The present article aims to examine the heat and mass distribution in a free convection flow of electrically conducted, generalized Jeffrey nanofluid in a heated rotatory system. The flow analysis is considered in the presence of thermal radiation and the transverse magnetic field of strength B_{0}. The medium is porous accepting generalized Darcy’s law. The motion of the fluid is due to the cosine oscillations of the plate. Nanofluid has been formed by the uniform dispersing of the Silver nanoparticles in regular engine oil. The problem has been modeled in the form of classical partial differential equations and then generalized by replacing time derivative with Atangana–Baleanu (AB) time-fractional derivative. Upon taking the Laplace transform technique (LTT) and using physical boundary conditions, exact expressions have been obtained for momentum, energy, and concentration distributions. The impact of a number of parameters on fluid flow is shown graphically. The numerical tables have been computed for variation in the rate of heat and mass transfer with respect to rooted parameters. Finally, the classical solution is recovered by taking the fractional parameter approaching unity. It is worth noting that by adding silver nanoparticles in regular engine oil, its heat transfer rate increased by 14.59%, which will improve the life and workability of the engine.
Fluids that show non-linear relation between shear stress and shear strain are termed as non-Newtonian fluids e.g., castor oil, engine oil, colloidal solutions, and blood, etc. The non-Newtonian fluid has different rheological behavior than Newtonian fluids and has very substantial uses in the area of engineering, medical sciences, industries, electrochemistry, and mechanics [1]. Due to the mentioned significance, it gathered great attention from researchers and scientists. Jeffrey fluid model has two generalized parameters λ1 and λ2. The models of Newtonian and second-grade fluid can also be recovered from it by neglecting the effects of their generalized parameters. Due to the above-stated facts, the Jeffrey fluid model has been dealt with by many researchers [2–5]. After further examination, researchers reported that the thermal transport properties of the considered regular fluid are low. To boost the heat transport properties of regular fluids, researchers and scientists used the idea of suspended solid particles in the assumed conventional base fluid. As solid particles possess high heat transport properties as compared to regular fluids, it consequently enhances the heat transport properties of the base fluid. Keeping in view the above-addressed features of the solid particles, Maxwell [6], in 1873, used micro-sized solid particles in regular fluids. However, the significant blemish of microfluid was smaller scale estimated particles that typically stuck in the microchannel that block the apparatus. Besides, microparticles settled quickly in the base liquid and made issues of clogging, pipe disintegration, and surface wear. To circumvent the flaws of micro-sized solid particles in 1995, Choi et al. [7], introduced the idea of dispersing of nano-sized solid particles in conventional fluids to augment the thermal transport features of the conventional fluids. The resultant fluid that is obtained from the suspension of nanoparticles in the base fluid is referred to as nanofluid. Nanoparticles have widespread uses in different zones, for example, cooling power, vitality sparing, and medicate conveyance, and so forth. Different studies have been reported on the uses of nanoparticles [8–10]. From the analysis, the researchers came to the point that nanoparticles are more stable and increase the conductive properties than micro size particles. Researchers also highlighted that nanoparticles decreases the problem of sedimentation and increase the stability of the regular fluid. Ali et al. [11] discussed the impact of silver nanoparticles in regular engine oil. Several other studies regarding nanoparticles applications can be seen in [12–15]. By keeping in view the importance of nano-particles.
The study of magnetic properties of those fluids which allow electric current is known as magneto-hydrodynamic. Saltwater, engine oil, electrolytes, kerosene oil are examples of MHD fluid. The core work on MHD has given by Alfvén [16], and for this contribution, he got Nobel Prize in 1970. Magnetohydrodynamic or MHD for short, is the study of these motions in the case where the fluid can be treated as a continuum. On the basis of the above applications, Hayat et al. [17] developed a series form solution via the Homotopy analysis method for second-grade fluid. They considered the fluid in between two porous walls under the consideration of binary chemical reaction [18] studied the Maxwell fluid flow in a rotatory coordinate system. Seth et al. [19] also discussed unsteady electrically conducted fluid in a rotating channel and observed that fluid flow shows acceleration/deceleration when the magnetic field is varied. Abelman et al. [20] found a numerical solution for the time-independent flow of 3rd-grade fluid. Ali et al. [21] investigated the effect of the magnetic field on fractionalized Brinkman-type fluid flowing in a heated system. The authors modeled their mathematical model by inserting relative constitutive relations, and for their solution, they used the tool of Laplace transformation. For the interpretations of inserted parameters, they sketched the graphs. In their study, they highlighted that the magnetic parameter decline the velocity profile due to Lorentz forces occurs in the fluid. Khan et al. [22] inspected the effect of wall shear stress on conducting fluid flow for Newtonian fluid with variable ramped temperature.
