A Novel Hybrid Model for Cu–Al2O3/H2O Nanofluid Flow and Heat Transfer in Convergent/Divergent Channels

In the present study, our aim is to present a novel model for the flow of hybrid nanofluids in oblique channels. Copper and aluminum oxide have been used to obtain a novel hybrid nanofluid. The equations that govern the flow of hybrid nanofluids have been transformed to a set of nonlinear equations with the implementation of self-similar variables. The resulting system is treated numerically by using coupled shooting and Runge–Kutta (R-K) scheme. The behavior of velocity and temperature is examined by altering the flow parameters. The cases for narrowing (convergent) and opening (divergent) channels are discussed, and the influence of various parameters on Nusselt number is also presented. To indicate the reliability of the study, a comparison is made that confirms the accuracy of the study presented.


Introduction
Heat transfer in regular fluids can be improved passively by altering flow situation, different flow parameters (like convective boundary condition, partial slip, etc.), or by enhancing fluid thermal conductivity. To improve the heat transfer in fluids several theoretical techniques have been proposed. Thus, researchers believed that the thermal conductivity of carrier fluids can be improved by a colloidal suspension of nano-or larger-sized particles of metals and their oxides because liquids have less thermal conductivity compared to that of metals. Thus, numerous experimental and theoretical studies on the colloidal suspension of fluids and solid particles of metals have been conducted since the larger size of metal particles and density are not helpful in preventing the mixture of particles and fluids because the lack of stability of such a colloidal composition produces extra flow resistance. Hence, suspension of large-sized metal particles in the fluids has not been popular until now.   Energies 2020, 13, x; doi: FOR PEER REVIEW www.mdpi.com/journal/energies this, only u component of the velocity is responsible for the flow of the fluid. In addition, flow is of the symmetric type. Figure 1 demonstrates the physical theme of the hybrid nanofluid flow between convergent/divergent channels. The shape factors [30] of different nanoparticles, namely spherical, bricks, cylinder, platelets, and blades are depicted in Figure 2.  Governing equations that describe the flow of hybrid nanofluids in an oblique channel in the presence of viscous dissipation are as follows [31]: Supporting conditions at the wall and central line are: where T w is temperature at the wall; U is the velocity at the central line; ρhn f , µhn f , ρC p hn f show effective density, dynamic viscosity, and heat capacitance, respectively; p denotes the pressure; and k hn f is the hybrid thermal conductance. Table 1 presenting the effective models for nanofluids characteristics. Table 1. Effective models for hybrid nanofluid characteristics.

Properties Nanofluid Hybrid Nanofluid
Density Viscosity Thermal conductivity here: where, ‫ג‬ = m − 1 and thermophysical properties [32] are tabulated in Table 2. The following models used for hybrid nanofluids are depicted in Table 1 [32].
The particular values of density, heat capacity and thermal conductivity of the host liquid and tiny particles defined in Table 2.
For a particular flow problem, the feasible similarity transformations are defined as: Energies 2020, 13, 1686 After entreating the suitable derivatives from self-similar parameters in Equations (2) and (3) and also using the suitable partial differentiation of similarity variables, governing partial differential equations were reduced into the following nonlinear flow model: In Equations (9) and (10), α denotes converging/diverging parameters; Pr is the Prandtl number; and Ec is the Eckert number. These parameters are: Further, the Reynold's number is defined in the following manner: The suitable boundary conditions in dimensionless form for velocity and temperature are as follows: F(η =0 ) = 1, F(η =1 ) = 0 and F (η =0 ) = 0 (13) The nondimensional expressions for shear stress and local heat transfer rate are:

Solution of the Problem
Under consideration hybrid nanofluid model is nonlinear for such model, it is tedious performed closed form of solution or not even exist. Thus, we focused on numerical treatment of the model. For this cause, we adopted Runge-Kutta scheme [26][27][28]. In order to apply afore said technique, firstly reduce the present model into a system of first ordinary differential equations (ODEs). The following substitutions were considered for this purpose.
The particular model can be written as: Energies 2020, 13, 1686 6 of 13 Now, entreating the described transformations we have: The corresponding initial conditions are in the following way.
Now, the model is treated by Mathematica 10.0.

