Flow Characteristics of Heat and Mass for Nanofluid under Different Operating Temperatures over Wedge and Plate

Background and Purpose: Nanofluids are a new class of heat transfer fluids that are used for different heat transfer applications. The transport characteristics of these fluids not only depend upon flow conditions but also strongly depend on operating temperature. In respect of these facts, the properties of these fluids are modified to measure the temperature effects and used in the governing equations to see the heat and mass flow behavior. Design of Model: Consider the nanofluids which are synthesized by dispersing metallic oxides (SiO2, Al2O3), carbon nanostructures (PEG-TGr, PEG-GnP), and nanoparticles in deionized water (DIW), with (0.025–0.1%) particle concentration over (30–50 °C) temperature range. The thermophysical properties of these fluids are modeled theoretically with the help of experimental data as a function of a temperature and volume fraction. These models are further used in transport equations for fluid flow over both wedge and plate. To get the solution, the equations are simplified in the shape of ordinary differential equations by applying the boundary layer and similarity transformations and then solved by the RK method. Results: The solution of the governing equation is found in the form of velocity and temperature expressions for both geometries and displayed graphically for discussion. Moreover, momentum and thermal boundary layer thicknesses, displacement, momentum thicknesses, the coefficient of skin friction, and Nusselt number are calculated numerically in tabular form. Finding: The maximum reduction and enhancement in velocity and temperature profile is found in the case of flow over the plate as compared to the wedge. The boundary layer parameters are increased in the case of flow over the plate than the wedge.

Many investigations have been carried out to see the behavior of thermophysical properties of nanofluids for different applications by using different types of nanoparticles [19][20][21][22][23]. In the list of thermophysical properties, viscosity plays an important role in the transport of mass and convective heat transfer. The viscosity of nanofluid is affected not only by shear rate but also by operating temperature, nanoparticle concentration, type of nanoparticles and their sizes, etc. Numerous studies have been conducted on the behavior of viscosity of nanofluids. Moghaddam et al. [24] studied the viscosity of graphene/glycerol nanofluids at a 6.32 shear rate, 20 • C temperature, and different particle concentrations. It increases by increasing nanoparticle concentration and decreases by enhancing temperature. Chen et al. [25] described the rheological properties of TiO2/EG nanofluids. The results exhibit the Newtonian at a 0.5-10 4 shear rate and found that viscosity is independent of the temperature. Rashin and Hemalatha [26] investigated the viscosity of CuO/coconut oil nanofluids. Their experiments showed the non-Newtonian behavior at a low shear rate (0-2.5%) concentration under different temperatures. Khedkar et al. [27] studied the viscosity of Fe 3 O 4 /paraffin at 0.01-0.1% concentration. Their experimental results showed that the viscosity is enhanced by increasing nanoparticle concentration whereas it shows Newtonian behavior at a high shear rate and non-Newtonian at a lower. Halelfadl et al. [28] studied the viscosity of CNT/water nanofluids at a high shear rate under different temperature conditions. The results showed that the nanofluids performed a non-Newtonian behavior at high nanoparticle concentration and Newtonian at lower nanoparticle concentration. Later, Chen et al. [29] studied the rheological properties of TiO 2 /EG nanofluids at different nanoparticle concentrations and temperatures. The nanofluids show a non-Newtonian property at 2% particle concentration under different temperatures. Numburu et al. [30] investigated the rhetorical property of SiO 2 /EG and SiO 2 /water nanofluids at −35-50 • C temperature. It is found that the nanofluid exhibited Newtonian properties at high temperatures and non-Newtonian properties at low temperatures. Kulkarni et al. [31] reported the viscosity of Al 2 O 3 /EG, CuO/EG, and SiO 2 /EG nanofluids under −35-50 • C temperature ranges. It is reported that viscosity reduces exponentially by increasing temperature. Yu et al. [32] observed the effects of the viscosity of ZnO/EG nanofluids. The results detected Newtonian behaviors at low particle concentrations and non-Newtonian behaviors at higher particle concentrations under different temperature conditions.
