4. Graphical discussion

A fractional model for the outflow of the second-grade fluid with nanoparticles over an isothermal vertical plate is studied. The coupled partial differential equations with Caputo-Fabrizio time-fractional derivatives are solved analytically via Laplace transform method. Furthermore, the influence of different embedded parameters such as α, ϕ, β, M, t Gr, Gm and Sc is shown graphically.

Figures 2–7 depict the effect of α on vð Þ ξ; τ for two different values of time. It is clear from the figures that for smaller value of τ, when (τ = 0.2) fractional velocity is larger than classical velocity and for larger value of τ, when (τ =2) fractional velocity is less than classical velocity. Clearly, increasing values of α decrease vð Þ ξ; τ .

Figure 2 represents the influence of β on both the velocity and microrotation profiles. A decreasing behaviour is observed for increasing values of β in both cases. In this figure, the comparison of second-grade fluid velocity with Newtonian fluid velocity is plotted. It is

obvious that the boundary layer thickness of second-grade fluid velocity is greater as compared to boundary layer thickness of Newtonian fluid. More clearly, the velocity of second-

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Figure 4. Velocity profile for different values of Gr when M ¼ 1, ϕ ¼ 0:02, Gm ¼ 2, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

Figure 3 shows the influence of ϕ on the flow. It was found that the velocity of fluid decreases with the increase in ϕ due to the increase in viscosity. Because by increasing volume fraction,

Figures 4 and 5 show the influence of thermal Grashof number Gr and mass Grashof number Gm on velocity and microrotation. Increasing values of both of these parameters are responsible for the rise in buoyancy forces and reducing viscous forces, which result in an increase in

Figure 6 depicts the MHD effect on velocity. In this type of flows, magnetic force results in achieving steady state much faster than the non-MHD flows. Moreover, increasing values of M enhances the Lorentz forces, as a result decelerates the fluid velocity. Figure 7 illustrates variations in velocity for different values of Schmidt number, Sc. It shows that velocity decreases

the fluid becomes more viscous, which leads to a decrease in the fluid velocity.

grade fluid is smaller than Newtonian fluid.

fluid velocity and magnitude of microrotation.

Figure 3. Velocity profile for different values of ϕ when M ¼ 1, Gr ¼ Gm ¼ 2, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

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4. Graphical discussion

266 Microfluidics and Nanofluidics

Clearly, increasing values of α decrease vð Þ ξ; τ .

graphically.

A fractional model for the outflow of the second-grade fluid with nanoparticles over an isothermal vertical plate is studied. The coupled partial differential equations with Caputo-Fabrizio time-fractional derivatives are solved analytically via Laplace transform method. Furthermore, the influence of different embedded parameters such as α, ϕ, β, M, t Gr, Gm and Sc is shown

Figures 2–7 depict the effect of α on vð Þ ξ; τ for two different values of time. It is clear from the figures that for smaller value of τ, when (τ = 0.2) fractional velocity is larger than classical velocity and for larger value of τ, when (τ =2) fractional velocity is less than classical velocity.

Figure 2 represents the influence of β on both the velocity and microrotation profiles. A decreasing behaviour is observed for increasing values of β in both cases. In this figure, the comparison of second-grade fluid velocity with Newtonian fluid velocity is plotted. It is

Figure 3. Velocity profile for different values of ϕ when M ¼ 1, Gr ¼ Gm ¼ 2, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

Figure 4. Velocity profile for different values of Gr when M ¼ 1, ϕ ¼ 0:02, Gm ¼ 2, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

obvious that the boundary layer thickness of second-grade fluid velocity is greater as compared to boundary layer thickness of Newtonian fluid. More clearly, the velocity of secondgrade fluid is smaller than Newtonian fluid.

Figure 3 shows the influence of ϕ on the flow. It was found that the velocity of fluid decreases with the increase in ϕ due to the increase in viscosity. Because by increasing volume fraction, the fluid becomes more viscous, which leads to a decrease in the fluid velocity.

