Heat Transfer Improvement in MHD Natural Convection Flow of Graphite Oxide / Carbon Nanotubes-Methanol Based Casson Nanoﬂuids Past a Horizontal Circular Cylinder

: This numerical investigation intends to present the impact of nanoparticles volume fraction, Casson, and magnetic force on natural convection in the boundary layer region of a horizontal cylinder in a Casson nanoﬂuid under constant heat ﬂux boundary conditions. Methanol is considered as a host Casson ﬂuid. Graphite oxide (GO), single and multiple walls carbon nanotubes (SWCNTs and MWCNTs) nanoparticles have been incorporated to support the heat transfer performances of the host ﬂuid. The Keller box technique is employed to solve the transformed governing equations. Our numerical ﬁndings were in an excellent agreement with the preceding literature. Graphical results of the e ﬀ ect of the relevant parameters on some physical quantities related to examine the behavior of Casson nanoﬂuid ﬂow were obtained, and they conﬁrmed that an augmentation in Casson parameter results in a decline in local skin friction, velocity, or temperature, as well as leading to an increment in local Nusselt number. Furthermore, MWCNTs are the most e ﬃ cient in improving the rate of heat transfer and velocity, and they possess the lowest temperature.


Introduction
The significance of convection from a cylindrical geometry lies in the fact that they are used in numerous physical and engineering applications; these wide usages made researchers direct their great efforts towards this problem. Blasius and Frossling [1,2] solved the momentum and energy equations of combined convection from a horizontal cylinder, respectively. Merkin [3,4] obtained the exact solution for free and combined convection past a horizontal cylinder. Since then, this topic has been extended to include many cases in Newtonian or non-Newtonian fluids, in mixed, forced, or free convection, and many other cases. Merkin and Pop [5] addressed the free convection of viscous fluid about a circular cylinder with prescribed heat flux. Nazar et al. [6] investigated the combined convection flow of micropolar fluid around a circular cylinder. Anwar et al. [7] illustrated the combined

Problem Description
The two-dimensional steady state boundary layer flow of CNTs/GO-Methanol based Casson nanofluid under MHD impact from a horizontal circular cylinder of radius a has been considered.
Further, uniform surface heat flux w q is taken into account. Figure 1  ξ ≈ , and the distance normal to the surface of the cylinder, respectively. The system of dimensional partial differential equations that govern our problem is: with dimensional boundary conditions which are given by [43]: The properties of nanofluid are defined by [45]: In order to nondimensionalization, the following variables were employed [43]: The system of dimensional partial differential equations that govern our problem is: with dimensional boundary conditions which are given by [43]: The properties of nanofluid are defined by [45]: In order to nondimensionalization, the following variables were employed [43]: where is the Grashof number.
By employing Equation (6) into Equations (1)-(4), we obtain the following dimensionless system: and the dimensionless boundary conditions are: Here  is the magnetic parameter. By using the following transformation [43]: Equations (8)-(10) are reduced to: 1 Pr Subject to: at the lower stagnation point of the cylinder (ξ ≈ 0), we obtained the following ODEs: 1 Pr and the boundary conditions become: Two physical quantities are highlighted in the current work, specifically the local skin friction coefficient C f and local Nusselt number Nu (given by [43,46]): where By using Equation (7) and the boundary condition in Equation (15), C f and Nu can be expressed as follows:

Numerical Solution
The Keller box method was first addressed by Keller [47]. This method gained an eminence when Jones [48] employed it to solve boundary layer problems. Cebeci and Bradshaw [49] explained the Keller box technique in detail. This method was used in this work to construct a numerical solution.

