1. Introduction

The idea of fractional order calculus is as old as traditional order calculus. The pioneering systematic studies are devoted to Riemann-Liouville and Leibniz [1]. The subject is growing day by day and its applications have been utilized in different fields, for example, viscoelasticity, bioengineering, biophysics and mechatronics [2]. The applications of non-integer order calculus have also been encountered in different areas of science despite mathematics and physics drastically [3–5]. In fluid dynamics, the fractional order calculus has been broadly used to describe the viscoelastic behaviour of the material. Viscoelasticity of a material is defined it deforms evince both viscous and elastic behaviour via storage of mechanical energy and simultaneous behaviour. Mainardi [6] examined the connections among fractional calculus, wave motion and viscoelasticity. It is increasingly seen as an efficient tool through which useful generalization of physical concepts can be obtained. Hayat et al. [7] studied the periodic unidirectional flows of a viscoelastic fluid with the Maxwell model (fractional). Qi and Jin [8] analyzed the unsteady rotating flows of viscoelastic fluid with the fractional Maxwell model between coaxial cylinders. Many other researchers used the idea of fractional calculus and published quite number of research papers in some reputable journals [9–11].

Oxides, such as copper ð Þ CuO<sup>2</sup> and titanium oxides ð Þ TiO<sup>2</sup> , can be used as an additive to lubricants. The combustion of fossil fuels produces injurious gases (CO and NO) that cause air pollution and global warming. To save natural recourses and produce environment-friendly products, currently, nanomaterials are used to enhance the fuel efficiency of the oils [25].

Magnetite Molybdenum Disulphide Nanofluid of Grade Two: A Generalized Model with Caputo-Fabrizio Derivative

http://dx.doi.org/10.5772/intechopen.72863

259

Among the different types of nanomaterial, there is one called molybdenum disulphide nanomaterial MoS2, used very rarely in nanofluid studies. Although MoS<sup>2</sup> nanoparticles are not focused more, they have several interesting and useful applications. Applications of molybdenum disulphide can be seen in MoS2-based lubricants such as two-stroke engines, for example, motorcycle engines, automotive CV and universal joints, bicycle coaster brakes, bullets and ski waxes [26]. Moreover, the MoS<sup>2</sup> has a very high boiling point and many researchers have investigated it as a lubricant The first theoretical study on MoS2-based nanofluid was performed by Shafie et al. [27], where they studied the shape effect of MoS<sup>2</sup> nanoparticles of four different shapes (platelet, cylinder, brick and blade) in convective flow of fluid in a channel filled with

By keeping in mind the importance of MoS<sup>2</sup> nanoparticles, this chapter studies the joint analysis of heat and mass transfer in magnetite molybdenum disulphide viscoelastic nanofluid of grade two. The concept of fractional calculus has been used in formulating the generalized model of grade-two fluid. MoS<sup>2</sup> nanoparticles of spherical shapes have been used in engine oil chosen as base fluid. The problem is formulated in fractional form and Laplace transform together. CF derivatives have been used for finding the exact solution of the problem. Results are obtained in tabular and graphical forms and discussed for rheolog-

Let us consider heat and mass transfer analysis in magnetite molybdenum disulphide nanofluid of grade two with viscosity and elasticity effects. MoS<sup>2</sup> nanoparticles in powder form of spherical shape are dissolved in engine oil chosen as base fluid. MoS<sup>2</sup> nanofluid is taken over an infinite plate placed in xy-plane. The plate is chosen in vertical direction along x-axis, and y-axis is transverse to the plate. Electrically conducting fluid in the presence of uniform magnetic B<sup>0</sup> is considered which is taken normal to the flow direction. Magnetic Reynolds number is chosen very small so that induced magnetic field can be neglected. Before the time start, both the fluid and plate are stationary with ambient temperature T<sup>∞</sup> and ambient concentration C∞. At time

, both the plate and fluid starts to oscillate in its own direction with constant amplitude U

Under these assumptions, the problem is governed by the following system of differential

<sup>0</sup>u þ g ρβ<sup>T</sup> 

> ∂2 T

nfð Þþ T � T<sup>∞</sup> g ρβ<sup>C</sup>

<sup>∂</sup>y<sup>2</sup> , (2)

nfð Þ C � C<sup>∞</sup> , (1)

saturated porous medium.

ical parameters.

t=0+

equations:

ρnf ∂u <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>μ</sup>nf

2. Solution of the problem

and frequency ω. Schematic diagram is shown in Figure 1.

