A comparative experimental investigation of dynamic viscosity of ZrO2/DW and SiC/DW nanofluids: Characterization, rheological behavior, and development of new correlation

In general, A nanofluid is a substance in which solids and fluids are mixed. The nano-powder of zirconium oxide (ZrO2) and silicon carbide (SiC) was dispersed into the distilled water (DW) using the widely adopted two-step technique. A Brookfield viscometer was used to measure the viscosity of the nanoparticles of ZrO2/DW and SiC/DW, where the temperature ranged between 20 and 60 °C and different solid volume fractions of 0.025, 0.05, 0.075, and 0.1 % were used. An examination of the mono nanofluids of ZrO2/DW and SiC/DW was conducted to assess their rheological behaviour. The findings of the experiments revealed that the Newtonian behaviour did not change when the nano-powder was added. Increasing the solid volume fraction of the nanoparticles and lowering the temperature resulted in the sample's dynamic viscosity being augmented. Hence, as the temperature rose, nanoparticles had a more observable impact on the viscosity. Furthermore, the findings showed that the increase in the ZrO2/DW nanofluid's viscosity peaked at 226.3 %, whereas for the SiC/DW nanofluid, it was 110.5 %. Additionally, according to the results of the experiments, new correlations capable of predicting the investigated nanofluids' viscosity in relation to solid concentration and temperature has been suggested. The study's results could motivate expanded utilization of nanofluids by researchers working on energy applications.