Graphical Results and Discussion
The     Figure 6 investigates the opposite behavior of the thermal field. It is clear that for negative α, fluid temperature decreases in the region −0.5 ≤ η ≤ 0 and in 0 ≤ η ≤ 0.5, the temperature profile starts decelerating. In the case of Cu/water, rapid reduction in the temperature occurs. The reverse alterations are examined from −1.0 ≤ η ≤ −0.5 and 0.5 ≤ η ≤ 1.0. On the other hand, the opening channel favors the fluid temperature. The temperature of Cu-Al2O3/H2O increases very rapidly. In addition, the thermal field of water containing copper nanoparticles increases, but upturns in the temperature are quite slow in    Figure 6 investigates the opposite behavior of the thermal field. It is clear that for negative α, fluid temperature decreases in the region −0.5 ≤ η ≤ 0 and in 0 ≤ η ≤ 0.5, the temperature profile starts decelerating. In the case of Cu/water, rapid reduction in the temperature occurs. The reverse alterations are examined from −1.0 ≤ η ≤ −0.5 and 0.5 ≤ η ≤ 1.0. On the other hand, the opening channel favors the fluid temperature. The temperature of Cu-Al2O3/H2O increases very rapidly. In addition, the thermal field of water containing copper nanoparticles increases, but upturns in the temperature are quite slow in The influence of volumetric fraction on the fluid temperature for converging and diverging channels is depicted in Figure 7a and 7b, respectively. It is noted that the volumetric fraction ϕ2 shows reverse variations in the fluid temperature Θ(η). For nanoparticles of higher volume fraction, temperature starts to decrease. Decreasing effects of ϕ2 are quite prominent in the portion −0.5 ≤ η ≤ 0.5. Fluid near the channel walls shows almost negligible variations in the temperature for both Cu/water and Cu-Al2O3/H2O nanofluids, and temperature reduction is very rapid as compared to that of Cu-Al2O3/H2O nanofluid. In the case of converging channel (Figure 7a), the volume fraction of the nanoparticles favors the fluid temperature. At the middle line and in the locality of the middle line, fluid temperature increases quickly and for Cu-Al2O3/H2O, increasing behavior of dimensionless temperature Θ(η) is very rapid.
Very fascinating behavior of Eckert number Ec on the temperature field was observed from −1.0 ≤ η ≤ 1.0 for opening and narrowing walls. These effects are demonstrated in Figures 8a and 8b, respectively. In the presence of Eckert number, temperature of Cu/water and Cu-Al2O3/H2O nanofluids increases very clearly. It was examined that for α < 0 and α > 0, Eckert number favors the temperature profile. In converging channel, the temperature field is flatter in the vicinity of the middle line of the channel in comparison with the opening case. The variations in the fluid temperature due to increasing Reynold's number Re are depicted in Figures 9a and 9b, respectively for narrowing and opening channels. It was examined that Reynold's number opposes the fluid temperature in the convergent case. In the divergent case, fluid temperature (for both Cu/water and Cu-Al2O3/H2O) increases sharply.   Alterations in heat transfer rate are of great importance due to many industrial and technological uses. The various flow parameters play a significant role in local heat transfer rate (Figures 10-12). From Figure 10 it is obvious that more heat transfers at the walls for the narrowing channel. For the diverging case these influences reverse. Water containing Cu-Al2O3 has the capability to transfer more heat at the channel walls compared to that of Cu/water. On the other hand, Eckert number favors the heat transfer rate in both opening and narrowing channels. These effects are demonstrated in Figures 11a and 11b, respectively. Further, Reynold's number affects the heat transfer rate reversely in both situations as shown in Figure 12. Table 3 and Table 4   Alterations in heat transfer rate are of great importance due to many industrial and technological uses. The various flow parameters play a significant role in local heat transfer rate (Figures 10-12). From Figure 10 it is obvious that more heat transfers at the walls for the narrowing channel. For the diverging case these influences reverse. Water containing Cu-Al2O3 has the capability to transfer more heat at the channel walls compared to that of Cu/water. On the other hand, Eckert number favors the heat transfer rate in both opening and narrowing channels. These effects are demonstrated in Figures 11a and 11b, respectively. Further, Reynold's number affects the heat transfer rate reversely in both situations as shown in Figure 12. Table 3 and Table 4   favors the heat transfer rate in both opening and narrowing channels. These effects are demonstrated in Figures 11a and 11b, respectively. Further, Reynold's number affects the heat transfer rate reversely in both situations as shown in Figure 12. Table 3 and Table 4 compare the current results with existing literature under certain conditions for −F''(0). For ϕ_1 = ϕ_2 = 0, our flow model transformed into the model of conventional fluids and our results are parallel to existing ones which confirms the reliability of the study.      