Newton's Method
The following linearized tridiagonal system was obtained by applying Newton's method to the previous system that consists of Equations (25)- (29): Processes 2020, 8, 1444 Pr

The Block Tridiagonal Matrix
The matrix form of the previous linearized tridiagonal system is: Processes 2020, 8, 1444 The boundary conditions in Equation (30) are satisfied exactly without iteration, because of these suitable values being kept in every iterate, we suppose that δF The elements of matrices are: The system in Equation (41) is solved by the Lower-Upper (LU) factorization method. Numerical calculations are carried out via MATLAB software version 7. The convergence criterion is assumed to be the wall shear stress z(ξ, 0) as recommended by Cebeci and Bradshaw [49], hence, the calculations processes iterate until satisfying the convergence criterion, and stopped when δz

Results and Discussion
With a view to achieving a fine insight for significant features of the flow and the heat transfer characteristics, numerical computations for various values of nanoparticles volume fraction, Casson and magnetic parameters were carried out, and its representation was performed graphically via MATLAB software.

Validation of Results
To validate our numerical procedure and to confirm that our approach is suitable for the examination issue, comparisons for local skin friction coefficient Gr 1/5 C f and surface temperature θ w (ξ, 0) with numerical findings in the prior literature has been made, the obtained outcomes achieved a close agreement with previously published results by Merkin and Pop [5], Molla et al. [43] and Alkasasbeh et al. [44] as shown in Tables 1 and 2. Table 1.