∂3 u <sup>∂</sup>t∂y<sup>2</sup> � <sup>σ</sup>nf <sup>B</sup><sup>2</sup>

> ρcp nf ∂T <sup>∂</sup><sup>t</sup> <sup>¼</sup> knf

∂2 u <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>α</sup><sup>1</sup>

Several versions of fractional derivatives are now available in the literature; however, the widely used derivatives are the Riemann-Liouville fractional derivatives and Caputo/fractional derivative [12, 13]. However, the researchers were facing quite number of difficulties in using them. For example, the Riemann-Liouville derivative of a constant is not zero and the Laplace transform of Riemann-Liouville derivative contains terms without physical significance. Though the Caputo fractional derivative has eliminated the short fall of Riemann-Liouville derivative, its kernal has singularity point. Ali et al. [14] reported the conjugate effect of heat and mass transfer on time fraction convective flow of Brinkman type fluid using the Caputo approach. Shahid et al. [15] investigated the approach of Caputo fractional derivatives to study the magnetohyrodynamic (MHD) flow past over an oscillating vertical plate along with heat and mass transfer. Recently, Caputo and Fabrizio (CF) have initiated a fractional derivative with no singular kernel [16]. However, Shah and Khan [17] analyzed that heat transfer analysis in a grade-two fluid over an oscillating vertical plate by using CF derivatives. Ali et al. [18] studied the application of CF derivative to MHD free convection flow of generalized Walter's-B fluid model. Recently, Sheikh et al. [19] applied CF derivatives to MHD flow of a regular second-grade fluid together with radiative heat transfer.

However, the idea of fractional calculus is very new in nanoscience, particularly in nanofluid also called smart fluid [20]. In this study, we have applied the fractional calculus idea more exactly, the idea of CF derivatives to a subclass of differential type fluid known as the secondgrade fluid with suspended nanoparticles in spherical shape of molybdenum disulphide (MoS2). Generally, the purpose of nanoparticles when dropped in regular fluid/base fluid/host fluid is to enhance the thermal conductivity of the host fluid. The inclusion of nanomaterial not only increases the thermal conductivity but also increases the base fluid viscosity (Wu et al. [21], Wang et al. [22], Garg et al. [23] and Lee et al. [24]). For this purpose, several types of nanomaterials, such as carbides, oxides and iron, and so on, are available in the market with their specific usage/characteristics and applications. For example, nanomaterial can be used as a nanolubricants, friction reductant, anti-wear agent and additive to tribological performance. Oxides, such as copper ð Þ CuO<sup>2</sup> and titanium oxides ð Þ TiO<sup>2</sup> , can be used as an additive to lubricants. The combustion of fossil fuels produces injurious gases (CO and NO) that cause air pollution and global warming. To save natural recourses and produce environment-friendly products, currently, nanomaterials are used to enhance the fuel efficiency of the oils [25].

Among the different types of nanomaterial, there is one called molybdenum disulphide nanomaterial MoS2, used very rarely in nanofluid studies. Although MoS<sup>2</sup> nanoparticles are not focused more, they have several interesting and useful applications. Applications of molybdenum disulphide can be seen in MoS2-based lubricants such as two-stroke engines, for example, motorcycle engines, automotive CV and universal joints, bicycle coaster brakes, bullets and ski waxes [26]. Moreover, the MoS<sup>2</sup> has a very high boiling point and many researchers have investigated it as a lubricant The first theoretical study on MoS2-based nanofluid was performed by Shafie et al. [27], where they studied the shape effect of MoS<sup>2</sup> nanoparticles of four different shapes (platelet, cylinder, brick and blade) in convective flow of fluid in a channel filled with saturated porous medium.

By keeping in mind the importance of MoS<sup>2</sup> nanoparticles, this chapter studies the joint analysis of heat and mass transfer in magnetite molybdenum disulphide viscoelastic nanofluid of grade two. The concept of fractional calculus has been used in formulating the generalized model of grade-two fluid. MoS<sup>2</sup> nanoparticles of spherical shapes have been used in engine oil chosen as base fluid. The problem is formulated in fractional form and Laplace transform together. CF derivatives have been used for finding the exact solution of the problem. Results are obtained in tabular and graphical forms and discussed for rheological parameters.
