Atangana–Baleanu fractional model for the flow of Jeffrey nanofluid with diffusionthermo effects: applications in engine oil
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Abstract
The present article investigates the effects of diffusionthermo, thermal radiation, and magnetic field of strength \(B_{0}\) on the time dependent MHD flow of Jeffrey nanofluid past over a porous medium in a rotating frame. The plate is assumed vertically upward along the xaxis under the effect of cosine oscillation. Silver nanoparticles are uniformly dispersed into engine oil, which is taken as a base fluid. The equations which govern the flow are transformed into a time fractional model using Atangana–Baleanu time fractional derivative. To obtain exact expressions for velocity, temperature, and concentration profiles, the Laplace transform technique, along with physical initial and boundary conditions, is used. The behaviors of the fluid flow under the impact of corresponding dimensionless parameters are shown graphically. The variations in Nusselt number and Sherwood number of relative parameters are found numerically and shown in tabular form. It is worth noting that the rate of heat transfer of engine oil is enhanced by 15.04% when the values of volume fraction of silver nanoparticles vary from 0.00 to 0.04, as a result the lubricant properties are improved.
Keywords
Jeffrey’s nanofluid Rotating frame Diffusionthermo Engine oil Atangana–Baleanu fractional derivativeAbbreviations
 u
Fluid velocity in the xdirection (m/s)
 \(I_{1} (\cdot )\)
Bessel function of the first kind
 v
Fluid velocity in the ydirection (m/s)
 \({}^{AB}\wp _{t}^{\alpha } (\cdot)\)
Atangana–Baleanu timefractional operator
 t
Time (s)
 \(q_{r}\)
Radiative heat flux (KW/m^{2})
 g
Acceleration due to gravity (m/s^{2})
 p
Laplace transform parameter
 \(k_{nf}\)
Thermal conductivity of nanofluids (W/mK)
 \(k^{*}\)
Mean absorption coefficient (1/s)
 \(\rho _{nf}\)
Nanofluid density (kg/m^{3})
 \(\beta _{T}\)
Thermal expansion coefficient (1/K)
 σ
Electric conductivity (s^{3}A^{2}/kgm^{3})
 \(B_{0}\)
Magnetic field (kg/s^{2}A)
 \(D_{nf}\)
Mass diffusivity (m^{2}/s)
 \(\phi _{1}\)
Porosity
 \(\rho _{s}\)
Density of the solid particle (kg/m^{3})
 \(Ha = \frac{\sigma \beta _{0}^{2}\nu }{\rho U_{0}^{2}}\)
Magnetic parameter
 \(\lambda _{i}\)\(i = 1,2\)
Material parameters of Jeffrey fluid
 \(k = \frac{k^{*}U_{0}^{2}}{\phi _{1}\nu } \)
Permeability of porous medium
 \(U_{0}\)
Characteristic velocity (m/s)
 \(R_{d} = \frac{16\sigma T^{3}}{3k_{f}k_{1}}\)
Radiation parameter
 α
Fractional order
 \(\Pr = \frac{\nu _{f}}{\alpha _{f}}\)
Prandtl number
 \(F = u + iv\)
Complex velocity
 \(\mu _{nf}\)
Dynamic viscosity of nanofluid (kg/ms)
 φ
Nanoparticles volume fraction
 \(Gr = \frac{\nu g\beta _{T} ( T_{w}  T_{\infty } )}{U_{0}^{3}}\)
Thermal Grashof number
 \(c_{p}\)
Specific heat \(( J/kgK )\)
 \(Gm = \frac{\nu g\beta _{C} ( C_{w}  C_{\infty } )}{U_{0}^{3}}\)
Mass Grashof number
 T
Temperature of the fluid \(( K )\)
 \(r = \frac{\varOmega \nu }{U_{0}^{2}}\)
Rotation parameter
 θ
Dimensionless temperature
 \(Sc = \frac{\nu }{D_{f}}\)
Schmidt number
 ϕ
Dimensionless concentration
 A
Constant with \(( s^{  1} )\) and shows amplitude
 \(K_{T}\)
Thermal diffusion ratio
 \(C_{s}\)
Concentration susceptibility (kg/m^{3})
 \(D_{f}\)
Dufour number
 \(D_{m}\)
Coefficient of mass diffusivity (m^{2}/s)
 \(T_{w}\)
Wall temperature (K)
 \(T_{\infty } \)
Ambient temperature (K)
 \(C_{w}\)
Wall concentration (kg/m^{3})
 \(C_{\infty } \)
Ambient concentration (kg/m^{3})
 \(H ( t )\)
Heaviside function
1 Introduction
Fluids that have both viscous and elastic behaviors are referred to as viscoelastic fluids, for example, polymers, castor oil, engine oil, etc. Viscoelastic fluid has very significant applications in the field of medicine, automobiles, polymers solutions, electrochemistry, and mechanics [1]. Due to contrasting simulations compared to Newtonian fluids, Navier–Stokes equations are no longer reliable to describe the rheology of viscoelastic fluids. Due to vast implementation in many areas, it got great attention of the scholars so that various models have been formulated. One of the wellknown models among them is Jeffrey fluid model, which deals with the time derivative instead of the convective derivative. Second grade fluid model and viscous fluid model can be deduced from it by letting their parameters tend to zero. Keeping in mind the above mentioned facts, Jeffrey model was considered by many scholars like Hayat et al. [2] who studied Jeffrey fluid in the presence of thermal radiation. They developed velocity and temperature field via HPM and also showed variations in the fluid behavior due to embedded parameters. In another study, Hayat et al. [3] discussed the phenomenon of heat transfer of Jeffrey fluid over a moving surface, in which Jeffrey fluid model was considered as a rheological model. The authors studied Jeffrey fluid in the presence of thermal radiation effect and solved the governing equations by using HAM. Turkyilmazoglu et al. [4] investigated the heat transfer phenomena of Jeffrey fluid flow near the stagnation point. Furthermore, Elahi et al. [5] examined the simultaneous effects of magnetohydrodynamic and partial slip on peristaltic flow of Jeffrey fluid in rectangular duct. The closedform solution for the velocity field was obtained by a separation of variables procedure. In the above studies, low thermal conductivity was reported by the researchers for the considered fluids. To circumvent this issue, the idea of suspension of nanosized particles was adopted by the researchers and scholars. The first successful attempt was done by Eastman [6] in 1995. Eastman showed 40% enhancement in the thermal conductivity of ethylene glycol, when copper nanoparticles were dispersed at 0.3% volume fraction in ethylene glycol. Inspired from the work of Eastman and Choi, many researchers did their work on the above mentioned idea; see, e.g., Dinvarad et al. [7], Mohyud din et al. [8], Parekh and Lee [9], and Loganath [10]. They have noticed that nanoparticles are responsible for the enhancement of thermal conductivity and viscous forces. They also observed that nanofluids are more stable and do not have the sedimentation problem.
Magnetohydrodynamic flow in a rotating frame has huge beneficial applications in various phenomena like cosmic and geographic flow, Earth rotation, formation of galaxies, circulation of oceans, electromagnetic pumping, turbines and power generation, etc. Motivated by the above tremendous applications in mentioned fields, many researchers worked on the rotating phenomenon. The influence of silver nanoparticles on Jeffrey fluid flow was discussed by Atirah et al. [11]. They highlighted that velocity shows acceleration/deceleration at both primary and secondary positions when the rotation parameter varies. They also observed great variation in heat transfer rate when nanoparticles were uniformly dispersed in the fluid. Singh et al. [12] discussed the convective flow in the presence of transverse magnetic field past over an accelerated porous plate in a rotating channel. They highlighted the impact of several parameters like suction/injection, Prandtl number, and rotation parameter on flow behavior and also have shown the results graphically. Seth et al. [13] discussed the Couette flow under the effect of inclined magnetic field in a rotatory channel. They observed that the angle of inclination of applied magnetic field is responsible for deceleration of primary and secondary velocity throughout the channel. Ali et al. [14] studied the magnetohydrodynamic flow of a Brinkmantype nanofluid in a rotating disk with Hall effect. They observed a 6.35% increase in the rate of heat transfer when MoS_{2} nanoparticles were dispersed uniformly in the considered fluid. Furthermore, Ali et al. [15] studied the different shapes of MoS_{2}, namely (platelets, cylinders, bricks, and blades) taking engine and kerosene oil as base fluids in a rotating frame.
