# Preparation Techniques of TiO\({}_{2}\) Nanofluids and Challenges: A Review
## 4 Viscosity  
Earlier research has been carried out to enhance the thermal conductivity of nanofluids, but it is not the only parameter that affects the heat transfer. To implement nanofluids in practical heat transfer applications, viscosity is another property that requires great attention. Factors such as pumping power [105], convection heat transfer and pressure drop [87; 106] in laminar flow are greatly influenced by the viscosity of nanofluids, hence a brief review on viscosity is discussed in this literature. The viscosity of the nanofluid depends on the quantity and effects of different parameters such as nanoparticle volume fraction [2; 67; 86; 105; 107], base fluid, particle size [83], nanolayer thickness, particle shape, dispersion technique, pH value, temperature [66; 83; 86; 105], the effect of clustering and Brownian motion of nanoparticles. Different instruments, like a viscometer, rheometer etc. have been used to measure the viscosity of nanofluids and the measured data has been compared to well-known correlations, which are discussed below. [PERSON] [108] developed the first model to measure the viscosity of nanofluids in 1906. The spherical shape of nanoparticles and less volume fraction (\(\varphi<0.02\%\)) were the factors that limited the formulated model of [PERSON]. Moreover, the discussed model did not consider the structure and particle to particle interaction within the solution. \[\mu_{nf}=(1+2.5\varphi)\mu_{bf} \tag{8}\]  
In 1952, [PERSON] [109] extended the work of [PERSON] by reducing the limitations of nanoparticles volume concentration to some extent. The developed model of [PERSON] was more effective and  
Figure 17: Thermal conductivity of TiO\({}_{2}\) nanofluid as a function of particle concentration (various studies). acceptable for volume fractions less than 4%. Equation 9 represents the empirical formula of [PERSON]'s model. \[\mu_{nf}=\frac{\mu_{bf}}{\left(1-\varphi\right)^{2.5}} \tag{9}\]  
[PERSON]'s [110] model considered the effect of Brownian motion of the spherical nanoparticles and introduced the new model by modifying the [PERSON] equation. The correlation of this developed model is expressed as:  
\[\mu_{nf}=\left(1+2.5\varphi+6.2\varphi^{2}\right)\mu_{bf} \tag{10}\]  
[PERSON] [111] proposed his power law model in 1970 which was considered more appropriate for nanoparticles more than 2% by volume fraction. Equation (11) denotes the [PERSON] model. \[\mu_{nf}=\left[(1+1.5\varphi)e^{\frac{\varphi}{\left(1-\varphi m\right)}} \right]\mu_{bf} \tag{11}\]  
where \(\varphi_{m}\) and \(\varphi_{p}\) represent the maximum volume fraction and nanoparticles volume concentration, respectively. The above models were outdated and failed to show reasonable agreement with the experimental results for higher concentrations of nanoparticles (above 1 %vol.) [66; 101; 104], because these models considered the viscosity of nanofluid as a function of particle concentration and base fluid viscosity only and did not consider the effect of temperature, which has a strong connection with viscosity [66; 83; 86; 105; 112]. However, the effect of temperature on the viscosity of TiO\({}_{2}\) is contradictory according to some reports [113; 114]. More recently, researchers have worked to develop more accurate models by introducing different parameters and factors, these are outlined in Table 2.  
Figures 18 and 19 shows the variation in viscosity as a function of temperature and particle concentration [66; 67; 82; 99; 100; 101; 105; 112; 117; 118], respectively. In Figure 18 the error bars represent the upper and lower values of viscosity with particle concentration at the specific temperature, while in Figure 19 the ends of the bars showed the maximum and minimum values of viscosity with temperature at the specific value of particle concentration, for the particular study.