Rheological Modeling of Metallic Oxide Nanoparticles Containing Non-Newtonian Nanofluids and Potential Investigation of Heat and Mass Flow Characteristics

Nanofluids have great potential due to their improved properties that make them useful for addressing various industrial and engineering problems. In order to use nanofluids on an industrial scale, it is first important to discuss their rheological behavior in relation to heat transfer aspects. In the current study, the flow characteristics of nanofluids are discussed using a mathematical model that is developed by fundamental laws and experimental data. The data are collected in the form of viscosity versus shear rate for different homogeneous ethylene glycol- (EG) based nanofluids, which are synthesized by dispersing 5–20% nanoparticle concentrations of SiO2, MgO, and TiO2 with diameters of (20–30 nm, 60–70 nm), (20 nm, 40 nm), and (30 nm, 50 nm), respectively. The data are fitted into a rheological power-law model and further used to govern equations of a physical problem. The problem is simplified into ordinary differential equations by using a boundary layer and similarity transformations and then solved through the numerical Runge–Kutta (RK) method. The obtained results in the form of velocity and temperature profiles at different nanoparticle concentrations and diameters are displayed graphically for discussion. Furthermore, displacement and momentum thicknesses are computed numerically to explain boundary-layer growth. The results show that the velocity profile is reduced and the temperature profile is raised by increasing the nanoparticle concentration. Conversely, the velocity profile is increased and the temperature profile is decreased by increasing the nanoparticle diameter. The results of the present investigation regarding heat and mass flow behavior will help engineers design equipment and improve the efficacy and economy of the overall process in the industry.

Non-Newtonian nanofluids that were synthesized by water or EG revealed shearthinning behavior [43][44][45][46][47]. Shear-thinning fluid is a kind of non-Newtonian fluid wherein viscosity declines by the rise of the shear rate. There are several mathematical models in the literature used to investigate the rheological behavior of such fluids. In the list of models, the power-law model demonstrates the relation between viscosity and shear rate. It is very popular in various disciplines, such as the biosciences and food and processing reservoir engineering [6][7][8][9][10][11][12][13][14]. Specifically, it is widely used in fluid flow problems under different conditions, and it is even used as a working principle in different kinds of rheometers.
In previous numerical studies, a deficiency was found in the theoretical models that can guess the exact behavior of nanofluids. The researchers used the Newtonian model for homogeneous nanofluids, which does not apply to the experimental behavior of all cases. No one study is available where homogeneous nanofluids deal with the non-Newtonian model. In the current study, we used a non-Newtonian power-law fluid model according to the trend of an experimental study. The parameters of the model are expressed as the function of nanoparticles, and we developed new mathematical relations to these parameters. By using these relations that govern equations, the physical problem became complex because every parameter is a function of the volume fraction, and solving this problem is not easy.
In this study, our goal is to investigate the flow behavior of non-Newtonian nanofluids, which are synthesized by the dispersion of metallic oxides in EG. For the flow, wedge shape is adopted as the geometry for our problem, which is favorable for accelerating or decelerating the fluids. The mathematical problem for flow is developed by the fundamental equations of fluid mechanics and modifies its parameters in view of the experimental evidence. The results from these equations are obtained in the form of velocity and temperature profiles and displayed in graphical form for discussion.

Nanofluid Modeling
In this section, the mathematical models of physical properties are developed by using experimental data for three homogenous nanofluids: SiO 2 -EG, MgO-EG, and TiO 2 -EG. The experimental data for nanofluids, which contain 5%, 10%, 15%, and 20% nanoparticle concentrations and (20-30 nm, 60-70 nm), (30 nm, 50 nm), and (20 nm, 40 nm) nanoparticle diameters, respectively, are collected for the study [1]. We have chosen the nanoparticles of materials SiO 2 , MgO, and TiO 2 for the nanofluid because these materials are used in manufacturing on a large scale at industry level. EG is used as a base fluid because it can be utilized within sufficiently large temperature ranges. For rheological behavior, the powerlaw equation is applied for nanofluid modeling in the formation of the viscosity-shear rate relationship as follows [49]: where µ n f is named as the consistency coefficient and n is the power index, which is justifiable according to the Newtonian or non-Newtonian behavior of fluids. When n = 1, the fluid exhibits Newtonian behavior, and the fluid shows shear-thinning behavior when n < 1. The rheological behavior of said nanofluids at different nanoparticle concentrations is displayed in Figures 1-3.   The curve-fitting technique is used to calculate the values of the two empirical parameters µ n f and (n) of Equation (1) for different nanofluids at different nanoparticle concentrations. For the curve fitting, we used the FindFit package from Mathematica, which is used for fitting in non-linear models. Afterward, these parameters are expressed in a second-order polynomial of the nanoparticle concentration φ. The equation of the power-law index and consistency index is defined as follows: where A 1 , A 2 , B 1 , and B 2 are constants calculated by using the curve-fitting technique in Table 1.  Therefore, the viscosity model of Equation (1) is written as The co-relation models for other physical properties, such as heat capacitance ρC p n f , effective density ρ n f , and thermal conductivity k n f , are defined by [50]. In the above equations, the thermal conductivity of nanoparticle k npl with a layer around the particle is defined by where γ * = k layer /k np is the ratio of the layers of the nanoparticle to the thermal conductivities of the nanoparticle and β * = h layer /R p is the ratio of the layer's height to the radius of the nanoparticle.