The effect of heat and mass transfer phenomena occur due to the differences in temperature and concentration. In modern technology, heat and mass transfer phenomena play a key role, especially in engineering, and due to this reason most researchers are attracted to further investigate this topic. Moreover, heat and mass transfer have a wide range of industrial and practical life applications, like freezedrying food. Blums [16] as well as Incropera and De Witt [17] discussed in details heat and mass transfer with several thermal and concentration applications. The phenomenon of heat and mass transfer is very important because of many physical uses in science and modern technology, as discussed in the book of Nield and Bejan [18]. In heat and mass transfer phenomena, the transfer of thermal energy from one system to another is happening not only due to temperature gradient but also due to concentration gradient. The transfer of thermal energy due to concentration gradient is referred to as diffusionthermo (Dufour effect). Already in 1952 Chapman and Cowling [19] developed the effect of diffusionthermo on the transport of heat and mass from kinetic molecular theory of gases. Based on the above effects, different investigations have been carried out, e.g., Reddy et al. [20] discussed numerically the effect of diffusionthermo and thermal diffusion in the presence of Hall current in a rotating frame for the fluid flowing in a porous medium. They highlighted that Dufour effect is responsible for the rise in the velocity and temperature field. Kafoussias et al. [21] analyzed the Dufour and Soret effects on a mixed convective flow along with temperaturedependent viscosity. The impact of Dufour and Soret numbers on a timeindependent mixed convective flow of heat and mass transfer flowing over a semiinfinite plate under the influence of magnetic field was discussed by Alam et al. [22]. Zafer and William [23] studied the effect of diffusionthermo and thermal diffusion on a natural convection flow over a vertical surface. They assumed helium–air mixture as a fluid and solved the governing equations analytically.
The concept of a fractional derivative was presented about 300 years ago, and it is known as the natural generalization of the ordinary calculus because it includes the derivatives and integrals of noninteger order. The idea of fractional calculus is based on a question [24] which was asked by L’Hospital in 1695 and addressed to Leibnitz about his notation that he used for the nth derivative of a function in his research publication, namely, what would happen if we take \(n=1/2\)? Leibniz responded that it would be an apparent paradox, but since then fruitful results were drawn. In the beginning, they did not get that much attention from mathematicians due to an abstract approach. But for the last three decades fractional calculus has shown remarkable development, and it changes from a pure mathematical formulation to different applied fields like bioengineering, physics, rheology, viscoelasticity, biophysics, and electrochemistry [25]. Especially, it has been proved that fractional calculus is a useful tool to deal with viscoelastic behavior [26]. The concept of fractional order calculus is used by various researchers in their work, and we refer to [27, 28, 29, 30, 31] for details. We note that there are some differences in the use of the involved operators. To successfully deal with the problem of a singular kernel, a new operator with exponential function for the fractional derivative was presented by Caputo and Fabrizio [32] in 2015. Using the concept of CF fractional derivative, Sheikh et al. [33] studied a generalized second grade fluid in a porous medium. Furthermore, Ali et al. [34] used timefractional derivative for the influence of magnetic field on the blood flow of Casson fluid. Ali et al. [34] analyzed the twophase blood flow of magnetic particles using CF fractional operators. Saqib et al. [35] studied a free convection flow of generalized Jeffrey fluid using CF fractional model. Although the existing fractional derivatives have been efficiently used in real world problems, there are still many things to be improved. For example, in the case of CF derivative [32], the operator is nonsingular, unlike in [24], but still it has a problem of nonlocality. Therefore, to fix this nonlocality problem, Atangana and Baleanu [29, 36] in 2016 used the generalized MittagLeffler function and proposed a new operator for a fractional derivative having nonlocal and nonsingular kernel. Now the Atangana–Baleanu definition will be more significant when discussing real world problems and will also have a great advantage while using Laplace transform method to solve physical problems with initial conditions. By using the idea of AB fractional derivative, Sheikh et al. [37] developed exact an expression for the velocity distribution of Casson fluid. Variation in the velocity profile was also displayed graphically for various parameters. Some other interesting work based on time fractional derivative can be found in [38, 39, 40, 41, 42, 43, 44].
Inspired by the above literature review, the objective of the present investigation is to analyze the electrically conducted mixed convection flow of generalized Jeffrey nanofluid in a rotating frame past over an infinite oscillating plate saturated in a porous medium along with thermal radiation and diffusionthermo. Tiwari and Das nanofluid model [45], along with Boussinesq approximation [46], is used for the development of the governing equations of the considered phenomenon. After that the governing equations are reduced to Atangana–Baleanu fractional model. Exact expressions for the velocity, heat, and mass distributions are obtained by using the Laplace transform method. To check the influence of pertinent parameters on the velocity profile, heat and mass distributions, exact expressions are plotted graphically. Variations in Sherwood and Nusselt numbers are also expressed in tabular form. It is worth noting that the rate of heat transfer is enhanced by 15.04% when the values of volume fraction of silver nanoparticles vary from 0.00 to 0.04.
2 Preliminaries
Definition 1
([47])
Definition 2
([29])
The Atangana–Baleanu fractional derivative operator is known to be helpful and is frequently used to discuss the real world phenomena. The AB time fractional derivative has huge beneficial applications when the Laplace transform technique is used to solve fractional differential equations.
2.1 Mathematical formulation of the problem
The thermophysical properties of the base fluid and nanosized particles
Properties  ρ  \(C_{p}\)  K  β  Pr 