Due to multidimensional features, the non-integer order calculus is fascinating to the attention of scientists and researchers [23–25]. Fractional calculus is an important and fruitful tool for describing many systems, including memory effects. In the last few years, fractional calculus is used for many purposes in various fields, such as electrochemistry, transportation of water in ground level, electromagnetism, elasticity, diffusion, and in conduction of heat process [26]. In 2015, Caputo et al. [27] worked together in the field of non-integer order calculus and presented a new expression for the fractional non-integer order derivative with the non-singular kernel. So keeping in sight the importance of the CF operator, many researchers used the CF operator in their studies such as [28] examined the MHD flow of Walter’s-B fluid by using the non-singular operator of CF. Several other studies regarding the uses of CF operators can be found in [29–31]. But despite many properties, the kernel of the CF operator found local, and due to this deficiency, it was not remain reliable for describing the memory effect. So to fill the gap and solve this issue occurred in the CF operator, in the year 2016, [32] modified the operator of CF derivative by utilizing the Mittag–Leffler function and made its kernel non-singular and non-local. AB fractional operator has the beauty that it gives information about the original function, and one can also achieve the original function as well by letting the parameter of the fractional operator tends to zero. So due to the worth mentioned properties and advantages of AB fractional operator, [33] inspected the influence of thermo-diffusion on Jeffrey nanofluid in a rotating frame. They formed nanofluid by dispersing the metallic silver nanoparticles in regular Mobil oil. They have been acquired that velocity boosts up for increasing the magnitude of the fractional parameter. Jan et al. [34] discussed the impact of molybdenum disulfide nano-size particles in engine oil using the Atangana–Baleanu fractional model. Saqib et al. [35] acquired an exact expression for generalize Casson nanofluid, using AB fractional derivative.
Keep in mind the above literature review; we find a gap that no one has investigated Jeffrey nanofluid flow in a rotating frame with time-dependent temperature and concentration in initial and boundary conditions. To fill this gap, we considered fractionalized Jeffrey fluid flow in a rotating system. The governing equations containing the system of PDEs are developed by following the assumption of Boussinesq’s approximation [36]. The classical model of the considered problem has been fractionalized via Atangana–Baleanu fractional operator. The mathematical tool of the Laplace transform technique has been utilized to acquire the solution mathematical model. The involvement of the constructive parameters on fluid motion, heat distribution, and concentration profile has been sketched. Change in the rate of heat and mass transfer was found numerically and computed in tabular form.
Mathematical Modeling and Solution
Consider a Cartesian coordinate system for unsteady electrically conducted free convection flow of an incompressible Jeffrey nanofluid past over an infinite rigid plate saturated in a porous medium under the effect of transverse magnetic field B_{o}. The plate is taken vertically upward in the direction of the x-axis. The rotation with constant angular velocity “Ω” of the fluid is assumed about z-axis, which is normal to the plate, as shown in Fig. 1.
The geometry of the flow
At time t = 0 both the plate and fluid are immobile. At t > 0 the plate is disturbed with cosine oscillation and transmits the disturbance to the fluid, which causes the fluid motion. Temperature and concentration of the fluid at time t = 0^{+} is T∞+(TW-T∞)At and C∞+(CW-C∞)At respectively. Where T∞ and C∞ represent surrounding temperature and concentration, respectively. T_{W} and C_{w} represent wall temperature and wall concentration, respectively. Keeping in view, the above consideration applying Boussinesq approximation and using Tiwari and Das nanofluid model, the momentum, energy, and concentration equations are governed by:
In dimensinoalization process the constants we get:
γ=ΩνU02,λ=λ2U02ν,1K=ϕ1νK1*U02,Ha=σ*β02νρU02,Gr=gβTν(Tw-T∞)U03,Gm=gβCν(Cw-C∞)U03,Gr1=a2Gr,Gm1=a6Gm,Pr=νfαf,Rd=16σ*T3kfk1*,Sc=νDf,m1=(1-φ)+φ(ρcp)s(ρcp)f,m2=ks+2kf-2φ(kf-ks)ks+2kf+φ(kf-ks),a1=((1-φ)2.5{(1-φ)+φρsρf})-1,a2=1ρnf((1-φ)ρf+(βT)s(βT)f),a3=(1-φ)+φρsρf,a4=m2+Rdm1Pr,a5=Sc1-φ,a6=1ρnf((1-φ)ρf+(βC)s(βC)f),
The term γ,λ,K,Ha,Gr,Gm,Pr,Rd, and Pr represent rotation parameter, Jeffrey fluid parameter, Permeability parameter, Hartman number, thermal Grashoff number, Mass Grashoff number, radiation, and Prandtl number, respectively.
