A Weibull Distribution: Flow and Heat Transfer of Nanofluids Containing Carbon Nanotubes with Radiation and Velocity Slip Effects

Department of Engineering Education, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor, Malaysia Department of Mechanical and Manufacturing Engineering, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor, Malaysia University of Duhok, College of Science, Department of Computer Sciences, Zakho Street 38, 1006 AJ Kurdistan Region, Duhok, Iraq


Introduction
Originally discovered in 1995, nanofluids are a class of fluids that have been attracting significant attention of researchers in various fields. Owing to their advantages, nanofluids have been implemented in various industrial sectors, such as energy and biomedical fields. Choi and Eastman [1] reported that the thermal conductivity of nanofluids can be enhanced by dispersing nanosized particles in the fluid. Moreover, it was discovered that the flow of the base fluid improved by suspending nanoparticles in it [2]. Alwaeli et al. [3] also reported that the addition of alumina and carbon black nanoparticles improved the cooling effect on the solar panels. Meanwhile, carbon nanotubes (CNTs) have become one of the most effective materials, owing to their ability to enhance the thermal characteristics of the fluid, high electrical conductivity, unique optical transmission, and high tensile strength. ey can also increase the entropy generation [4]. CNTs are rolled-up graphene sheets arranged in a cylindrical shape. ey are of two types: single-walled (SWCNTs) and multiwalled (MWCNTs) [5]. It was observed that the boundary layer separation could be delayed if suction effects on the CNTs' nanoparticle volume fraction are provided [6]. According to Naganthran et al. [7] and Ahmad et al. [8], CNTs have higher thermal conductivity. When dispersed in the base fluid, they can accelerate the rate of the heat transfer and subsequently increase the base fluid's thermal conductivity. Using Buongiorno's concept, Khan et al. [9] analysed the effects of the Brownian motion and thermophoresis on the CNTs' flow, using Xue's model. Anuar et al. [10] reported the enhanced performance of CNTs, owing to the existence of the velocity slip in boundary conditions. Nanoparticles exhibit very unique physical and mechanical properties [11]. Graphene nanoparticles, for example, are used in many applications, especially in nanotechnology [12] and more widely in cooling technology [13,14]. As such, the study of rheological characteristics of the hybrid nanofluids is of considerable importance in order to determine the effects of two or more nanoparticles in a base fluid [15,16]. e important outcome of such a hybrid nanoparticle is to enhance the thermal properties of nanofluids in heat transfer applications [17][18][19][20]. ere are many widely used methods to enhance heat transfer. For instance, nanofluids are used to increase the heat transfer rate [21,22]. Additionally, suitable changes in the geometrical conditions, cross-sectional area, and composition of microchannels [23,24] can also significantly improve the heat transfer performance.
In previously reported studies, the researchers focused only on studying the boundary layer flow of nanofluids, without considering the slip effect. As a result, Bhattacharyya et al. [25] explored the presence of slip effect and found that, as the value of the velocity slip parameter increased, it resulted in a decrease of the boundary layer thickness. Using this concept, Bachok at al. [26] applied this study to a moving plate, using dispersed copper nanoparticles in a base fluid. Later, Imtiaz et al. [27] analysed the effects of CNTs by employing the thermal radiation effect in the modelling systems and discovered that single-wall CNTs offer a larger heat transfer rate. Many intriguing properties of nanofluids have been discussed in the literature presented here. Recently, as scientists are showing significant interest in heat transfer properties, the utilization of nanofluids in real-life situations, in this aspect, seems promising. To achieve this, fundamental research in the mathematical aspect of the thermal conductivity of nanofluids is needed in order to understand the mechanisms that can change the nanofluid behaviour. erefore, the objective of this study is to investigate and highlight the effects of thermal radiation and slip parameter of SWCNTs and MWCNTs in the case of a moving plate. is is performed by employing a theoretical mathematical model, previously introduced by Tiwari and Das [28]. eir model presented the effects of nanoparticle volume fraction in influencing the viscosity of the nanofluid.

