CaputoTimeFractionalModelBasedonGeneralizedFourier’s and Fick’s Laws for Jeffrey Nanofluid: Applications in Automobiles

Department of Mathematics, City University of Science and Information Technology, Peshawar, Khyber Pakhtunkhwa, Pakistan Computational Analysis Research Group, Ton Duc &ang University, Ho Chi Minh City, Vietnam Faculty of Mathematics and Statistics, Ton Duc &ang University, Ho Chi Minh City, Vietnam Department of Mathematics, College of Science Al-Zulfi, Majmah University, Al-Majmah 11952, Saudi Arabia


Introduction
e study of non-Newtonian fluid is very significant because it has vast applications in many areas, especially in the engineering and industrial sectors [1][2][3][4]. One of the wellknown models of non-Newtonian fluid is the Jeffrey fluid model which has both the property of viscosity and elasticity, and therefore it comes in the class of viscoelastic fluid. Engine oil, castor oil, and polymers are few examples of Jeffrey fluid. e Jeffrey fluid model has the facility to customize the time derivative on behalf of the convective derivative. e viscous fluid and second-grade fluid models can be recovered by taking Jeffrey fluid parameters equal to zero. Due to the above-stated applications, many scholars considered the Jeffrey model. Hayat et al. [5] studied the incompressible flow of viscoelastic Jeffrey fluid on a stretching sheet. For the solution of velocity and temperature field, the authors used the Homotopy method. e heat transfer phenomena of viscoelastic Jeffrey fluid at the stagnation point were discussed by Turkyilmazoglu and Pop [6]. Ellahi and Hussain [7] observed the instantaneous behavior of partial slip and MHD effects on the wave-like flow of Jeffrey fluid. e separation of variables technique is used to get the closed-form solutions. Mabood et al. [8] examined the two-dimensional steady incompressible flow of Jeffrey fluid over a stretching sheet. To get solutions, the Runge-Kutta order four method has been applied. Moreover, Qasim [9] studied the Jeffrey fluid with mass and heat transfer in the occurrence of a heat source/heat sink. To get the exact solution, the author used the power series method. Two-phase dusty non-Newtonian fluid flow together with the impact of free convection and power law has been examined on a vertical surface by Siddiqa et al [10].
Cooling and lubricity are significant in several industries, especially in transportation and energy production. However, conventional fluids such as ethylene glycol, water, and oil have very low thermal conductivity, which is the main problem to innovation in thermal management and energy efficiency. To avoid the intrinsic limitation of convectional heat transfer fluid, a new thought of nanofluid was introduced by Choi and Eastman [11]. Due to various applications, many researchers used different nanoparticles to improve the base fluid's thermal and mechanical properties. Ali et al. [12] examined the efficiency of EO in the response of silver nanoparticles together with the effect of diffusionthermo in a heated rotatory system. Moreover, Kole and Dey [13] studied the influence of gear oil and dispersed copper oxide nanoparticles. ey observed that nanofluid's viscosity is enhanced 3 times of the base fluid with the volume fraction of 0.025. Tesfai et al. [14] empirically examined the application of controlling thermal systems and used graphene oxide and graphene nanoparticles. Sheikholeslami and Rokni [15] analyzed the simulation of the nanofluid heat transfer under the impact of magnetic field. e effects and significance of thermophoresis and Brownian motion in natural convection flow of nanofluid have been examined by Haddead et al. [16], Parekh and Lee [17], Dinvarad et al. [18], Mohyuddin et al. [19], Loganath [20], and Ferrouillat et al. [21]. ey observed that nanoparticles are responsible for enhancing the thermal and mechanical properties of the base fluids. ey also detected that nanofluids are more stable and do not have a sedimentation problem.
According to Eric Temple Bell [22], "Calculus is considered the most powerful tool" for scientific thought. Fractional calculus is a calculus with a noninteger order derivative.
is was originated from a letter written by Leibniz in 1695 to Marquis de L'Hospital [23]. Later on, many mathematicians have been attracted and started work on this new topic. Euler, Riemann, Liouville, Laplace, Grunewald, Letniker, and so on worked on the fractional calculus. In the 18 th century, the Riemann-Liouville definition [24] of fractional derivative was mostly used definition. e deficiency that occurred in the Riemann-Liouville definition was removed by Caputo and presented a novel definition for fractional derivative [25]. However, it still contains the problem of singularity. e problem of singularity was fixed by Caputo-Fabrizio by giving a novel definition based on exponential function [26]. Nowadays, the implementation of fractional calculus is not limited to mathematics problems only but also contributes to solving the problems in many sectors like elasticity, chaos, diffusion, and polymerization. Fractional calculus is a very effective and efficient tool for elaborating heredity and the memory effect of the phenomena. In the last few years, remarkable development has been done using fractional calculus [27][28][29][30]. Bearing in mind the above-stated significance, many researchers contribute their potentials in the area of fractional calculus. Alizadeh et al. [31] discussed the transient response of the parallel circuit with the nonlocal derivative of the Caputo-Fabrizio. e authors used the Laplace transformed technique for their analysis. For the examination of the ground water pollution, Atangana and Alqahtani [32] used the time-space model of the Caputo-Fabrizio fractional derivative. Dokuyucu et al. [33] used the fractional derivative for the investigation of the tumor dealing model. Atangana and Alqahtani [32] discussed the ground level water problem using CF fractional derivative with the local and nonsingular kernel. Moreover, Ahmad et al. [34] discussed the existence and uniqueness of φ− Caputo pantograph differential equation. Keeping in view the applications of Caputo-Fabrizio fractional derivative, Doungmo Goufo [35] used the CF derivative for the analysis of Kortweg-de Vries-Burgers equation. With the help of CF derivative, Hristov [36]

