Introduction

One of the perspectives for solving conservation equations for nanofluids is the single-phase method. In this method, the thermophysical properties of the nanofluid replace the thermophysical properties of the base fluid [1,2,3]. The single-phase model is used by some investigations [4,5,6,7]. This shows the impact of thermophysical properties of the nanofluid, especially its viscosity, on heat transfer. [8,9,10]

Among the thermophysical properties of nanofluids, viscosity indicates fluid resistance. Therefore, viscosity has determined the performance of energy and heating systems. [11,12,13]

In industrial equipment and scientific research, where heat transfer is in the forms of forced convection and natural convection, the viscosity of nanofluids plays a crucial role in determining the flow regime, pumping power, pressure drop, and workability of systems [14,15,16,17].

The first viscosity model for suspensions containing metal particles was introduced by Einstein in 1906 [18]. Later on, viscosity models proposed by Brinkman [19], Batchelor [20], and other equations started to be used to model the heat transfer of nanofluids. However, these correlations each have weaknesses, including the inability to estimate the viscosity of nanofluids in a wide range of temperatures and concentrations used in heat transfer.

In the experiments conducted by Duangthongsuk and Wongwises [21] on the behavior of TiO2 and water nanofluid, they presented a correlation by applying the effects of base fluid viscosity and volume fraction variables on the viscosity model. In their model, nanofluid’s temperatures and volume fraction ranged between 15 to 35 °C and 0.2 to 2%, respectively.

In an experimental test performed by Esfe and Saedodin [22] on the viscosity of ZnO nanofluid with ethylene glycol-based fluid at a temperature between 25 and 50 °C and a volume fraction of 0.25 to 5%, the viscosity model with the variables of temperature and volume fraction and the viscosity of the base fluid presented that the ratio of mean variation of the model data compared to other the experimental values ​​was less than 2%.

Sharifpur et al. [23] also introduced a viscosity model based on the data derived from experiments using \({\text{Al}}_{2} {\text{O}}_{3}\) and glycerin nanofluid with an accuracy of 0.9495. In their viscosity model, in addition to the variables of temperature and volume fraction and the viscosity of the base fluid, the effect of the thickness of the nanoparticles is also taken into account. It is used for nanofluids in the temperature range of 20 to 70 °C and volume fraction of 0 to 5%, and diameter of nanoparticles about 19 to 160 nm. But Aberoumand et al. [24] presented a viscosity model for oil-silver nanofluid that depends only on the variables of volume fraction and viscosity of the base fluid and is valid for nanofluids at temperatures between 25 and 60 °C and volume fraction of 0 to 2%.

In an experimental study performed by Akbari et al. [25] on Si \({\text{O}}_{2}\) and ethylene glycol nanofluids in the temperature range of 30 to 50 °C and volume fraction of 0.5 to 3%, using temperature and volume fraction components and the viscosity of the base fluid proposed a viscosity model for the nanofluid. Li and Zuo [26] had proposed a viscosity model for a nanofluid including TiO2 nanoparticles and a mixture of water-based fluid and ethylene glycol at a temperature between 20 to 50 °C and a volume fraction of 0.25 to 1%.

Yu et al. [27] also proposed a new viscosity model based on the data derived from experiments using multi-walled carbon nanotubes [MWCNT] and water nanofluids. In their model, in addition to the role of temperature, mass fraction, and base fluid viscosity, the effect of shear rate on viscosity variations is also considered. This model is valid in a temperature range of 275 to 283 Kelvin, and mass fraction range of 0.1 to 0.6 percent, and a shear rate of 10 to 1000 s−1. According to the experiment performed by Yan et al. [28] on a hybrid nanofluid with multi-walled carbon nanotube [MWCNT] nanoparticles and TiO2 with a base fluid of ethylene glycol at 25 to 55 °C and a volume fraction of 0.05 to 1%, the viscosity model with volume fraction and non-dimensional temperature components has been presented with an accuracy of 0.995.

Figure 1 is plotted to know the year of publication of the evaluated correlations in the present study. Therefore, it can be concluded that researchers have considered the study and presentation of models for the viscosity of nanofluids in recent years.

Fig. 1
figure 1

Year of publication of evaluated correlations in articles

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Figure 2 is plotted to indicate the temperature range at which the viscosity models are valid. The correlations separated according to the temperature range they cover in different temperature ranges that differ by 10 degrees. (Temperature difference of less than five °C in the classification has been neglected.)

Fig. 2
figure 2

The temperature ranges are covered by the viscosity models

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According to Fig. 2, 21.4 and 23.2% of the temperature range cover 30 and 40°, respectively, and only 23.2% of the correlations in the 50° temperature range can estimate the viscosity of the nanofluid.

Similar to the temperature diagram, the viscosity models were separated into 1% by volume (mass) fraction intervals relative to the concentration range in which they are valid, and Fig. 3 shows that only 19.6% of the correlations can cover the concentration range of 2 percent.

Fig. 3
figure 3

The concentration ranges are covered by the viscosity models

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As the authors worked on the thermal conductivity of nanofluid based on present studies and introduced a new general model named MAG. [29] So this study must be done about nanofluid viscosity.

In this study, the correlations presented for the viscosity of nanofluids were wholly reviewed and investigated thoroughly in terms of compliance with the physics and viscosity of nanofluids. In the following, the relationship between viscosity and variables of temperature and volume (mass) fraction of nanofluids was evaluated according to the statistical test of variance. Moreover, all the correlations presented for nanofluid viscosity were investigated with the sensitivity analysis test to identify the variable with the most significant effect on the viscosity model. Finally, two general models for water-based nanofluids and ethylene glycol were presented to predict the viscosity behavior of nanofluids.

Strategy

Analysis of variance

Statistics is a broad field of mathematics that studies how data collection, summary, and conclusion are studied. Here, the status of variables related to the viscosity of nanofluid investigated using statistical science based on probability theory and mathematics.

