Introduction

Nanofluids are an evolution in fluid science. NPs have unique mechanical, thermal, magnetic and electrical properties. Actually, nanofluids are obtained by suspending particles with sizes below 100 nm in common fluids such as water and oil. By adding insignificant amounts of nanoparticles into the base fluids, an improvement in thermophysical properties is considered. Nanofluids are the term used by Choi (1995) for first time. The purpose of nanofluids is a significant increment in thermal properties of fluids by adding small amounts of nanoparticles. So thermophysical properties such as thermal conductivity, viscosity and density were examined. In recent decades, many studies have been implemented to measure and study the thermophysical properties of nanofluids like viscosity [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26], thermal conductivity [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60] and thermal convection [61,62,63,64,65,66,67,68,69,70,71,72]. Nanofluids are used in various fields such as air-conditioning [73], solar cell [74], automotive [75], nuclear reactors [76], lubrication [77], electrical systems [78], heat exchangers [79] and microchannels [80].

One of the important parameters in fluids is viscosity, which has important role in calculating Reynolds number, Prandtl number and heat transfer coefficient. Many studies have been conducted on the viscosity of nanofluids. Studies show that different parameters such as VF, particle size, temperature, base fluid properties, surfactant, etc., are affected on viscosity of nanofluids. Asadi et al. [81] examined the influence of temperature and particle volume concentration on the dynamic viscosity of MWCNT-MgO/SAE50 hybrid nanofluid. Results indicated that with increase in the temperature and concentration, the dynamic viscosity decreases and increases, respectively. They presented an empirical correlation for dynamic viscosity of nanofluid as a function of temperature and particle volume concentration. Hemmat et al. [82] examined the influence of temperature and particle volume concentration on the dynamic viscosity of SWCNTs-ZnO/EG nanofluid. This study determined that dynamic viscosity increases significantly by changing VF from 0.25 to 5%, while diameter of NP is 18 nm and temperature is 50 °C, whereas it does not change with increase in temperature. As well as, they presented an empirical correlation for dynamic viscosity of nanofluid as a function of temperature and particle volume concentration. Soltani and Akbari [83] examined the effect of temperature and particle volume concentration on the dynamic viscosity of MWCNT-MgO/EG hybrid nanofluid. This study conducted in volume concentration from 0 to 1% and temperature from 30 to 60 °C. Results of this study showed that hybrid nanofluid has the behavior such as Newtonian fluid in mentioned VFs and temperatures. The experiment indicated that with increase in the SVF from 0.1 to 1%, the relative viscosity increases up to 168%. Hemmat et al. [84] performed an experimental study on the dynamic viscosity of Mg(OH)2-EG nanofluids in VF range of 0.1–2.0% under the temperatures ranging from 23 to 55 °C. The results showed that the viscosity rises with growth in the concentration values and relative viscosity increases with rise in temperature. They also suggested a new correlation. Many studies have been done to predict a new correlation for nanofluid viscosity. Some correlations for predicting nanofluids viscosity are presented in Table 1.

Table 1 Some of recent new correlations for viscosity
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In some cases, experts have tried to improve viscosity index of engine lubricants. Hemmat Esfe et al. [90] conducted an experimental study on a nanolubricant enriched by MWCNT and ZnO NPs to control viscosity of nanofluids after adding nanoparticles.

In regard to importance of the viscosity of oil, the viscosity of the hybrid nano-oils would be of great importance. Adding NPs into hybrid fluids of nano-oils in different temperatures and VFs has a major impact. In this study, rheological behavior of MWCNT-MgO (50–50%)/SAE50 hybrid nano-oil under the temperature and concentration variations examined exactly for first time. As well as, a new correlation was suggested to estimate relative viscosity of hybrid nano-oil.

Experimentation

Sample preparation

In this study, MWCNT and MgO NPs were blended in 50:50 volume percent and they were dispersed in SAE50 oil with VFs of 0%, 0.125%, 0.25%, 0.5%, 0.75% and 1% by the use of two-step method. MWCNT/MgO properties are presented in Table 2. For the purpose of identifying specifications of nanofluids, it is necessary to disperse nanoparticles in base oil uniformly and gain a stable nanofluid. To create stable nanofluid samples, a magnetic stirrer device was used for 2 h. An ultrasonic device (Ultrasonic Homogenizer Development of Ultrasonic Technology, Iran) was applied for 3 h to create a great dispersion and stop formation clusters of NPs. In this way, a stable nanofluid was made and deposition was not observed with the naked eye.

Table 2 MWCNT NP properties
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The size was measured using X-ray diffraction (XRD). The results of XRD pattern for MgO and MWCNT nanoparticles are displayed in Fig. 1.

Fig. 1
figure 1

XRD pattern of MgO and MWCNT NPs

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Dynamic viscosity measurement

Dynamic viscosity of nano-oils samples of MWCNT-MgO (50–50%)/SAE50 hybrid nanofluids with SVFs of 0.0625, 0.125, 0.25, 0.75 and 1% was measured at temperatures between 25 and 50 °C by using Brookfield viscometer. Brookfield viscometer properties are listed in Table 3. SAE50 oil was used for calibrating the viscometer at room temperature. In order to ensure the results of the experiments, each experiment with different shear rates, temperatures and VFs was repeated several times, and then, the average of the measured data was recorded.

