Shear in non-Newtonian liquids



The article What is shear in liquids introduces general rules for fluid flow and relations of viscosity to shear rate and shear stress. It also defines the difference between Newtonian and non-Newtonian fluids. This article gives an overview of the non-Newtonian fluids behavior.


Newtonian fluids albeit Newtonian flow characteristics, i.e., at constant pressure and temperature, shear stress is proportional to shear rate and the dynamic viscosity, which is constant. Industrial growth gave rise to many new substances, especially of multiphase nature.  Foams, emulsions, suspensions, slurries, polymer solutions and others do not conform to the Newtonian postulate of the flow behavior. These fluids have a complex relation of viscosity as a function of shear rate. There is a large range of different behavior types for such fluids, but they are generally called non-Newtonian fluids.



Non-Newtonian fluid behavior    


A simple representation of the diversity in fluid flow behavior can be demonstrated by plotting a shear stress vs shear rate for various types of liquids (Fig. 1).


Figure 1. – Flow behavior for different types of fluids.

Figure 1. – Flow behavior for different types of fluids.


The deviation from the Newtonian behavior can occur when the plotted curve does not pass through the origin and/or does not follow the linear relationship between shear rate and shear stress. In these circumstances, viscosity, as a shear property, becomes a function of the flow conditions for non-Newtonian fluids. To make the viscosity property even more complex, some fluids experience time-dependent viscosity, i.e., the apparent viscosity of the fluid changes with time as a fluid continues to experience shearing.



Time-independent fluid behavior


The relation between shear stress, shear rate, and apparent viscosity characterizes this category of fluids. For Newtonian fluids, viscosity is a constant of proportionality between shear rate and shear stress. Viscosity of time independent non-Newtonian fluids is a function of shear rate, but at the same time it has no memory of kinematic history. This means that while shear rate is applied, apparent viscosity would change (increase or decrease), but it will be the same as original when the fluid is rested.


Time independent non-Newtonian fluids can be categorized as:


  • Shear-thinning or pseudoplastic – apparent viscosity of these fluids decreases with increasing shear rate (e.g. ketchup,quicksand, blood).
  • Shear-thickening or dilatant – apparent viscosity of these fluids increases with increasing shear rate (e.g. cornstarch in water or oobleck).
  • Visco-plastic with or without shear-thinning behavior – those fluids are characterized by the existence of a threshold stress which must be exceeded prior the fluid flow (e.g. drilling mud, tooth paste). This stress is called yield stress, as the substance behaves as an elastic solid if externally applied
    stress is less than the yield stress. Once this threshold is exceeded the fluid will start to flow. It can either exhibit Newtonian properties with constant
    value of viscosity (bingham plastics) or as shear-thinning fluids (visco-plastic).



Time-dependent fluid behavior


Apparent viscosity of some substances is not only function of applied shear rate, but also depends on the duration of time over which the fluid has been subjected to shearing. Depending on how substances react to shear over time, they can be categorized as:


  • Thixotropic fluids – at constant shear rate these fluids will decrease their apparent viscosity with time (e.g. red mud suspension). The time needed to reach the equilibrium value of shear stress for a given shear rate will decrease with increasing the rate of applied shear. If shear rate applied to these fluids is gradually increased at a certain rate and then decreased at the same rate, a hysteresis loop of shear rate – shear stress diagram will be obtained as shown in Fig 2.
  • Rheopectic fluids – these fluids show inverse behavior compared to thixotropic fluids, i.e., their apparent viscosity increases with time at constant shear rate (e.g. some types of printer ink). The hysteresis loop in a shear rate-shear stress diagram will also be inverted, respectively.


The shape and the area of a hysteresis loop depends on the experimental conditions. It will vary with changes in shear rate and the maximum value of shear rate. General interpretation of such behavior can be formulated as the fluid structure undergoes the breakdown with time. The number of structures that can break reduces with time, thus the rate of viscosity change approaches zero. The simultaneous reverse process of structure rebuilding is possible for certain fluids. In this situation, after a certain period, a state of equilibrium is established, where rates of structure breakdown and re-build are balanced.


Figure 2. – Hysteresis loops for thixotropic and rheopectic fluids.

Figure 2. – Hysteresis loops for thixotropic and rheopectic fluids.



Visco-elastic behavior


As already mentioned for bingham plastics, some substances require applied stress which exceeds the yield stress before they start to flow. This is common behavior for most solid materials. Solids have elastic properties if the applied stress is lower than yield stress. If the stress exceeds this yield value the material will start to creep. On the opposite side, Newtonian fluids possess completely viscous behavior. Visco-elastic materials show both elastic and viscous characteristics under certain conditions, i.e., materials show fluid-like behavior in one situation and solid-like behavior in another.


For the ease of visualization, mechanical analogues of visco-elastic fluid behavior are proposed. They represent a set of a viscous dumper and a purely elastic spring. In the Maxwell model, they are connected in series while in the Kelvin-Voight model – in parallel.


Figure 3. – Mechanical representation of Maxwell model (a) and Kelvin-Voight model (b).

Figure 3. – Mechanical representation of Maxwell model (a) and Kelvin-Voight model (b).



One interesting feature of visco-elastic materials is the memory effect. This means that materials are able to come back to the original state when the shear stress is removed. Pure viscous fluids have no memory, while elastic solids have perfect memory as long as it is within the yield stress conditions.  The visco-elastic fluids have some memory, which is characterized by a relaxation time. The parameter to estimate the memory of the materials is the dimensionless Deborah number (De):


De = \frac{t_{{c}}}{t_{{p}}}              (1)



t_{{c}} - stress relaxation time

t_{{p}} - time scale of observation.

At low Deborah numbers, the material has fluid-like behavior, with De→0 the material is a pure viscous fluid. At high Deborah numbers, the material exhibit more non-Newtonian fluid behavior, with more dominating elastic than viscous characteristics. At De→∞ the material shows purely elastic behavior.



Shear characteristics of non-Newtonian liquids


As described earlier shear forces affect non-Newtonian fluids in different ways. It is therefore crucial to understand fluid flow behavior. In many industrial processes, non-Newtonian fluids have to be transported through the pipe network. The intensity of the fluid flow determines the shear rates. Process equipment such as pumps and valves can also cause high shear rates to the flow.


High shear rates lead to reduced apparent viscosity for some fluids. Lower viscosity results in lower pressure drop during fluid transport. Conversely, other fluids become more resistant to flow with increased viscosity at increased shear rates.


High shear rates can also cause breakdown of the fluid structures, which leads to reduced viscosity. In practice, this can be applied to food industry, where some products may lose its properties, like taste, color and consistency, if certain shear thresholds are exceeded. Similar observation can be related to polymer solutions used in the oil industry for improved oil recovery. The polymer is added to the water in order to increase its viscosity for more efficient sweeping of the oil during the flooding of the reservoir. Such polymer solutions are normally shear sensitive, i.e., if high shear rates are applied, long polymer molecules break down and the fluid mixture loses its high viscosity. This effect is called shear degradation. Degraded polymer solution reduces the efficiency of reservoir flooding.


From an engineering point of view, it is important to know fluid properties and their behavior at various shear rates. Some fluids will benefit from high shear during transport and handling, while for others it might be critical for fluid qualities to maintain low shear mode.



Additional reading


Chhabra, R.P., 2006. Bubbles, drops and particles in non-Newtonian Fluids. CRC, Boca Raton, FL.