ANOVA test or analysis of variance is a subset of statistical science, which analyzes and compares the means of different statistical groups and determines the effect of independent variables on the dependent variable. This method has been introduced by the famous statistician and geneticist "R. Fisher."[30]

This method tries to estimate the differences between several statistical populations. In other words, using the mean index in statistical populations, we will be able to express the characteristics of the population; thus, if the mean of one group is different from other groups in society, we conclude that the statistical populations are not the same. In the one-way analysis of variance, the null hypothesis indicates that the mean of the experimental groups is equal to each other. The opposite assumption is that at least one of the means is different from the others; if the null hypothesis is confirmed, it will be accepted that there is no difference between the means of the groups, and the variable has no role in the correlation. Therefore, to better understand the correspondence of variables on nanofluid viscosity, it is necessary to perform variance analysis. [31]

Thus, by groups that will be created in terms of temperature and concentration variables for each equation and with the help of a one-way ANOVA test, the results presented in Table 1 are obtained.

Table 1 Experimental correlation of nanofluid viscosity and their analysis
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In some of the correlations proposed by researchers due to the lack of attention to the response power of the correlation in the temperature range affecting the viscosity of nanofluids and also because the appropriate relationship is not used in correlations to express the relationship between temperature and viscosity of nanofluid and the correlation are not able to predict the nanofluid viscosity at sensitive temperature and do not express the role and importance of temperature variables in nanofluid viscosity models.

One of the issues that researchers had not been considered in presenting correlations is the effect of terms on the viscosity model. For example, Dalkılıça et al. [32] presented Eq. (1) for nanofluid viscosity by experimental investigation of the viscosity of a hybrid nanofluid containing graphene and SiO2 nanoparticles in a water-based nanofluid in the temperature range of 15 to 60 °C and a volume fraction of 0.001 to 0.02%.

$$\mu_{\rm nf} = \left[ {1.00527 \times \left( { T^{0.00035} } \right) \times \left( {1 + \varphi } \right)^{9.36265} \times \left( {\frac{{\varphi_{{\rm w, G}} }}{{\varphi_{{{\rm w, SiO}_{2} }} }}} \right)^{ - 0.028935} } \right] \mu_{\rm bf}$$
(1)

In the case, where φ = 1 and other components are a constant value, if the lower and upper-temperature limits are set in Eq. (1), the range of viscosity changes will be less than 0.04%. Therefore, Eq. (1) is not dependent on temperature, but the researcher had given in the equation, and this is not considered significant by the researcher.

In addition to the temperature variable, the concentration variable also has an undeniable role in the viscosity models of nanofluids. So that with increasing the concentration of nanofluid, the viscosity of nanofluid increases significantly; therefore, in most viscosity models, the prominent role of the concentration variable was considered by researchers. However, in some correlations recently published by researchers, the correlations have been measured in the range of inappropriate volume (mass) fractions and only at low concentrations. On the other hands, due to the use of irrational relationships to express the relationship between the viscosity of nanofluid and the concentration of nanoparticles, viscosity models cannot respond commensurately with the expected concentration range in heat transfer.

By conducting experimental tests on nanofluids of Al2O3 and ethylene glycol, Li et al. [33] presented the new viscosity model in compliance with the trend of temperature variation from 25 to 60 °C and a mass fraction of 0 to 2% in Eq. (2).

$$\mu_{\rm nf} = - 334.9\varphi_{\rm w}^{4.044} \left( \frac{1}{T} \right)^{10.03} + 296.8\left( \frac{1}{T} \right)^{0.7795} - 6.841$$
(2)

Concentration variable in term \(- 334.9\varphi_{\rm w}^{4.044} \left( \frac{1}{T} \right)^{10.03}\) less than 0.01% affects the nanofluid viscosity. Therefore, the viscosity model of Li et al. does not have the expected dependence on the concentration variable.

By evaluating the correlations presented for nanofluid viscosity, the situation of temperature and volume (mass) fraction variables in the correlations is determined. The results showed if there is a significant relationship between variables and nanofluid viscosity.

Therefore, it is necessary to consider the effects of variables on the viscosity models of nanofluids in the temperature range and volume (mass) fraction of heat transfer.

Physical analysis of correlations

Viscosity models at different dimensions in this part of the research are studied. First, the structure of the proposed correlations to calculate the viscosity has been examined and then evaluated for analyzing complex, heterogeneous, or ambiguous components. Meanwhile, in another section, based on the experimental studies of researchers in the temperature range and volume (mass) fraction, the data extracted from the correlations are evaluated. Also, the accuracy of the experimental model for the case where the concentration of nanofluid is considered zero with the viscosity of base fluid has been investigated. Finally, the correlations have been measured in terms of the laws governing the physics of nanofluids and the changes due to an increase or decrease in temperature and concentration of nanofluids.

According to Table 1 of the Term section, in most of the experimental relations studied for calculating viscosity, it is observed that mathematical expressions and terms do not interfere with the calculation of the viscosity of nanofluids. Thus, these expressions have only caused the complexity and inefficiency of empirical relations, which increases the possibility of errors in the viscosity calculations of nanofluids.

Given the above, Alarifi et al. [34] studied the viscosity of hybrid nanofluids, which are composed of a mixture of MWCNT and TiO2 nanoparticles in oil, and they presented a new model by Eq. (3) for the viscosity of the nanofluid by examining the effects of temperature and concentration on the viscosity of the nanofluid. According to the equation, a trigonometric ratio has been used to express the relationship between concentration and viscosity. Therefore, a disproportionate function in relationships is not necessary and can only cause problems in calculations.

$$\mu_{\rm nf} = 2.936T + \frac{{2e^{4} }}{{1.68 + T - \left( {1.68\varphi } \right)}} - 448.8 - {\text{tan}}\left( {\left( {1.68\varphi } \right) - 1.68} \right)$$
(3)