Graphical Results and Discussion
The range of the parameters for the computational simulations have been considering as: nanoparticles volume fraction (0.1 ≤ χ ≤ 0.2), magnetic parameter (0.5 ≤ M ≤ 3) and Casson parameter (γ > 0). Table 3 displays the thermo-physical properties of methanol and nanoparticles.  Figures 2 and 3 reveal the impact of nanoparticle volume fraction χ on temperature and velocity, respectively. It is found from these figures that a rise in χ leads to enhance the temperature and velocity; this occurs because an increment χ improves the convection from the cylinder to methanol and the energy transmission, therefore increases the temperature and velocity. This can be demonstrated when the thermal conductivity of nanofluids increases as the solid nanoparticles enlarge which are having a great thermal conductivity than the base fluid. Hence, the heat transfer from the base fluid to solid nanoparticles is greater and magnifies the temperature of the nanofluid. Further, it is depicted that the nanofluid temperature increases sufficiently with increasing values of χ for SWCNTs as well as for GO than in the case of MWCNTs. Substantially, this is due to the cause that a rise in χ indications to a growth in the thermal conductivity of SWCNTs/Go nanofluid, and hereafter the viscosity of the thermal boundary layer elevates. Figures 4 and 5 illustrate that the nanoparticle volume fraction χ is directly proportional to both local skin friction coefficient Gr 1/5 C f and local Nusselt number. Augmentation in the value of χ improves thermal conductivity and density of methanol, thereby enhancing Gr −1/5 Nu and Gr 1/5 C f . This is due to the fact that, with increasing the values of nanoparticle volume fraction parameter, both the momentum and thermal boundary layer thickness grows as mentioned in Figures 4 and 5. This means the skin friction and the heat transfer rate intensifies in the nanofluids area when the volume fraction of nanoparticle χ augments.  Table 3. Thermo-physical properties of nanoparticles and methanol [45,50,51].  3 reveal the impact of nanoparticle volume fraction χ on temperature and velocity, respectively. It is found from these figures that a rise in χ leads to enhance the temperature and velocity; this occurs because an increment χ improves the convection from the cylinder to methanol and the energy transmission, therefore increases the temperature and velocity. This can be demonstrated when the thermal conductivity of nanofluids increases as the solid nanoparticles enlarge which are having a great thermal conductivity than the base fluid. Hence, the heat transfer from the base fluid to solid nanoparticles is greater and magnifies the temperature of the nanofluid. Further, it is depicted that the nanofluid temperature increases sufficiently with increasing values of χ for SWCNTs as well as for GO than in the case of MWCNTs. Substantially, this is due to the cause that a rise in χ indications to a growth in the thermal conductivity of SWCNTs/Go nanofluid, and hereafter the viscosity of the thermal boundary layer elevates.           Figures 6 and 7 evidence that with rising the Casson parameter γ , the temperature and velocity decrease. Actually, growing values of γ generate resistance forces that act to inhibit the fluid's velocity. Physically, a greater and smaller value of γ corresponds to Newtonian and non-Newtonian fluids, respectively (i.e., γ decreases the yield stress). However, it agrees with the stated physical analogy, which is due to the temperature injected that influences the nanofluid temperature.  Physically, a greater and smaller value of γ corresponds to Newtonian and non-Newtonian fluids, respectively (i.e., γ decreases the yield stress). However, it agrees with the stated physical analogy, which is due to the temperature injected that influences the nanofluid temperature. Figures 8 and 9 portray the effect of the Casson parameter γ on both local skin friction coefficient Gr 1/5 C f and local Nusselt numb Gr −1/5 Nu. It can be observed that at higher values of the Casson parameter, the yield stress decreases, causing a decrease in skin friction, while the reverse happens with the Nusselt number. This is because, as mentioned above, greater values of γ are accompanied by lower in the yield stress of the Casson fluid, which causes a reduction in rheological characteristics. Therefore, the flow approaches closer to Newtonian conduct and the fluid is able to shear slower along the cylinder surface. Consequently, the Nusselt number is found to enhance as γ is boosted. This occurs with the above data on temperature distribution as explored in Figure 7.  Figures 6 and 7 evidence that with rising the Casson parameter γ , the temperature and velocity decrease. Actually, growing values of γ generate resistance forces that act to inhibit the fluid's velocity. Physically, a greater and smaller value of γ corresponds to Newtonian and non-Newtonian fluids, respectively (i.e., γ decreases the yield stress). However, it agrees with the stated physical analogy, which is due to the temperature injected that influences the nanofluid temperature. shear slower along the cylinder surface. Consequently, the Nusselt number is found to enhance as γ is boosted. This occurs with the above data on temperature distribution as explored in Figure 7.  happens with the Nusselt number. This is because, as mentioned above, greater values of γ are accompanied by lower in the yield stress of the Casson fluid, which causes a reduction in rheological characteristics. Therefore, the flow approaches closer to Newtonian conduct and the fluid is able to shear slower along the cylinder surface. Consequently, the Nusselt number is found to enhance as γ is boosted. This occurs with the above data on temperature distribution as explored in Figure 7. In Figures 10 and 11, the influence of the magnetic field M on both temperature and velocity are elaborated. It is noted here that Lorentz force formed by the growth of the magnetic field decelerates velocity and enhances the temperature. In fact, a magnetic field yields a drag-like force; namely, the Lorentz force. This force acts in the opposite direction, which results in the reduction in velocity, and the fluid temperature boosts by magnifying the strength of the magnetic field.  In Figures 10 and 11, the influence of the magnetic field M on both temperature and velocity are elaborated. It is noted here that Lorentz force formed by the growth of the magnetic field decelerates velocity and enhances the temperature. In fact, a magnetic field yields a drag-like force; namely, the Lorentz force. This force acts in the opposite direction, which results in the reduction in velocity, and the fluid temperature boosts by magnifying the strength of the magnetic field.

