# Caputo Time Fractional Model Based on Generalized Fourier’s and Fick’s Laws for Jeffrey Nanofluid: Applications in Automobiles

Authors: Jawad Ahmad; Farhad Ali; Saqib Murtaza; Ilyas Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This article aims to examine Jeffery nanofluid with joint effects of mass and heat transfer in a horizontal channel. The classical model is transferred to the Caputo fractional model by using the generalized Fourier’s and Fick’s laws. The nanofluids are formed by dispersing two different nanoparticles, silver and copper, into a based fluid. A novel transformation has been applied to the mass and energy equation and then solved by using the sine Fourier and the Laplace transformation jointly. The exact solution is given in terms of a special function, that is, the Mittag-Leffler function. The Sherwood number and Nusselt number are calculated and displayed in the tabular form. The effect of embedded parameters on the velocity, concentration, and temperature profile is discussed graphically. It is noted that the heat transfer rate of EO is improved by 28.24% when the volume fraction of Ag nanoparticles is raised from 0.00 to 0.04.


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## Body

## 1. Introduction

The study of non-Newtonian fluid is very significant because it has vast applications in many areas, especially in the engineering and industrial sectors [1–4]. One of the well-known 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. The Jeffrey fluid model has the facility to customize the time derivative on behalf of the convective derivative. The 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. The 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. The 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 diffusion-thermo in a heated rotatory system. Moreover, Kole and Dey [13] studied the influence of gear oil and dispersed copper oxide nanoparticles. They 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. The 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]. They observed that nanoparticles are responsible for enhancing the thermal and mechanical properties of the base fluids. They 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. This 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 18th century, the Riemann–Liouville definition [24] of fractional derivative was mostly used definition. The 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. The 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 [27–30]. 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. The 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] discussed the transient heat diffusion equation together with impact of nonsingular kernel fading memory. The author used the Cattaneo constitutive equation with Jeffrey’s kernel. Koca and Atangana [37] utilized the nonlocal fractional derivative operator of Caputo–Fabrizio for the inspection of the Cattaneo–Hristov model together with the effect of elastic heat diffusion.The present paper investigates the MHD flow of Jeffrey nanofluid in a horizontal channel with mass and heat transfer. Unlike the published work, the constitutive equations are transformed to fractional model by using generalized Fick’s and Fourier’s laws. A novel transformation has been applied to the energy and concentration equation and then solved by using the sine Fourier and the Laplace transformation jointly. The obtained general solutions satisfy all the conditions, which show the validity of the obtained general solutions.