Based on an experimental test on the viscosity behavior of SiO2 nanoparticles dispersed in a mixture of water and ethylene glycol, Ruhani et al.[35] proposed a viscosity model in Eq. (4), valid in the temperature range of 25 to 50 °C and a volume fraction of 0.1 to 1.5%. The temperature variable in the viscosity model is both in the position of the power function and is powered by the exponential function in terms of position. Therefore, using such functions one after the other is not justified and causes the calculations to be complex.

$$\mu_{\rm nf} = \left[ {2.030 - \left( {931.616 \times \varphi^{0.9305} \times 5.4597 \times T^{ - 3.4574} } \right)} \right.\left. { - \exp \left( { - 0.0028 \times \varphi^{2.1421} \times T^{1.0133} } \right)^{2} } \right] \mu_{\rm bf}$$
(4)

Yan et al. [28] have presented Eq. (5) for the hybrid nanofluid of MWCNT and TiO2 in ethylene glycol in the temperature range of 25 to 55 °C and at the volume fraction between 0.05 and 1%;

$$\begin{gathered} \mu_{\rm nf} = [0.90463 + 280.20104\varphi + 0.25734\theta + 368.05239\theta \varphi \hfill \\ \quad - 28643.68399\varphi^{2} - 0.012051\theta^{2} - 1968.73612\varphi^{2} \theta \hfill \\ \quad- 235.04729\theta^{2} \varphi + 2.09629 \times 10^{6} \varphi^{3} - 0.099694\theta^{3} ] \mu_{\rm bf} \hfill \\ \end{gathered}$$
(5)

Equation (5) has many terms that entering the equation for subsequent heat transfer calculations may be associated with many errors. On the other hands, by reducing the terms of the equation with increasing the accuracy, the equation becomes easier to use. Therefore, Eq. (5) can be presented more simply as Eq. (6).

$$\mu_{\rm nf} = \left( {0.0029741 + \varphi^{1.07982} } \right) \times \left( {T^{{ - 1.{25573}}} } \right) \times 311157$$
(6)

Equation (6) has been obtained by the nonlinear regression method from experimental data of Yan et al. [28] for hybrid nanofluid viscosity.

According to Fig. 4, the proposed model in Eq. (6), while having higher accuracy than Eq. (5), has a simpler form compared to Eq. (5). Figure (4) plotted at a volume fraction of 0 to 1% and a temperature of 30 °C for nanofluids.

$$\begin{gathered} \mu_{\rm nf} = a_{1} + a_{2} T + a_{3} \varphi + a_{4} T^{2} + a_{5} T\varphi + a_{6} \varphi^{2} + a_{7} T^{3} + a_{8} T^{2} \varphi + a_{9} T\varphi^{2} \hfill \\ + a_{10} \varphi^{3} + a_{11} T^{3} \varphi + a_{12} T^{2} \varphi^{2} + a_{13} T\varphi^{3} + a_{14} \varphi^{4} \hfill \\ \end{gathered}$$
(7)
Fig. 4
figure 4

Comparison of the results of Yan et al.'s correlation [28] and the currently proposed correlation with experimental data

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Huminic et al. [36] proposed Eq. (7) for La2O3 and water nanofluids in the temperature range of 293 to 323 K and a volume fraction between 0 and 0.03%.

According to Eq. (7), there are many components in the correlation that are similar to Eq. (5), and the experimental data provided by Huminic et al. are used, and Eq. (8) is presented as follows.

$$\mu_{\rm nf} = \left( {0.458474 + \varphi^{1.10104} } \right) \times \left( \frac{T}{323} \right)^{ - 0.636006} \times 1.5773$$
(8)

According to Fig. 5, Eq. (8), simplicity has accuracy more than 2% more than Eq. (7).

Fig. 5
figure 5

Comparison of the results of Huminic et al.'s [36] correlation and the proposed correlation with experimental data

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A closer look reveals similar cases in which researchers try to present complex correlations; however, using the same experimental data, simple and sometimes linear equations can be given with much higher accuracy than the desired equations, so Eqs. (6) and (8) can replace Eqs. (5) and (7).

While the models proposed for nanofluid viscosity by Esfe et al. [37, 38] presented in Eq. (9) and (10), there are several terms in the viscosity model; therefore, the presence of multiple terms in the equations is not necessary and makes the equations more complex, and these multiple components in viscosity calculation will lead to increased computational error.

$$\begin{gathered} \mu_{\rm nf} = 679.7806 + 259.62463\varphi - 33.64131T - 0.045393\dot{\gamma } - 6.0695 \times \varphi T \hfill \\ \quad \quad - 0.00031841\varphi \dot{\gamma } + 0.00129007T\dot{\gamma } - 236.23287\varphi^{2} + 0.65796T^{2} + 2.31776E - 06\dot{\gamma }^{2} \hfill \\ \quad \quad + 1.55167\varphi^{2} T + 0.049085\varphi T^{2} - 9.63258E - 8T\dot{\gamma }^{2} + 94.57115\varphi^{3} \hfill \\ \quad \quad - 0.0051586T^{3} + 0.0000000001\dot{\gamma }^{3} \hfill \\ \end{gathered}$$
(9)
$$\begin{gathered} \mu_{\rm nf} = 688.46 + 347.09\varphi - 33.12T - 0.04\dot{\gamma } - 7.36\varphi T - 0.0087\varphi \dot{\gamma } + 0.0014T\dot{\gamma } \hfill \\ \quad - 305.24\varphi^{2} + 0.61T^{2} + 1.49 \times 10^{ - 6} \dot{\gamma }^{2} + 0.0001\varphi T\dot{\gamma } + 0.46\varphi^{2} T + 0.0014\varphi^{2} \dot{\gamma } \hfill \\ { } \quad + 0.065\varphi T^{2} + 1.87 \times 10^{ - 7} \varphi \dot{\gamma }^{2} - 7.25 \times 10^{ - 6} T^{2} \dot{\gamma } - 7.33 \times 10^{ - 8} T\dot{\gamma }^{2} + 169.62\varphi^{3} \hfill \\ \quad - 0.0043T^{3} + 0.00000000011\dot{\gamma }^{3} \hfill \\ \end{gathered}$$
(10)

In the continuation of reviewing the results and data extracted from viscosity models, it has been observed that sometimes the equations at zero concentration and in a certain range of temperature and concentration have an unusual response. The results of this analysis are presented in Table 1 of the Data section.

