Statistical Modeling for Nanofluid Flow: A Stretching Sheet with Thermophysical Property Data

This paper reports the use of a numerical solution of nanofluid flow. The boundary layer flow over a stretching sheet in combination of two nanofluids models is studied. The partial differential equation that governs this model was transformed into a nonlinear ordinary differential equation by using similarity variables, and the numerical results were obtained by applying the shooting technique. Copper (Cu) nanoparticles (water-based fluid) were used in this study. This paper presents and discusses all numerical results, including those for the local Sherwood number and the local Nusselt number. Additionally, the effects of the nanoparticle volume fraction, Brownian motion Nb, and thermophoresis Nt on the performance of heat transfer are discussed. The results show that the stretching sheet has a unique solution: as the nanoparticle volume fraction φ (φ = 0), Nt (Nt = 0.1), and Nb decrease, the rate of heat transfer increases. Furthermore, as φ (φ = 0) and Nb decrease, the rate of mass transfer increases. The data of the Nusselt and Sherwood numbers were tested using different statistical distributions, and it is found that both datasets fit the Weibull distribution for different values of Nt and rotating φ.


Introduction
In recent years, many researchers have investigated stretching plates, which are used in industry for materials such as lubricants and glass fibers. The theory of flow over a stretching plate was first proposed by Crane [1]. Researchers [2] have also analyzed heat transfer over a stretching sheet with a permeable surface. Related studies involving a stretching surface were conducted by Grubka and Bobba [3], Ali [4], Wang [5], and Hayat et al. [6]. Convective heat transfer is a very important property of nanofluids, and the addition of nanoparticles has been found to improve the thermal conductivity. Widely used types of nanoparticles include monotubes, carbide nanoparticles, and metal nanoparticles, where v is the velocity of the component in the y-direction, and u is the velocity of the component in the x-direction. T w , T ∞ , and T are the surface temperature, ambient temperature, and temperature, respectively. C is the nanoparticle volume fraction, C ∞ is the nanoparticle volume fraction far from the plate, and C w is the nanoparticle volume fraction at the plate. D T is the coefficient of thermophoretic diffusion, and D B is the coefficient of Brownian diffusion. The ratio of heat capacity is τ = (ρC p ) s /(ρC p ) f , in which (ρC p ) s represents the heat capacity of the nanoparticle, and (ρC p ) f represents the heat capacity of the fluid. Further, α nf is the nanofluid thermal diffusivity, μ nf is the nanofluid viscosity, and ρ nf is the nanofluid density. These parameters were previously described by Oztop [28]. The same assumption also applies to the velocity of the fluid U ∞ (x). The governing equations for a stretching sheet, where a > b, are as follows.
where v is the velocity of the component in the y-direction, and u is the velocity of the component in the x-direction. T w , T ∞ , and T are the surface temperature, ambient temperature, and temperature, respectively. C is the nanoparticle volume fraction, C ∞ is the nanoparticle volume fraction far from the plate, and C w is the nanoparticle volume fraction at the plate. D T is the coefficient of thermophoretic diffusion, and D B is the coefficient of Brownian diffusion. The ratio of heat capacity is τ = (ρC p ) s /(ρC p ) f , in which (ρC p ) s represents the heat capacity of the nanoparticle, and (ρC p ) f represents the heat capacity of the fluid. Further, α nf is the nanofluid thermal diffusivity, µ nf is the nanofluid viscosity, and ρ nf is the nanofluid density. These parameters were previously described by Oztop [28].
where the parameter ϕ is the nanoparticle volume fraction, (ρC p ) nf is the heat capacity of the nanofluid, k nf is the thermal conductivity of the nanofluid, k f is the thermal conductivity of the fluid, k s is the thermal conductivity of the solid, ρ f is the fluid density, µ f is the fluid viscosity, and ρ s is the density of the solid. According to Abu-Nada [25], the term k nf is used for spherical nanoparticles, and its value is negligible for other shapes. Next, given the constraints in Equation (5), the similarity solution of Equations (1)-(4) is computed; Equation (4) was proposed by Buongiorno [15]. We introduce the similarity transformations below.
where θ(η) is a dimensionless variable for temperature, and φ(η) is a dimensionless variable for nanoparticle concentration. The boundary conditions are 1 Pr subject to the boundary conditions The Brownian motion parameter is the thermophoresis parameter is the Lewis number is the stretching parameter is ε = a/b, the skin friction coefficient is

and the local Reynolds number is
The data for the Nusselt and Sherwood numbers were tested using the Akaike information criterion (AIC). This test is used to find the goodness of fit and determine the distribution that best fits the data. Table 1 lists the different statistical distributions. For each distribution in the table, the AIC was calculated, and the best distribution was identified from the AIC values.

Distribution Cumulative Distribution Function, F(x)
Lognormal where erf is the complete error function; σ is the shape of the distribution; x is the value used to evaluate the function; µ is the expected value for the normal distribution. Weibull where x is the value used to evaluate the function; α is the scale parameter; β is the shape parameter. Rayleigh where x is the value used to evaluate the function; σ is the shape of the distribution. Exponential where x is the value used to evaluate the function; θ is the scale parameter. Gamma where Γ(α) is the incomplete gamma function; x is the value used to evaluate the function; α is the shape parameter; β is the scale parameter.
Inverse Gaussian x is the value used to evaluate the function; Φ denotes the distribution function of the standard normal; µ is the mean; λ is the shape parameter.

