Thin or viscous – different rheological models

Most people have heard of the term viscosity and know that a viscous substance exhibits high viscosity. Viscosity is an adequate word for describing the state of an egg white when we crack an egg, for example. But when an egg is boiled and it solidifies, viscosity becomes irrelevant for describing the consistency of the boiled egg. In that case, it is more relevant to describe how elastic the boiled egg is when we bite into it. So, different models are needed for different situations. 

Allen Pipkin’s overview of different rheological models is often used to determine the right model for a specific fluid and situation; see Figure 2 [1] The x-axis displays the Deborah number, which is the ratio between the time it takes for a fluid to relax or flow and how long we observe the flow. So, we find water at the left and pudding at the right of the diagram. The y-axis displays the degree of deformation – how much we deform the fluid.

At the far left of the diagram are fluids that have a constant viscosity. These are called Newtonian fluids. The Newtonian model works when we have a fluid that relaxes quickly in relation to observation time, like water and other beverages. In our ordinary physical world, however, we rarely perceive things that happen faster than tenths of a second, and the Deborah number is therefore low when we apply it to beverages. On the far right of the diagram, we see the opposite phenomenon. Here, we have fluids that would be perceived as extremely viscous or, even more likely, as solid materials. These are called elastic materials. Examples of elastic foods are flaky biscuits and hard cheese. All construction materials also fall under elastic materials.

To understand high Deborah numbers, let’s image a bookshelf that we load with heavy books. The bookshelf will give way under the weight of the books, but will spring back when we remove them. However, if we think of a cheap bookshelf that we loaded with heavy books ten years ago, we can sometimes note a residual deformation when we remove the books. The shelf curves downwards even after we remove the books. This shows that the long observation time of ten years is longer than the relaxation time of the shelf material. In the Pipkin diagram, this means that the Deborah number is indeed high, but does not extend into the area for elastic materials.

Figure 2. Overview of different models describing material properties of viscous fluids. Freely reproduced from [1] and [2].

In reality many fluids end up in the middle area, especially food and other biological fluids. Because these fluids are both viscous and elastic, they are called viscoelastic. Many processes, such as pumping, eating, swallowing and stirring, involve deforming and reshaping the fluid, so the predominant area in Figure 2 is non-linear viscoelasticity. This is where most fluids lie. Unfortunately there are few simple and well-defined measurement methods for non-linear viscoelasticity, although much research is taking place. On the other hand, well-established methods are available for measuring linear viscoelasticity, called small amplitude oscillatory shear (SAOS) tests, since linear viscoelastic fluids do not deform more than that they can linearly return to their original state. Even though few processes are linear, the methods are important for enabling us to study melting, gel formation and other transitions.

It is now time to review what material properties we can measure for different fluids under different conditions. But first, we need a suitable way to express forces and deformation.

Stress and strain

A flow is caused by forces and causes fluids to deform. A large surface produces greater resistance than a small one, and in order to ignore geometry we can measure the resistance as a stress rather than a force and express it as force per unit area. And instead of absolute deformation we can use a dimensionless deformation called strain. The strain caused by a force acting parallel to a surface is known as shear strain, and the strain caused when a material is subject to tension due to elongation is called tensile strain. Since viscosity depends on how fast a flow is, we also need to express the speed of the strain as shear rate in shear and extension rate in extension.

A normal flow causes the fluid to deform in shear mode. To understand this term, we can first study simple shear in Figure 3a. Shearing means that a force F acts along a plane, while under elongation it acts perpendicular to a plane (Figure 3d). The fluid velocity is zero at the lower stationary plane and v at the upper plane in Figure 3a. Because the velocity is constant, deformation increases linearly from the lower to the upper plane and the shear rate  becomes

Similarly, in the pipe flow in Figure 3b we see that the velocity is greatest in the centre of the pipe and that it decreases to zero at the pipe wall. If we divide the flow into thin layers and straighten the layers, we can understand the analogy of simple shear in Figure 3a. In the pipe flow, however, the velocity varies quadratically over the cross section of the pipe. The shear strain is the derivative of the velocity (equation 1) and therefore varies linearly, from zero in the centre of the pipe to the maximum value at the pipe wall.