Introduction
In recent years, scientific researchers have been seeking novel ways of enhancing the heat transfer ability of different systems designed to move heat.In this regard, nanofluids are regarded as being a particularly efficient method of enhancing different thermal systems' thermal performance.Nanofluids constitute standard fluids (e.g., engine oil, water, and ethylene glycol) in which ultrafine particles are suspended.Such novel fluids have recently been the subject of increased focus among scientists due to their capability to facilitate perceptible transfer of heat compared with standard fluids.There is widespread agreement that a characteristic of novel nanofluids that has particular significance when applied in engineering and industrial contexts is their level of viscosity.consequently, a significant body of literature has described the importance of viscosity in the implementation of nanofluids [1,2].However, considering that when nanoparticles are added to fluids, their dynamic viscosity is increased, this enhancement also has a direct impact on the velocity field, friction forces and the ultimate ability of different kinds of thermal systems to transfer heat.Nanofluids can be differentiated from their base fluids with respect to their thermophysical characteristics, specifically thermal conductivity, and viscosity, which have received the most attention.Investigating fluid flow systems that incorporate nanofluids requires an understanding of the viscosity of nanofluids.In general, the addition of nanoparticles into the base fluid causes its viscosity to increase [3][4][5] [3-5] [3][4][5].
Today, as technology is developed across various fields of industry, it is unavoidable that industrial applications will be used that have increased power and thermal load; resultantly, state-of-the-art systems designed to optimise cooling are needed to a greater extent.Generally, the optimization of thermal systems is performed by enlarging the surface area, which causes the size and volume of machinery to be increased; hence, it is necessary to develop systems of cooling with increased efficiency to resolve this issue [6][7][8] [6-8] [6][7][8].As standard fluids have reduced thermal conductivity, the rate of heat transferred is reduced, which is regarded as being highly problematic.A summarized review of the previous studies on the dynamic viscosity of nanofluids is presented in Table 1.
Throughout the last decade, several subfields within nanoscience have achieved notable progress and innovations paralleling the developments made by the higher scientific community.Zirconium (Zr) has unique characteristics that make it very useful for several markets.Zirconium nanoparticles have several uses as nanocatalysts, nanosensors, adsorbents, and various other forms of nanomaterials.In addition, zirconium considers benefits in many applications including chemical industry, medical devices, optoelectronics, energy storage, fuel cells, water purification, and military industry.Silicon (Si) offers an important part as an element in a diverse range of nanomaterial heterostructures, with a wide range of interesting characteristics.The utilization of their absorptive characteristics provides them with importance in the field of papermaking.It can operate as a cohesive element in the production of plastics and concrete.Furthermore, they are non-toxic and stable materials having many applications in biomedicine, light-emitting applications, energy, electronic field, and photocatalysts.Nanoparticles of (Si) and (Zr) are widely available, and their increased specific surface area and hydrophilic properties suggest they could find widespread application in industrial settings.Various approaches have been developed for this to be accomplished, and a recent technique that has attracted particular interest involves nanofluids.Certainly, the development of nanofluids represents significant progress in the area of heat transfer, and the pioneer in this field was Choi et al. [22].Due to the fact that the size of the substances dispersed throughout the base fluids are measured in nanometres in comparison to additives whose measurements are millimetres or micrometres, such colloids offer multiple advantages including reduced erosion and pressure drop, they are more thermally conductive and stable, and the fact that the viscosity is not increased to the same extent means that less power is required to pump them [23][24][25].The advantages related to the utilization of zirconium oxide and silicon carbide nanoparticles are explained as follows.
• The addition of ZrO 2 and SiC nanoparticles into base fluids enhances the thermal physical properties of the base fluid due to the superior qualities exhibited by the nanoparticles compared to the base fluid.• Zirconium dioxide (ZrO 2 ) and silicon carbide (SiC) nanoparticles can be widely obtained and easy to synthesize.
• The utilization of ZrO 2 and SiC nanoparticles has favorable economic sustainability due to their low cost.
• These nanoparticles have better stability in water as compared to several other types of nanoparticles.
• No evidence of toxicity or flammability was discovered during the utilization of these nanoparticles, indicating their environmentally friendly nature.• ZrO 2 /DW and SiC/DW nanofluid have shown significant thermophysical properties for a wide range of applications, including heat exchangers, renewable energy, electronics cooling, and thermal storage.
Literature is enriched with studies on the viscosity of nanofluids.In this context, Bahrami et al. [26] conducted an experimental study to comprehend the changes in the viscosity of the hybrid nanofluid with fluctuations in temperature and solid volume fraction.The nanofluid was composed of water/ethylene-glycol base fluid with dispersed silicon dioxide/carbon nanotubes.It became evident that nanofluid showed greater viscosity at greater solid volume fractions and showed lower viscosity at increasing temperatures.They also examined the effects of different shear rates on the behavior of nanofluid and observed the Newtonian behavior of the base fluid and the non-Newtonian behavior of nanofluid at all concentrations.The water-based nanofluids were considered by Toghraie et al. [27] in their research; they used nanofluids with magnetic nanoparticles of Fe 3 O 4 and examined their performance in heating and cooling systems.They revealed that the nanofluid depicted a considerable rise of 129.7 % in its viscosity when a 3 % volume of nanoparticles was incorporated into the base fluid.They also put forward an equation to determine the viscosity of water-based magnetic nanofluid Fe 3 O 4 .The value evaluated from the equation was almost the same as that evaluated practically.Another notable work was done by Etaig et al. [28] who used nanofluids for the evaluation of the new effective viscosity model.The model simulations indicated a rise in the effective viscosity of the nanofluid at higher concentrations of nanoparticles.They found an increase in the nanofluid's viscosity at higher concentrations of the nanoparticles in a nanofluid.Considering temperature changes, the viscosity increased as the temperature lowered.Hemmat Esfe [29]experimented to determine the viscosity of the MWCNT (25 %)/MgO (75 %)/10W40 NF nanofluid under the subsequently given conditions; solid volume fraction between 0 and 1 %, temperature settings between 5 and 55 • C and shear rate between 6665 and 11997 s − 1 .He discovered that nanofluid depicted greater dynamic viscosity with a rise in solid volume fraction and also depicted a decline in dynamic viscosity with a reduction of temperature.They noted the highest value of viscosity equalling 26 % at temperature settings of 45 • C and nanoparticle concentration of 1 %, and the lowest viscosity of 5 % at a temperature of 15 • C and nanoparticle concentration of 0.05 %.Afrand et al. [30].demonstrated that when Fe 3 O 4 nanoparticles are dispersed into water, the Newtonian behaviour of the base fluid remained constant.Additionally, they found that decreasing the temperature and increasing the volume fraction resulted in the amplification of the nanofluid viscosity.Pastoriza-Gallego et al. [31].conducted a study in which the effects of the size of particles, solid weight fraction and temperature on CuO/water nanofluids' viscosity were investigated.Weight fractions with a maximum of 10 % and a temperate range between 10 • C and 50 • C were used in the experiments.According to their findings, as the solid weight fraction increased, the nanofluids' viscosity increased in parallel, whereas it decreased as the temperature and nanoparticle size increased.In the study of Azmi et al. [32], the viscosity of SiO2/water nanofluid with solid volume fractions varying between 0.5 % and 4 % as well as a temperature of 30 • C was measured.Their results demonstrated that the increase in viscosity reached 50 % with a nanoparticle volume fraction of 4 %.In the study of Duangthongsuk and Wongwises [33], the viscosity and temperature-dependent thermal conductivity TiO 2 -water nanofluids was measured.They demonstrated that an increase in the concentration of particles caused both variables to increase to a level that exceeded that of the base fluids.Non-Newtonian nanofluids' rheological properties were analysed in the study of Hojjat et al. [34].
