# Non-Newtonian Models

## Overview:

The ‘Non-Newtonian models’ define how the viscosity and behaviour of a fluid are defined for Non-Newtonian fluids. The most commonly used Non-newtonian models that are available on the SimScale platform are described in brief detail.

### Non-Newtonian Fluid

In a non-Newtonian fluid, the local shear stresses and the local shear rates in the fluid have a non-linear relation, where a proportionality constant can not be defined. Therefore, the ‘Viscosity’ is not a fixed scalar but a variable. Further, it is also important to note that the viscosity can be dependent on the shear rate or the time history of shear rate.

Some examples are fluid substances like ketchup, custard, toothpaste, starch suspensions, paint, blood, and shampoo etc.

Fig. 1 shows the general types of Non-Newtonian fluids and their stress-strain behaviour.

The 3 main types of Non-newtonian fluids are briefly described as follows:

### I- Pseudo-plastic or Shear-thinning fluids:

The ‘Pseudo-plasticity’ or ‘Shear-thinning’ behaviour is described as a decrease in the viscosity of the fluid with an increase in the shear rate. Some common substances that undergo shear-thinning are ketchup, nail polish, silicone oil, and whipped cream.

### II- Dilatant or Shear-thickening fluids:

The ‘Dilatant’ or ‘Shear-thickening’ behaviour is then described as being opposite to ‘Shear-thinning’. That means an increase in the fluid viscosity with an increase in the shear rate. Hence also called as ‘thickening’ fluids. A common example of a thickening or dilatant fluid substance is a corn-starch solution in water.

### III- Visco-Plastic fluids:

For this type of behaviour, there is a minimum stress threshold, called ‘yield stress’, that must be exceeded for the fluid to flow or deform. A visco-plastic fluid material will deform elastically ( like a rigid body) if the external stress is less than the yield stress value. When the external stress is greater than the yield stress, the relation between stress and strain rate may be linear or non-linear. [2]

• A Visco-plastic fluid with a linear behaviour is then called a “Bingham plastic fluid” that has a constant viscosity (coefficient of proportionality) and constant yield stress.

## Non-Newtonian Models

The available mathematical models that describe the relationship between shear stress and shear rate of non-Newtonian fluids are as follows:

Important

Please note that all models use the ‘kinematic viscosity’, $$u$$ with units $$m^2/s$$

## 1. Power Law Model

A ‘Power-Law’ fluid model is a type of generalized Non-Newtonian fluid model. It gives a basic relation for viscosity, $$\nu$$, and the strain rate $$\dot{\gamma}$$. In this model, the value of viscosity can be bounded by a lower bound value, $$\nu_{min}$$, and an upper bound value $$\nu_{max}$$.

The relation is given as:
$$\nu=\kappa\cdot\dot{\gamma}^{n-1} [1]$$

Where,

• $$\kappa$$ is the flow consistency index (SI units $$m^2/s$$),
• $$\dot{\gamma}$$ is the strain-rate (SI units $$s^{-1}$$),
• $$n$$ is the flow behaviour index.

Based on the flow behavior index, $$n$$:

• if $$0 < n < 1$$: The fluid shows ‘Pseudoplastic or Shear thinning’ beahaviour. Here a smaller value of $$n$$ means, a greater degree of shear-thinning.
• if $$n=1$$: The fluid shows Newtonian behaviour
• if $$n<1$$: The fluid shows ‘Dilatant or shear thickening’ behaviour with a higher value of $$n$$ resulting in greater thickening.

Important

Power Law is the simplest model that approximates the behavior of a non-Newtonian fluid. Its limitations are that it is valid over only a limited range of shear rates. Therefore the values of $$k$$ and $$n$$ are dependant on the range of shear rates taken into account [2]. Yet, ‘Power Law’ is the most commonly and widely used model dealing with process engineering applications.

## 2. Bird-Carreau Model

This is a four-parameter model that is valid over the complete range of shear rates. For cases where there are significant variation from the power-law model i.e. at very high and very low shear rates, it becomes essential to incorporate the values of viscosity at zero shears, $$\nu_{0}$$ and at infinite shear, $$\nu_{∞}$$ into the formulation. [2]

For this model, the viscosity, $$\nu$$, is related to the shear rate, $$\dot{\gamma}$$, by the following equation:

$$\nu =\nu_{\infty}+(\nu_{0}-\nu_{\infty})\ast [1+(\kappa\cdot \dot{\gamma})^{a}]^{(n-1)/a} [1]$$

Where,

• $$a$$ has a default value of $$2$$
• $$\nu_{0}$$ is the viscosity at zero shear rate
• $$\nu_{\infty}$$ is the viscosity at infinite shear rate
• $$\kappa$$ is the ‘relaxation time’ in units of seconds
• $$n$$ is the ‘Power Index’
• At low shear rate Carreau fluid behaves as a Newtonian fluid and at a high shear rate as a power-law fluid. [2]

Important

This model is mostly used for food, beverages and also blood flow applications.

## 3. Cross-Power Law Model

The ‘Cross-Power law’ model is also a four parameter model that covers the entire shear rate range.

The model formulation is given as:
$$\nu =\nu_{\infty}+\frac{(\nu_{0}-\nu_{\infty})}{1+(m\cdot \dot{\gamma})^{n}} [1]$$
where,

• $$n$$ has a default value of $$2$$
• $$\nu_{0}$$ is the viscosity at zero shear rate
• $$\nu_{\infty}$$ is the viscosity at infinite shear rate
• $$\dot{\gamma}$$ is the shear rate
• $$m$$ is in units of seconds.

Here, ‘$$n$$’ (<1) and ‘$$m$$’ are two fitting parameters and $$\nu_{0}$$, is the viscosity value at very low shear rate and $$\nu_{\infty}$$ at a very high shear rate.

• This model is reduced to Newtonian fluid behavior as $$m$$ approached value zero.

## 4. Herschel-Bulkley Model

The ‘Herschel–Bulkley’ fluid is also a generalized, non-linear model of non-Newtonian fluids. This model combines the behaviour of Bingham and power-law fluids in a single relation. For very low strain rates, the material behaves as a very viscous fluid with viscosity $$\nu_{0}$$. After a minimum value of strain-rate corresponding to a threshold stress $$\tau_{0}$$, the viscosity is represented by the power law relation [2].

The model formulation is given as:
$$\nu = min(\nu_{0}, \tau_{0}/\dot{\gamma} + \kappa\cdot \dot{\gamma}^{n-1}) [1]$$
Where,

• $$n$$ is the ‘Power/Flow Index’
• $$\kappa$$ is the ‘Consistancy Index’ with units
• $$\tau_{0}$$ is the yield stress
• $$\nu_{0}$$ is the viscosity at zero shear rate

Further, if $$\tau > \tau_{0}$$ the Herschel-Bulkley fluid behaves as a fluid.

• if   $$0<n<1$$: The fluid shows ‘Pseudoplastic or Shear thinning’ beahaviour.
• if   $$n=1$$ and $$\tau_{0}=0$$: The fluid shows Newtonian behaviour
• if   $$1<n$$: The fluid shows ‘Dilatant or shear thickening’ behaviour.

### References

[1] https://cfd.direct/openfoam/user-guide/transport-rheology.

[2] Continuum Mechanics-Progress in Fundamentals and Engineering Applications, Dr. Yong Gan.

## Disclaimer

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