critical shear rate, .

The Herschel-Bulkley law for viscosity is

(10–13)

where is the yield stress, is the critical shear rate, is the consistency factor, and is the power-law index.

Like the Bingham law, the Herschel-Bulkley law is commonly used to describe materials such as concrete, mud, dough, and toothpaste, for which a power-law viscosity after a critical shear stress is a reasonable assumption. In addition to the transition behavior between a flow and no-flow regime, the Herschel-Bulkley law exhibits a shear-thinning behavior that the Bingham law does not.

A modified Herschel-Bulkley law is also available:

(10–14)

Compared to the standard Herschel-Bulkley law, the modified Herschel-Bulkley law is an analytic expression, which means that it may be easier for Ansys Polyflow to calculate, leading to a more stable solution. The integer value 3 that appears in the argument of the exponential term has been selected so that the standard and modified Herschel-Bulkley laws exhibit the same behavior above the critical shear rate, .

The log-log law for viscosity is

(10–15)

where is the zero-shear-rate viscosity and , , and are the coefficients of the polynomial expression.

This viscosity law is purely empirical, but sometimes provides a better fit to experimental data than the others. Nevertheless, you should pay special attention to the coefficients you specify for the log-log law, as detailed below.

The function is a parabola in the () space. Depending on the values of the polynomial coefficients, the viscosity may decrease as the shear rate approaches zero, which does not reflect physical behavior. Moreover, for high shear rates, the slope of the curve may be less than , which is also not physical. When you are using the log-log law, you must therefore ensure that the range of shear rates in your application lies within the range of physically acceptable shear rates for the law. This is accomplished by careful specification of the polynomial coefficients.

The Carreau-Yasuda law for viscosity is

(10–16)

where	= zero-shear-rate viscosity
	= infinite-shear-rate viscosity
	= natural time (that is, inverse of the shear rate at which the fluid changes from Newtonian to power-law behavior)
	= index that controls the transition from the Newtonian plateau to the power-law region
	= power-law index

The Carreau-Yasuda law is a slight variation on the Bird-Carreau law (Equation 10–8). The addition of the exponent allows for control of the transition from the Newtonian plateau to the power-law region. A low value (a < 1) lengthens the transition, and a high value (a > 1) results in an abrupt transition.