They reached the conclusion that temperature and the concentration of particles influence the rheological attributes of such fluids.The thermal conductivity and viscosity of nanofluids in which carbon nanotubes with multiple walls were contained were studied by Phuoc et al. [35].The findings showed that increases in the thermal conductivity were not related to the viscosity of the base fluid, suggesting that enhanced thermal conductivity is not significantly influenced by the particle velocity.A nanofluid based on ethylene glycol containing particles of SiO 2 was investigated in the study of Rudyak et al. [36] for the purpose measuring the viscosity coefficient.They reported that a nanofluid's viscosity coefficient is highly dependent on temperature.Sadri et al. [37] demonstrated the ability of carbon nanotubes (CNTs) to enhance the thermal performance of standard working fluids.Their results showed that sonication time is a factor that influences the thermal conductivity and viscosity of such nanofluids.In the research of Li et al. [38], ZnO nanofluids based on ethylene-glycol were analysed in terms of their viscosity and thermal conductivity.They reached the conclusion that as expected, an increased concentration of ZnO nanoparticles caused the nanofluid's viscosity to increase, whereas it decreased as the temperature rose.Jarahnejad et al. [39] explored how the dynamic viscosity of nanofluids based on water that contained nanoparticles of titania (TiO 2 ) and alumina (Al 2 O 3 ) were effected by factors such as temperature as well as nanoparticle size and concentration.Their findings indicated that in general, the viscosity of the nanofluids that contained nanoparticles of TiO 2 nanoparticles was greater compared to those containing Al 2 O 3 when the loading was the same.Abbasi et al. [40] investigated the effects of temperature and concentration on the flow behaviour and viscosity of nanofluids containing TiO 2 nanoparticles, pristine MWCNTs, oxidised MWCNTs, and decorated MWCNTs were impacted by concentration and temperature in terms of the viscosity and flow characteristics.Their findings revealed that nanofluids' properties are dependent on concentration and temperature, while a decrease in the viscosity of the produced nanofluids was observed as the temperature rose and the concentration decreased.According to data from previous studies, a novel correlation was proposed by Meybodi et al. [41] on the basis of an exhaustive database of the viscosity data of Al 2 O 3 , TiO 2 , SiO 2 , and CuO nanofluids based on water.Their outcomes indicated that the developed correlation generated predictions that higher levels of agreement than values produced from experiments compared to past models, particularly in cases involving increased values of temperature, volumetric concentration and viscosity.Hemmat Esfe et al. [42] investigated how nanoparticle load and temperature impacted the viscosity of a nanofluid containing CuO, where the base fluid consisted of ethylene glycol.The results of their investigation indicated that the nanofluid relative viscosity value for the nanofluid containing 1.5 vol% CuO nanoparticles at 50 • C peaked at 82.46 %.Additionally, the developed a model design to predict the viscosity of a nanofluid and demonstrated that the data obtained from experimentation only deviated from those generated by the model by approximately 4 %.The results of the experiments also showed that a significant decrease in the dynamic viscosity occurred as the temperature of the nanofluid increased.Additionally, Murshed et al. [43] conducted measurements on nanofluids containing TiO 2 and Al 2 O 3 to determine their thermal conductivity and dynamic viscosity.They reached the conclusion that the nanofluids had significantly increased values for these parameters compared to the unaltered base fluids.Furthermore, they detected significant increases in the dynamic viscosity and thermal conductivity values when the volume fraction of the nanoparticles increased from 0.01 to 0.05 vol%.Chandrasekar et al. [44]  precipitation facilitated by microwaves.Subsequently, a sonicator was used to disperse the particles in distilled water.An increase in the nanofluids' viscosity was observed proportional to the volume fraction of the nanoparticle.Conventional models have been designed that are capable of predicting nanofluid viscosity, which largely agreed with the results produced the experiments.In the study of Yiamsawas et al. [45], a nanofluid comprised of Al 2 O 3 and water was evaluated to determine its viscosity when both the particle concentration and temperature were increased.The nanoparticles' volume fraction ranged between 1 % and 8 % as the temperature increased from 15 • C to 60 • C. The concluded that as the temperature rises, the viscosity is lowered.Jeong et al. [46] conducted experiments to investigate the viscosity of nanofluids containing ZnO whose nanoparticles were largely shaped in the form of rectangles or sphered with range of nanoparticle volume fractions between 0.05 and 5.0 vol%.The findings showed that as the volume concentration increased, the nanofluids' viscosity increased proportionally from 5.3 % to 68.6 % in comparison to water, which comprised the base fluid.
According to prior researches, the conclusion can be drawn that a nanofluid's viscosity can be directly influenced by multiple factors including temperature, methods of suspension, solid volume fraction, the shape of the particle, surfactant, kind of nanoparticles and the base fluid.The viscosity of nanofluids has been the focus of numerous studies, which can be referenced to improve the comprehension of nanofluids' viscosity characteristics.In order for nanofluids' rheological behaviour to be researched, it is firstly necessary to determine their viscous behaviour from Newtonian and non-Newtonian perspectives.The findings of certain researchers have indicated the base fluid's rheological behaviour is not changed when nanoparticles are added.
In the present study, the preparation of various samples of ZrO 2 /DW and SiC/DW mono nanofluids was conducted utilising the twostage approach for the purpose of investigating the rheological behaviour of the nanofluids, while the XRD pattern, TEM, EDX, FESEM, DLS, UV-vis, and Zeta potential methods were used to measure its structural characteristics and stability.Measurement of the investigated nanofluids' viscosity was performed in various solid concentrations (between 0.025 % and 0.1 %) and temperatures (between 20 • C and 60 • C).Furthermore, according to data obtained from the experiments, new correlations for predicting the dynamic viscosity of ZrO 2 /DW and SiC/DW nanofluids with respect to solid concentration and temperature have been suggested.The study also highlighted the significance of comprehending the nanofluid viscosity to get better insight into the working of fluid flow systems.In short, the incorporation of nanoparticles into the base fluid yields nanofluid with higher Viscosity.

