Rheology is defined as ‘the science of the deformation and flow of matter’, and it is concerned with relationships between stress, strain, rate of shear and time.
The details for the various rheological models discussed here may be found in Reed (1995). The simplest flow behaviour is that of the Newtonian liquid which is characterized by a direct proportionality between the shear stress and the velocity gradient, otherwise known as the shear rate, with its intercept at the origin:
τ = η. (2.1) γ&
where τ is the shear stress (Pa), η is the coefficient of viscosity (Pa s) and γ is shear rate (rad/s). The coefficient of viscosity indicates the resistance to flow due to internal friction in the fluid.
&
In suspensions containing non-attracting anisometric particles, laminar flow may orientate the particles such that resistance to shear decreases. This means that the stress required to increase the shear rate by an increment diminishes with increasing shear rate. This behaviour is often described by an empirical power law equation,
τ = K. γ& n (2.2) where K is the consistency index and n < 1 is the shear thinning constant which indicates the departure from Newtonian behaviour. Apparent viscosity is often used to describe non-Newtonian fluids. It is the viscosity of the fluid if its behaviour is Newtonian in nature, i.e. shear stress is zero when there is no flow. The apparent
γ τ
&
viscosity ηa is obtained from dividing the shear stress by the rate of shear, i.e. ηa = . This is similar to Equation (2.1). When apparent viscosity decreases with increasing shear rate, the behaviour is said to be shear thinning or pseudoplastic. The apparent viscosity of a power law material is obtained by dividing Equation (2.2) with the shear rate : γ&
ηa = K. γ& n -1 (2.3) On the other hand, the power law with n > 1 approximates the flow behaviour of moderately concentrated suspensions containing large agglomerates, and concentrated, deflocculated particles. This phenomenon is known as shear thickening.
For a material of this nature, the apparent viscosity increases with an increase in the shear rate. Power law materials have no yield point.
There is another type of material that contains suspension of bonded particles which requires a finite stress called the yield stress τ0 to initiate flow. Beyond the yield stress, the material flows with a constant viscosity known as the plastic viscosity ηp. This material is called a Bingham plastic and its flow behaviour can be described by the Bingham equation:
τ = τ0 + ηp. (2.4) γ&
where τ0 is the yield stress and ηp is the plastic viscosity. The apparent viscosity ηa of a Bingham material is obtained by dividing Equation (2.4) with the shear rate : γ&
γ τ0
&
ηa = ( ) + ηp (2.5) The apparent viscosity of the Bingham material is higher when the yield stress is higher and decreases with increasing shear rate, as illustrated in Fig.2.1. Figure 2.1 (a) a shows that the apparent viscosity, which is given by the gradient of the dashed lines
in bold, is higher for a material with a higher yield value of τ1 than one with a lower yield value of τ2, at the same shear rate . Also, in the same Fig.2.1 (b), it is shown that the gradient of the dashed line in bold decreases as the shear rate increases from to . Although the Bingham material flows with a constant plastic viscosity after the applied shear stress exceeds the yield stress, Equation (2.5) indicates that the Bingham material is shear thinning with respect to the apparent viscosity. The equation also shows that the plastic viscosity is the viscosity limit for a Bingham material at a high shear rate. Hence, for both the shear thinning systems (i.e. power law with n < 1, and Bingham material), the apparent viscosity decreases with the increase of shear rate, and two parameters are required to characterise the viscous behaviour. Figure 2.2 shows how the shear stress varies with the shear rate in each of the three rheological models discussed above.
γ&
γ&1 γ&2
γ τ1
τ2
Shear rate Shear
stress
a
γ2
γ1
Shear rate Shear
stress
b
τ
γ & γ &1 γ &2
Fig.2.1 – The apparent viscosity of a Bingham material is higher for higher yield stress (a) and decreases with increasing shear rate (b)
Fig.2.2 – Various rheological models showing variation of shear stress with shear rate (Reed, 1995)
A more general model is the Herschel-Bulkley model:
τ = τ0 + K. γ& n (2.6) This equation provides shear thinning behaviour after the stress exceeds the yield stress, like the Bingham equation, but provides for a non-linear dependence of shear stress on shear rate as described by the power law equation.
The rheological behaviour described above was assumed to be independent of the shear history and shearing time. For some materials the apparent shear resistance and viscosity at a particular shear rate may decrease with shearing time (Fig.2.3). This behaviour, called thixotropy, is commonly observed for shear thinning materials when the orientation and coagulation of particles change with time during shear flow. For a thixotropic material with a yield stress, the apparent yield stress is higher after the suspension has been at rest and a particle structure has reformed. This higher apparent yield stress after a period of rest is often called the gel strength, or the static yield stress. The static yield stress will be discussed in Section 2.2 (page 17).
Fig.2.3 – Shear stress decreases with shear flow at constant shear rate, which indicates thixotropic behaviour (Reed, 1995)