For example, Huminic et al. [36] presented the viscosity model for La2O3 and water nanofluids in the temperature range of 293–323 K and at a volume fraction between 0 and 0.03 contrary to the researcher claims, the correlation is not able to respond at zero concentration.

Finally, according to studies by researchers on the viscous behavior of nanofluids, variables of temperature and concentration independently and directly affect the nanofluid viscosity, so that with increasing temperature, the viscosity of nanofluid decreases, and by adding nanoparticles to the base fluid, the viscosity of nanofluid increases. Therefore, at this stage, the variation trend of viscosity values ​​at different temperatures and concentrations according to the physical laws governing nanofluid viscosity is investigated. The results of this analysis are presented in Table 1 of the Trend section.

According to the presented issue, in the viscosity model proposed by Esfe and Esfandeh [39] for the viscosity of nanofluids, including oil and hybrid particles, the variation trend of viscosity values ​​with increasing concentration is contrary to the physical laws governing nanofluid viscosity.

Table 1 exhibits the correlations presented for nanofluid viscosity by various researchers from 2009 to 2020, along with statistical analysis and physical analysis. Statistical analysis of ANOVA test was performed, and terms related to physical examination [Data, Term, Trend] entirely have been presented for each experimental equation.

In addition, the validity range of the equations and the overall conclusion have been presented by the correlation evaluation.

Sensitivity analysis with Monte Carlo test

The sensitivity analysis method has been used to continue the statistical study process of relationships and know the position of variables in correlations. According to the general definition in statistics, sensitivity analysis is the study of the effectiveness of output variables from a set of assumed input variables in a statistical model.

As a result, the researcher can determine how changes in a component affect the model's output. Therefore, in the continuation of the statistical study article, the sensitivity analysis will give a deeper look at viscosity models with the variables of base fluid viscosity, temperature, and concentration. [80]

Therefore, to obtain a complete conclusion about the performance of viscosity models and the effectiveness of variables, only the correlations in which the variables have the expected dependence on the viscosity equations in terms of variance test are examined.

In the present study, to analyze viscosity models, also a method known as the Monte Carlo test is used.

Variables with little effect and little change on the equations are displayed as flat lines in the graph. So the more curved lines show the more dependence of the variable on the equation.

Li and Zou [26] introduced the viscosity model of Eq. (11) for nanofluids consisting of \({\text{Al}}_{2} {\text{O}}_{3}\) nanoparticles and water-based nanofluid, and Saeedi et al. [55] proposed the viscosity model of Eq. (12) for Ce \({\text{O}}_{2}\) nanoparticles dispersed in ethylene glycol.

$$\mu_{\rm nf} = \left[ {781.4 \times T^{ - 2.117} \times \varphi^{0.2722} + \frac{0.05776}{{T^{ - 0.7819} \times \varphi^{ - 0.04009} }} + 0.511 \times \varphi^{2} - 0.1779 \times \varphi^{3} } \right] \mu_{\rm bf}$$
(11)
$$\mu_{\rm nf} = 0.838\varphi^{0.188} T^{0.089} \mu_{\rm bf}^{1.1}$$
(12)

The test results in Figs. 6 and 7 show that the curvature of the green line is greater than that of the blue and purple lines, and the presence of the variable of the viscosity of the base fluid in correlation is more important than other variables. In addition, concentration and temperature, respectively, have an influential role in the equations.

Fig. 6
figure 6

Result of sensitivity analysis on the correlation proposed by Li and Zou [26]. (green color: base fluid viscosity variable/ blue color: volume fraction variable/ purple color: temperature variable)

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Fig. 7
figure 7

Result of sensitivity analysis on the correlation proposed by Saeedi et al. [55]. (green color: base fluid viscosity variable/ blue color: volume fraction variable/ purple color: temperature variable)

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Equation (13) presents Nabil et al. [62] viscosity model for the hybrid nanofluid of TiO2 and SiO2 in a mixture of water and ethylene glycol.

$$\mu_{\rm nf} = \left[ {37\left( {0.1 + \frac{\varphi }{100}} \right)^{1.59} \left( {0.1 + \frac{T}{80}} \right)^{0.31} } \right] \mu_{\rm bf}$$
(13)

In Fig. 8, the green line has higher curvature than the blue and purple lines. Therefore, the sensitivity analysis results showed that the variables of base fluid viscosity, concentration, and temperature significantly affect Eq. (13), respectively.

Fig. 8
figure 8

Result of sensitivity analysis on the correlation proposed by Nabil et al. [62]. (green color: base fluid viscosity variable/ blue color: volume fraction variable/ purple color: temperature variable)

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The variables of base fluid viscosity, temperature, and concentration are available in the above viscosity equations. The variable of base fluid viscosity applies the value of the base fluid viscosity in proportion to the reference temperature in the viscosity model. Therefore, the dependence of the viscosity equations on the viscosity variable of the base fluid expresses the relationship of the equations on temperature. On the other hands, the sensitivity analysis results show that the variable of the base fluid viscosity has a more contribution in estimating the viscosity of nanofluid than other variables. Therefore, considering the mentioned conditions, it is concluded that the temperature factor indirectly has a more significant effect on viscosity equations than other variables.