In the literature related to nanofluids, the behavior of thermal conductivity is investigated widely due to heat transfer's applications, and found that the behavior of conduction depends on various factors such as temperature, nanoparticle shape, size, and type [33,34]. Teng et al. [35] investigated the impact of a particle's size and temperature on the thermal conductivity of Al 2 O 3 /H 2 O nanofluids. The results exhibit that the thermal conductivity is increased with increasing nanoparticles concentration and temperature. Chandrasekar et al. [36] observed that the thermal conductivity of Al 2 O 3 /water nanofluids increased by increasing nanoparticle concentration under room temperature. Sundar et al. [37] predicted the behavior of thermal conductivity and viscosity of Al 2 O 3 /EG-Water nanofluids on different particle concentrations (0.3-1.5%) at temperatures range (20-60 • C). The results specified that the thermal conductivity of nanofluids improves with increasing nanoparticle concentrations and temperatures. Mahbubul et al. [38] studied the behavior of the thermal conductivity of Al 2 O 3 /R141b nano-refrigerant and found an enhancement in thermal conduction by increasing nanoparticle concentration and temperature. Mostafizur et al. [39] investigated the thermal conductivities of SiO 2 /methanol, Al 2 O 3 /methanol, and TiO 2 /methanol nanofluids. It was concluded that the thermal conductivity is increased for all nanofluids but found higher for Al 2 O 3 /methanol nanofluids as compared to the other two nanofluids. Das et al. [40] studied the thermal conductivity in different ranges of temperature for five distinct nanofluids which are prepared by dispersion of SiO 2 , Al 2 O 3 , TiO 2 , CuO, and ZnO nanoparticles in propylene glycol-water. The improvement in thermal conductivity of all nanofluids by enhancing temperature and nanoparticle concentration is found. Murshed et al. [41] investigated the thermal conductivity of TiO 2 /DI H 2 O nanofluids. Their experiments show the enhancement in thermal conductivity by increasing particle concentration (0.5-5%) at room temperature. Duangthongsuk and Wong wises [42] detected the behavior of thermal conductivity of TiO 2 /H 2 O nanofluids. The thermal conductivity of nanofluids increased by nanoparticle concentration as well as increased temperature.
In the above studies, it is found that the nature of fluid whether it is Newtonian or non-Newtonian depends on the behavior of viscosity. The behavior of viscosity is not only changed by nanoparticles but also depends on operating temperature. Similarly, the thermal conductivity of the nanofluid not only increased by nanoparticle concentration but also increased by increasing temperature. Keeping in mind these facts, the rheological properties of four different nanofluids such as SiO 2 /DIW, Al 2 O 3 /DIW, PEG-TGr/DIW, and PEG-GnP/DIW are modeled as a function of nanoparticle concentration and operating temperature in the current study. For modeling, experimental data is picked at 0.025%, 0.05%, 0.075%, and 0.1% nanoparticle concentration under 30 • C, 40 • C, and 50 • C temperature range [1]. Further, these models are used in transport equations to see the boundary layer flow over two different geometries such as wedge and plate. The whole investigation is divided into different sections. After introductions in Section 1, the mathematical models are established based on experimental data to discuss the thermophysical properties and parameters of schematic nanofluids in the form of graphs and tables respectively in Section 2. In Section 3, the mathematical problem for flow is developed by using continuity, momentum, and energy equations. In Section 4, physical parameters such as momentum and thermal boundary layers thickness, momentum and displacement thicknesses, coefficient of skin friction, and Nusselt number are modeled. The numerical solution of the problem is obtained using the RK method and gets the solutions in the form of velocity and temperature functions. In next Section 6, attained results are displayed in graphical and tabular form for discussion. In last Section 7, the significant outcomes are concluded.

