2. Solution of the problem

1. Introduction

258 Microfluidics and Nanofluidics

research papers in some reputable journals [9–11].

of a regular second-grade fluid together with radiative heat transfer.

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

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

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. 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 t=0+ , both the plate and fluid starts to oscillate in its own direction with constant amplitude U and frequency ω. Schematic diagram is shown in Figure 1.

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

$$\rho\_{\eta\circ} \frac{\partial u}{\partial t} = \mu\_{\eta\circ} \frac{\partial^2 u}{\partial y^2} + \alpha\_1 \frac{\partial^3 u}{\partial t \partial y^2} - \sigma\_{\eta\circ} B\_0^2 u + g \left(\rho \beta\_T\right)\_{\eta\circ} (T - T\_\circ) + g \left(\rho \beta\_C\right)\_{\eta\circ} (\mathbb{C} - \mathbb{C}\_\simeq), \tag{1}$$

$$\left(\rho c\_p\right)\_{\eta\zeta} \frac{\partial T}{\partial t} = k\_{\eta\zeta} \frac{\partial^2 T}{\partial y^2} \,. \tag{2}$$

Figure 1. Schematic Diagram of the flow.

$$\frac{\partial \mathcal{C}}{\partial t} = D\_{\eta^f} \frac{\partial^2 \mathcal{C}}{\partial y^2} \,' \tag{3}$$

where ϕ describes the volume fraction of nanoparticles. The subscripts s and f stands for solid nanoparticles and base fluid, respectively. The numerical values of physical properties of

Engine oil 863 2048 0.1404 0.00007 MoS2 <sup>5</sup>:<sup>06</sup> � <sup>10</sup><sup>3</sup> <sup>397</sup>:<sup>21</sup> 904.4 <sup>2</sup>:<sup>8424</sup>

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

t, <sup>θ</sup> <sup>¼</sup> <sup>T</sup> � <sup>T</sup><sup>∞</sup> Tw � T<sup>∞</sup>

K�<sup>1</sup> Cp Jkg�<sup>1</sup>

, <sup>Φ</sup> <sup>¼</sup> <sup>C</sup> � <sup>C</sup><sup>∞</sup>

<sup>∂</sup>τ∂ξ<sup>2</sup> � <sup>M</sup>1<sup>v</sup> <sup>þ</sup> Grϕ2<sup>θ</sup> <sup>þ</sup> Gmϕ3Φ, (5)

3ð Þ σ � 1 ϕ ð Þ� <sup>σ</sup> <sup>þ</sup> <sup>2</sup> ð Þ <sup>σ</sup> � <sup>1</sup> ,

ρf

<sup>ρ</sup><sup>f</sup> <sup>ν</sup><sup>2</sup> is the non-dimensional second-grade parameter,

, a<sup>4</sup> <sup>¼</sup> Sc

<sup>1</sup> � <sup>ϕ</sup> ,

<sup>U</sup><sup>3</sup> ð Þ Tw � T<sup>∞</sup> is the

βCs βCf ,

, a<sup>1</sup> <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>þ</sup> <sup>ϕ</sup> <sup>ρ</sup><sup>s</sup>

Cw � C<sup>∞</sup>

<sup>∂</sup>ξ<sup>2</sup> , (6)

<sup>∂</sup>ξ<sup>2</sup> : (7)

(8)

:

<sup>K</sup>�<sup>1</sup> <sup>β</sup> � <sup>10</sup>�<sup>5</sup> <sup>K</sup>�<sup>1</sup>

261

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

nanoparticle and base fluid are mentioned in Table 1. Introducing the following dimensional less variables

Model <sup>ρ</sup> kgm�<sup>3</sup> Cp Jkg�<sup>1</sup>

Table 1. Numerical values of thermophysical properties.