Esfe et al. [48] have presented Eq. (14) for the viscosity of hybrid nanoparticles of MWCNT and Ti \({\text{O}}_{2}\) in a mixture of water-based nanofluid and ethylene glycol;

$$\mu_{\rm nf} = 6.35 + 2.56\varphi - 0.24T - 0.068\varphi T + 0.905\varphi^{2} + 0.0027T^{2}$$
(14)

According to the sensitivity analysis results presented in Fig. 9, the effect of the temperature is greater than the concentration in the equation because the curvature of the green line is more significant.

$$\begin{gathered} \mu_{\rm nf} = 688.46 + 347.09\varphi - 33.12T - 0.04\dot{\gamma } - 7.36\varphi T - 0.0087\varphi \dot{\gamma } + 0.0014T\dot{\gamma } \hfill \\\quad \quad - 305.24\varphi^{2} + 0.61T^{2} + 1.49 \times 10^{ - 6} \dot{\gamma }^{2} + 0.0001\varphi T\dot{\gamma } + 0.46\varphi^{2} T + 0.0014\varphi^{2} \dot{\gamma } \hfill \\ \quad \quad { } + 0.065\varphi T^{2} + 1.87 \times 10^{ - 7} \varphi \dot{\gamma }^{2} - 7.25 \times 10^{ - 6} T^{2} \dot{\gamma } - 7.33 \times 10^{ - 8} T\dot{\gamma }^{2} + 169.62\varphi^{3} \hfill \\ \quad \quad - 0.0043T^{3} + 0.00000000011\dot{\gamma }^{3} \hfill \\ \end{gathered}$$
(15)
Fig. 9
figure 9

Result of sensitivity analysis on the correlation proposed by Esfe et al. [48]. (green color: temperature variable/ blue color: concentration variable)

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Esfe et al. [38] presented the nanofluid viscosity model for the MWCNT and \({\text{Al}}_{2} {\text{O}}_{3}\) hybrid nanoparticles dispersed in oil in Eq. (15).

The sensitivity analysis results in Fig. 10 showed that the green line has higher curvature than the purple and blue lines, so the effect of the temperature is greater than the concentration in the equation.

Fig. 10
figure 10

Result of sensitivity analysis on the correlation proposed by Esfe et al. [37]. (green color: temperature variable/ blue color: volume fraction variable/purple color: variable of shear rate)

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According to the results and the role of temperature and concentration in Eq. (15), the shear rate of nanofluid is also effective in calculating the viscosity of nanofluid.

The results of sensitivity analysis of the previous two equations show that when the variable of base fluid viscosity is not present in the viscosity equations, conditions are created that the effect of temperature factor is directly applied in the viscosity equations, and thus, the temperature variable has a more influential role than other variables.

Equation (16) presents Li et al. [79] model for the viscosity of SiC and water nanofluids.

$$\mu_{\rm nf} = \left[ {1.07879 + 0.45546\varphi + 0.4051\varphi^{2} - 0.2871\varphi^{3} } \right] \mu_{\rm bf}$$
(16)

Based on the sensitivity test results in Fig. 11 and given the curvature of the green line, the viscosity of base fluid has a higher contribution to the concentration variable in calculating the viscosity of the nanofluid.

Fig. 11
figure 11

Result of sensitivity analysis on the correlation proposed by Li et al. [79]. (green color: base fluid viscosity variable/ blue color: concentration variable)

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There is a base fluid viscosity variable \(\mu_{\rm bf}\) in most of the correlations reviewed here and in the known nanofluid viscosity models, and this factor determines the viscosity value of the base fluid relative to the reference temperature in the viscosity models, so the effects of temperature through the base fluid viscosity variable are considered in viscosity models. Therefore, the high dependence of the viscosity models on the variable of base fluid viscosity is due to the dependence of the viscosity models on temperature.

It is concluded that the effects of temperature are not directly considered in the correlations, and the \(\mu_{\rm bf}\) component is not an independent variable, which can cause problems.

According to the issues mentioned above and based on the sensitivity analysis results performed on viscosity models, the temperature factor plays a decisive role in viscosity models. Therefore, the results show the inherent dependence of nanofluid viscosity on temperature.

Overall analysis of empirical correlations

Table 1 is statistically and physically examined the prediction correlations of nanofluid viscosity. In the statistical study, variance analysis for temperature and volume (mass) fraction of nanofluid has been performed, and the results have been presented in the column related to the statistical study. In addition, in physical examination, three factors of Data, Term, and Trend have been considered. The Data column in Table 1 examines the experimental correlations' ability to respond at zero concentration to reach the value of the base fluid viscosity. In the Term column of Table 1, the results have been mentioned regarding the existence of numerous and irrational terms for estimating the viscosity of the mentioned experimental correlations. The compliance and the role of the variables introduced in the empirical correlations presented in Table 1 relative to the physics governing the viscosity of the nanofluid are evaluated in the Trend column of Table 1.

In evaluating viscosity relationships, it was observed that there are relationships that have good conditions in the physical examination but are not statistically similar. Also, reverse conditions for equations are possible. Therefore, it was decided to report the relationships with good status in two physical and statistical states in the total section.

By examining all the correlations in Table 1 and their statistical analysis and examining the physical performance of the correlations and the validity of each equation, it can be concluded that 53.6% of the equations are statistically valid. Also, 73.2% of the equations have accuracy and simplicity in terms of performance; in total, 35.7% of the equations in both physical and statistical states have an acceptable condition (Table 2).

Table 2 Status of statistical and physical analysis of all equations of Table 1
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The results of the evaluation of the experimental correlations presented in Table 1 for nanofluids based on water and ethylene glycol, which are widely used in the field of research, have been developed. Accordingly, in a comprehensive study on the correlations of Table 1 for water-based nanofluids, it is statistically and physically determined that statistically, 38.9% of the correlations are acceptable, 83.3% are physically reliable correlations, and a total of 27.7% of the correlations are acceptable. The results of this study are presented in Table 3.

Table 3 Status of evaluation of viscosity equations based on water-based nanofluid
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Suppose the examination for water-based nanofluid is performed again for ethylene glycol-based nanofluids, as shown in Table 4. In that case, 54.5% of the correlations are statistically acceptable, 54.5% are physically reliable, and a total of 27.2% of the ethylene glycol-based nanofluid equations in both physical and statistical states had an acceptable condition.

Table 4 Status of evaluation of viscosity equations based on ethylene glycol-based nanofluid
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Thus, the statistically and physically acceptable correlations for the water- and ethylene glycol-based nanofluid are presented in Tables 5 and 6.