Material
In the present research, nanofluids for ZrO 2 /DW and SiC/DW samples at volume fractions of 0.025, 0.05, 0.075, and 0.1 % were created by dispersing nanoparticles of ZrO 2 and SiC into the distilled water.Table 2 present summary of the thermophysical characteristics of nanoparticles of ZrO 2 and SiC, respectively, which were sourced from US Research Nanomaterials.The images of zirconium oxide and silicon carbide nanoparticles are shown in Fig. 1.

Nanofluid preparation
A two-step approach was used to obtain samples of ZrO 2 /DW and SiC/DW nanofluids.Subsequent to weighting, nanoparticles of ZrO 2 and SiC at the intended solid volume fraction (0.025, 0.05, 0.075, and 0.1 %) were added to purified water.It was necessary to firstly calculate the necessary mass amounts before the various nanofluid fractions could be prepared.For the purpose of calculating the nanoparticle and base fluid values and the respective amounts needed to prepare various volume fractions, Eq. (1) was employed.A digital scale (A&D WEIGHING, GE-320, USA) with high sensitivity was used for measuring the materials' weight.When the experiment started, computation of the mass amounts was performed in terms of the nanoparticles' density.Subsequent to the addition of nanoparticles to the water, the mixture was stirred using a magnetic stirrer (HS-12, HU) for approximately 1 h.Afterward, suspensions were entered into an ultrasonic processor (PS 30A 6L, Germany) for the process of breaking down the agglomeration among the particles as well as preventing sedimentation, ensuring the particles were uniformly dispersed and the suspension was stable; this process lasted approximately 3-4 h.Fig. 2 shows preparation process of ZrO 2 /DW and SiC/DW nanofluids.

Dynamic light scattering (DLS) analysis
After being dispersed into liquid phase, the synthesised nanoparticles' hydrodynamic size distribution was measured with dynamic light scattering (DLS).The respective particle size distributions for the nanoparticles of silicon carbide and zirconium oxide according to number, volume, and intensity are shown in Fig. 3.As a result of the data analysis, according to the DLS measurements, the nanoparticles of SiC and ZrO 2 had respective sizes is less than 100 nm, where the distribution of sizes was limited.