Fractional Model
To fractionalized the ordinary model of the considered problem given in Eqs. (9)–(11) to AB fractional model, we utilized the operator AB℘tα instead of ∂∂t(.), and get:
where
δ0=11-α,δ1=αδ0,δ2=δ0a4,δ3=a5δ0,b1=1+a1λk(1+λ1),b2=a11+λ1,b3=a1λδ01+λ1,b4=Haa3+a1k(1+λ1)+2iγ,δ4=b2+b3,δ5=b2δ1,δ6=b1δ0+b4,δ7=δ1b4,
using conditions stated in Eq. (20) the solutions of Eqs. (17)–(19) will be:
where
ϕ*(z,p,m1,m2,m3,m4)=1pα+m1 exp(-zm2pα+m3pα+m4),ϕ1*(z,p,m1,m2,m3,m4)= exp(-zm4m1pα+m2pα+m3),ϕ2*(z,p,m1,m2,m3,m4,m5)=1pα+m1 exp(-zm5m2pα+m3pα+m4),
and
δ8=δ2δ5-δ1δ6-δ7δ2δ4-δ6,δ9=δ1δ7δ2δ4-δ6,δ10=δ82,δ11=(δ10)2+δ9,δ12=δ10+δ11,δ13=δ10-δ11,δ14=δ12δ12δ13,δ15=(δ1-δ13)2δ12-δ13,δ16=(δ1-δ13)2δ13(δ13-δ12),δ17=δ3δ5-δ1δ6-δ7δ3δ4-δ6,δ18=δ1δ7δ3δ4-δ6,δ19=δ172,δ20=(δ19)2+δ18,Γ1=δ19+δ20,Γ2=δ19-δ20,Γ3=δ12Γ1Γ2,Γ4=(δ1-Γ1)2Γ1(Γ1-Γ2),Γ5=(δ1-Γ2)2Γ2(Γ1-Γ2),ℜ1=Gr1δ14,ℜ2=Gr1δ15,ℜ3=Gr1δ16,ℜ4=Gr1Γ3,ℜ5=Gr1Γ4,ℜ6=Gr1Γ5.
Inverting the Laplace transform on Eqs. (21)–(23), we get:
where
ϕ1*(z,t,m1,m2,m3,m4)=1π∫ 0∞∫ 0∞Ψ1(z,u,c1,c2,c3,c4) exp(tr1-r2r1α cosαπ)sin(r2r1α sinαπ)dr1dr2,ϕ2*(z,t,m1,m2,m3,m4,m5)=1π∫0∞∫0∞Ψ2(z,u,c1,c2,c3,c4,c5) exp(tr1-r2r1α cosαπ)× sin(r2r1α sinαπ)dr1dr2,ϕ*(z,t,m1,m2,m3,m4)=1π∫ 0∞∫ 0∞Ψ1(z,u,c1,c2,c3,c4) exp(tr1-r2r1α cosαπ)sin(r2r1α sinαπ)dr1dr2,
in which
Ψ1(z,t,c1,c2,c3,c4)=e-zc4c1-zc2-c1c32c4πt∫ 0∞1r2 exp(-z24c42r2-c1r2)⋅I1(2(c2-c1c3)r2t)dr2.Ψ2(z,t,c1,c2,c3,c4,c5)=e-c1t--zc5c1-zc3-c2c42c5π∫0∞∫0te-c1tt exp(c1t-c4t--z24c5r2-c2r2)×I1(2(c3-c2c4)r2t)dtdr2.Ψ3(z,t,c1,c2,c3,c4)=e-c1t-zc2-zc3-c2c42c5π∫0∞∫0te-c1ttexp(c1t-c4t--z24r2-c2r2)×I1(2(c3-c2c4)r2t)dtdr2.
And
£-1[1p2-α]=d(t)=1Γ(2-α)t1-α
Classical Exact Expression
For α→1 and using the following property,
limα→1AB℘tαF1(z,t)=limα→1L−1[L{AB℘tαF1(z,t)}]=L−1{limα→1pαF¯1(z,p)−F1(z,0)(1−α)pα+α}=L−1{pF1¯(z,p)−F1(z,0)}=L−1[L{F1/(z,t)}]=F1/(z,t),n the exact expression for velocity distribution obtained in Eq. (24) reduced to
where
b5=b3a5,d4=a5b2-b12b5,d5=(b2a5-b1)2+4b4b52b5,d6=Gm1b5.
Nusselt Number and Sherwood Number
In the non-dimensional form, the Nusselt number and Sherwood number are presented as:
Nu=-knfkf∂θ∂z|z=0
Sh=-Dnf(∂ϕ∂z)z=0.