Thermo-Physical
In Figures 12 and 13, both the skin friction coefficient and local Nusselt number declined with increasing values of the magnetic parameter M. This is attributable to the fact that as the intensity of the magnetic field rises, the motion of the fluid is inhibited as a result of the generation of Lorentz force. This leads to the argument that an applied magnetic field tends to heat the nanoliquid, and thus heat transfer from the cylinder reduces, and then both local skin friction coefficient Gr 1/5 C f and local Nusselt number Gr −1/5 Nu reduce. All of this happens due to Lorentz force in the visualization of the transverse applied magnetic field opposing the transport phenomena and slowing down the fluid movement. No doubt, the magnetic can be utilized as a beneficial agent for controlling the flow and heat transfer characteristics. Moreover, the graphical findings revealed that MWCNTsmethanol generated the highest heat transfer rate, skin friction, and velocity, as well as it had the lowest temperature, this is due to the unmatched thermal properties that possess MWCNTs.
In Figures 10 and 11, the influence of the magnetic field M on both temperature and velocity are elaborated. It is noted here that Lorentz force formed by the growth of the magnetic field decelerates velocity and enhances the temperature. In fact, a magnetic field yields a drag-like force; namely, the Lorentz force. This force acts in the opposite direction, which results in the reduction in velocity, and the fluid temperature boosts by magnifying the strength of the magnetic field. In Figures 12 and 13, both the skin friction coefficient and local Nusselt number declined with increasing values of the magnetic parameter M. This is attributable to the fact that as the intensity of the magnetic field rises, the motion of the fluid is inhibited as a result of the generation of Lorentz force. This leads to the argument that an applied magnetic field tends to heat the nanoliquid, and thus heat transfer from the cylinder reduces, and then both local skin friction coefficient Gr Nu − reduce. All of this happens due to Lorentz force in the visualization of the transverse applied magnetic field opposing the transport phenomena and slowing down the fluid movement. No doubt, the magnetic can be utilized as a beneficial agent for controlling the flow and heat transfer characteristics. Moreover, the graphical findings revealed that MWCNTs -methanol generated the highest heat transfer rate, skin friction, and velocity, as well as it had the lowest temperature, this is due to the unmatched thermal properties that possess MWCNTs. In Figures 12 and 13, both the skin friction coefficient and local Nusselt number declined with increasing values of the magnetic parameter M. This is attributable to the fact that as the intensity of the magnetic field rises, the motion of the fluid is inhibited as a result of the generation of Lorentz force. This leads to the argument that an applied magnetic field tends to heat the nanoliquid, and thus heat transfer from the cylinder reduces, and then both local skin friction coefficient Gr Nu − reduce. All of this happens due to Lorentz force in the visualization of the transverse applied magnetic field opposing the transport phenomena and slowing down the fluid movement. No doubt, the magnetic can be utilized as a beneficial agent for controlling the flow and heat transfer characteristics. Moreover, the graphical findings revealed that MWCNTs -methanol generated the highest heat transfer rate, skin friction, and velocity, as well as it had the lowest temperature, this is due to the unmatched thermal properties that possess MWCNTs.

Conclusions
In the current examination, the magnetohydrodynamics natural convection flow of Graphite oxide/Carbon nanotubes-Methanol based Casson nanofluids past a horizontal curricular cylinder is presented. Tiwari-Das's model is employed to consider the volume fraction of nanoparticles in the computational simulations. On the other hand, uniform heat flux is taken into account. However, the outcomes exposed that the multiple walls carbon nanotubes produce the most effective heat transfer performance, it also gave the highest velocity and lowest temperature to the host Casson liquid.
Temperature and velocity profiles confirmed that a rise in χ leads to enhancing temperature and velocity, while with rising the Casson parameter γ , the temperature and velocity decrease. The

Conclusions
In the current examination, the magnetohydrodynamics natural convection flow of Graphite oxide/Carbon nanotubes-Methanol based Casson nanofluids past a horizontal curricular cylinder is presented. Tiwari-Das's model is employed to consider the volume fraction of nanoparticles in the computational simulations. On the other hand, uniform heat flux is taken into account. However, the outcomes exposed that the multiple walls carbon nanotubes produce the most effective heat transfer performance, it also gave the highest velocity and lowest temperature to the host Casson liquid. Temperature and velocity profiles confirmed that a rise in χ leads to enhancing temperature and velocity, while with rising the Casson parameter γ, the temperature and velocity decrease. The magnetic field M had a positive effect on the temperature and an inverse effect on the velocity. Augmentation in the value of χ improved Gr −1/5 Nu and Gr 1/5 C f , both the Gr −1/5 Nu and Gr 1/5 C f declined with increasing values of the magnetic parameter. We also found that Gr 1/5 C f is a decreasing function of Casson parameter γ while Gr −1/5 Nu is an increasing function of it.