Figure 3c shows the flow through a contraction in the pipe. The fluid is forced to extend in the direction of the flow in order to pass through the contraction. This gives rise to a strain, and we can similarly calculate a tensile stress, still as a force per unit are, and a strain rate  as dv_z/dz. Since the flow rate also varies with the pipe diameter, the fluid is exposed to shearing at the same time.

Shear viscosity

Viscosity expresses resistance to flow and is a relevant parameter for the left part of the Pipkin diagram (Figure 2). A viscous fluid thus has a higher viscosity than a more liquid fluid, such as syrup compared with water. In most cases, we are trying to deform the fluid in shear (Figure 3a-b), like the flow in a pipe, but a fluid has an even greater resistance to stretching (Figure 3c). The extension occurs when we try to squeeze the fluid through a contraction, like when we swallow or force a fluid out of a nozzle. The viscosity is always at least three times greater in extension than in shear mode. There is therefore both a shear viscosity and an extensional viscosity, which however are not equal. 

Now, we can first define the shear viscosity η

where η is expressed in units of Pascal-seconds, Pa·s=Ns/m², σ is the shear stress in units of Pa=N/m² and  is the shear rate expressed in units of s^-1. Both the shear viscosity and the extensional viscosity depend on how quickly we deform the fluid rather than on how much we deform it. Shear viscosity is used much more frequently than extensional viscosity, and in everyday language viscosity is usually synonymous with shear viscosity. The same applies in this text, and shear or tension will only be indicated when relevant to the context.  

The shear rate varies depending on the process. In slow flows like sedimentation the shear rate is low, typically ≤10^-4 s^-1. But it is high in processes involving flow in thin layers, such as when we apply skin cream or coat paper, typically ≥10⁴. However, many processes including chewing, swallowing, stirring and pipe flows fall somewhere between these extremes, usually 0.1–1000 s^-1. See Figure 4 for common shear rate ranges.

Viscosity also varies greatly – from water (10^-3 Pa·s at 20°C) to molten plastic (10³ Pa·s) and asphalt (10⁸ Pa·s), see Figure 4 [3]. All these levels are approximate because we will later see that viscosity depends on several different parameters. Equation 2 describes the simplest model for a fluid, a Newtonian fluid for which the viscosity is constant and independent of shear rate. As mentioned earlier, this applies for oils but is otherwise unusual for food and beverages. The viscosity usually depends on the shear rate, as well as temperature and other parameters. The main parameters influencing viscosity are presented here individually.

Temperature and pressure effects on viscosity

Viscosity decreases with increasing temperature. It is due to individual elements or particles in the fluid attracting each other across flow lines. As the temperature increases, the increased thermal energy causes an increase in molecular mobility and counteracts the attraction. Temperature dependence can often be described using a simple Arrhenius-type equation

where η denotes the shear viscosity and T temperature. A and B are constants.

Temperature dependence has many implications. If we want to measure exact viscosity, it is crucial to keep the temperature constant. For ±1 percent accuracy in the viscosity value, we typically need to keep the temperature within ±0.3°C. The temperature effect is something we observe in everyday life. We heat the bottle of syrup in the microwave oven to more easily get it out and we melt plastics to shape them. The temperature effect is also noticed when eating as the viscosity of a fluid food such as a soup heated from 20°C to 37°C typically is halved.

Viscosity increases with increasing pressure, but extreme pressure is needed before we can observe it. The effect of pressure is important in processes like off-shore oil drilling far below the seabed (~200 times atmospheric pressure) and in lubrication of internal combustion engines (~10,000 times atmospheric pressure). But for medical applications and food, the pressure effect is negligible.