Viscosity Model
Consider the following viscosity model as Here, T is the temperature, .
γ are temperature and shear rate depended functions. The µ 1 (T) is taken as exponential form whereas power law model is taken for µ 2 . γ which are defined as Here n, µ n f and C 1 are curve fitting parameters. The numerical values of these parameters are obtained by fitting the Equation (2) to experimental data [1]. The best-fitting results are presented in Figures 1-4. In these figures, dots represent the experimental data [1] and graph illustrates the Equation (2).       In respect of Figures 1-4, the values of curve fitting parameters , nf  n and 1 C are listed in Table 1.    In respect of Figures 1-4, the values of curve fitting parameters , nf  n and 1 C are listed in Table 1.  In respect of Figures 1-4, the values of curve fitting parameters µ n f ,n and C 1 are listed in Table 1.
The results of curve fitting are displayed in  Table 2.   Table 1.  Table 1.    Table 1.  Table 1.  Table 1.

Thermal Conductivity Model
Consider the thermal conductivity model on the pattern of Equation (1) as  Table 1.  Table 1.  Table 1.

Thermal Conductivity Model
Consider the thermal conductivity model on the pattern of Equation (1) as  Table 1.

Thermal Conductivity Model
Consider the thermal conductivity model on the pattern of Equation (1) as where, k and C 2 are curve fitting parameters. The values of these parameters are obtained by fitting the Equation (4) for experimental data [1]. The results of curving fitting are displayed in  and the values of curve fitting parameters are displayed in Table 3.
Micromachines 2022, 13, x where, k and 2 C are curve fitting parameters. The values of these parameter tained by fitting the Equation (4) for experimental data [1]. The results of curvin are displayed in Figures 9-12 and the values of curve fitting parameters are disp  Table 3.   Micromachines 2022, 13, x where, k and 2 C are curve fitting parameters. The values of these parameters tained by fitting the Equation (4) for experimental data [1]. The results of curvin are displayed in Figures 9-12 and the values of curve fitting parameters are disp  Table 3.     ( ) Al2O3 PEG-TGr P     Moreover, the values of the parameters in Table 3 are fitted into Equation (3) and their results are presented in Figures 13-16.  Table 3.  Table 3.  Table 3.  Table 3. Figure 14. Fitting of the Equation (4) for parameters listed in Table 3.  Table 3.  Table 3.  Table 3.

Density and Heat Capacity Models
The co-relation models for density ( ) nf  and heat capacity ( ) p nf C are developed by fitting the polynomial of 1st-degree Equation (5) for experimental data [1] ( ) The results of curve fitting are displayed in Figures 17 and 18.  Table 3.

Density and Heat Capacity Models
The co-relation models for density ( )   Table 3.

Density and Heat Capacity Models
The co-relation models for density (ρ n f ) and heat capacity C p n f are developed by fitting the polynomial of 1st-degree Equation (5) for experimental data [1] The results of curve fitting are displayed in Figures 17 and 18.  In view of Figures 17 and 18, the values of the co-efficient of Equation (5) are displayed in Table 4.  In view of Figures 17 and 18, the values of the co-efficient of Equation (5) are played in Table 4. In view of Figures 17 and 18, the values of the co-efficient of Equation (5) are displayed in Table 4.

Heat and Mass Flow Modeling
Consider the steady state and an incompressible boundary layer fluid flow propagating over two different geometries (Plate and Wedge). The fluid at the wall flowed with u w (x) = bx m velocity and flowed with u e (x) = cx m velocity in the free stream region as seen in Figure 19. The relationship between the Falkner-Skan power law parameter (m) and the wedge's angle β = Ω/π is stated as

Heat and Mass Flow Modeling
Consider the steady state and an incompressible boundary layer fluid flow propagating over two different geometries (Plate and Wedge). The fluid at the wall flowed with  Under the boundary layer approximation, the continuity, momentum, and energy equations can be written as 0 xy uv += Figure 19. Flow Structure over geometry.
Geometry exhibited plate-shaped when m = 0 and wedge when m > 0. The temperature at the wall and away from the wall is maintained with constant T w and T ∞ i.e., (T w > T ∞ ) respectively.
Under the boundary layer approximation, the continuity, momentum, and energy equations can be written as ρC p n f u T x + v T y = ∂ y (k T y ) + µ n f u y 2 (9) with the boundary conditions For simplicity, introduced the similarity transformations [43] as After the substitution of Equation (11) into Equation (6)- (8), we obtain the following non-dimensional equations Here, Pr = is the local Grashof number, and is the Eckert number.

Displacement Thickness
The displacement thickness is written as By using Equation (11), it is written as

Momentum Thickness
Momentum thickness is described as By using Equation (11), it is illustrated as

Skin Friction Coefficient
The skin friction coefficient is defined as After applying Equation (11), we get

Nusselt Number
Nusselt number is written in the following form Here h is a convective heat transfer coefficient. In view of Equation (11), Equation (21) is shaped as

Solution Technique
The solution of Equations (12) and (13), with respect to Equation (14), 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 the system of first-order differential equations as along boundary conditions Here Ω 1 and Ω 2 are unknown boundary conditions. To evaluate the accuracy of the results, the values of f (0) and −θ (0) against parameters β and Pr are compared with existing limited results [44,45] in Tables 5 and 6. The velocity and temperature distribution in the numeric form are displayed in Table 7 at φ = 0.025%. Table 7. Numeric values of velocity and temperature distribution at φ = 0.025%.

Result and Discussion
In  Table 8.