Figures 4 and 5 show the influence of thermal Grashof number Gr and mass Grashof number Gm on velocity and microrotation. Increasing values of both of these parameters are responsible for the rise in buoyancy forces and reducing viscous forces, which result in an increase in fluid velocity and magnitude of microrotation.

Figure 6 depicts the MHD effect on velocity. In this type of flows, magnetic force results in achieving steady state much faster than the non-MHD flows. Moreover, increasing values of M enhances the Lorentz forces, as a result decelerates the fluid velocity. Figure 7 illustrates variations in velocity for different values of Schmidt number, Sc. It shows that velocity decreases

Figure 5. Velocity profile for different values of Gm when M ¼ 1, Gr ¼ 2, ϕ ¼ 0:02, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

when Sc value increases. The effect of Schmidt number on velocity is identical to that of the magnetic parameter. The influence of phase angle ωτ on the velocity profile is shown in Figure 8. The velocity is showing fluctuating behaviour.

that skin friction increases when there is an increase in α, Gr, Gm, M, Sc, ωτ and τ:but it is noticed that for increasing value of β and ϕ, skin friction decreases. It is due to the fact that when ϕ increases, it gives rise to lubricancy of the oil. Table 3 represents the effect of α, ϕ and τ on Nusselt number. As values of α, ϕ and τ increase, Nusselt number decreases. From Table 4, it is clear that when Sc increases, Sherwood number increases, and an increase

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Figure 6. Velocity profile for different values of M when ϕ ¼ 0:02, Gr ¼ Gm ¼ 2, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

Unsteady MHD flow of generalized second-grade fluid along with nanoparticles has been analyzed. The exact solution has been obtained for velocity, temperature and concentration profile via the Laplace transform technique. The effects of various physical parameters are

studied in various plots and tables with the following conclusions:

in τ decreases Sherwood number.

5. Conclusion remarks

In order to show the effect of α, τ and ϕ on the temperature profile in Figure 9, it is found that temperature increasing with increasing value of ϕ: Figure 10 shows the effect of α and τ on temperature profile. This figure shows the effect of α on the temperature profile for two different values of τ: For smaller value of τ τð Þ ¼ 0:2 , classical temperature is less than fractional temperature, and for larger value, when τ ¼ 2, then the graph shows opposite behaviour. Figure 11 shows the comparison of present solution with published result of Sheikh et al. [19]. It is noted that in the absence of porosity and radiation, the present result is similar to those obtained in [19. See Figure 9], which shows the validity of our obtained results.

Variations in skin friction, Nusselt number and Sherwood number are shown in Tables 2–4. The effect of β, α, Gr, Gm, M, Sc,ϕ, ωτ and τ on the skin friction is studied in Table 2. It is found

Magnetite Molybdenum Disulphide Nanofluid of Grade Two: A Generalized Model with Caputo-Fabrizio Derivative http://dx.doi.org/10.5772/intechopen.72863 269

Figure 6. Velocity profile for different values of M when ϕ ¼ 0:02, Gr ¼ Gm ¼ 2, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

that skin friction increases when there is an increase in α, Gr, Gm, M, Sc, ωτ and τ:but it is noticed that for increasing value of β and ϕ, skin friction decreases. It is due to the fact that when ϕ increases, it gives rise to lubricancy of the oil. Table 3 represents the effect of α, ϕ and τ on Nusselt number. As values of α, ϕ and τ increase, Nusselt number decreases. From Table 4, it is clear that when Sc increases, Sherwood number increases, and an increase in τ decreases Sherwood number.

#### 5. Conclusion remarks

when Sc value increases. The effect of Schmidt number on velocity is identical to that of the magnetic parameter. The influence of phase angle ωτ on the velocity profile is shown in Figure 8.

Figure 5. Velocity profile for different values of Gm when M ¼ 1, Gr ¼ 2, ϕ ¼ 0:02, Pr ¼ 5, Sc ¼ 5 and β ¼ 0:3.