## 2. Mathematical Modeling

We considered the laminar and unsteady flow of Jeffrey nanofluid in a horizontal channel in which both the plates are separated by distancel. The flow is assumed to be in the direction of x-axis with ambient temperature Ta and ambient concentration Ca. Initially, both the plates and fluid are at rest. At the time t=0+, the upper plate temperature and concentration levels are increased to Ta+Tp−TaAt and Ca+Cp−CaBt, respectively. The physical geometry of the problem is given in Figure 1.Figure 1
Flow regime of problem.The constitutive equations which govern the flow using Boussinesq’s approximation [38, 39] are given by(1)ρnf∂uy,t∂t=μnf1+λ11+λ2∂∂t∂2u∂y2−σnfB02uy,t+ρβT1nfg1T−Ta+ρβc1nfg1C−Ca,(2)ρCpnf∂T1y,t∂t=−∂qy,t∂y,(3)qy,t=−knf∂T1y,t∂y,(4)∂Cy,t∂t=−∂γ1y,t∂y,(5)γy,t=−Dnf∂Cy,t∂y,with physical condition as follows:(6)uy,0=0,Ty,0=0,Cy,0=0,u0,t=0,T0,t=Ta,C0,t=Ca,ul,t=0,Tl,t=Ta+Tp−TaAt,Cl,t=Ca+Cp−CaBt.The following dimensionless variables are used for dimensional analysis:(7)u1=uU0,ξ=yl,τ=νl2t,θ=T1−T1aTp−Ta,ϕ=C1−C1aCp−Ca,λ=qlknfTp−Ta,δ=γlDnfCp−Ca,Aτ=Al2νt,Bτ=Bl2νt.Using above-stated dimensionless variables, equations (1)–(6) become(8)∂u1∂τ=m01+λ11+λ3∂∂τ∂2u1∂ξ2−M0u1+Gr0θ+Gm0ϕ,(9)∂θ∂τ=1a0∂λξ,t∂ξ,(10)λξ,t=−∂θ∂ξ,(11)∂ϕ∂τ=1b0∂δξ,τ∂ξ,(12)δξ,τ=−∂ϕξ,τ∂ξ,(13)u1ξ,0=0,θξ,0=0,ϕξ,0=0,u10,τ=0,θ0,τ=0,ϕ0,τ=Ca,u1,τ=0,θ1,τ=Aτ,ϕ1,τ=Bt.The following are the dimensionless parameters and some constants after the dimensionalization process:(14)Gr=gl2βTTp−TaU0νb0,Gr0=Grφ0,M=δB02l2μ,Gm=gβcl2Cp−CaU0ν,Sc=νDnf,M0=Mφ0,Gmo=Gmφ0,λ3=λ2νl2,ao=φ1prφ2,pr=νρcpkf,b0=Sc1−φ,ϕ0=1−φ+φρsρf,m0=1−φ2.51−φ+φρsρf−1,ϕ1=1−φ+φρcpsρcpf,ϕ2=ks−2kf−2φkf−ksks+2kf+φkf−ks.A fractional model is developed using the generalized Fick’s and Fourier laws as follows:(15)λξ,τ=−c℘τ1−α∂θξ,τ∂ξ,0<α≤1,(16)δξ,τ=−℘τ1−αc∂ϕξ,τ∂ξ,0<α≤1,where ℘ταc(.) is the Caputo time operative which is as follows:(17)℘ταcfy,t=1Γ1−α∫0t1f1y,st1−s1−α1ds=ηα1t1∗fy,t1,0<α≤1,where ηαt=t−α/Γ1−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(18)∂θξ,τ∂t=1a0℘τ1−αc∂2θξ,τ∂ξ2,(19)∂ϕξ,τ∂t=1b0℘τ1−αc∂2ϕξ,τ∂ξ2.Equations (18) and (19) can be written as follows:(20)℘ταθξ,τ=1a0c∂2θξ,τ∂ξ2,(21)℘ταϕξ,τ=1b0c∂2ϕξ,τ∂ξ2.The thermo-mechanical properties of the considered base fluid and solid nanoparticles are given in Table1.Table 1
Thermo-mechanical properties of the base fluid and nanoparticles [12, 40].
PropertiesρCpkβPrEngine oil86320480.14040.000076000Cu89333854011.67×10−5—Ag105002354291.89×10−5—


### 2.1. Solution of Energy Equation

Using the following transformation:(22)Χξ,τ=θξ,τ−ξAτ,on equation (20), we get(23)℘τ1−αcΧξ,τ+ξc℘τ1−αAτ=1a0∂2Χξ,τ∂ξ2,according to equation (22), and the initial and boundary conditions are as follows:(24)Χξ,0=0,Χ0,τ=0,Χ1,τ=0.Taking the Laplace transformation on equation (23), we get(25)sαΧ¯ξ,s+ξsαA¯s=1a0d2Χ¯ξ,sdξ2.Now, applying sine Fourier transform to equation (25), we get(26)sαΧ¯F1n1,s−−1nnπsαA¯s=−1a0Χ¯Fn1,sn1π2.After simplification, we arrived at(27)Χ¯F1n1,s=sA¯s−1nn1πsα−1sα+n1π2/a0.Inverting the essential transformation of equation (26), we get(28)Χξ,τ=2∑n=1∞−1sinξnπnπ∫0τAτ−tEα,α+1−nπ2a0tαdt.Consequently, the exact solution of equation (20) is as follows:(29)θξ⋅τ=Χξ,τ+ξAτ.