The influence of opening (diverging case) and narrowing (converging case) channels on the fluid velocity F(η) are incorporated in Figure 3a,b, respectively. A very fascinating behavior of converging/diverging parameter α is observed. It can be seen that when the channel walls become narrow, the fluid flowing area between the walls decreases and as a result fluid velocity starts increasing. The variations in fluid velocity above (0 ≤ η ≤ 1) and below (−1 ≤ η ≤ 0) the central line are similar because the flow is of a symmetric nature. For Cu/water and Cu-Al 2 O 3 /H 2 O almost similar velocity behavior is examined. At the middle part of the channel (η = 0) the velocity field coincides for both Cu/water and Cu-Al 2 O 3 /H 2 O nanofluids. It is observed that fluid velocity increases very slowly near the upper and lower portion of the channel. The converging parameter α affects the velocity field significantly in the area −0.5 ≤ η < 0 and 0 < η ≤ 0.5. The influence of increasing α on the velocity profile F(η) is depicted in Figure 3b. As for the opening channel (diverging case), the flowing area increases which causes a decrease in fluid motion. For Cu/water, velocity drops promptly in comparison with Cu-Al 2 O 3 /H 2 O. The pattern of fluid velocity is parabolic at the middle of the channel. Near the upper and lower channel walls, the velocity of both Cu/water and Cu-Al 2 O 3 /H 2 O decreases very rapidly. Near the walls, back flow produces for higher α.
For flow characteristics of nanofluids, the volumetric fraction plays the role of back bone. The influence of this important parameter on fluid motion is portrayed in Figure 4. It noticed that the convergent channel volumetric fraction favors the fluid motion, but these variations are almost inconsequential for Cu/water and Cu-Al 2 O 3 /H 2 O. A more flattened velocity profile is observed in the middle of the channel. In the case of the opening channel (Figure 4b), the volumetric fraction opposes the fluid motion. In the locality of the center line, the velocity profile is more steepened and almost similar for Cu/water and Cu-Al 2 O 3 /H 2 O. For Cu/water, the reduced velocity is quite rapid as compared to Cu- The dimensionless physical quantities, namely Reynold's number, which is the ratio of inertial to viscous forces, is a key point to analyze whether the flow is laminar or turbulent. The effects of this important physical parameter in the flow regimes are demonstrated in Figure 5a The influence of volumetric fraction on the fluid temperature for converging and diverging channels is depicted in Figure 7a,b, respectively. It is noted that the volumetric fraction φ 2 shows reverse variations in the fluid temperature Θ(η). For nanoparticles of higher volume fraction, temperature starts to decrease. Decreasing effects of φ 2 are quite prominent in the portion −0.5 ≤ η ≤ 0.5. Fluid near the channel walls shows almost negligible variations in the temperature for both Cu/water and Cu-Al 2 O 3 /H 2 O nanofluids, and temperature reduction is very rapid as compared to that of Cu-Al 2 O 3 /H 2 O nanofluid. In the case of converging channel (Figure 7a), the volume fraction of the nanoparticles favors the fluid temperature. At the middle line and in the locality of the middle line, fluid temperature increases quickly and for Cu-Al 2 O 3 /H 2 O, increasing behavior of dimensionless temperature Θ(η) is very rapid.
Very fascinating behavior of Eckert number Ec on the temperature field was observed from −1.0 ≤ η ≤ 1.0 for opening and narrowing walls. These effects are demonstrated in Figure 8a,b, respectively. In the presence of Eckert number, temperature of Cu/water and Cu-Al 2 O 3 /H 2 O nanofluids increases very clearly. It was examined that for α < 0 and α > 0, Eckert number favors the temperature profile. In converging channel, the temperature field is flatter in the vicinity of the middle line of the channel in comparison with the opening case. The variations in the fluid temperature due to increasing Reynold's number Re are depicted in Figure 9a,b, respectively for narrowing and opening channels. It was examined that Reynold's number opposes the fluid temperature in the convergent case. In the divergent case, fluid temperature (for both Cu/water and Cu-Al 2 O 3 /H 2 O) increases sharply.
Alterations in heat transfer rate are of great importance due to many industrial and technological uses. The various flow parameters play a significant role in local heat transfer rate (Figures 10-12). From Figure 10 it is obvious that more heat transfers at the walls for the narrowing channel. For the diverging case these influences reverse. Water containing Cu-Al 2 O 3 has the capability to transfer more heat at the channel walls compared to that of Cu/water. On the other hand, Eckert number favors the heat transfer rate in both opening and narrowing channels. These effects are demonstrated in Figure 11a,b, respectively. Further, Reynold's number affects the heat transfer rate reversely in both situations as shown in Figure 12.
Tables 3 and 4 compare the current results with existing literature under certain conditions for −F"(0). For φ_1 = φ_2 = 0, our flow model transformed into the model of conventional fluids and our results are parallel to existing ones which confirms the reliability of the study.

Conclusions
The colloidal hybrid flow regimes between the opening and narrowing channels are significant from medical and industrial perspectives. Therefore, Cu-Al 2 O 3 /H 2 O is contemplated between the particular geometry. For mathematical treatment of the model, the R-K technique was adopted and painted significant results for multiple parameters. It was detected that for multiple Re values, the velocity declines abruptly and back flow is detected near the walls. The velocity enhances for high fraction factor and Re in narrowing flow situations. For more dissipative colloidal mixture, the temperature abruptly enhances for opening flow situation. A decline in the heat transfer rate is detected for α and Re in highly dissipative cases for opening and narrowing cases. A comparison between the presented results with scientific literature confirmed the reliability of the analysis.