Flow Modeling
We considered the boundary-layer fluid flow of nanofluids over a moving wedge, which has incompressible and steady-state properties. The wedge is moved with the velocity u w (x) = cx m , whereas u ∞ (x) = bx m is the velocity of the nanofluid over an inviscid region, as shown in Figure 6. On the basis of the results in Figures 1-3, the power-law model is used in momentum Equation (9). Under the boundary-layer approximation, the continuity, momentum, and energy equations are written as [51,52] subjected to the boundary conditions In the above equations, u and v are the velocity component parts in x and y directions. T w is the temperature on the wedge's surface, and T ∞ is the temperature away from the surface (T w > T ∞ ).
The relationship between the Falkner-Skan power-law parameter (m) and the wedge's angle β = Ω/π is stated as For simplicity, we introduced the similarity transformations as follows: Substituting Equation (14) into Equations (9)-(12), we obtain the following equations: with the boundary conditions Here, the modified Prandtl number is Pr , and the Reynold number is

Displacement Thickness
Displacement thickness is recognized as the vertical distance that is produced by the absent mass flow rate due to the boundary-layer phenomena. The expression for displacement thickness is written as Dimensionless displacement thickness is composed as

Momentum Thickness
Momentum thickness is the height of an imaginary stream, which is transmitted by the loss of the momentum flow rate due to the boundary-layer phenomena and is described mathematically as Dimensionless momentum thickness is illustrated as

Skin Friction
The skin-friction coefficient is a dimensionless parameter that represents the shear stress at the wall. It is expressed as The wall shear stress τ w can be written as After applying transformation Equation (13), we receive

Nusselt Number
The Nusselt number is a dimensionless parameter that represents the convective heat-transfer rate at the wall. It is written in the following form: Here, h is a convective heat-transfer coefficient that can be written as After applying transformation Equation (13), it is written as

Solution Technique
The solution of Equations (14) and (15) with respect to Equation (16) is obtained by using the RK method. The method is executed in the following manner: Let f = F 1 , θ = G 1 , and convert Equations (14) and (15) into a system of first-order differential equations as along the initial conditions Here Ω 1 and Ω 2 represent unknown boundary conditions. To evaluate the accuracy of the results, the values of f (0) and −θ (0) against the parameters of β and Pr are compared with the existing limited results [53,54] in Tables 2 and 3.

Result and Discussion
In this segment, the results under the influence of the governing parameters are displayed for discussion. The first set of results is displayed graphically in the form of velocity and temperature profiles related to three homogenous metallic oxide nanofluids under the impacts of different nanoparticle concentrations and diameters. To see the effects of these parameters, the values of the geometry's parameters, for example, wedge angle Ω = π/6, wedge speed u w = 0.01, and free-stream velocity u ∞ = 0.04, are taken as fixed, and other parameters, such as the Reynold number and the modified Prandtl number, are varied according to different nanoparticle concentrations and diameters, which are listed in Tables 4 and 5. It is seen that the Reynold number is varied due to the variation in inertial force. The inertial force is a function of the power-law index, which becomes less than unity when the nanoparticle concentration is increased in the case of the MgO -EG and TiO 2 -EG nanofluids. The modified Prandtl number is a function of the Reynold number and is varied by it.

Velocity Profiles
The results of the velocity profiles for the SiO 2 -EG, MgO-EG, and TiO 2 -EG nanofluids under the impact of distinct nanoparticle concentrations are illustrated in Figures 7-9. It is seen that the velocity profile for all specified nanofluids is reduced with the increase in nanoparticle concentration. In the current situation, the viscosity of present nanofluids is enhanced due to the increase in nanoparticle concentration, which causes the reduction in velocity. The trend of viscosity and velocity profile for the current study is found to be similar in existing studies [1,[55][56][57][58][59][60]. These graphs assert that the dominant effects are found in the velocity profile of the SiO 2 -EG nanofluid due to maximum viscosity as compared with the MgO-EG and TiO 2 -EG nanofluids.   The velocity profiles of the schematic nanofluids under the effect of two distinct nanoparticle diameters are shown in Figures 10-12. It is clear that the velocity profile is increased by increasing the nanoparticle diameter due to the decline in the viscosity of the nanofluids. In addition, the dominant impact of this parameter on the profile is found to be more pronounced in the MgO-EG nanofluid than in the other nanofluids.

