**5. Analysis of results**

The dimensionless differential equations of non-coaxial rotating nanofluid flow with associated boundary and initial conditions are analytically solved using the method of Laplace transform to obtain the closed form solutions of heat transfer. Further analysis for the role of dimensionless time *t*, Grashof number *Gr*, volume fraction of nanoparticles *ϕ*, porosity parameter *K*, magnetic field parameter *M* and amplitude of disk *U* on velocity and temperature distributions as well as Nusselt number and skin friction are presented in figures and tables. The profiles are plotted with the physical value of parameters as Pr ¼ 6*:*2, *Gr* ¼ 0*:*5, *M* ¼ 0*:*2,*K* ¼ 2*:*0, *ϕ* ¼ 0*:*02, *U* ¼ 2*:*0 and *t* ¼ 0*:*2. The values are same unless for the investigated parameter of the profile. Since the rotating nanofluid is part of the problem, the results are discussed by presenting the graph of velocity profile in real and imaginary parts, specifically describes the primary *f* and secondary *g* velocities. The velocity profiles are demonstrated in **Figures 2**–**7** and the temperature profiles are illustrated in **Figures 8** and **9**. From these profiles, it is found that all the obtained results satisfy both boundary and initial conditions. SWCNTs and MWCNTs have an identical nature of fluid flow and heat transfer.

**Figure 2** depicts the plotting of *f* and *g* profiles with varying *t* values. Overall, the velocity of both SWCNTs and MWCNTs rises over time. As *t* increases, the buoyancy force becomes more effective and functions as an external source of energy to the flow, causing the velocity of fluid to increase. **Figure 3** illustrates the variation of *f* and *g* profiles for SWCNTs and MWCNTs cases under the effect of *Gr*. It is essential to note that *Gr* is an approximation of the buoyancy force to the viscous force exerting on the flow. Hence, an increase of *Gr* suggests to the domination of

**Figure 2.** *Profile of f and g for varied values of t:*

**Figure 3.** *Profile of f and g for varied values of Gr:*

**Figure 4.** *Profile of f and g for varied values of M.*

**Figure 5.** *Profile of f and g for varied values of K.*

**Figure 6.** *Profile of f and g for varied values of ϕ.*

**Figure 7.** *Profile of f and g for varied values of U.*

*Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon… DOI: http://dx.doi.org/10.5772/intechopen.100623*

**Figure 8.** *Profile of T z*ð Þ , *t for varied values of t.*

**Figure 9.** *Profile of T z*ð Þ , *t for varied values of ϕ.*

buoyancy force and reduces the viscosity of fluid. Thus, growing *Gr* leads to an augment of fluid velocity.

On the other hand, **Figure 4** discloses the nature of fluid flow in response to *M*. For both SWCNTs and MWCNTs cases, the figure suggests that amplifying *M* decreases *f* and *g* profiles. This impact is owing to the fact that a greater *M* value increases the frictional forces acting on the fluid, commonly known as the Lorentz force. Consequently, the fluid encounters substantial resistance along the flow and its velocity decreases. Next, the contribution of *K* in SWCNTs and MWCNTs nanofluids for both *f* and *g* profiles are displayed in **Figure 5**. It suggests that *K* value increases linearly with the velocities for both SWCNTs and MWCNTs. Noting that porosity is also greatly affected by the permeability of a medium, where it determines the ability of a medium to enable the fluid to flow through it. Then, the increasing values of *K* cause the medium to be more permeable and the fluid can easily pass through the medium. Therefore, it increases both *f* and *g* profiles.

**Figure 6** reveals the consequences of *ϕ* on *f* and *g* profiles in the cases of SWCNTs and MWCNTs. It shows that increasing *ϕ* values result in the increment of *f* profiles and fluctuating trend of *g* profiles. This suggests significant advantages of non-coaxial rotation in CNTs, especially in industrial and medical applications. In line with a general finding, an analysis proceeding in cancer treatment has reported that the CNTs with higher velocity have been used to reach the tumor's site. Besides, referring to **Figure 7**, it is noticed that ascending *U* also has a positive impact on velocity profiles for both CNTs suspensions, where the velocity ascends linearly with the values of *U*. As *U* increases, this proposes to the creation of external sources, which are used to enhance the thrust force acting in the fluid flow. Thus, the velocity fluid elevates with increasing *U*.

Furthermore, the temperature profiles *T z*ð Þ , *t* under the impacts of *t* and *ϕ* are displayed graphically in **Figures 8** and **9**. It reveals that increment of *t* and *ϕ* contributes to a rise in nanofluid temperature for both types of CNTs case and followed by the magnification of thermal boundary layer. Physically, the addition of sufficient *ϕ* of CNTs can improve nanofluid's thermal conductivity. The more CNTs being inserted, the higher the thermal conductivity, which unsurprisingly improves the ability of fluid to conduct heat. Therefore, a growth of temperature profile is exhibited for increasing *ϕ*. The comparison of physical behavior for SWCNTs and MWCNTs are clearer when referring to the zooming box of each graph. Overall, **Figures 2**–**7** reveal that the velocity profile of MWCNTs case is more significant compared to the velocity of SWCNTs. This behavior is agreed to the thermophysical features in **Table 1**, where MWCNTs have low density, which also being a key factor for the increase of velocity profiles. Meanwhile, from **Figures 8** and **9**, SWCNTs have provided a prominent effect on temperature profiles as it is affected by a high thermal conductivity property.