<sup>U</sup> , <sup>ξ</sup> <sup>¼</sup> <sup>U</sup>

<sup>ν</sup> y, <sup>τ</sup> <sup>¼</sup> <sup>U</sup><sup>2</sup>

ν

∂3 v

Prϕ<sup>4</sup> λnf

> ∂Φ <sup>∂</sup><sup>τ</sup> <sup>¼</sup> <sup>1</sup> a4 ∂2 Φ

vð Þ¼ ξ; 0 0 θ ξð Þ¼ ; 0 0 Φð Þ¼ ξ; 0 0 vð Þ¼ 0; τ cos ωτ θð Þ¼ 0; τ 1 Φð Þ¼ 0; τ 1 vð Þ¼ ∞; τ 0 θð Þ¼ ∞; τ 0 Φð Þ¼ ∞; τ 0,

<sup>a</sup>1, M<sup>1</sup> <sup>¼</sup> <sup>M</sup>ϕ<sup>1</sup>

βTs βTf

 s ρcp f

where Re is the Reynolds number, <sup>β</sup> <sup>¼</sup> <sup>α</sup>1U<sup>2</sup>

a1

, <sup>λ</sup>nf <sup>¼</sup> knf

, ϕ<sup>1</sup> ¼ 1 þ

, <sup>ϕ</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>ρ</sup><sup>f</sup> <sup>þ</sup> ϕρ<sup>s</sup>

kf

<sup>ρ</sup><sup>f</sup> <sup>U</sup><sup>2</sup> shows the Hartmann number (magnetic parameter), Gr <sup>¼</sup> <sup>g</sup>νβTf

∂θ <sup>∂</sup><sup>τ</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup> θ

<sup>v</sup> <sup>¼</sup> <sup>u</sup>

∂v <sup>∂</sup><sup>τ</sup> <sup>¼</sup> <sup>1</sup> Re ∂2 v <sup>∂</sup>ξ<sup>2</sup> <sup>þ</sup> <sup>β</sup> a1

into Eqs. (1)–(4), we get

where

<sup>M</sup> <sup>¼</sup> <sup>σ</sup><sup>f</sup> <sup>B</sup><sup>2</sup> 0ν

Re <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>2</sup>:<sup>5</sup>

<sup>ϕ</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>ρ</sup><sup>f</sup> <sup>þ</sup> ϕρ<sup>s</sup>

<sup>ϕ</sup><sup>4</sup> <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>þ</sup> <sup>ϕ</sup> <sup>ρ</sup>cp

where ρnf , σnf , μnf β<sup>T</sup> nf , β<sup>C</sup> nf , knf , ρCp nf , Dnf are the density, electrical conductivity, viscosity, thermal expansion coefficient, coefficient of concentration, thermal conductivity, heat capacity and mass diffusivity of nanofluid. α<sup>1</sup> shows second two parameters, and g denotes acceleration due to gravity.

The appropriate initial and boundary conditions are

$$\begin{cases} u(y,0) = 0 & T(y,0) = T\_{\leadsto} & \mathsf{C}(y,0) = \mathsf{C}\_{\leadsto} \\ u(0,t) = \mathsf{L}\mathsf{H}(t)\cos\omega t & T(0,t) = T\_{w} & \mathsf{C}(0,t) = \mathsf{C}\_{w} \\ u(\csc\theta, t) = 0 & T(\csc\theta, t) = T\_{\leadsto} & \mathsf{C}(\csc\theta, t) = \mathsf{C}\_{\leadsto} \\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$$

For nanofluids, the expressions for ρnf , μnf , ρβnf , ρcp nf are given by:

$$
\mu\_{\eta'} = \frac{\mu\_f}{\left(1-\phi\right)^{2.5}}, \quad \rho\_{\eta'} = \left(1-\phi\right)\rho\_f + \phi\rho\_{s'} \quad \left(\rho\beta\_T\right)\_{\eta'} = \left(1-\phi\right)\left(\rho\beta\right)\_f + \phi\left(\rho\beta\right)\_{s'}
$$

$$
\begin{split}
\left(\rho\beta\_C\right)\_{\eta'} &= \left(1-\phi\right)\left(\rho\beta\right)\_f + \phi\left(\rho\beta\right)\_{s'}\left(\rho c\_p\right)\_{\eta'} = \left(1-\phi\right)\left(\rho c\_p\right)\_f \\ &+ \phi\left(\rho c\_p\right)\_{s'} \quad \sigma\_{\eta'} = \sigma\_f\left(1+\frac{3(\sigma-1)\phi}{(\sigma+2)-(\sigma-1)\phi}\right),
\end{split}
$$