Measuring dynamic viscosity
In the current study, a Brookfield DV2TRVTBG rotational viscometer was used for measuring the ZrO 2 /DW and SiC/DW nanofluids' viscosity as well as for identifying their rheological characterization when influenced by varying parameters.Such viscometers function by taking measurements of fluid's resistance caused by the torque from the spindle placed into the fluid.Additionally, to ensure the temperature remained stable as well as to determine how nanofluids' viscosity is impacted by temperature, a temperature bath with significant accuracy was linked with the viscometer.Hence, this facilitated the process of conducting analysis of the nanofluids' rheological properties and viscosity according to the effects of shear rate and temperature at varying concentrations.Therefore, via the connection between the viscometer and temperature bath, the parameters of the experiment were fixed such that the temperature ranged between 20 and 60 • C to measure the nanofluids' dynamic viscosity at different solid volume fractions.All measurements were performed using a ULA spindle that was a small for sampling adapter system.Furthermore, calibration of the viscometer was performed prior to taking measurements using water as the base fluid at an ambient temperature.An image of the Brookfield viscometer is depicted in Fig. 4.

Characterization
The structural properties of SiC and ZrO 2 nanoparticles are measured by applying X-ray diffraction.The XRD instrument is used to  obtain the angle of the incident (theta), and angle of reflection to maintain a fixed focusing distance by modifying the relative places of the reflected X-ray, sample surface and incident X-ray.Therefore, there will be an angle of 2θ between the reflected and incident X-ray.In contrast, the degree of crystallinity of the exact plane is shown by the peak intensity since crystallinity is a relative term and not fixed.X-ray diffraction spectra of SiC and ZrO 2 are shown in Fig. 5-c and 5-g, respectively.There is an intense peak (diffraction peaks) at 2θ = 35.78• is attributed to silicon carbide and 2θ = 28.29 • for zirconium oxide.Low-intensity peaks can also be observed in the XRD plot.
The samples film of SiC and ZrO 2 dry nanoparticles are subjected to a ray of high-energy electrons for accurate measurement of nanoparticle size and shape.The electrons move through the sample and produce an image on the screen.The presence of near spherical shaped for ZrO 2 and cubic shape for SiC nanoparticles are confirmed by the TEM images.The TEM images also indicate nanoparticles to be characterized with a 20 nm average size for ZrO 2 (Fig. 5-e) and rang size between 45 and 65 nm for SiC nanoparticles (Fig. 5-a).
Field emission scanning electron microscopy (FESEM) is one of the most suitable methods that can be used to discover the morphological properties of nanoparticles.FESEM images with the magnification scale of 200 nm can be seen in Fig. 5-b and 5-f that as expected, there is a layered structure of silicon carbide and zirconium oxide nanoparticles with nano-size.silicon carbide and zirconium oxide nanoparticles are enabled by the planar surface and interconnected surface to generate conduction paths by generating networks.In addition, a spherical shape of ZrO 2 nanoparticles is shown in Fig. 5-f with a size 20 nm.Also, Fig. 5 b show morphological cubic of SiC nanoparticles with size between 45 and 65 nm.Particle sizes at the nanoscale were stressed by both FESEM images.Fig. 5d and 5-h shows the chemical compositions of SiC and ZrO 2 , respectively, the determination of which was made through EDX analysis.The results indicated that 53.26 % carbon, and 46.74 % silicon were present in the SiC while 67.88 % zirconium, 32.12 % oxygen were detected in ZrO 2 .

Stability of nanofluid
Initially, a zeta potential test is performed to check the stability of the prepared samples of Sic and ZrO 2 .Next, the samples are poured inside an ultrasonic device where it is ensured that particles are uniformly dispersed in the sample by treating it at 20 kHz frequency for 40 min.The resultant nanofluid shows higher stability.After about 15 days, the nanofluid samples with different volume concentrations (0.025 %− 0.1 %) were checked for stability revealing adequate stability.Fig. 6 depicts the values of zeta potential test conducted on the given samples of SiC and ZrO 2 nanofluids.The zeta potential test allows evaluation of the disparity in the electrical potential of the slipping layer and that observed at distance from the particle.The density of particles is directly associated with zeta potential.The surface charge on the particles results in colloid stability.The particles experience a repulsion force between them due to similarly charged loads and thus, they remain apart from each other leading to higher stability and weaker agglomeration.The results indicated stable nanofluid as evident form the values of zeta potential shown in this figure.The criteria suggest that nanofluids with zeta potential values between 30 and 50 mV are considered stable.