*The significance of bold emphasis used in Table 2 is for the comparison of the effects for varied values of the particular parameters. For each parameter, the changes of skin friction values are compared among the bold values of parameters.*

#### **Table 2.**

*Values of primary τ<sup>p</sup> and secondary τ<sup>s</sup> skin friction for SWCNTs and MWCNTs.*


*The significance of bold emphasis used in Table 3 is for the comparison of the effects for varied values of the particular parameters. For each parameter, the changes of Nusselt number values are compared among the bold values of parameters.*

#### **Table 3.**

*Values of Nusselt number Nu for SWCNTs and MWCNTs.*

*Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon… DOI: http://dx.doi.org/10.5772/intechopen.100623*

**Tables 2** and **3** show the results of skin friction (*τ<sup>p</sup>* and *τs*) and Nusselt number *Nu* for various parameters on both cases SWCNTs and MWCNTs. According to **Table 2**, it shows that both *τ<sup>p</sup>* and *τ<sup>s</sup>* of SWCNTs and MWCNTs rise when the strength of *M* higher. These effects cause the surface to produce high friction drag due to the maximization of wall shear stress. On the contrary, as *Gr*,*K* and *t* increase, both suspension of SWCNTs and MWCNTs report a diminution in *τ<sup>p</sup>* and *τs*. This shows that augmentation of *Gr*, *K* and *t* have reduced the friction between fluid and surfaces which lead the velocity to increase. Meanwhile, as *ϕ* and *U* increase, both suspension of SWCNTs and MWCNTs report a growth of *τ<sup>p</sup>* and a diminution in *τs*. From **Table 3**, it shows that *Nu* for both CNTs cases decrease as the values of *t* increase. However, when involving high *ϕ*, both SWCNTs and MWCNTs have large *Nu* which also implies to have a great of heat transfer rate. This effect is also directly affected by the reduction of nanofluid heat capacitance as *ϕ* increases. Overall, for **Table 3**, it is found that SWCNTs case have high value of *Nu* compared to MWCNTs, due to its reduction of heat capacitance. This effect also signifies for a better heat transfer process that can be used in several engineering and industrial system.

#### **Figure 10.**

*Comparison of f profiles from present results in Eq. (18) with the published work by Mohamad et al. [40] in Eq. (53).*

#### **Figure 11.**

*Comparison of g profiles from present results in Eq. (18) with the published work by Mohamad et al. [40] in Eq. (53).*


#### **Table 4.**

*Comparison of exact and numerical solution of f profiles for SWCNTs and MWCNTs with t* ¼ 0*:*2, *Gr* ¼ 0*:*5, *M* ¼ 0*:*2, *K* ¼ 2, *ϕ* ¼ 0*:*02, *U* ¼ 2, *Pr* ¼ 6*:*2*.*


#### **Table 5.**

*Comparison of exact and numerical solution of g profiles for SWCNTs and MWCNTs with t* ¼ 0*:*2, *Gr* ¼ 0*:*5, *M* ¼ 0*:*2, *K* ¼ 2, *ϕ* ¼ 0*:*02, *U* ¼ 2, *Pr* ¼ 6*:*2*.*

The accuracy of the obtained solution is verified by comparing solution in Eq. (18) with the solution obtained by Mohamad et al. [40] in Eq. (53). The comparison is conducted by letting magnetic parameter and nanoparticle volume fraction *M* ¼ *ϕ* ¼ 0, and porosity parameter *K* ! ∞ in the present solution for both types of CNTs and letting phase angle *ω* ¼ 0 and amplitude of disk oscillation *U* ¼ 2 in the published work. This comparison shows that *f* and *g* profiles for both present and previous works are identical to each other as clearly presented in **Figures 10** and **11**, which thus proves that the accuracy of obtained solution is verified. Meanwhile, another verification is also carried out to verify the validity of present solution by comparing the values of velocity profiles from the present work with the numerical values solved by numerical Gaver-Stehfest algorithm [53, 54]. **Tables 4** and **5** observe that the results of *f* and *g* profiles from the exact solution in Eq. (18) and the results from numerical solution are in excellent agreement.

#### **6. Summary with conclusion**

The unsteady non-coaxial rotation of water-CNTs nanofluid flow in a porous medium with MHD effect is analytically solved for the exact solutions by applying the Laplace transform method. The temperature and velocity profiles with various values of parameter for the immersion of SWCNTs and MWCNTs are plotted graphically and analyzed for their effects. From the discussion, significant findings emerge:

*Analysis of Heat Transfer in Non-Coaxial Rotation of Newtonian Carbon… DOI: http://dx.doi.org/10.5772/intechopen.100623*