Table 3 Properties of used viscometer (CAP 2000+)
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Results and discussion

In this study, the viscosity variations of MWCNT-MgO (50–50%) hybrid nanofluid are examined experimentally in SVFs between 0.0625 and 1% and in temperatures between 25 and 50 °C.

Newtonian behavior

In fluid mechanics, a non-Newtonian fluid is a fluid in which viscosity depends on shear rate. For determination of Newtonian, non-Newtonian and Bingham behavior usually uses Ostwald–de Waele equation that is considered as follows:

$$\tau = m\dot{\gamma }^{\rm n}$$
(1)

Figure 2 displays n as a function of different temperatures for all samples. A value n is close to unity indicates nanofluid behavior close to Newtonian behavior. By enhancing SVF, n values decreases. That means non-Newtonian behavior is clearer in concentrated samples.

Fig. 2
figure 2

Effect of temperature and VF on n index

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In Fig. 3, consistency index (m) of hybrid nanofluids is displayed against temperature. The results show that the random motion of NPs in the base fluid increases by rising VF and van der Waals force causes major nanoclusters which prevent motion of fluid layers on each other. Therefore, m index of nanofluids increases to create a more viscous fluid. It has been reported that intermolecular interactions decrease with increase in temperature and it cause to decrease consistency index. It should be noted that some visible fluctuations in amounts of consistency index have no important reason. In this figure, the trend of changes is more important than fluctuations.

Fig. 3
figure 3

Effect of temperature and VF on m index

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Effective parameters on viscosity

The viscosity of fluid is the measure of its resistance to relative motion. On the other hand, the viscosity is known as an inhibition force and measure of the frictional properties of fluid. The viscosity of fluids is due to intermolecular force which is the van der Waals force.

In Fig. 4, variations of MWCNT-MgO (50–50%) hybrid nanofluid versus shear rate are presented for different VFs. Temperature increment leads to viscosity reduction in nanofluids. Low viscosity variations at higher temperatures of SAE50 base oil and MWCNT-MgO nanofluid are more significant in all volume fractions. The variation of viscosity versus shear rate illustrates non-Newtonian behavior of nanofluids.

Fig. 4
figure 4

The variation of viscosity versus shear rate at different temperatures and VFs

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Concentration of NPs effect on viscosity at a certain temperature is investigated, and results are depicted in Fig. 5. In a specific temperature, while shear rate enhances and VF reduces, the viscosity of nanofluid decreases. It seems that intermolecular force decreases by increase in temperature and viscosity of nanofluid reduces accordingly. The variation trends illustrate dependency of viscosity to its shear rate and a non-Newtonian behavior of nanofluids.

Fig. 5
figure 5

The variation of viscosity versus shear rate at different VFs and a constant temperature

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Figure 6 displays the variation of relative viscosity of MWCNT-MgO (50–50%)/SAE50 nanofluid versus temperature in different VFs. As seen in Fig. 6, increase or decrease in shear rate at a constant temperature leads no noticeable change in relative viscosity. Considering low variations in relative viscosity of nanofluids by increasing temperature in a constant VF, the independency of the relative viscosity changes to temperature is possible. The optimum concentration of dispersed NPs in oil depends on the expected application by user. For applications that pressure drop in operating cycle is not important or applications with only heat transfer goals, higher SVFs are suitable. On the other hand to reach a fluid with improved thermal characteristics and also minimum enhanced pressure drop in comparison with basefluid, users can select lower solid VFs like 0.0625% or 0.125%. As it is clear in Fig. 6, SVF of 0.0625% caused viscosity reduction in comparison with pure basefluid that gives users a fluid with modified thermal characteristics and also reduced viscosity that causes pumping cost saving.

Fig. 6
figure 6

Relative viscosity versus shear rate

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Figure 7 displays the variation of viscosity of MWCNT-MgO (50–50%)/SAE50 nanofluid versus different VFs. The results show that by rising temperature, the intermolecular van der Waals force decreases and it causes low viscosity. In higher temperatures, the variation of viscosity is fewer and it means that the viscosity has low dependency on effective parameters such as VF and temperature. The random motion of NPs in the base fluid increases by rising volume fraction and possibility of agglomeration in nanofluids increases. Therefore, the variation of viscosity is significant in higher volume fractions. The results show that the viscosity of nanofluid decreases to 329.6% by rising temperatures in a constant volume fraction of 1% and shear rate 200 1 s−1. As well as, the viscosity of nanofluid increases to 10.66% by rising VFs at a constant temperature of 25 °C and shear rate 200 1 s−1.