Validation of measurements
Validity testing is essential in experimental investigations because of the variety of possible causes of inaccuracy.Pure water's dynamic viscosity was measured with a Brookfield rotational viscometer at different temperatures, and outcomes were compared with the values found in the ASHRAE handbook [47].The results are displayed in Fig. 7, which indicates an acceptable level of agreement between the measured values and the reference (ASHRAE Handbook).This finding provides confirmation of the measuring instruments' high level of accuracy.

Rheological behavior
The Prandtl and Reynolds number of nanofluids is influenced by the viscosity.Hence, it is a critical thermophysical characteristic for convective heat transfer, lubrication and pumping power.For the purpose of evaluating the rheological properties of ZrO 2 /DW and SiC/DW nanofluids, measurements of the nanofluids' viscosity were taken at varying shear rates.A fluid's rheological behaviour is based on the given Eq. ( 2).
Where γ, τ, and μ denote the shear rate (1/s), dynamic viscosity (Pa⋅s), and shear stress (Pa), respectively.Based on this equation, a fluid is defined as being Newtonian in cases where the shear stress is a linear function of the shear rate.In Fig. 8, the shear stress is shown as a function of shear rate for mono nanofluids of ZrO 2 /DW and SiC/DW where the solid volume fraction is 0.1 % and the temperatures vary.As the changes in shear stress as result of the shear rate show a linear pattern, it can be deduced that the ZrO 2 /DW and SiC/DW nanofluids exhibit Newtonian characteristics, which supports the standard rule proposed by Venerus et al. [48].This is a fundamental requirement for the use of nanofluid in thermal systems for applications like convection.Fig. 9 demonstrates the changes in dynamic viscosity according to the shear rate at various temperatures and volume fraction of 0.1 % for the nanofluids of ZrO 2 /DW and SiC/DW.As the dynamic viscosity remains constant with regard to the shear rate, this suggests that the nanofluids of ZrO 2 /DW and SiC/DW behave in a Newtonian manner.

Dynamic viscosity
The viscosity of a fluid is a thermophysical characteristic and manifests in cases were slipping among the layers of fluid happens.It is the characteristic due to which a fluid offers resistance to relative motion among molecules of the fluid.Viscosity is more evident when the layers of the fluid experience movements.The mechanism underling the creation of viscosity in fluids is van der Waals forces among molecules.When this occurs, frictional forces are formed according to the value of viscosity.The strength of the friction force rises in line with the viscosity.The addition and dispersion of nanoparticles among the molecules of the base fluid results in tighter slippage of the fluid layers.Hence, the strength of the frictional force formed among the layers of fluid within the nanofluid is enhanced.
In Fig. 10, the changes in dynamic viscosity according to temperature at various volume fractions are shown for the ZrO 2 /DW and SiC/DW nanofluids.As illustrated in the figure, as the temperature rises, the viscosity of the liquid is lowered.This is due to the fact that the interactions between molecules and the van der Waals forces are weakened when the temperature increases.It is an intriguing    finding that when the temperature remains the same, when the solid concentration increases, the nanofluid's dynamic viscosity also increases accordingly.The rise is more discernible when temperatures are reduced as opposed to increased.Based on the findings, when the volume fractions are increased, the temperature has a more discernible impact on the viscosity of the nanofluid.Indeed, when the volume fraction is elevated, there is an increased likelihood that agglomeration will occur among solid particles within the base liquid, the bonds connecting particles break more frequently, and the changes in viscosity are more pronounced, in comparison to situations where the temperature is reduced.
In Fig. 11, the changes in dynamic viscosity for the mono nanofluids of ZrO 2 /DW and SiC/DW according to the volume fraction at various temperatures are shown.As illustrated in the figure, as the volume fraction rises, nanofluids viscosity increases accordingly.This is due to the fact that the addition of nanoparticles to the base fluid causes the molecules of the liquid and the solid particles to interact to a greater extent, thus causing the viscosity to increase.Nanoclusters with greater magnitude are formed when the proportion of solid particles within a particular volume of base liquid is increased because of the greater van der Waals forces among them.The resultant nanoclusters restrict layers of water from moving easily on each other, which causes the viscosity to rise to a large extent.Additionally, van der Waals forces improve the viscosity according to the volume when the temperature is reduced.The reason for this is that when the temperature is increased, particles are capable of overcoming van der Waals forces, which causes the rate at which the viscosity increases to decelerate as the volume fraction rises, in comparison to situations where the temperature is reduced.On the other hand, minimum increases of 39.5 % and 13.6 % were observed for the ZrO 2 /DW water and SiC/DW nanofluids, respectively, when the solid concentration was 0.025 % and temperature was 20 • C. Another remarkable finding was that when the solid concentration was 0.025 % and the temperature was set at 20, 30, 40,50,and 60 • C, the dynamic viscosity of the ZrO 2 /DW nanofluid was under 89 %, whereas it was approximately 26 % for the SiC/DW nanofluid; this shows the potential of these nanofluids for application in industrial and engineering settings due to the increased importance of pumping strength at reduced pressures.It should be noted that in the case of the ZrO 2 /DW nanofluid, the dynamic viscosity increase was maximised and minimised at 226.3 % and 39.5 %, whereas for the SiC/DW nanofluid, its maximum and minimum increase were 110.5 % and 13.6 %, which manifested at sold concentrations and temperatures of 0.025 % and 20 • C for the former, and 0.1 % and 60 • C for the latter.