In order to show the effect of α, τ and ϕ on the temperature profile in Figure 9, it is found that temperature increasing with increasing value of ϕ: Figure 10 shows the effect of α and τ on temperature profile. This figure shows the effect of α on the temperature profile for two different values of τ: For smaller value of τ τð Þ ¼ 0:2 , classical temperature is less than fractional temperature, and for larger value, when τ ¼ 2, then the graph shows opposite behaviour. Figure 11 shows the comparison of present solution with published result of Sheikh et al. [19]. It is noted that in the absence of porosity and radiation, the present result is similar to those obtained in [19. See Figure 9], which shows the validity of our obtained

Variations in skin friction, Nusselt number and Sherwood number are shown in Tables 2–4. The effect of β, α, Gr, Gm, M, Sc,ϕ, ωτ and τ on the skin friction is studied in Table 2. It is found

The velocity is showing fluctuating behaviour.

results.

268 Microfluidics and Nanofluidics

Unsteady MHD flow of generalized second-grade fluid along with nanoparticles has been analyzed. The exact solution has been obtained for velocity, temperature and concentration profile via the Laplace transform technique. The effects of various physical parameters are studied in various plots and tables with the following conclusions:

Figure 7. Velocity profile for different values of Sc when M ¼ 1, Gr ¼ Gm ¼ 2, Pr ¼ 5, ϕ ¼ 0:02 and β ¼ 0:3.

Figure 9. Temperature profile for different values of ϕ when Pr ¼ 5.

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Figure 10. Temperature profile for different values of time parameter.

Figure 8. Velocity profile for different values of ω.

Magnetite Molybdenum Disulphide Nanofluid of Grade Two: A Generalized Model with Caputo-Fabrizio Derivative http://dx.doi.org/10.5772/intechopen.72863 271

Figure 9. Temperature profile for different values of ϕ when Pr ¼ 5.

Figure 7. Velocity profile for different values of Sc when M ¼ 1, Gr ¼ Gm ¼ 2, Pr ¼ 5, ϕ ¼ 0:02 and β ¼ 0:3.

Figure 8. Velocity profile for different values of ω.

270 Microfluidics and Nanofluidics

Figure 10. Temperature profile for different values of time parameter.

• With increase in volume friction ϕ of nanofluid, lubricancy of the fluid increases.

• In limited cases, the obtained solutions reduced to the solution of Sheikh et al. [19].

1 Computational Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City,

2 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City,

3 Department of Mathematics, City University of Science and Information Technology,

4 Basic Engineering Sciences Department, College of Engineering Majmaah University,

[1] Nonnenmacher TF, Metzler R. On the Riemann-Liouville fractional calculus and some

[2] Sabatier J, Agrawal OP, Machado JT. Advances in fractional calculus (Vol. 4, no. 9).

• The velocity profile shows different behaviour for fractional parameter for different values

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

, Nadeem Ahmad Sheikh<sup>3</sup>

, Syed Aftab Alam Jan<sup>3</sup>

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273

• An increase in second-grade parameter β leads to a decrease in fluid velocity.

, Ilyas Khan<sup>4</sup>

of time.

Author details

Vietnam

Vietnam

Farhad Ali1,2,3\*, Madeha Gohar3

\*Address all correspondence to: farhad.ali@tdt.edu.vn

Table 4. Effect of various parameters on the Sherwood number.

recent applications. Fractals. 1995;3(03):557-566

Peshawar, Khyber Pakhtunkhwa, Pakistan

Dordrecht: Springer; 2007

and Muhammad Saqib3

Majmaah Saudi Arabia

References

Figure 11. Comparison of this study with Sheikh et al. [19], when <sup>1</sup> <sup>k</sup> ¼ R ¼ 0.


Table 2. Effect of various parameters on the skin friction.


Table 3. Effect of various parameters on the Nusselt number.

Magnetite Molybdenum Disulphide Nanofluid of Grade Two: A Generalized Model with Caputo-Fabrizio Derivative http://dx.doi.org/10.5772/intechopen.72863 273


Table 4. Effect of various parameters on the Sherwood number.