Models for shear viscosity

There are a plethora of models that describe how shear stress is due mainly to shear rate and yield stress. We have already come across the simplest model in equation 2, where the shear stress is proportional to the shear rate by a constant, that is, the viscosity η. The models are usually named after their authors and the most common can be described, with increasing complexity, as

where 𝜎=shear stress, Ý=shear rate, 𝜎_y=yield stress, η₀=zero shear viscosity, η_∞=upper plateau viscosity, and constants: n=flow index or power-law index, K=consistency index, and λ and m are constants. See also Figure 6.

There are many more models used for specific fluids or, according to tradition, in specific areas. However, the Power law model is often sufficient, since many applications and processes rely mainly on medium-high shear rates (~0,1–1 000 s^-1) and describe flows where any yield stress is exceeded. Models with more parameters are required for more complex flows as well as for low or high shear rates where the viscosity levels off in the respective Newton plateau; compare Figure 4. Also note that n ≠ m in the power law and Carreau models. 

The behaviour of the fluid affects the velocity profile when the fluid flows in pipes. Figure 6c shows the velocity profile of a few types of fluids. The parabolic profile flattens as the fluid becomes more shear thinning, and a similar behaviour is obtained when the fluid has a yield stress. This is usually called plug flow and means that a large portion of the fluid flows at about the same velocity and is sheared only along the pipe wall. In practical contexts, this is often positive. When food undergoes heat treatment, most of the fluid stays in the pipe for the same amount of time and becomes neither underheated (microbiological risks) nor overheated (deteriorating quality).

Figure 6. a) Viscosity curves and b) shear rate curves (flow curves) for the described models. c) Velocity profiles in pipe flow for the same models.

Extensional viscosity

Like shear viscosity, extensional viscosity is the ratio of tensile stress to extension rate, but the conditions are different. Extensional viscosity therefore warrants its own description.

The extensional viscosity is always at least three times greater than the shear viscosity of regular flows. For Newtonian fluids and for extremely slow flows, the ratio is exactly 3, while it can be significantly greater for elastic fluids. This ratio is called the Trouton ratio. It is especially relevant to take extensional viscosity into account when the fluid is elastic, as well as when the geometry of the flow is changed. A change in geometry can be a transition from one pipe dimension to another, flow through a nozzle, or the transition from mouth to throat during swallowing. An elastic fluid is a fluid that can store parts of the deformation energy like a coiled spring. If we want to imagine a more everyday fluid, toy slime (Figure 1) is close at hand, and for food some types of fermented milk products like ropy yoghurt. Because the energy can be stored, the flow depends on what the fluid has been exposed to earlier. It has a flow memory, and therefore behaves differently from a non-elastic fluid for which the deformation energy is completely converted into heat.

Unlike shear flow, there is a maximum for extensional flow because the fluid eventually breaks. This limit depends on the properties of the fluid and is not fully investigated scientifically. Extension is limited in many flows, as shown by the flow through the contraction in our example in Figure 3c. Like shear viscosity, extensional viscosity can vary with the extension rate and be extension-thinning or extension-thickening, even if these expressions are not as common and also not as accepted as for shear flows. In fact, a fluid may well be shear-thinning and extension-thickening at the same time. These behaviours are not connected, and even if we know that a fluid is shear-thinning we cannot predict whether it is extension-thinning or extension-thickening. 

The extensional viscosity increases with the extension itself and does not reach a plateau or steady-state in the same way as the shear rate. It is not fully understood whether there is a theoretical plateau for high extensions or if the extensional viscosity passes a peak before the fluid breaks. Here, high extension mean that the fluid is typically stretched more than a thousand times its original length. This is of less interest for practical processes in which the extension is significantly lower. In practical applications, the effects of fluid elasticity are more interesting. A small elasticity acts cohesively, which is useful during firefighting when spraying water for a long period or in fountains, or for preventing droplets from bouncing off leaves when crop dusting fields. For swallowing, it has also been found that elasticity promotes safe swallowing [5]. The reason is believed to be the same as for the other applications mentioned, that is, that elasticity contributes to fluid cohesivity and sticking together during swallowing thus preventing it from entering into the airways.

Mats Stading