Table 5 Acceptable correlations for the viscosity of water-based nanofluid
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Table 6 Acceptable correlations for the viscosity of ethylene glycol-based nanofluid
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Proposing a viscosity model and its validation

Preliminary analysis

By studying the correlations proposed by the researchers, the relationship of variables with the viscosity of nanofluid was determined. Therefore, it was confirmed by the analysis of variance that temperature and concentration variables play a decisive role in the relationship between the viscosity of nanofluids. In addition, the results of the statistical tools of sensitivity analysis showed that the viscosity of nanofluid is directly dependent on the temperature factor.

Given that the volume fraction variable of nanoparticles affects nanofluid's viscosity, and the rate of viscosity changes relative to the volume fraction of nanoparticles depends on the type of the base fluid.

On the other hands, it is crucial for the viscosity of the nanofluid with water-based nanofluid and ethylene glycol, which does not have limited use in terms of temperature, volume fraction, and especially particle material. Therefore, in the present study, two models with very high accuracy for nanofluids with water-based nanofluid in a wide range of volume fractions and temperature have been presented, and this important has been done for ethylene glycol-based nanofluid.

Evaluating the experimental studies on the viscosity of nanofluids in proportion to temperature and concentration, approximately the physical conditions of more than 1200 experimental data were examined, and dispersed viscosity values ​​were observed for the experimental data under the same physical conditions. Therefore, to increase the accuracy of the proposed correlation, a group of articles has been removed, and articles with appropriate and centralized experimental data have been selected to provide the correlation.

Figure 12 is plotted in terms of temperature and volume fraction of the water-based nanofluid to know the physical condition of the experimental data used. [21, 43, 64, 78, 81,82,83,84]

Fig. 12
figure 12

Three-dimensional representation of the viscosity dispersion of nanofluids with water-based nanofluid. [21, 43, 64, 78, 81,82,83,84]

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Figure 12 shows that the congestion of experimental data for the viscosity of water-based nanofluid at concentrations less than 1%, and the temperature range of 30 to 40 °C is higher.

Also, the dispersion of experimental data to present the viscosity model is shown in Fig. 13 in terms of temperature and volume fraction of ethylene glycol-based nanofluid. [55, 85,86,87,88]

Fig. 13
figure 13

Three-dimensional representation of the viscosity dispersion of nanofluids with ethylene glycol-based nanofluid. [55, 85,86,87,88]

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Also, according to Fig. 13, at low concentrations and the temperature range of 30 to 50 °C, the viscosity of ethylene glycol-based nanofluid has higher congestion.

Proposed correlation

Experimental viscosity correlations proposed by previous researchers cover a limited temperature range and volume fraction. Most experimental correlations proposed for nanofluid viscosity are unable to estimate the base fluid viscosity. On the other hands, in the analysis of variance, it was found that most of the mentioned correlations do not depend on the independent variables of those correlations. The sensitivity analysis results also showed that the factor of temperature directly affects the viscosity of nanofluid, and the variable of temperature has a more significant contribution in estimating the viscosity of nanofluid than other variables. In such conditions, to eliminate these shortcomings, a model has been presented for estimating the viscosity of water- and ethylene glycol-based nanofluid entitled BAG, Barkhordar-Armaghani-Ghasemiasl. The summary of the statistical and physical study of the BAG model is given in Table 7. According to the variance analysis, the temperature variable and the volume fraction have the appropriate P Values for the BAG model.

Table 7 BAG models for nanofluids viscosity
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The results of estimating the viscosity of water- and ethylene glycol-based nanofluid based on the BAG model in Table 7 compared to the experimental data in Figs. 12 and 13 based on R2 of water-based nanofluid are 97.01% and for ethylene glycol-based nanofluid are 96.08%.

Evaluation of BAG viscosity model

For assessing the validity of the accuracy of the presented correlations, it is necessary to compare the obtained results with the conventional and selected correlations. For this purpose, some conventional correlations in articles are introduced as follows. Einstein [1] was the first to introduce a microfluidic viscosity model for suspensions containing metal particles in 1906. This correlation applies to the viscosity of microfluid with spherical particles at a volume fraction of less than 5%. This model is given in Eq. (17).

$$\mu_{\text {nf}} = \left( {1 + 2.5\varphi } \right)\mu_{\text {bf}}$$
(17)

Brinkman [19] proposed a new model in 1952 according to Einstein’s model. This correlation is suitable for suspensions with a volume fraction of less than 4%. This correlation is given in Eq. (18), used in most studies by researchers.

$$\mu_{\text {nf}} = \left( {\frac{1}{{\left( {1 - \varphi } \right)^{{2.{ }5}} }}} \right)\mu_{\text {bf}}$$
(18)

In 1977, Batchelor [20] proposed a viscosity model for single-phase suspensions based on the Brownian motion of particles. Moreover, Eq. (19) is derived according to the Einstein equation and the existence of spherical particles.

$$\mu_{\text {nf}} = \left( {1 + 2.5\varphi + 6.2\varphi^{2} } \right)\mu_{\text {bf}}$$
(19)

In recent studies, Wang et al. [89] performed experiments on Al2O3 nanofluids separately for water and ethylene glycol-based nanofluid. Equations (20) and (21) are obtained for water-based Al2O3 nanofluid and ethylene glycol-based Al2O3 nanofluid.

$$\mu_{\rm nf} = \left( {1 + 7.3\varphi + 123\varphi^{2} } \right)\mu_{\text{bf}}$$
(20)
$$\mu_{\rm nf} = \left( {1 + 4.6\varphi + 6.7\varphi^{2} } \right)\mu_{\text{bf}}$$
(21)

In another study, Chen et al. [90] presented Eq. (22) for nanofluid viscosity. This correlation has been used in numerous previous articles.

$$\mu_{\rm nf} = \left( {1 + 10.6\varphi + \left( {10.6\varphi } \right)^{2} } \right)\mu_{\text{bf}}$$
(22)

Ho et al. [91] then performed an experimental experiment based on convection heat transfer and examined the variation trend of viscosity with increasing nanofluid concentration and presented Eq. (23).

$$\mu_{\rm nf} = \left( {1 + 4.93\varphi + 222.4\varphi^{2} } \right)\mu_{\rm bf}$$
(23)

Then, the results of estimating the viscosity of water-based nanofluid based on the BAG I model in row 1 of Table 7 have been evaluated with the conventional and selected correlations in Eqs. (17) to (23) according to the experimental data in Fig. 12.