Relative dynamic viscosity (RDV)
According to the measurements, the definition of the "relative viscosity" is based on the ratio between the dynamic viscosity of nanofluids of ZrO 2 /DW and SiC/DW and the nanofluids' dynamic viscosity (distilled water), expressed in Eq. ( 3).In Figs. 12 and 13, the relative dynamic viscosity in relation to temperature and solid volume fraction for the nanofluids of ZrO 2 /DW and SiC/DW is shown.The figures clearly show that as the solid volume fraction is increased, the relative viscosity is increased.This pattern was also supported in past studies.As shown in Figures, the relative viscosities increased marginally as the temperature rose, and this was more observable when the solid volume fractions also increased.In the solid volume fraction range from 0.025 % to 0.05 %, and with temperature of 60 • C, the relative viscosity of ZrO 2 /DW nanofluid changed the most by 33 %, whereas for SiC/DW, it changed by 15 %.Furthermore, augmenting the particle ration with a volume fraction of 0.1 % to approximately 72 % and 67 % for the ZrO 2 /DW and SiC/DW nanofluids, respectively, caused the relative viscosity to change by the greatest amount.This result implies that volume fraction as opposed to temperature has a greater influence on nanofluids' viscosity.

Proposed correlation
When nanofluids applied in engineering, it is frequently necessary to demonstrate thermophysical characteristics as equations or by utilising mathematical programmes for the properties of the nanofluid to be used.In the current research, the fitting method along with Levenberg-Marquardt algorithm by Sigma Plot program was applied to the data obtained from the experiments, which facilitated the development of a novel model capable of predicting the dynamic viscosity of the investigated nanofluids with respect to solid concentration and temperature.New correlations for the prediction of the dynamic viscosity of nanofluids as functions of solid concentration (φ) and temperature (T) is developed through Eqs. ( 4) and ( 5) for ZrO 2 /DW and SiC/DW, respectively, as shown below.
For the purpose of investigating the suggested correlations accuracy, it is necessary to define a specific parameter known as the Margin of Deviation, as expressed in Eq. (6).
In Eq. ( 6) above, the indices Exp and Pred represent experimental data and the predictions of the suggested correlations, respectively.

Comparison between previous theoretical models and experimental outcomes
Researchers have presented several theoretical models to estimate the dynamic viscosity of nanofluids.In this study, three commonly models, namely Einstein [49] and Wang [50], have been decided to evaluate their accuracy in forecasting the dynamic viscosity of the studied nanofluid.The outcomes obtained from these models are then compared with the experimental data.Fig. 18 shows the comparison between the theoretical models that were chosen and the experimental data.It is notable that each of these models lacks the capability to accurately estimate the dynamic viscosity of the nanofluid.Therefore, it can be concluded that it is necessary to provide a model that has the capability to predict with precision the dynamic viscosity of the nanofluid within an acceptable range.Thus, it is required to propose a model that can accurately predict the dynamic viscosity of ZrO 2 /DW and SiC/DW mono nanofluid.Likewise, an experimental data set was compiled from the outcomes of previous researchers and is shown in Table 3 to verify the validity of the correlations in this work.This table compares different kinds of nanofluid, that had different kinds of nanoparticles, solid volume fraction, and temperature range.These results demonstrate that the proposed correlation can evaluate the experimental results of other research at different temperatures and volume solid fractions.