Fig. 7
figure 7

Viscosity versus temperature

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Proposed correlation

RSM method is used as a fast and economic method for prediction of viscosity of nanofluids based on mathematics. According to this method, a three variable mathematical correlation is proposed as a function of shear rate, SVF and temperature. By this formula, researchers can predict the viscosity of MWCNT-MgO (50–50%)/SAE50 nanofluid without needing any experimental setup.

$$\mu_{\rm nf} = 2240.02 + 346.13\varphi - 112.92T - 0.055\dot{\gamma } - 11.03\varphi T - 0.002\varphi \dot{\gamma } + 0.001T\dot{\gamma } - 109.11\varphi^{2} + 2.11T^{2} + 3.01E - 06\dot{\gamma }^{2} + 4.19E - 05\varphi T\dot{\gamma } + 1.71\varphi^{2} T - 0.0004\varphi^{2} \dot{\gamma } + 0.09\varphi T^{2} + 1.03E - 07\varphi \dot{\gamma }^{2} - 1.67E - 05T^{2} \dot{\gamma } - 9.50E - 09T\dot{\gamma }^{2} + 14.97\varphi^{3} - 0.013T^{3}$$
(2)

The coefficients of correlation for desired nanofluid are shown in Table 4. The P value less than 0.05 shows non-removable parameters in proposed correlation. By removing these coefficients, the order of correlation is eliminated. As shown in F values, there is only 0.01% probability of variance in the correlation, which indicates a valid relation in determining viscosity of nanofluid. Also Table 5 gives some information about the accuracy of the proposed model.

Table 4 Analysis of proposed correlation parameters
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Table 5 Analysis of variance for proposed correlation
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To check the accordance of predicted results by RSM method with empirical results, Fig. 8 is drawn up. As seen in Fig. 8, experimental data are laid on the bisector line or at least deviating from it. According to Fig. 8, dynamic viscosity data are predicted well by proposed correlation (2) and the correlation error is acceptable.

Fig. 8
figure 8

Matching analysis of correlation results with experimental results

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Figure 9 displays mean square errors of two shear rates in different VFs and temperatures. As shown in Fig. 9, maximum deviation is less than 3% that shows high accuracy of the correlations.

Fig. 9
figure 9

Accuracy control of predicted results

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Sensitivity analysis

Sensitivity analysis is performed to analyze the effect of adding unwanted amounts of NPs to base fluids (because of laboratory errors and human mistakes) on viscosity of nanolubricant. Change by 10% in each certain SVF and investigating the effect of this change on viscosity is studied for the introduced nanolubricant. Equation (3) was used for sensitivity analysis:

$${\text{Sensitivity}}\,{\text{of}}\,{\text{dynamic }}\,{\text{viscosity(\% )}} = \left( {\frac{{\left( {\mu_{\text{nf}} } \right)_{{{\text{After }}\,{\text{change}}}} }}{{\left( {\mu_{\text{nf}} } \right)_{{{\text{Base }}\,{\text{condition}}}} }} - 1} \right) \times 100$$
(3)

Sensitivity of viscosity in different temperatures and shear rates versus different VFs is illustrated in Fig. 10. At temperatures higher than 30 °C, the sensitivity of viscosity and their shear rates increases by adding VF and considering 10% extra NP to the sample. Increasing VF at a constant temperature resulted in increase in sensitivity of nanofluids. Increasing NPs collisions with base fluid molecule in higher VFs causes it to happen. The sensitivity rises more at 45 °C and 50 °C. Because of high sensitivity of viscosity at higher temperatures, more accuracy for supplying nanofluid is necessary.

Fig. 10
figure 10

Sensitivity analysis of viscosity

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Conclusions

In present experimental work, rheological behavior of MWCNT-MgO (50–50%) hybrid nanofluid based on SAE50 oil was examined experimentally. For this purpose, nanofluid samples were prepared in six solid VF under the temperatures ranging from 25 to 50 °C. According to results, adding NPs to base fluid causes viscosity dependency to shear rate and it is a sign of non-Newtonian behavior. Therefore, MWCNT-MgO (50–50%) hybrid nanofluid shows a non-Newtonian behavior. The examination of experiments shows below results:

  1. 1.

    The results show that the viscosity of nanofluid decreases to 329.6% by rising temperatures in a constant VF of 1% and shear rate 200 1 s−1. As well as, the viscosity of nanofluid increases to 10.66% by rising VFs at a constant temperature of 25 °C and shear rate 200 1 s−1.

  2. 2.

    Increase or decrease in shear rate at a constant temperature leads no noticeable change in relative viscosity.

  3. 3.

    In higher temperatures, the variation of viscosity is fewer and it means that the viscosity has low dependency on effective parameters such as VF and temperature. This trend causes a better balance in the variation of viscosity versus temperature.

  4. 4.

    By enhancing VF, power law index decreases and nanofluids behavior would become closer to non-Newtonian behavior.

  5. 5.

    Dynamic viscosity of nanofluid is predicted by the mathematical model well. Maximum deviation is less than 3% that shows high accuracy of the correlations.

  6. 6.

    Sensitivity of viscosity at higher temperatures is more than lower temperatures, because the sensitivity of viscosity to different VFs at higher temperatures rises.

  7. 7.

    SVF of 0.0625% causes 2–4% viscosity reduction in comparison with pure 5W50.

  8. 8.

    Concentrations of 0.0625% and 0.125% cause no excess pressure drop in operating cycle, and in some cases, it lowers the pumping cost.