In Fig. 14, parts a and b, the results for the nanofluid viscosity have been plotted at the temperature of 20 and 50 °C and a variable volume fractions of 0 to 3%, respectively. In the results, where scattered data are available, the values estimated by the BAG I model are more accurate than the points where the data are most concentrated.

Fig. 14
figure 14

Comparison of BAG I model with conventional correlations in Eqs. (17) to (23)

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Given that the present study uses a variety of experimental data, the experimental data used in this study have been extracted from several sources; also, the existence of data scatter in constant physical conditions seems reasonable. Therefore, in such cases, it is expected that BAG models can predict the values in which the data are more focused.

According to Fig. 14, parts c and d, the results for the nanofluid viscosity have been plotted at volume fractions of 0.1 and 1% and a variable temperature of 20–60 °C, respectively. The results indicated the BAG I model accurately predicts nanofluid viscosity according to the trend of temperature changes.

In another analysis, the BAG I model with the acceptable correlations in Table 1 for water-based nanofluid has been examined based on the experimental data in Fig. 12. However, the accuracy range of the acceptable correlations in Table 1 is not the same as the range of experimental data in Fig. 12, but to quantitatively express the estimation of the considered correlations relative to the BAG I model, which can estimate over a wide range of temperature 5–60 °C and volume fraction 0–3%. The results in the same physical conditions in Fig. 14 is also shown in Fig. 15; as can be seen, the results obtained from the BAG I model relative to the acceptable correlations in Table 1 are in good agreement with the experimental data.

Fig. 15
figure 15

A comparison of the BAG I model with acceptable correlations appeared in Table 1

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RMSE measures the error rate of two datasets. This parameter compares the predicted values ​​and the experiment's values ​​with each other, and the lower value leads to the lower error of the model. Thus, RMSE is an appropriate tool to compare correlations.

Based on Eqs. 24 and 25, to determine the error of the equations in predicting the experimental viscosity values, the "root mean square error," or RMSE index, has been used. Also, the accuracy of the equations in estimating the experimental viscosity values ​​is expressed by the R-squared index.

Table 8 shows the RMSE values ​​obtained for the conventional correlations and acceptable correlations in Table 1 and the BAG I model on the experimental points for estimating the viscosity of the water-based nanofluid.

$${\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {\mu_{\text{pre}} - \mu_{\text{exp}} } \right)^{2} }}{N}}$$
(24)
$$R^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {\mu_{\text{pre}} - \mu_{\text{exp}} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{N} \left( {\mu_{\text{pre}} - \overline{{\mu_{exp} }} } \right)}}$$
(25)
Table 8 The RMSE value of the BAG I model compared to other correlations for water-based nanofluids
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Findings based on the RMSE value indicate that the BAG I model has a 35.82% lower performance error than the best correlation presented by the researchers to estimate the viscosity of water-based nanofluids.

In addition, the diagram of the results of the R2 coefficient for the conventional and acceptable correlations of Table 1 and the BAG I model is presented in Fig. 16. Besides the reasonable accuracy of other correlations in estimating nanofluid viscosity, the BAG I model has higher accuracy than other correlations in estimating nanofluid viscosity according to the trend of nanofluid viscosity changes.

Fig. 16
figure 16

The value of R2 in the BAG I model compared to other correlations for water-based nanofluids

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In most studies on the viscosity of nanofluids, especially the conventional correlations, the correlations cannot predict the viscosity of nanofluids with the ethylene glycol-based nanofluid. One of the reasons for the weakness of these correlations is the high concentration of ethylene glycol-based nanofluid viscosity relative to water. On the other hands, usual correlations have been optimized for the viscosity of low concentration nanofluids. Therefore, conventional correlations do not respond proportionally to the nanofluid's viscosity with the ethylene glycol-based nanofluid. With these interpretations, the BAG II model for the viscosity for the nanofluids with ethylene glycol-based nanofluid has high accuracy for estimating viscosity.

Then, the results of estimating the viscosity of ethylene glycol-based nanofluid based on the BAG II model in row 1 of Table 7 have been evaluated with the conventional and selected correlations in Eqs. (17) to (23) according to the experimental data in Fig. 13.

In Fig. 17, parts a and b, the viscosity of the nanofluid is, respectively, at 30 and 50 °C and the variable concentration. In sections c and d, the results for the nanofluid viscosity have been plotted at volume fractions of 0.2 and 0.8% and variable temperature, respectively.

Fig. 17
figure 17

Comparison of BAG I model with conventional correlations in Eqs. (17) to (23)

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As expected, because the conventional correlations for the viscosity of nanofluids with the water-based nanofluid have been optimized, they cannot estimate the viscosity of nanofluids with ethylene glycol-based nanofluid. Considering the experimental data, the viscosity values predicted by the BAG II relation are much more accurate than the conventional relations.

In another analysis, the BAG II model with the acceptable correlations in Table 1 for ethylene glycol-based nanofluids has been investigated based on the experimental data in Fig. 13.

However, the accuracy range of the acceptable correlations in Table 1 is not the same as the range of experimental data in Fig. 13, but to quantitatively present the estimation of the considered correlations relative to the BAG II model, which can estimate in a wide range of temperature 20 to 60 °C and volume fraction 0 to 2%. Also, the results are presented in Fig. 18.