Governing Equations.
Consider a steady, two-dimensional horizontal moving plate, which is placed in a nanofluid containing SWCNTs and MWCNTs. e plate has a constant velocity U w . e temperature of the wall, T w , is expected to remain uniform, whereas the temperature, T ∞ , of the ambient fluid is assumed constant. A schematic of the model is shown in Figure 1. Taking into account the thermal radiation effect, the following equations can be written (as indicated in [20,21]): Solving equations (1)-(3), the following conditions are derived: where u and v are the velocity elements (in x and y directions, respectively), T is the temperature in the boundary layer (in Kelvin), q r is the radiative heat flux, μ is the viscosity, and ρ and α are the density and thermal diffusivity of the nanofluid. Here, ξ 1 denotes the slip factor and is defined as ξ 1 � ξ ��������� Re w + Re ∞ , where ξ is the primary length and Re w and Re ∞ are the Reynolds numbers.
It is important to have a basic understanding about the thermophysical properties of nanofluids, such as thermal diffusivity, density, dynamic viscosity, heat capacity, and thermal conductivity, before we develop a thermal system. For example, thermal conductivity is an important factor in determining the heat transfer effectiveness. Moreover, the dynamic viscosity indicates that the resistance of the fluid directly affects the pressure and decreases it, for example, in pumping power (PP) systems. Meanwhile, the effectiveness of the working fluid is determined by the heat capacity, which can be inferred from the fact how the generated heat moves away from the heat source, in the fluid. Table 1 lists the thermal diffusivity, density, dynamic viscosity, heat capacity, and thermal conductivity of nanofluids. e relations given in the second column indicate that these parameters are suitable and quite important to determine the effect of CNTs on the flow and heat transfer of the system. Table 2 lists the physical properties of water (base fluid), SWCNTs, and MWCNTs. Applying the Rosseland approximation [24,25], the thermal radiation is expressed as with σ * and k * are Stefan-Boltzmann's and average absorption's coefficients, respectively. Omitting the higher order, T can be expressed as Equation (6) is differentiated over T, and then resubstitute the results in equation (5); therefore, equation (3) now becomes To solve the set of partial equations (1)-(8) and taking into account the equations in Table 1, we applied the similarity transformation [31], which is given by the following equation: Here, U represents the composite velocity U � U w + U ∞ , θ the dimensionless temperature, and ψ is the stream function. Here, the terms u and v are defined as u � (zψ/zy) and v � − (zψ/zx), which satisfy continuity equation (1).
is leads us to 1 1 Pr with the radiation parameter N R � ((4σ * T 3 ∞ )/(k * k)) and Prandtl number Pr � (] f /α f ). In accordance with equations (9) and (10), converted equation (4) can be represented as where the velocity ratio parameters, c and β, are expressed as It is necessary to identify the skin friction coefficient C f and the local Nusselt number Nu x , which are expressed as where τ w and q w are defined as Table 1: Parameters defined in the work [29].

Parameters
Governing equations ermal diffusivity where k nf � thermal conductivity of nanofluids k f � thermal conductivity of the fluid k CNT � thermal conductivity of carbon nanotubes Table 2: e physical properties of fluids and CNTs [30].

Physical properties
Base fluid Carbon nanotubes To derive an expression for equations (13) and (14), equation (8) is used. So, we can write where Re x � (Ux/] f ) is the local Reynolds number.

Anderson-Darling Test Statistic.
e data for the reduced skin friction were further analysed using the Anderson-Darling (AD) statistic. e goodness of fit is determined to verify which distribution in the skin friction data fits well.
e AD test was selected over other tests because of two reasons: (1) the AD test is one of the best goodness-of-fit tests for a small size of the sample, and (2) the AD test is, most often, used in practice. To check which distribution in the data would follow, we need to apply the hypothesis tests that will be discussed later in this paper. e null hypothesis states that the data would follow a specific population as stated in Table 3. However, the alternative hypothesis states that the data would not follow a specific population.
First, the test statistic AD * must be computed based on AD, but to conclude which best fits the data, the AD * test statistic must be significantly lower than the others. Hence, we conclude that the skin friction data were drawn from a specific population. e AD * test statistic is illustrated in Table 4, and the formula to compute the AD is given as follows: where n is the sample size, i runs from 1 to n (calculated when the skin friction data are sorted in the ascending order), and F(x i ) is a CDF for the specified distribution function.