$$
\sigma = \frac{\sigma\_s}{\sigma\_f}, \frac{k\_{\eta'}}{k\_f} = \frac{\left(k\_s+2k\_f\right)-2\phi\left(k\_f-k\_s\right)}{\left(k\_s+2k\_f\right)+\phi\left(k\_f-k\_s\right)}, \quad D\_{\eta'} = \left(1-\phi\right)D\_f.\tag{4a}
$$

Magnetite Molybdenum Disulphide Nanofluid of Grade Two: A Generalized Model with Caputo-Fabrizio Derivative http://dx.doi.org/10.5772/intechopen.72863 261


Table 1. Numerical values of thermophysical properties.

where ϕ describes the volume fraction of nanoparticles. The subscripts s and f stands for solid nanoparticles and base fluid, respectively. The numerical values of physical properties of nanoparticle and base fluid are mentioned in Table 1.

Introducing the following dimensional less variables

$$
\sigma = \frac{\mu}{\mathcal{U}}, \quad \xi = \frac{\mathcal{U}}{\nu} y, \quad \tau = \frac{\mathcal{U}^2}{\nu} t, \; \theta = \frac{T - T\_{\circ \circ}}{T\_w - T\_{\circ \circ}}, \quad \Phi = \frac{\mathcal{C} - \mathcal{C}\_{\circ \circ}}{\mathcal{C}\_w - \mathcal{C}\_{\circ \circ}}.
$$

into Eqs. (1)–(4), we get

$$\frac{\partial \upsilon}{\partial \tau} = \frac{1}{\text{Re}} \frac{\partial^2 \upsilon}{\partial \xi^2} + \frac{\beta}{a\_1} \frac{\partial^3 \upsilon}{\partial \tau \partial \xi^2} - M\_1 \upsilon + Gr \phi\_2 \theta + Gm \phi\_3 \Phi\_{\prime} \tag{5}$$

$$\frac{\text{Pr}\phi\_4}{\lambda\_{\eta f}}\frac{\partial\theta}{\partial\tau} = \frac{\partial^2\theta}{\partial\xi^2},\tag{6}$$

$$\frac{\partial \Phi}{\partial \tau} = \frac{1}{a\_4} \frac{\partial^2 \Phi}{\partial \xi^2} \,. \tag{7}$$

$$\begin{aligned} v(\xi,0)&=0 & \quad &\Theta(\xi,0)=0 & \quad &\Theta(\xi,0)=0\\ v(0,\tau)&=\cos\omega\tau & \quad &\Theta(0,\tau)=1 & \quad &\Theta(0,\tau)=1\\ v(\circ,\tau)&=0 & \quad &\Theta(\circ,\tau)=0 & \quad &\Theta(\circ,\tau)=0,\end{aligned} \tag{8}$$

where

∂C <sup>∂</sup><sup>t</sup> <sup>¼</sup> Dnf

nf , knf , ρCp 

<sup>1</sup> � <sup>ϕ</sup> <sup>2</sup>:<sup>5</sup> , <sup>ρ</sup>nf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>ρ</sup><sup>f</sup> <sup>þ</sup> ϕρs, ρβ<sup>T</sup>

<sup>¼</sup> ks <sup>þ</sup> <sup>2</sup>kf

ks þ 2kf

<sup>f</sup> þ ϕ ρβ s , ρcp 

, σnf ¼ σ<sup>f</sup> 1 þ

� <sup>2</sup><sup>ϕ</sup> kf � ks

<sup>þ</sup> <sup>ϕ</sup> kf � ks

nf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> ρβ

þϕ ρcp s

where ρnf , σnf , μnf β<sup>T</sup>

260 Microfluidics and Nanofluidics

<sup>μ</sup>nf <sup>¼</sup> <sup>μ</sup><sup>f</sup>

ρβ<sup>C</sup> 

> <sup>σ</sup> <sup>¼</sup> <sup>σ</sup><sup>s</sup> σf , knf kf

due to gravity.