Engine oil  863  2048  0.1404  0.00007  600 
Ag (silver)  10,500  235  429  0.0000189  – 
2.2 Atangana–Baleanu fractional model
2.3 Exact solutions
2.4 Nusselt number
2.5 Sherwood number
3 Results and discussion
This section deals with the interpretation of the obtained results. After transforming the classical model to Atangana–Baleanu time fractional model, exact expressions were obtained via the Laplace transform technique. In order to highlight the effects the relative parameters, namely, Dufour effect Df, Schmidt number Sc, thermal Grashof number Gr, mass Grashof number Gm, rotation parameter r, Hartman number Ha, permeability parameter K, material parameters of Jeffrey fluid λ and \(\lambda _{1}\) on the velocity, temperature, and concentration profile, plots were drawn using MATHCAD.
Variation in Nusselt number
α  φ  Rd  Df  t  Nu  % 

0.2  0.00  0.5  1.5  2  0.698  – 
0.2  0.01  0.5  1.5  2  0.731  4.72 
0.2  0.02  0.5  1.5  2  0.766  9.74 
0.2  0.03  0.5  1.5  2  0.791  13.32 
0.2  0.04  0.5  1.5  2  0.803  15.04 
Variations in Sherwood number
α  Sc  T  φ  Sherwood number  % 