### 2.2. The Solution of Mass Equation

Using the following transformation:(30)ℜξ,τ=ϕξ,τ−ξBτ,on equation (21), we get(31)℘τ1−αcℜξ,τ+ξc℘τ1−αBτ=1b0∂2ℜξ,τ∂ξ2.According to equation (30), the transformed conditions are as follows:(32)ℜξ,0=0,ℜ0,τ=0,ℜ1,τ=0.Taking the Laplace on both sides of equation (30), we get(33)sαℜ¯ξ,s+ξsαB¯s=1b0d2ℜ¯ξ,sdξ2.Now, taking sine Fourier transform to equation (33), we get(34)sαℜ¯Fn,s−−1nnπsαB¯s=−1b0ℜ¯Fn,snπ2,and after simplification, we arrived at(35)ℜ¯Fn,s=sB¯s−1nnπsα−1sα+nπ2/b0.Inverting the essential transformation of equation (35), we get(36)ℜξ,τ=2∑n=1∞−1sinξnπnπ∫0τBτ−tEα,α+1−nπ2b0tαdt.Consequently, the final exact solution of equation (21) is as follows:(37)ϕξ,τ=ℜξ,τ+ξBτ.


### 2.3. Solution of the Velocity Distribution

Taking the Laplace of equation (8), we get(38)su1ξ,s=m01+λ11+λ3sd2u¯1dξ2−M0u¯1+Gr0θ¯+Gm0ϕ¯.Applying sine Fourier transform to equation (36) and after simplification, we obtained the following:(39)u1Fn,s=Gr01+λ1s+A01+λ1+λ3m0nπ2θ¯Fn,s+Gm01+λ1s+A01+λ1+λ3m0nπ2ϕ¯Fn,s,where A0=m0nπ2+M01+λ1/1+λ1+λ3m0nπ2; incorporating the value of θ¯Fn,s and ϕ¯Fn,s in equation (39) and taking the Laplace inverse, we get (40)u1Fn,τ=Gr0A1−1nnπexpA0τ∗∫0τAτ−tEα,α−1−nπ2a0tαdt−Aτ+Gm0A1−1nnπexpA0τ∗∫0τBτ−tEα,α−1−nπ2a0tαdt−Bτ,where A1=1+λ1/1+λ1+λ3m0nπ2.Now, applying the inverse sine Fourier transform of equation (40),(41)u1ξ,τ=2∑n=1∞Gr0A1−1nnπexpA0τ∗∫0τAτ−tEα,α−1−nπ2a0tαdt−Aτsinnπξ+2∑n=1∞Gr0A1−1nnπexpA0τ∗∫0τBτ−tEα,α−1−nπ2a0tαdt−Bτsinnπξ,where E1a,b⋅ is the Mittag-Leffler function.


### 2.4. Skin Friction

For Jeffery nanofluid, skin friction is given by(42)Sf=11+λ11+λ2∂∂t∂u1∂ξ|ξ=0.


### 2.5. Nusselt Number

For industrialists, the Nusselt number is a significant quantity, which is defined as follows:(43)Nn=−knfkf∂θξ,τ∂ξ|ξ=1.


### 2.6. Sherwood Number

In nondimensional form, the Sherwood number is defined as follows:(44)Sn=−Dnf∂ϕξ,τ∂ξ|ξ=1.


