A quest for a model of non-colloidal suspensions with Newtonian matrices

Abstract

It would be convenient to have a model, albeit approximate, of particle-laden materials (suspensions) that would not need large amounts of computing and/or experimentation to implement for design purposes. There are now adequate models of the pure matrix fluid behaviour, but there are no such models for suspensions with large particles (non-colloidal suspensions). One of the obstacles has been the single-minded devotion to shearing motions of suspensions; experience with the matrix modelling has shown that it is not possible to formulate widely usable models if only shear is considered. Here some new results of axially symmetric elongational tests on suspensions are compared with shearing data. Some suggestions for modelling these and other observations based on using strain rate and strain in a modified Reiner-Rivlin constitutive equation are presented. The model generally works quite well, but it does not predict the positive storage modulus seen in small and medium amplitude oscillatory shear flows.

Appendix. Conversion from torque to shear viscosity

In parallel plate viscometry, one has that the torque M on the plates is

$$ M=2\pi {\int}_0^R\eta \dot{\gamma}{r}^2 dr $$

(18)

where η is the viscosity, R is the plate radius, and $\dot{\gamma}$ is the local shear rate.

We suppose that the shear rate at the rim is constant ($\dot{\gamma}$ _o) and that the shear strain at the rim is γ _o. One has

$$ \dot{\gamma}={\dot{\gamma}}_{\mathrm{o}}r/R\kern0.5em \mathrm{and}\kern0.5em \gamma ={\gamma}_{\mathrm{o}}r/R $$

(19)

Changing variables in Eq. 18 using 19, one has

$$ M=2\pi \kern0.5em {\dot{\gamma}}_{\mathrm{o}}\frac{R^3}{\gamma_0^4}{\int}_0^{\gamma_0}\eta \left(\gamma \right){\gamma}^3 d\gamma $$

(20)

where we assume the viscosity is a function of the local shear strain—the local shear rate is constant in time and the viscosity is assumed to be independent of the rate. Since most of the torque comes from the larger radius region, this does not lead to large errors (Shaw and Liu 2006).

Differentiating 20 with respect to γ _o, we find

$$ \frac{dM}{d{\gamma}_0}=-\frac{4M}{\gamma_0}+\frac{2\pi {\dot{\gamma}}_0{R}^3\ }{\gamma_0}\eta \left({\gamma}_0\right) $$

(21)

or

$$ M+\frac{\gamma_0}{4}\frac{dM}{d{\gamma}_0}=\frac{\pi }{2}{\dot{\gamma}}_0{R}^3\ \eta $$

(22)

If we denote the Newtonian result for the torque when the viscosity is constant (η _o) as M _o, so that

$$ M\mathrm{o}=\frac{\pi }{2}{\eta}_o\ {\dot{\gamma}}_o{R}^3 $$

(23)

we find

$$ \frac{\eta \left(\gamma \right)}{\eta_0}=\frac{M}{M_0}+\frac{\gamma_0}{4}\frac{d\left(\frac{M}{M_0}\right)}{d{\gamma}_0} $$

(24)

From the Gadala-Maria and Acrivos (1980) work, we have approximately, in the range 0 < γ < 3.5, the fitted result for the 50% suspension shown in Fig. 2.

Hence, one can find the value of the steady-state viscosity using Eq. 24 as a function of strain.

Table 4 shows this result and also the value of the Hencky strain (ε _H) corresponding to the shear strain (γ _o ) where one has

$$ {\varepsilon}_{\mathrm{H}}=\ln\ \left(\frac{\gamma }{2}+\sqrt{1+\frac{\gamma^2}{4}}\right) $$

(25)

The final column in Table 4 subtracts the ‘unwinding’ period ($\dot{\gamma t\sim }0.5,\mathrm{equivalent} \mathrm{to}$ ε _H = 0.247) after shear reversal from the total strain.

In a similar way, one can analyse the results of Gadala-Maria and Acrivos (1980) for the 30 and 40% suspensions. The resulting correction factors which enable us to estimate the steady-state elongational viscosity are given in Fig. 7.

Table 4 $\frac{\eta }{\eta_0}$as a fraction of γ _o and the Hencky rim strain ε _H, data of Fig. 2 (50% suspension)