Results and Discussion
This portion of the article devote to the physical arguments and detailed descriptions of the outcomes. Using relative constitutive equations, the governing system of PDEs is formulated. Nano-liquid has been made by the distribution of nano-silver in Mobil oil. The classical model of electrically conducted, incompressible Jeffrey nano-liquid is transformed into AB fractional model. Exact solutions have been gotten for the governing equations. To check the impact of inserted parameters on fluid motion, various sketches are portrayed.
The impact of Gr and Gm has been sketched in Figs. 2 and 3. Both the sketches show enchantment in the velocity profile for larger values. This happened due to a decrease in the density and increase in the bouncy forces which leads to speed up the velocity. The effect of Jeffrey fluid parameter λ and volume fraction φ can be reported from Figs. 4 and 5. When the values of the material parameter λ and volume fraction of nanoparticles φ rise, the viscous forces in the fluid increase; as a result, the friction force between the fluid particles increases, which leads to slow down the velocity of the fluid. It is concluded from Fig. 6 that for larger values of magnetic parameter Ha, the momentum of the fluid decelerates. A rise in the magnitude of Ha tends to increase the Lorentz forces which works as a drag force and retards the fluid motion. Moreover, the effect of angular rotation γ is drawn in Fig. 7. It can be noted from the sketch that velocity decreases for a large value of γ. From the mathematical relation of γ it can be seen that γ it has a direct variation with viscous forces of the fluid so larger the magnitude of γ the fluid motion decrease.
Effect of thermal Grashof number <italic>Gr</italic> on the velocity profileEffect of mass Grashoff number <italic>Gm</italic> on the velocity profileEffect of retardation time parameter <inline-formula id="ieqn-30"><alternatives><inline-graphic xlink:href="ieqn-30.png"/><tex-math id="tex-ieqn-30"><![CDATA[$ \lambda $]]></tex-math><mml:math id="mml-ieqn-30"><mml:mi>λ</mml:mi></mml:math></alternatives></inline-formula> on the velocity profileEffect of volume fraction <inline-formula id="ieqn-31"><alternatives><inline-graphic xlink:href="ieqn-31.png"/><tex-math id="tex-ieqn-31"><![CDATA[$ \varphi $]]></tex-math><mml:math id="mml-ieqn-31"><mml:mi>φ</mml:mi></mml:math></alternatives></inline-formula> on the velocity profileEffect of magnetic parameter <italic>Ha</italic> on the velocity profileEffect of rotation parameter <inline-formula id="ieqn-32"><alternatives><inline-graphic xlink:href="ieqn-32.png"/><tex-math id="tex-ieqn-32"><![CDATA[$ \gamma $]]></tex-math><mml:math id="mml-ieqn-32"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula> on the velocity profile
Variation in the Sherwood numbers has been discussed in Tab. 2. As the volume fraction φ increases, the rate of mass distribution decrease. By introducing nano-silver in the mobile oil bring enhancement in the density; as a result, the fluid becomes denser, which leads to slow down the rate of mass distribution.
Variations in Sherwood number
φ
Sc
α
t
Sh
%
0.00
5
0.2
1
0.048
—
0.01
5
0.2
1
0.045
6.6
0.02
5
0.2
1
0.040
17.77
0.03
5
0.2
1
0.037
24.44
0.04
5
0.2
1
0.035
28.88
Tab. 3 shows a change in the rate of heat transfer of Engine oil with Silver nanoparticles. It can be seen from the table that by increasing the volume fraction of Ag nanoparticles, the heat transfer rate has been increased by 14.59%. It is physically true that by adding silver nanoparticles in regular engine oil, its mechanical properties e.g., improving the load-carrying capacity, adsorption and deposition, and minimization of skin friction which will consequently improve the workability of the engine.
Variation in Nusselt number
φ
Rd
α
t
Nu
%
0.00
2
0.2
1
0.185
−
0.01
2
0.2
1
0.191
3.24
0.02
2
0.2
1
0.198
7.02
0.03
2
0.2
1
0.205
10.81
0.04
2
0.2
1
0.212
14.59
Concluding Remarks
In the present work, the role of nano-silver on the life span and mechanical properties of engine oil has been investigated. For the mentioned analysis, Jeffrey nanofluid model has been assumed in a rotating frame. The Laplace transform technique has been used to gain the solution of the governing equations. The key and worthy features that are observed from the study analysis are presented below:
Enhancing the order of the fractional parameter α slows down the flow motion.
Both the parameters Gr and Gm boost up the fluid motion.
For higher values of λ,γ,Sc and φ motion of the fluid slow down.
The opposite behavior is noticed for MHD and Permeability.
As the volume fraction increase from 0.00 to 0.04, an efficiency of 14.59% in the heat transfer rate and 28.8% efficiency was reported for the mass transfer rate.
Funding Statement: The author(s) received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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