## 2.1. Solution of Energy Equation

Using the following transformation:(22)Χξ,τ=θξ,τ−ξAτ,on equation (20), we get(23)℘τ1−αcΧξ,τ+ξc℘τ1−αAτ=1a0∂2Χξ,τ∂ξ2,according to equation (22), and the initial and boundary conditions are as follows:(24)Χξ,0=0,Χ0,τ=0,Χ1,τ=0.Taking the Laplace transformation on equation (23), we get(25)sαΧ¯ξ,s+ξsαA¯s=1a0d2Χ¯ξ,sdξ2.Now, applying sine Fourier transform to equation (25), we get(26)sαΧ¯F1n1,s−−1nnπsαA¯s=−1a0Χ¯Fn1,sn1π2.After simplification, we arrived at(27)Χ¯F1n1,s=sA¯s−1nn1πsα−1sα+n1π2/a0.Inverting the essential transformation of equation (26), we get(28)Χξ,τ=2∑n=1∞−1sinξnπnπ∫0τAτ−tEα,α+1−nπ2a0tαdt.Consequently, the exact solution of equation (20) is as follows:(29)θξ⋅τ=Χξ,τ+ξAτ.


## 2.2. The Solution of Mass Equation

Using the following transformation:(30)ℜξ,τ=ϕξ,τ−ξBτ,on equation (21), we get(31)℘τ1−αcℜξ,τ+ξc℘τ1−αBτ=1b0∂2ℜξ,τ∂ξ2.According to equation (30), the transformed conditions are as follows:(32)ℜξ,0=0,ℜ0,τ=0,ℜ1,τ=0.Taking the Laplace on both sides of equation (30), we get(33)sαℜ¯ξ,s+ξsαB¯s=1b0d2ℜ¯ξ,sdξ2.Now, taking sine Fourier transform to equation (33), we get(34)sαℜ¯Fn,s−−1nnπsαB¯s=−1b0ℜ¯Fn,snπ2,and after simplification, we arrived at(35)ℜ¯Fn,s=sB¯s−1nnπsα−1sα+nπ2/b0.Inverting the essential transformation of equation (35), we get(36)ℜξ,τ=2∑n=1∞−1sinξnπnπ∫0τBτ−tEα,α+1−nπ2b0tαdt.Consequently, the final exact solution of equation (21) is as follows:(37)ϕξ,τ=ℜξ,τ+ξBτ.


## 2.3. Solution of the Velocity Distribution

Taking the Laplace of equation (8), we get(38)su1ξ,s=m01+λ11+λ3sd2u¯1dξ2−M0u¯1+Gr0θ¯+Gm0ϕ¯.Applying sine Fourier transform to equation (36) and after simplification, we obtained the following:(39)u1Fn,s=Gr01+λ1s+A01+λ1+λ3m0nπ2θ¯Fn,s+Gm01+λ1s+A01+λ1+λ3m0nπ2ϕ¯Fn,s,where A0=m0nπ2+M01+λ1/1+λ1+λ3m0nπ2; incorporating the value of θ¯Fn,s and ϕ¯Fn,s in equation (39) and taking the Laplace inverse, we get (40)u1Fn,τ=Gr0A1−1nnπexpA0τ∗∫0τAτ−tEα,α−1−nπ2a0tαdt−Aτ+Gm0A1−1nnπexpA0τ∗∫0τBτ−tEα,α−1−nπ2a0tαdt−Bτ,where A1=1+λ1/1+λ1+λ3m0nπ2.Now, applying the inverse sine Fourier transform of equation (40),(41)u1ξ,τ=2∑n=1∞Gr0A1−1nnπexpA0τ∗∫0τAτ−tEα,α−1−nπ2a0tαdt−Aτsinnπξ+2∑n=1∞Gr0A1−1nnπexpA0τ∗∫0τBτ−tEα,α−1−nπ2a0tαdt−Bτsinnπξ,where E1a,b⋅ is the Mittag-Leffler function.


## 2.4. Skin Friction

For Jeffery nanofluid, skin friction is given by(42)Sf=11+λ11+λ2∂∂t∂u1∂ξ|ξ=0.


