# 3.8: The Relative or Apparent Viscosity

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Einstein (1905) published an analysis for the viscosity of dilute suspensions. The result of this analysis is an equation giving the relation between the apparent dynamic viscosity and the volumetric concentration of the solids. The concentrations however are limited to low concentrations.

\[\ \mu_{\mathrm{r}}=\frac{\mu_{\mathrm{m}}}{\mu_{\mathrm{l}}}=1+2.5 \cdot \mathrm{C}_{\mathrm{vs}}\]

Thomas (1965) collected data regarding the relative viscosity from 16 sources. The particle materials included polystyrene, rubber latex, glass and methyl methacrylate. The results are shown in Figure 3.8-1. In all studies, either the density of the suspending medium was adjusted or the viscosity of the suspending medium was sufficiently large that settling was unimportant. Examination of the experimental procedure used in these studies shows no basis for eliminating any of the data because of faulty technique; consequently, there must be at least one additional parameter that has not been accounted for. One parameter of importance is the absolute value of the particle diameter. For particles with diameters less than 1 to 10 microns, colloid-chemical forces become important causing non-Newtonian flow behavior. The result is a relative viscosity which increases as particle size is decreased, but which decreases to a limiting value as the shear rate is increased. For particles larger than 1 to 10 microns, the inertial effects due to the restoration of particle rotation after collision result in an additional energy dissipation and consequent increase in relative viscosity with increasing particle diameter.

In flow through capillary tubes, the increase in viscosity observed with large particle size suspensions is opposed by a decrease in viscosity caused by a tendency for particles to migrate toward the center of the tube as the particle diameter is increased. Examination of the data from which Figure 3.8-1 was prepared showed that in several cases the tests covered a sufficient range of shear rates or particle sizes that it was possible to extrapolate to conditions where particle size effects were negligible. For particles less than 1 micron diameter, the limiting value of the relative viscosity was obtained as the intercept of either a linear plot of **1/d **versus **μ**_{m}**/μ**** _{l} **or a linear plot of

**1/(du/dr)**versus

**μ**

_{m}

**/**

**μ**

_{l}

*.*For particles larger than 1 to 10 microns, the limiting value of the relative viscosity was obtained as the intercept of a linear plot of

**d**versus

**μ**

_{m}

**/μ**

_{l•}In the event that large particle size data were also available as a function of shear rate, the reduced particle size data were further corrected by plotting against

**1/(du/ dr).**Treatment of the suitable data in this manner gave a unique curve for which the maximum deviation was reduced from three- to six fold over that shown in Figure 3.8-1, that is, to ± 7 % at

**C**

_{vs}

**=**

**0.2**and to ±13 % at

**C**

_{vs}

**=0.5,**as is show in Figure 3.8-2.

Based on this Thomas (1965) derived an equation to determine the relative dynamic viscosity as a function of the concentration **C****vs**** **of the particles in the mixture.

\[\ \mathrm{\mu_{\mathrm{r}}=\frac{\mu_{\mathrm{m}}}{\mu_{\mathrm{l}}}=1+2.5 \cdot \mathrm{C}_{\mathrm{vs}}+10.05 \cdot \mathrm{C}_{\mathrm{vs}}^{2}+0.00273 \cdot \mathrm{e}^{16.6 \cdot \mathrm{C}_{\mathrm{vs}}}}\text{ with: }v_{\mathrm{m}}=\frac{\mu_{\mathrm{m}}}{\rho_{\mathrm{m}}} \Rightarrow v_{\mathrm{r}}=\frac{v_{\mathrm{m}}}{v_{\mathrm{l}}}=\frac{\mu_{\mathrm{m}}}{\mu_{\mathrm{l}}} \cdot \frac{\rho_{\mathrm{l}}}{\rho_{\mathrm{m}}}\] |

The Thomas (1965) equation can be used for pseudo homogeneous flow of small particles.

Figure 3.8-2 shows that the first two terms are valid to a volumetric concentration of about 6%. Adding the 3^{rd} term extends the validity to a volumetric concentration of about 25%. Adding the 4^{th} term extends the validity to a volumetric concentration of 60%, which covers the whole range of concentrations important in dredging applications.

Figure 3.8-3 shows experiments of Boothroyde et al. (1979) with Markham fines (light solids, high concentration) without using the Thomas (1965) viscosity. Figure 3.8-4 shows these experiments using the Thomas (1965) viscosity.

Figure 3.8-5 shows experiments of Thomas (1976) with iron ore (very heavy solids with SG of 4.5-5.3, medium concentration) without using the Thomas (1965) viscosity. Figure 3.8-6 shows these experiments using the Thomas (1965) viscosity.

In both cases the data points are above the ELM curves if the normal liquid viscosity is used. Using the Thomas (1965) viscosity correction places the data points very close to the ELM curves. Applying the Thomas (1965) viscosity gives a good result for the fines, as long as they behave like a Newtonian fluid.

The limiting particle diameter for particles influencing the viscosity can be determined based on the Stokes number. A Stokes number of **Stk=0.03 **gives a good first approximation. The velocity in the denominator can be replaced by **7.5·D**_{p}** ^{0.4} **as a first estimate of the LDV near operational conditions.

\[\ \mathrm{d}=\sqrt{\frac{\mathrm{Stk} \cdot 9 \cdot \rho_{\mathrm{l}} \cdot v_{\mathrm{l}} \cdot \mathrm{D}_{\mathrm{p}}}{\rho_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{ls}}}} \approx \sqrt{\frac{\mathrm{Stk} \cdot 9 \cdot \rho_{\mathrm{l}} \cdot v_{\mathrm{l}} \cdot \mathrm{D}_{\mathrm{p}}}{\rho_{\mathrm{s}} \cdot 7.5 \cdot \mathrm{D}_{\mathrm{p}}^{0.4}}}\] |

Figure 3.8-3 shows experimental data versus the DHLLDV Framework for uniform particles with the pure liquid viscosity. The data do not match the curve, but are much higher. Figure 3.8-4. Shows the experimental data versus the DHLLDV Framework for graded particles according to Boothroyde et al. (1979) and full Thomas (1965) viscosity. Now the data match the curve for graded particles.

Figure 3.8-5 shows the experimental data versus the DHLLDV Framework. Figure 3.8-6 shows these experiments using the Thomas (1965) viscosity based on the particle size distribution mentioned by Thomas (1976). The data points now match the DHLLDV Framework for graded particles and adjusted viscosity