Viscosity sensitivity
Sensitivity analysis is conducted to determine the extent to which changes in a specific model's output are caused by corresponding changes in the model's input factors (parameters or variables).For instance, when the model's parameters or input variables are changed slightly and the output is affected comparatively significantly, it is considered that the output has sensitivity to the parameters or variables.This type of analysis is generally conducted by performing a number of consecutive experiments where the designer of the model employs various input variables to identify how changing an input influences changes in the model's output.In the current study, sensitivity analysis was based on Eq. (7).Fig. 19 plots the viscosity sensitivity values against volume (φ) with a variation of solid volume fraction for the nanofluids of ZrO 2 /DW and SiC/DW.This indicates that the greatest change to sensitivity happened when the volume was highest, which equated to 4.55 % and 6.19 % for ZrO 2 /DW and SiC/DW, respectively.Moreover, in this figure it is shown that when the temperature remains the same and the volume fraction changes, the sensitivity increase is higher compared to when the temperature changed, and the volume remains the same.Hence, the conclusion can be made that the sensitivity of the volume fractions' objective function was greater compared to temperature, which should therefore be taken into account when preparing nanofluids, particularly when the volume fraction is 0.1 % such that an error reduction can be achieved when analysing rheological behaviour.

Conclusion
This study is aimed at determining the changes observed in dynamic viscosity of nanofluids of ZrO 2 /DW and SiC/DW with fluctuations in temperature and solid volume fraction.According to the findings of the experiments, the conclusions detailed below can be drawn.
• A rise in the temperature caused the nanofluid's viscosity to decrease.Van der Waals forces are weakened as the temperature rises.
Thus, the viscosity decreased when the temperature rose.• Newtonian behaviour was exhibited by the investigated nanofluids across the range of temperatures and solid volume fraction studied.• The results indicate that dynamic viscosity increased to the maximum level at a solid concentration of 0.1 % and temperature of 60 • C, where the increase was around 226.3 % for the nanofluid of ZrO 2 /water and 110.5 % for the nanofluid of SiC/water.• Through the application of fitting method to the data obtained from experiments, new correlations capable of predicting the investigated nanofluids' dynamic viscosity with respect to solid concentration and temperature has been developed.• The comparison of the laboratory data with the outcomes from the proposed correlation showed a 5.38 % margin of deviation and R 2 value of 98.52 % for ZrO 2 /DW and by 5.27 % and 99.11 %, respectively for SiC/DW nanofluids.

Data availability statement
Data will be made available on request.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 7 .
Fig. 7. Comparison between ASHRAE data on viscosity of water and experimental outcomes at different temperatures.

Fig. 10 .
Fig. 10.Differences of dynamic viscosity of nanofluids with solid volume fraction at different temperatures for (a) ZrO 2 /DW and (b) SiC/ DW nanofluid.

Fig. 11 .
Fig. 11.Differences of dynamic viscosity of nanofluids with temperatures at different solid volume fraction for (a) ZrO 2 /DW and (b) SiC/ DW nanofluid.

Fig. 14
show the margin of deviation values.The different values for the ratio of dynamic viscosity acquired from the experiments and correlation are compared in figure.Additionally, this figure presents the margins of deviation calculated from Eq. (6).Moreover, for the nanofluids of ZrO 2 /DW and SiC/DW the margin of deviation peaked at 5.38 % and 5.27 %, respectively.Furthermore, Fig.15compare the values from the experiments and the predictions for the respective nanofluids.As these figures indicate, most points are located on or near to the bisector and it is therefore not significantly distant, which suggests that the correlation is suitably accurate.Also, the figure suggests that the data from the experiments and the correlation results have a good level of agreement.Due to the significance of the estimations of thermal conductivity being accurate, Figs.16 and 17 compare the data from the experiments and the empirical correlation results for the respective nanofluids at a range of temperatures.These figures reveal that for the majority of the measurement values, there is either overlap between the points denoting the experimental and correlation results or they deviate slightly.This trend indicates that the suggested correlation is suitably accurate.

Table 1
A summary of studies for the dynamic viscosity of nanofluids.

Table 2
Physical and chemical properties of zirconium oxide and silicon carbide nanoparticles.

Table 3
The comparison between relative viscosity obtained by proposed correlation with experimental data obtained by other researchers.