*4.1. Velocity Distribution*

Figure 2 depicts the behavior of *d f*(*η*) *<sup>d</sup><sup>η</sup>* by varying the magnetic parameter *M* and thin film thickness parameter *α* of both CNTs (SWCNTs and MWCNTs) nanofluids. The impacts of these quantities on *d f*(*η*) *<sup>d</sup><sup>η</sup>* are very clear during the flow of both CNTs nanofluids. It can be noted that the larger magnitude of *M* reduced the fluid motion in both cases (SWCNTs and MWCNTs). Physically, such a situation arises as a result of a constantly applied magnetic field *B*<sup>0</sup> that can be induced current in inductive nanoliquid. It creates resistant forces called Lorentz forces, which reduce the liquid velocity. Finally, it is clear that *B*<sup>0</sup> is used to govern the boundary layer separation. Comparatively, a rapid fall in the velocity field is perceived in the case of SWCNTs as related to the MCWNTs. In Figure 2, the effect of *α* (thin film nanofluid parameter) is depicted for both sorts of CNTs nanofluids. It can be observed that by increasing the value of *α*, the fluid motion decelerates, because in this case, the mass of the fluid is enhanced. Actually, the tiny size of the film accelerates the velocity and less energy is required for fluid motion, for example, the flow in the pipe is much easier and faster than the flow in sea water. Moreover, in the case of MWCNTs, the velocity field is dominant when compared to SWCNTs in the present study.

**Figure 2.** *d f*(*η*) *<sup>d</sup><sup>η</sup>* distribution for varying *M* and *α*.

Figure 3 presents the velocity distribution *d f*(*η*) *<sup>d</sup><sup>η</sup>* for several values of *Gr* using SWCNTs- and MWCNTs-based nanofluid. The velocity *d f*(*η*) *<sup>d</sup><sup>η</sup>* elevates for both CNTs by maximizing the value of *Gr*. Similarly, the velocity field *d f*(*η*) *<sup>d</sup><sup>η</sup>* shows the slowing behavior for both CNTs (SWCNTs and MWCNTs), reducing the value of *Gr*. Actually, the ratio of the thermal buoyancy force in the direction of viscous force is termed the Grashoff number *Gr* Therefore, the basic reason for this is that in the absenteeism of buoyancy force, there is no motion of fluid. The present outline indicates that motion of liquid is occurring due to the buoyancy force and the liquid is stationary in the absence of this force. In addition, it is clear from the figure that SWCNTs are more dominant than MWCNTs.

Figure 4 elucidates the behavior of Casson parameter *β* and nanoparticle volume friction *ϕ* on *d f*(*η*) *<sup>d</sup><sup>η</sup>* for both SWCNTs and MWCNTs nanofluids. For the increasing values of *β*, the velocity distribution *d f*(*η*) *<sup>d</sup><sup>η</sup>* in the boundary layer is shown to be declining. It can be noted that accelerating the value of *β* implies condensing the yield stress of Casson liquid and therefore successfully assisting the motion of boundary film flow adjacent to the stretching surface of the cylinder. Furthermore, it is found that the Casson fluid is close to Newtonian fluid for the large value of *β* → ∞. Similarly, Figure 4 displays the impact of *ϕ* on the flow of nanoliquid for both nanoparticles (SWCNTs and MWCNTs). Obviously, it is perceived that the velocity distribution *d f*(*η*) *<sup>d</sup><sup>η</sup>* improves as the magnitude of *ϕ* increases

for both SWCNTs and MWCNTs nanofluids. Substantially, this occurs by inserting more particles *ϕ* in the thin fluid of the nanoliquid and increasing the strength of heat carriage and cohesive among the nanoliquid atoms, so that they become frail to halt the faster fluid flow since the heat transport in thin materials is faster than in thick materials. In addition, it is clear from the figure that the flow of SWCNTs is more dominant than MWCNTs. The velocity profiles *d f*(*η*) *<sup>d</sup><sup>η</sup>* of different magnitudes of the Reynolds number Re for both types of CNTs nanofluids are presented in Figure 5. Basically, Re is the ratio of inertial force toward the viscous force. It can be noted that the velocity profile *d f*(*η*) *dη* reduces as Re increases, so the velocity tends to be zero at a certain large space from the cylinder surface. Generally, the greater value of Re controlled the inertial force, which reduced the viscous force. Hence, for the larger magnitudes of the Reynolds number Re, the velocity of nanofluids reduces and the flow of fluids declines slowly to the ambient condition. The inertial forces are more influential forces and they do not permit the liquid atoms to flow. Strong viscous forces have a strong resistance to the flow of the liquids. Boundary layer flow of fluid motion decreases with strong inertial forces.