Fig. 18
figure 18

A comparison of the BAG II model with acceptable correlations appeared in Table 1

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According to parts a and b of Fig. 18, at the volume fraction range of 1–2% and temperatures of 30 and 50 °C, the BAG II model is significantly more accurately predicted than the other correlations.

Also, in Fig. 18, parts c and d, at volume fractions of 0.2 and 0.8% and in the temperature range of 20–60 °C, the BAG II model has mainly provided better results than other correlations.

Table 9 shows the RMSE values obtained for the conventional correlations and acceptable correlations in Table 1 and the BAG II model on the experimental points for estimating the viscosity of the ethylene glycol-based nanofluid.

Table 9 The RMSE value of the BAG II model compared to other correlations for ethylene glycol-based nanofluid
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Findings based on the RMSE value indicate that the BAG II model has a 49.84% lower performance error than the best correlation presented by the researchers to estimate the viscosity of ethylene glycol-based nanofluids.

To know the accuracy of the BAG II model for ethylene glycol in terms of R2 coefficient, Fig. 19 is plotted. The results show that the accuracy of the BAG II model is significantly higher than other correlations.

Fig. 19
figure 19

The value of R2 in the BAG II model compared to other correlations for ethylene glycol-based nanofluid

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The base fluid viscosity parameter \(\mu_{bf}\) is available in most of the correlations presented in Table 1 and the conventional correlations for calculating nanofluid viscosity. In this case, the viscosity of the base fluid plays the role of the variable temperature of the base fluid in addition to its role in the calculations so that the viscosity of the base fluid changes with the change of temperature. Therefore, the viscosity of the base fluid must also show the effect of temperature. Thus, by assuming the nanofluid type to be constant with temperature change, the base fluid viscosity in the nanofluid viscosity estimation correlation changes. On the other hands, in the nanofluid viscosity estimation correlations, the nanofluid temperature factor is not directly present in the above correlations. Therefore, these correlations alone are not able to estimate the nanofluid viscosity. Thus, when the nanofluid temperature is variable, the mentioned correlations increase the error probability in the calculations.

Also, while solving numerical problems of heat transfer due to temperature changes in the problem, it is sometimes impossible to change the base fluid's viscosity in the problem, which deviates the answer from the correct path. Therefore, the presence of a temperature variable in viscosity models is also felt here. Under such circumstances, the BAG model in Table 7 could be a turning point for other viscosity models in the future.

According to the above, another strength of BAG models is the ability to estimate the viscosity of the base fluid in the conventional temperature range used in heat transfer, which is less likely to provide a suitable response at zero concentration. To demonstrating the relationship ability of BAG, the values ​​predicted by the BAG relation are examined with the experimental values ​​of the viscosity of water-based nanofluid and ethylene glycol in Fig. 20, parts a and b, respectively.

Fig. 20
figure 20

Estimation of water- and ethylene glycol-based nanofluid viscosity using the BAG model

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Conclusions

The viscosity component plays a crucial role in heat transfer, especially convection heat transfer. The addition of nanoparticles to the base fluid is commonly considered to increase the viscosity rate. With the increase in the number of nanofluid viscosity models proposed, it has become necessary to review these models in the present study. We also evaluated the correlations in terms of physical compatibility with the viscosity of nanofluids and performed statistical tests of variance and sensitivity analysis on the viscosity models.

Finally, based on the weakness identified in previous models, through our statistical and correlation evaluations, two general equations for water- and ethylene glycol-based nanofluids were presented to predict the behavior of nanofluids. A summary of the results presented in this study is as follow;

  • The results of variance analysis on the viscosity correlations showed the non-dependence of 42.2% of the correlations on the temperature component and another 27.3% on the volume (mass) fraction component.

  • The volume (mass) fraction variables in nanofluid viscosity models are valid only in a certain range, and most correlations are not able to provide a solution at a zero volume (mass) fraction. Therefore, most nanofluid viscosity models do not cover the range of volume (mass) fractions used in heat transfer.

  • In some of the viscosity models introduced by researchers, correlations sometimes have complex and long terms unrelated to viscosity physics. The presence of such terms in viscosity models only increases the likelihood of errors in calculations. However, many of these correlations can be corrected and modified with simple and short models.

  • According to the study conducted physically and statistically on the viscosity models of nanofluids, only 53.6% of the correlations are statistically acceptable, and 73.2% of the correlations are physically reliable, and a total of 35.7% of the correlations have acceptable conditions.

  • Sensitivity analysis revealed the significant contribution of temperature component in estimating nanofluid viscosity. It is while the effect of temperature in the form of nanofluid viscosity models is not directly considered. Instead, the effect of temperature is determined by the independent variable of base fluid viscosity \(\mu_{\rm bf}\) in the desired viscosity models. Therefore, the viscosity models cannot estimate the nanofluid's viscosity in proportion to the temperature variation trend, and this factor can cause problems.

  • For modeling, the viscosity of nanofluids separated to water or ethylene glycol-based nanofluids, which is valid in a wide temperature and volume fraction range and function independent of the type of nanoparticles, BAG models were introduced.

  • The BAG models presented for nanofluid viscosity for water- and ethylene glycol-based nanofluid have 97.01% and 96.08% accuracy, respectively. Also, the RMSE value improved by 35.82% and 49.84% compared to the best correlation presented by the researchers for estimating the viscosity of water-based and ethylene glycol-based nanofluids, respectively.

  • Most of the viscosity models have been optimized for nanofluids with water-based nanofluids. However, the BAG model has the ability to estimate the viscosity of nanofluid with ethylene glycol-based nanofluid with much higher accuracy than other correlations.

  • Unlike other correlations, the results of BAG models showed that by changing the temperature of the nanofluid, BAG models maintain the ability to estimate the viscosity of the nanofluid accurately. Also, when the nanoparticle concentration is zero, the viscosity of the base fluid is well predicted.

Given that nanofluid with the oil base such as the applied fluids in industry and few correlations have been provided for oil-based nanofluids. Therefore, developing correlations for oil-based nanofluids is a challenge and an open field of research.