Figure 1. Schematic Diagram of the flow.

nf , β<sup>C</sup> 

The appropriate initial and boundary conditions are

For nanofluids, the expressions for ρnf , μnf , ρβnf , ρcp

∂2 C

ity, thermal expansion coefficient, coefficient of concentration, thermal conductivity, heat capacity and mass diffusivity of nanofluid. α<sup>1</sup> shows second two parameters, and g denotes acceleration

uð Þ¼ ∞; t 0 Tð Þ¼ ∞; t T<sup>∞</sup> Cð Þ¼ ∞; t C<sup>∞</sup> as y ! ∞, t > 0:

nf are given by:

nf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> ρβ

nf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> <sup>ρ</sup>cp

3ð Þ σ � 1 ϕ ð Þ� σ þ 2 ð Þ σ � 1 ϕ 

 f

, Dnf <sup>¼</sup> <sup>1</sup> � <sup>ϕ</sup> Df : (4a)

,

<sup>f</sup> þ ϕ ρβ s ,

u yð Þ¼ ; 0 0 T yð Þ¼ ; 0 T<sup>∞</sup> C yð Þ¼ ; 0 C<sup>∞</sup> y > 0, uð Þ¼ 0; t UH tð Þ cos ωt Tð Þ¼ 0; t Tw Cð Þ¼ 0; t Cw t > 0,

<sup>∂</sup>y<sup>2</sup> , (3)

(4)

nf , Dnf are the density, electrical conductivity, viscos-

$$\begin{array}{ll} \text{Re} = \left(1 - \phi\right)^{2.5} a\_{1\prime} & M\_{1} = \frac{M \phi\_{1}}{a\_{1}}, & \phi\_{1} = 1 + \frac{3(\sigma - 1)\phi}{(\sigma + 2) - (\sigma - 1)}, \\\\ \phi\_{2} = \left(1 - \phi\right)\rho\_{f} + \phi\rho\_{s}\frac{\beta\_{\text{Ts}}}{\beta\_{\text{Tf}}}, & \phi\_{3} = \left(1 - \phi\right)\rho\_{f} + \phi\rho\_{s}\frac{\beta\_{\text{Cs}}}{\beta\_{\text{df}}}, \\\\ \phi\_{4} = \left(1 - \phi\right) + \phi\frac{\left(\rho c\_{p}\right)\_{s}}{\left(\rho c\_{p}\right)\_{f}}, & \lambda\_{\text{nf}} = \frac{k\_{\text{nf}}}{k\_{f}}, \quad a\_{1} = \left(1 - \phi\right) + \phi\frac{\rho\_{s}}{\rho\_{f}}, & a\_{4} = \frac{\text{Sc}}{\left(1 - \phi\right)}. \end{array}$$

where Re is the Reynolds number, <sup>β</sup> <sup>¼</sup> <sup>α</sup>1U<sup>2</sup> <sup>ρ</sup><sup>f</sup> <sup>ν</sup><sup>2</sup> is the non-dimensional second-grade parameter, <sup>M</sup> <sup>¼</sup> <sup>σ</sup><sup>f</sup> <sup>B</sup><sup>2</sup> 0ν <sup>ρ</sup><sup>f</sup> <sup>U</sup><sup>2</sup> shows the Hartmann number (magnetic parameter), Gr <sup>¼</sup> <sup>g</sup>νβTf <sup>U</sup><sup>3</sup> ð Þ Tw � T<sup>∞</sup> is the thermal Grashof number, Gm <sup>¼</sup> <sup>g</sup>νβCf <sup>U</sup><sup>3</sup> ð Þ Cw � <sup>C</sup><sup>∞</sup> is the mass Grashof number, Pr <sup>¼</sup> ð Þ <sup>μ</sup>cp <sup>f</sup> <sup>k</sup> is the Prandtl number and Sc <sup>¼</sup> <sup>ν</sup> Df is the Schmidt number.