Temperature Profiles
The graphs of the temperature profiles for the nanofluids with respect to various nanoparticle concentrations are displayed in Figures 13-15. The figures show that the temperature is raised with the increase in nanoparticle concentration. The fact is that the temperature distribution is enhanced due to the increase in thermal conductivity and the decline in specific heat. The trend of this profile is also matched with the trend demonstrated in existing studies [55][56][57][58][59][60]. Moreover, the graphs disclose that the temperature of the SiO 2 -EG nanofluid is extensively affected by the enhancement of nanoparticle concentrations as compared with the other nanofluids.   The results of the temperature profiles for the schematic nanofluids under the influence of the nanoparticle diameter are illustrated in Figures 16-18. It is revealed that the temperature is reduced by enhancing the nanoparticle's diameter, as shown in all graphs. This is because of the increase in the Prandtl number that occurs by decreasing thermal diffusion.

Physical Parameters
The results of the boundary-layer parameters, such as the velocity and temperature boundary-region thicknesses, displacement thicknesses, and momentum thicknesses, are presented in Figures 19-22, whereas the results of the coefficient of skin friction and the Nusselt number are displayed in Figures 23 and 24.       Figures 19 and 20. It is noted that the thickness of the velocity boundary-layer region is enlarged with the enhancement of the nanoparticle concentration and increases at a position far from the origin. It is also observed that the velocity thickness of the SiO 2 -EG nanofluid is lowest at low concentrations but highest at high concentrations compared with both the MgO-EG and TiO 2 -EG nanofluids. For temperature boundary-layer thickness, it is perceived that the thickness is increased by enhancing the nanoparticle concentration. In addition, the temperature boundary-layer thickness is found to be smaller as compared with the thickness of the velocity boundary layer due to the dominant effects of mass diffusion as compared with thermal diffusion. It is further observed that the thermal boundary-layer thickness is found to be the greatest in the SiO 2 -EG nanofluid as compared with both the TiO 2 -EG and MgO-EG nanofluids. Figures 21 and 22 show the results of displacement and momentum thicknesses against different nanoparticle concentrations at distinct locations on the wedge's surface. The value of the displacement thickness is raised by enhancing the nanoparticle concentration and also increases along the distance of the wedge. It is seen that the maximum displacement is found in the case of the SiO 2 -EG nanofluid, and the lowest displacement is found in the TiO 2 -EG nanofluid. The momentum thickness indicates a reduction in the momentum of the nanofluid, and it is observed that the value is increased by raising the nanoparticle concentration.
The values of the coefficient of skin friction for the nanofluids against the nanoparticle concentration are displayed in Figure 23. It is to be noted that the value of the coefficient of skin friction is increased when the nanoparticle concentration is enhanced, and maximum enhancement is found in the case of the SiO 2 -EG nanofluid. The values of the Nusselt number with respect to the nanoparticle concentration are displayed in Figure 24. It is seen that the Nusselt number is decreased with the increase in nanoparticle concentration, and a maximum decline is found in the SiO 2 -EG nanofluid as well. In addition, the trend of the results for the coefficient of skin friction and the Nusselt number agrees with the trend of published studies [55][56][57][58][59][60].
The results of the coefficient of skin friction and the Nusselt number at different values of Re x are shown in Table 6. To calculate the results, the values of Re x in Table 4 are used. The results show the same trend as seen in Figures 23 and 24 because Re x is dependent on the value of the volume fraction.

Conclusions
In the present analysis, the boundary-layer fluid flow of three homogenous non-Newtonian nanofluids over a moving wedge is investigated. The mathematical results are presented graphically in the form of velocity and temperature profiles and further used to obtain the values of boundary-layer parameters. The main conclusions from the results are as follows:

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The profile of velocity is decreased and increased by raising the values of nanoparticle concentration and diameter, respectively.

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The profile of temperature is increased and decreased by enhancing the values of nanoparticle concentration and diameter, respectively.

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The velocity and temperature boundary-layer regions are increased by increasing the nanoparticle concentration.

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The displacement and momentum thicknesses are increased by the rise of nanoparticle concentration.

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The skin-friction coefficient is enhanced whereas the Nusselt number is decreased with the increase in nanoparticle concentration.