Results and Discussion
e obtained results are analysed in order to demonstrate the CNTs' flow in the influences of N R and β. With the help of the bvp4c solver in software MATLAB, the system of nonlinear ordinary equations (9) and (10) and conditions given in equation (11) are numerically solved. e solver bvp4c was programmed with a finite difference code that implements the 3-stage Lobatto IIIa formula. is effective solver required the users to have a set of initial guesses with a combination of the boundary layer thickness. Results are obtained when the boundary conditions are asymptotically fulfilled, and no errors are produced in MATLAB. e use of bvp4c codes solved the transformed momentum and energy equations (9) and (10), respectively, and boundary conditions given in equation (11), thus validating the numerical result. e obtained values for the present results and the data obtained from previously reported studies [10,32] are included in Table 5 for comparison. As evident, the present result and the data in the literature are found to be in agreement. Figures 2 and 3 elucidate the water-SWCNTs' variations, for different values of φ, from 0 to 0.2, when β � 0.2 and N R � 0.1. As it is seen, the dual solutions occur when the plate shifts to opposite directions (c c < c < 0), while an exact solution occurs when the plate moves in assisting flow conditions (c ≥ 0). However, no solution exists when c < c c . As seen, the increase in φ results in decreased f ″ (0) values. In Figure 2, it is evident that the effect of slip parameter and thermal radiation as well as CNTs added to the nanofluid made it more viscous. Hence, increasing φ will increase the fluid viscosity, thereby increasing the fluid's resistance to flow. Figure 3 indicates that the heat transfer rate decreases with increasing aided volume fraction of nanoparticles. e effect of radiation parameter transports heat to functional flow, and hence, the squeezed nanofluid flow temperature decreases.
e variation of C f Re 1/2 x with different ranges of φ and β is presented in Figures 4 and 5 as the plate travels in assisting flow. As highlighted in Figure 4, if the value of β increased in both SWCNTs and MWCNTs, the skin friction coefficient C f Re 1/2 x begins to depreciate. However, in Figure 5, Nu x Re − 1/2 x increases with the increase of φ. As seen, larger values of C f Re 1/2 x and Nu x Re − 1/2 x are seen by SWCNTs due to their larger density and thermal conductivity. ese are illustrated in Figures 4 and 5. High values of heat transfer rate and fluid friction factor for SWCNTs (compared with MWCNTs) are due to thermal boundary resistance between the CNT and the surrounding fluid. SWCNTs are highly hydrophobic because the particles are bounded together due to higher van der Waals force of attraction. e SWCNTs' flow and heat transfer are further verified by studying the effects of difference in ranges of φ on the velocity and temperature profiles. Figures 6 and 7 graphically illustrate the convergence series solution of the velocity and temperature profiles. From the figures, it is noted that these profiles asymptotically suffice the converted conditions in equation (11). e dual solutions can also be spotted from these figures. Moreover, the exact solutions have a boundary layer thickness thinner than that of the dual solutions. It also indicates that f′ is a decreasing function of φ, owing to an increase in φ. However, in Figure 7, with the presence of N R � 0.1 and β � 0.2, the thermal boundary layer thickness increases, which would decelerate the heat transfer. Figures 8 and 9 portray the effect of β on the velocity and temperature profiles. ese figures indicate that, as the value of β increases, the momentum thickness and thermal boundary layer thickness decrease. e influence of various values of thermal radiation on the temperature profile for water-based SWCNTs is also presented in Figure 10. It can be seen quite evidently that an increase in the value of N R leads to a higher boundary layer thickness.
Further tests are conducted using the Anderson-Darling test statistics. e skin friction data (from Figure 2) are utilized to perform this test. e parameters, for each distribution that was tested, are displayed in Table 6. Table 7 shows the AD test statistics for the reduced skin friction data. Overall, the lowest AD values are obtained from the same distribution, i.e., the Weibull distribution. It implies that, as the volume fraction of nanoparticles increases, AD values still acknowledge that the Weibull distribution is the optimum one, among the other two distributions (gamma and exponential). erefore, it can be concluded that the flow Table 3: Distribution test for f ″ (0).

Distribution
Cumulative distribution function (CDF), , where x i , δ, and β stand for the skin friction data, scale, and shape parameters, respectively. Exponential , where x i and θ stand for the skin friction data and the scale parameter, respectively.
Gamma G(x i ) � ((c Stac (δ, (x i /β)))/(Γ(δ))), where Γ(δ) is the gamma function, x i , δ, and β stand for the skin friction, scale, and shape parameters, respectively. c Stac (δ, (x i /β)) is the lower incomplete gamma function.      Mathematical Problems in Engineering enhancement depends on low values of nanoparticle volume fraction and slip parameter. However, the higher thermal radiation will enhance the temperature at the boundary layer. e data of the Weibull distribution should concur with the data from different orders of slip; hence, it is very important to fit the data using statistical modelling.

Conclusion
In the present work, a detailed analysis of the influences of slip and thermal radiation of carbon nanotubes' (SWCNTs and MWCNTs) flow over a moving plate is conducted. From the analysis, an exact solution was found when the plate moves in the assisting flow condition, while dual solutions appeared in the opposing flow condition. It is also found that an increment in volume fraction of nanoparticles at φ � 0.2 decreased the skin friction and heat transfer rate since it influences the boundary layer thickness. At β � 0.4, the skin friction decreases, whereas the heat transfer rate increases. In contrast, as N R increases to 0.7, the heat transfer rate decreases. In addition, for all other values of β � 0, 0.2, and 0.4, it is found that the SWCNTs are more efficient, as compared with MWCNTs, with regard to the skin friction coefficient and the Nusselt number. Moreover, these results verified that the skin friction data fit the Weibull distribution, which are found to be in good agreement with results of the previous study [33]. e slip and thermal radiation are also modelled based on the flow and heat enhancement of CNTs. However, to explain the flow and heat enhancement of CNTs in studies on fluid dynamics, further research in the domain of slip is needed.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.