Mathematical Modeling
We considered the laminar and unsteady flow of Jeffrey nanofluid in a horizontal channel in which both the plates are separated by distance l. e flow is assumed to be in the direction of x-axis with ambient temperature T a and ambient concentration C a . Initially, both the plates and fluid are at rest. At the time t � 0 + , the upper plate temperature and concentration levels are increased to T a + (T p − T a )A(t) and C a + (C p − C a )B(t), respectively. e physical geometry of the problem is given in Figure 1.
e constitutive equations which govern the flow using Boussinesq's approximation [38,39] are given by with physical condition as follows: e following dimensionless variables are used for dimensional analysis: Using above-stated dimensionless variables, equations (1)-(6) become e following are the dimensionless parameters and some constants after the dimensionalization process: A fractional model is developed using the generalized Fick's and Fourier laws as follows: where c ℘ α τ (.) is the Caputo time operative which is as follows: where η α (t) � (t − α /Γ(1 − α)) is the kernel of the singular power law. Utilizing the definition of Caputo time derivative, using equations (9), (11), (15), and (17), we arrived at Equations (18) and (19) can be written as follows: e thermo-mechanical properties of the considered base fluid and solid nanoparticles are given in Table 1.

Solution of Energy Equation.
Using the following transformation: on equation (20), we get according to equation (22), and the initial and boundary conditions are as follows:

Mathematical Problems in Engineering
Taking the Laplace transformation on equation (23), we get Now, applying sine Fourier transform to equation (25), we get After simplification, we arrived at Inverting the essential transformation of equation (26), we get Consequently, the exact solution of equation (20) is as follows:

e Solution of Mass Equation.
Using the following transformation: on equation (21), we get According to equation (30), the transformed conditions are as follows: Taking the Laplace on both sides of equation (30), we get Now, taking sine Fourier transform to equation (33), we get and after simplification, we arrived at Inverting the essential transformation of equation (35), we get Consequently, the final exact solution of equation (21) is as follows:

Nusselt Number.
For industrialists, the Nusselt number is a significant quantity, which is defined as follows: 2.6. Sherwood Number. In nondimensional form, the Sherwood number is defined as follows:

Results and Discussion
e incompressible flow of Jeffrey nanofluid in a bounded channel is elaborated in this article. e joint effect of mass and heat transfer has been studied. Fick's and Fourier's laws are used to develop the fractional model. To get the closedform solution, a novel technique is applied to change the equation into a simple form, and then Fourier's and Laplace transforms are used. e influence of embedded parameters on the velocity, temperature, and concentration distribution is presented in Figures 2-14.
Response of velocity profile against fractional order α is portrayed in Figure 2. In the classical order derivative, we have only one fluid layer for the analysis of considered fluid.
e key feature of using fractional order derivative is to obtain more than one fluid layer for the investigation of the fluid rheology as shown in Figure 2. Because of this salient advantage, the experientialists can compare their results with one of the layer which will be best fitted to their solution. Figure 3 interprets the comparative examination of the copper and silver nanoparticles on velocity field. From the figure, the magnitude of the velocity for silver nanofluid is greater than the magnitude of the velocity of copper nanofluid. It is because that the density of silver nanoparticle is (10497 kg/m 3 ) and the density of copper is (8940 kg/m 3 ). Figures 4 and 5 represent the influence of generalized Jeffery fluid parameters λ 1 and λ 2 on the velocity distribution. It is clear from the figure that for greater values of λ 1 , the velocity profile rises. is is because λ 1 is the time relaxation parameter, and due to the quick response of shear stresses, the fluid accelerates. In contrast, the greater value of λ 2 decreases the velocity distribution due to the delay response of shear stress. Figure 6 illustrates the impact of Gr (thermal Grashof number) on velocity distribution. If Gr rises, the velocity of the fluid also rises. Because of greater values of Gr, the difference between the temperature of plate and surrounding temperature increases which leads to decrease in the viscous forces, and due to this, the motion of the fluid boosts up. Figure 7 describes the influence of velocity distribution for distinct values of Gm (mass Grashof number). It is noted from the sketch that rising values of Gm also boost up the fluid velocity. It is because of greater values of Gm the difference between the surrounding concentration and concentration on the plate increases which consequently enhances the fluid motion. Figure 8 shows that fluid velocity falls by rising the magnitude of Sc (Schmidt number). is is true because the Schmidt number has direct relation with the viscosity of the fluid. e behavior of the velocity distribution in response of volume fraction φ is given Figure 9. It is noted from the sketch that velocity profile against φ fell down. It is physically true because nanoparticles enhance the viscous powers in the fluid; due to this, it delays the motion.
is outcome is of vital attentiveness. It is observed that hanging nanoparticles in engine oil improve the interconnected forces which can improve the life span and thermo-mechanical properties of the engine oil. Figure 10 shows variation in the velocity profile of the Jeffrey fluid against M (Hartman number. Higher values of M generate drag forces (Lorentz) which lead to suppress the motion of the fluid. Figure 11 displays the behavior of temperature profile against the fractional orderα. is is the beauty of the fractional derivative because it gives more than one temperature profile for the investigation as we discussed in Figure 2. Figure 12 shows the comparison of silver and copper nanoparticles on temperature distribution. e figure displays that the silver nanoparticle has a higher temperature as compared with the copper nanoparticle. The thermal conductivity of silver nanoparticle is (406 W/m · K) and thermal conductivity of copper nanoparticle is (385 W/m · K). Because of this difference, silver nanoparticle will conduct more temperature as compared with copper. Figure 13 reveals the performance of φ (volume fraction) on temperature distribution. As for higher value of φ, the absorption ability of the fluid rises, and due to this, the fluid temperature is improved. Figure 14 highlights the influence of fractional order α on concentration distribution. e same trend is observed as in temperature and velocity profile. Decay in the concentration profile has been observed for rises Sc (Schmidt number) which is revealed in Figure 15.   Furthermore, Table 2 represents the variation of skin friction in the counter of corresponding nondimensional parameters. Skin friction is calculated numerically using MATHCAD software. Table 3 shows that the silver nanoparticles are better than the copper nanoparticles for enhancement of heat transfer rate in engine oil, while Table 4 shows the variation in Sherwood numbers.

Conclusion Remarks
In this paper, a novel technique is used to establish the fractionalize model of Jeffrey nanofluid. To fractionalize the model, the generalized Fourier and Fick's laws are used. To get the closed-form solution, a novel transformation has been used and then solved by the Fourier sine and the Laplace transform techniques. e acquired results are drawn and displayed in tables. By the above results and discussions, the following key observations have been carried out from this work: (i) e fractional parameter α delivers more than one line as associated with the classical model. is influence describes the memory effects which is not possible to elaborate by the classical model. (ii) For the solution, the new transformation is more reliable. It is very simple to solve the fractional model by using these transformations. (iii) e velocity of the fluid rises by rising the value of Gr, Gm, and λ 1 . (iv) e fluid velocity drops by rising the value of Sc, M, λ 2 , and φ. (v) It is interesting to note that the heat transfer rate of engine oil is enhanced by 24.820% for Ag nanoparticles and 16.910% for Cu nanoparticles.

Data Availability
e data used to support the findings of this study are included within the article.

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