**Figure 3.** *d f*(*η*) *<sup>d</sup><sup>η</sup>* distribution for varying *Gr*.

**Figure 4.** *d f*(*η*) *<sup>d</sup><sup>η</sup>* distribution for varying *ϕ* and *β*.

**Figure 5.** *d f*(*η*) *<sup>d</sup><sup>η</sup>* distribution for varying Re.

#### *4.2. Thermal Distribution*

Similarly, the next set of Figure 6, Figure 7 displays the impact of flow quantities on the thermal profile Θ(*η*). Variation in the temperature profile Θ(*η*) for both types of CNTs nanofluids with magnetic parameter *M* and thickness variable *α* is shown in Figure 6. An intensification in the thermal field Θ(*η*) is perceived with a large value of *M* for both nanofluids because the high estimation of *M* produces the Lorentz forces, which increase the fraction force between the fluid molecules for SWCNTs and MWCNTs. This force favors and supports the temperature of fluids. Since the *M* magnetic field is executed vertically, with the growing magnitude of *M* magnetic field effect, the fluid is controlled and restricted. Additionally, the greater value of *α* decreases Θ(*η*) as the thin liquid coating is heated up quicker than the thick liquid coating. As a result, the thermal field Θ(*η*) cools down at high values of *α*. The reason for this is that with the thickness of the fluid film, the mass of the fluid increases, which consumes the amount of temperature. Heat enters fluid, and as a result, the environment is cooled down. Thick film fluid needs more heat compared to thin film fluid. Figure 7 demonstrates the performance of the Reynolds number Re and Pr on the thermal filed Θ(*η*) for SWCNTs and MWCNTs. The same behavior is noted in the variation of Re and Pr for both CNTs. It is seen that a higher measure of Re denigrates Θ(*η*), explained by the basic fact that a greater magnitude of Re results in extra inertial forces arising, which tightly bonds the particles of flow nanoliquids, and greater heat is enforced to break the contacts amongst the liquid particles. Additionally, the behavior of the Prandtl number Pr on the thermal field Θ(*η*) is presented in Figure 7. From the figure, it is shown that Θ(*η*) displays a falling act for a greater magnitude of Pr for both types of CNTs nanoparticles. Generally, a greater magnitude of Pr increases the thickness of the boundary layer, which boosts the cooling efficiency of the nanomaterial. This is because Pr is the ratio of motion diffusivity to thermal diffusivity. Those fluids which have the lowest Prandtl number Pr have good thermal conductivities; therefore, thick boundary layer structures are maintained for diffusing heat.

**Figure 7.** Θ(*η*) distribution for varying Re and Pr.

#### *4.3. Pressure Distribution*

Finally, in the set of Figures 8 and 9, we portray the variation in the key element of spray phenomena pressure distribution *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) in terms of different variables for SWCNTs and MWCNTs nanofluids. The effect of *ϕ* (volume fraction) and thickness parameter *α* on pressure distribution *p*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) is sketched in Figure 8. The higher values of *<sup>ϕ</sup>* lead to stronger pressure *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*); as a result, fraction forces are reduced and the concentration of nanoparticles is enhanced for both SWCNTs and MWCNTS nanofluids. Due to a higher concentration, the fluid becomes dense and the collision of molecules increases, exerting pressure on the wall of the cylinder. It has been noticed that the high-pressure phenomena have a vital role in blood flow, chemical reactions, and cooking easily. Furthermore, it can be observed that the pressure distribution *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) is enhanced for greater values of *α*. The large size of the film exerts a strong pressure *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) and high power is applied to diminish the stress of the thick film. The combined relationship of the film thickness and pressure created a massive force, which is compulsory for the body to move on a fluid surface. Figure 9 exhibits the effect of *M* and Re on *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) for SWCNTs and MWCNTs. From Figure 9, it can be obviously seen that less pressure is produced by a large magnitude of *M*. It can be seen that the pressure distribution *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) is weak due to Lorentz forces, which decrease the movement of fluid, and extra pressure is required. The magnetic field *M* is applied perpendicular to the flow of nanofluids. Therefore, the Lorentz forces capture the liquid in the boundary layer. To compete with the Lorentz forces due to the strong magnetic

field, the pressure must be high in order to cause motion of the fluid. Moreover, large quantities of the Reynolds number Re drop the pressure distribution *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) and a strong inertial effect is produced. Due to this inertial force, the fluid particles are packed closely and inflexibly and more pressure is imposed to overcome these forces.

**Figure 8.** *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) distribution for varying *ϕ* and *α*.

**Figure 9.** *<sup>p</sup>*−*pb <sup>μ</sup><sup>c</sup> <sup>f</sup>* (*η*) distribution for varying *M* and Re.

The certain mathematical values of CNTs (SWCNTs and MWCNTs) and human blood, depend on various thermo-physical characteristics, such as density *ρ*, thermal conductivity *k <sup>f</sup>* , and specific heat *cp*, as presented in Table 1. Also, Table 2 demonstrates the convergence analysis of the series solution for *f* (1) (velocity field) and Θ (1) (thermal field).

**Table 1.** Various mathematical values of thermophysical characteristics of CNTs of three base liquids [16].



**Table 2.** The convergence of the Homotopic results for different orders of estimation.

Table 3 demonstrates the statistical data of *f* (1) (surface drag force) HAM approximation at the 20th order for several values of related physical quantities, such as *M* = 0.3, Pr = 24, *Ec* = 1.5, *ϕ* = 0.01, *Gr* = 0.2, *α* = 1.4,Re = 0.3 for SWCNTs/MWCNTS nanofluids. In Table 3, it is shown that the magnitude of *f* (1) (surface drag force) intensifies for greater values of *ϕ*, *M*,Re for both SWCNTs and MWCNTs. The growing thickness of the nanoparticles enhances the resistance forces, which improve skin friction. The *M* also governs an opposing force named the Lorentz force and a greater magnitude of *M* upsurges the skin friction. This drop-in influence is fast using the SWCNTs as compared to the MWCNTs.

**Table 3.** The numerical values of the skin friction coefficient (*f* (1)).


Similarly, Table 4 is organized to explain *Nu* at the 20th order HAM estimate for different values of relevant model variables for both SWCNTs and MWCNTs nanoliquids. From Table 4, it can be clearly observed that the value of rate of heat transport accelerates for a high magnitude of both *ϕ*, Pr and declines for a higher value of the *Ec*. The *Ec* is related to the dissipation term and a larger magnitude of *Ec* enhances the thermal field. Therefore, the opposite result for the higher magnitude of the *Ec* verses *Nu* is perceived.

**Table 4.** The numerical values of the Nusselt number (−Θ (1)).