Velocity Profiles
Figures 20-23 illustrate the graphs of velocity profiles for SiO 2 /DIW, Al 2 O 3 /DIW, PEG-GnP/DIW, and PEG-TGr/DIW nanofluids under the influence of different nanoparticle volume fractions for both moving wedge and plate. It is seen that the values of viscosity of schematic nanofluids are amplified due to increasing the nanoparticle concentration. Given this evidence, the velocity profile of all schematic nanofluids over both geometries is decreased by raising the nanoparticle volume fraction. It is also observed that the profile of velocity is slowed down over a moving plate as compared to a moving wedge.  Tables 9 and 10. It is seen that the thicknesses of momentum and thermal boundary region are enlarged by increasing the nanoparticle concentration and further increased along the parallel distance of the geometry's wall. It is also seen that the momentum boundary layer thickness is greater than the thermal boundary layer thickness due to the dominant effects of viscosity as thermal diffusion. In addition, the boundary layer phenomena are produced more effectively on a plate as compared to a wedge.
In Tables 11 and 12, the values of displacement and momentum thicknesses of schematic nanofluids are obtained under the impact of nanoparticle concentration at distinct positions on the x axis − . The value of displacement thickness shows the reduction in mass flow rate whereas the value of momentum thickness illustrates the reduction in momentum flow rate in the boundary layer region. The value of displacement thickness is raised by enhancing nanoparticle concentration and also increased along the parallel distance of the wall. It is also seen that the values of displacement thickness for flow over the plate are found greater than the values for flow over the wedge. Similarly, the momentum thickness is increased by raising of nanoparticle volume fraction as seen in Table 12. Moreover, it is observed that momentum thickness is found higher in the case of flow over the plate as compared to wedge. Tables 13 and 14

Temperature Profiles
Figures 24-27 display the results of temperature profiles for all schematic nanofluids under the impact of different nanoparticle volume fractions over both geometries. It is detected from Figs. that the thermal conductivity is increased while specific heat is declined by increasing the nanoparticles concentration. In respect of changes in these properties, the distribution of temperature is increased. The prominent effects on temperature profile are found for PEG-GnP/DIW as compared to other nanofluids. Additionally, it is also seen that the temperature distribution is more raised in the case of a moving plate as compared to a moving wedge.

Physical Parameters
Tables 9-12 dictate the results of boundary layer parameters such as momentum and thermal boundary region's thicknesses, displacement thickness, and momentum thickness whereas the values of coefficient of skin friction and Nusselt number are illustrated in Tables 13 and 14 for both geometries.     The results of momentum and thermal boundary region's thicknesses of schematic nanofluids are computed numerically at a distinct location on the x − axis under the impact of nanoparticle volume fraction are listed in Tables 9 and 10. It is seen that the thicknesses of momentum and thermal boundary region are enlarged by increasing the nanoparticle concentration and further increased along the parallel distance of the geometry's wall. It is also seen that the momentum boundary layer thickness is greater than the thermal boundary layer thickness due to the dominant effects of viscosity as thermal diffusion. In addition, the boundary layer phenomena are produced more effectively on a plate as compared to a wedge.
In Tables 11 and 12, the values of displacement and momentum thicknesses of schematic nanofluids are obtained under the impact of nanoparticle concentration at distinct positions on the x − axis. The value of displacement thickness shows the reduction in mass flow rate whereas the value of momentum thickness illustrates the reduction in momentum flow rate in the boundary layer region. The value of displacement thickness is raised by enhancing nanoparticle concentration and also increased along the parallel distance of the wall. It is also seen that the values of displacement thickness for flow over the plate are found greater than the values for flow over the wedge. Similarly, the momentum thickness is increased by raising of nanoparticle volume fraction as seen in Table 12. Moreover, it is observed that momentum thickness is found higher in the case of flow over the plate as compared to wedge.
Tables 13 and 14 dictate the results of Nusselt number and coefficient of Skin friction for said nanofluids under the impact of nanoparticle concentration at a distinct location on the x − axis. The results demonstrate that the values of Nusselt number are enhanced by raising nanoparticle concentration due to the enhancement of thermal conductivity. Furthermore, the values of Nusselt number are also enhanced away from the origin along the x − axis. The values of Nusselt number are found higher when fluid flow over a wedge as compared to a plate. The coefficient of Skin friction is increased by raising nanoparticle concentration due to enhancement of viscous effects and is decreased along x − axis. The values of the coefficient of Skin friction are found larger in the case of flow over wedge as compared to the plate.

Conclusions
In the current investigation, the mathematical model for thermophysical properties of nanofluids is developed with help of experimental data and then used in transport equations to explore the boundary layer flow over plate and wedge. The results are obtained in the form of velocity and temperature and are further used to obtain the values of physical parameters. From the results, the following conclusions are exposed:

•
The velocity is reduced whereas the temperature is enlarged due to amplifying viscosity and thermal conductivity respectively by variation of nanoparticle volume fraction for both wedge and plate.

•
The velocity and temperature are more reduced and raised respectively in the case of flow over the plate as compared to a wedge.

•
The momentum and thermal boundary layers are increased by enhancing nanoparticle volume but are found maximum in case of flow over a plate.

•
The displacement and momentum thicknesses have followed the pattern of boundary layer thicknesses and are enhanced by variations in nanoparticle volume fraction.

•
The skin friction coefficient and Nusselt number are raised with an enhancement in nanoparticles volume fraction but the maximum is found in the case of a wedge as compared to a plate.