0.2  5  2  0.00  0.154  – 
0.2  5  2  0.02  0.142  7.79 
0.2  5  2  0.03  0.137  11.03 
0.2  5  2  0.04  0.120  22.07 
4 Concluding remarks

Increasing the fractional parameter α decreases the velocity.

Velocity profile increases with the increasing values of Df, Gr and Gm.

For higher values of λ, r, Sc and φ, the viscosity of the engine oil increases, which cause a rise in the boiling point of the engine oil. This will intensely increase the heat carrying capacity and lubrication properties of the oil.

For MHD and permeability parameters, the viscosity of the engine oil decreases, as a result the motion of the fluid retards.

Temperature profile is enhanced when increasing the value of Df, which is due an increase in the thermal conductivity and decrease in the specific heat capacity of the engine oil.

The rate of heat transfer increases by 15.04% when the volume friction is 0.04.

By increasing of volume friction of nanoparticles, the rate of mass distribution increases by 22.07%.
Notes
Acknowledgements
The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number No. (RGP20193).
Availability of data and materials
Not applicable.
Authors’ contributions
FA and IK modelled the problem, SM solved the problem and plotted the graphs, NAS wrote the manuscript. KSN did the final revision of the manuscript. All authors read and approved the final manuscript.
Funding
The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number No. (RGP20193).
Competing interests
The authors declare no conflict of interest.
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