## 2.5. Nusselt Number

For industrialists, the Nusselt number is a significant quantity, which is defined as follows:(43)Nn=−knfkf∂θξ,τ∂ξ|ξ=1.


## 2.6. Sherwood Number

In nondimensional form, the Sherwood number is defined as follows:(44)Sn=−Dnf∂ϕξ,τ∂ξ|ξ=1.


## 3. Results and Discussion

The incompressible flow of Jeffrey nanofluid in a bounded channel is elaborated in this article. The 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 closed-form solution, a novel technique is applied to change the equation into a simple form, and then Fourier’s and Laplace transforms are used. The influence of embedded parameters on the velocity, temperature, and concentration distribution is presented in Figures2–14.Figure 2
Deviation in velocity distribution for the distinct valueα.Figure 3
Comparison of silver and copper nanoparticles on the velocity profile.Figure 4
Outcome of time relaxation parameterλ1 on the velocity distribution.Figure 5
Impact of retardation time parameterλ2 on the velocity distribution.Figure 6
Impact of Gr on the velocity profile.Figure 7
Effect ofGm on the velocity distribution.Figure 8
Influence of Sc on the velocity profile.Figure 9
Impact of volume fractionφ on the velocity profile.Figure 10
Impact of the Hartman numberM on the velocity profile.Figure 11
Deviation in temperature distribution for the distinct value ofα.Figure 12
Evaluation of silver and copper nanoparticle on the temperature distribution.Figure 13
Influence of volume fractionφ on the temperature distribution.Figure 14
Variation in concentration profile for the distinct value ofα.Response of velocity profile against fractional orderα is portrayed in Figure 2. In the classical order derivative, we have only one fluid layer for the analysis of considered fluid. The 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 10497kg/m3 and the density of copper is 8940kg/m3. Figures 4 and 5 represent the influence of generalized Jeffery fluid parameters λ1 and λ2 on the velocity distribution. It is clear from the figure that for greater values of λ1, the velocity profile rises. This is because λ1 is the time relaxation parameter, and due to the quick response of shear stresses, the fluid accelerates. In contrast, the greater value of λ2 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). This is true because the Schmidt number has direct relation with the viscosity of the fluid. The 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. This 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. This 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. The figure displays that the silver nanoparticle has a higher temperature as compared with the copper nanoparticle. The thermal conductivity of silver nanoparticle is 406W/m⋅K and thermal conductivity of copper nanoparticle is 385W/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. The 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.Figure 15
Influence of the Schmitt number Sc on the concentration profile.Furthermore, Table2 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.Table. 2
Skin friction variation with corresponding embedded parameters.
αGrGmScMφλ1λ2Sf0.25.05.05.00.50.010.50.51.9150.35.05.05.00.50.010.50.51.9400.2105.05.00.50.010.50.51.8150.25.0105.00.50.010.50.51.8050.25.05.0100.50.010.50.52.0500.25.05.05.01.00.010.50.52.1500.25.05.05.00.50.020.50.52.4500.25.05.05.00.50.011.00.52.5010.25.05.05.00.50.010.51.01.801Table. 3
Variation in the Nusselt number (rate of heat transfer).
φαtNnAg%NnCu%0.000.21.01.780———0.010.21.01.8574.3251.8524.0440.020.21.01.9368.7641.9268.2020.040.21.02.20224.8202.08116.910Table. 4
Distinction in the Sherwood number (rate of mass transfer).
αtScφSn0.21.05.00.011.1240.31.05.00.011.0910.21.0100.011.8850.21.05.00.021.105


## 4. 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. The 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)
The fractional parameterα delivers more than one line as associated with the classical model. This 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)
The velocity of the fluid rises by rising the value ofGr,Gm, and λ1.(iv)
The fluid velocity drops by rising the value ofSc,M,λ2, and φ.(v)
It is interesting to note that the heat transfer rate of engine oil is enhanced by 24.820% forAg nanoparticles and 16.910% for Cu nanoparticles.



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