Inverse Gamma
where Γ(p) is the incomplete gamma function; x is the value used to evaluate the function; α is the shape parameter; β is the scale parameter.
The AIC measures the quality of statistical models for a sample set of data. The model that provides the lowest AIC value best fits the data. The formula for the AIC is where L is a model of the likelihood function, and k is the number of parameters.

Results and Discussion
The shooting technique was used to determine the numerical solutions for Equations (9)-(11) with the boundary conditions in Equation (12). The shooting method converts a boundary value problem (BVP) into an initial value problem (IVP). The main reason for using the shooting method is that it can establish the applicable initial conditions for a related IVP to generate the solution to the BVP. This Colloids Interfaces 2020, 4, 3 6 of 10 method was applied in the Maple programming language using the "dsolve" command and "shoot" implementation. The influences of Nb, Nt, and ϕ on the heat transfer rate were investigated for Cu nanoparticles. The values of the thermophysical properties are shown in Table 2. Table 2. Thermophysical properties of a nanofluid [18].

Physical Properties
Base Fluid Nanoparticle, Cu When ϕ = 0 for a regular fluid, the range of ϕ should be 0-0.2 [18]. The value of Pr (Prandtl number) is 6.2. Figure 2a,b depicts the variation in the local Nusselt number and Sherwood number with Nb for different Nt when ϕ = 0.1, Pr = 6.2, Le = 3, and ε = 1.2 for the stretching case. As clearly shown in Figure 2a, as Nb increases, the local Nusselt number decreases, while Figure 2b shows the opposite trend. Furthermore, Figure 3a,b shows that the local Nusselt number and local Sherwood number vary with Nb for different ϕ when Pr = 6.2, Nt = 0.1, Le = 3, and ε = 1.2 for the stretching case. These figures reveal that as Nb increases, the local Nusselt number and local Sherwood number decrease. It is important to note that the Brownian motion parameter Nb and thermophoresis Nt are related to the random motion of nanoparticles (Cu). For small values of Nb and Nt, the viscosity of the base fluid is weak, and the nanoparticles (Cu) tend to move easily among each other. Because of this phenomenon, the fluid is cooled faster, and the heat transfer rate increases.

Results and Discussion
The shooting technique was used to determine the numerical solutions for Equations (9)- (11) with the boundary conditions in Equation (12). The shooting method converts a boundary value problem (BVP) into an initial value problem (IVP). The main reason for using the shooting method is that it can establish the applicable initial conditions for a related IVP to generate the solution to the BVP. This method was applied in the Maple programming language using the "dsolve" command and "shoot" implementation. The influences of Nb, Nt, and φ on the heat transfer rate were investigated for Cu nanoparticles. The values of the thermophysical properties are shown in Table 2. When φ = 0 for a regular fluid, the range of φ should be 0-0.2 [18]. The value of Pr (Prandtl number) is 6.2. Figure 2a  On the basis of Figures 2 and 3, the data for the Nusselt and Sherwood numbers were further analyzed to obtain the statistical properties for the tested distributions. Tables 3 and 4 show the parameters for different distributions that were tested with the data.  On the basis of Figures 2 and 3, the data for the Nusselt and Sherwood numbers were further analyzed to obtain the statistical properties for the tested distributions. Tables 3 and 4 show the parameters for different distributions that were tested with the data.  On the basis of Figures 2 and 3, the data for the Nusselt and Sherwood numbers were further analyzed to obtain the statistical properties for the tested distributions. Tables 3 and 4 show the parameters for different distributions that were tested with the data.  On the basis of Figures 2 and 3, the data for the Nusselt and Sherwood numbers were further analyzed to obtain the statistical properties for the tested distributions. Tables 3 and 4 show the parameters for different distributions that were tested with the data.  On the basis of Figures 2 and 3, the data for the Nusselt and Sherwood numbers were further analyzed to obtain the statistical properties for the tested distributions. Tables 3 and 4 show the parameters for different distributions that were tested with the data.  On the basis of Figures 2 and 3, the data for the Nusselt and Sherwood numbers were further analyzed to obtain the statistical properties for the tested distributions. Tables 3 and 4 show the parameters for different distributions that were tested with the data.  Tables 5 and 6 show the Akaike information criteria for the Nusselt and Sherwood numbers. The results show that the lowest AIC values are those for the Weibull distribution. It was also found that as the values of Nt and ϕincrease, the AIC values still indicate that the Weibull distribution is optimal.

Conclusions
Nanofluid flow past a stretching sheet and the influences of parameters Nb, Nt, and ϕ were examined and studied. From this investigation, a unique solution was obtained for the stretching sheet. It was found that as Nb and Nt decrease, the rate of heat transfer increases, but the rate of mass transfer decreases. The case is different when ϕ decreases: Nb decreases, and the rates of heat and mass transfer decrease. These results show that for different Nt and ϕ, the Weibull distribution best fits both the Nusselt and Sherwood data.