Al 2 O 3 and γ Al 2 O 3 Nanomaterials Based Nano ﬂ uid Models with Surface Diffusion: Applications for Thermal Performance in Multiple Engineering Systems and Industries

: Thermal transport investigation in colloidal suspensions is taking a signi ﬁ cant research direction. The applications of these ﬂ uids are found in various industries, engineering, aerodynamics, mechanical engineering and medical sciences etc. A huge amount of thermal transport is essential in the operation of various industrial production processes. It is a fact that conventional liquids have lower thermal transport characteristics as compared to colloidal suspensions. The colloidal suspensions have high thermal performance due to the thermophysical attributes of the nanoparticles and the host liquid. Therefore, researchers focused on the analysis of the heat transport in nano ﬂ uids under diverse circumstances. As such, the colloidal analysis of H 2 O composed by γ Al 2 O 3 and Al 2 O 3 is conducted over an elastic cylinder. The governing ﬂ ow models of γ


Introduction
The analysis of heat transfer had been a major problem for engineers, industrialists, and scientists. For many industrial productions, a large amount of heat is essential to complete the production processes. Unfortunately, regular liquids are not capable to provide the required heat transfer. This major problem economically affected the industrialists and engineers. So, scientists and engineers thought to gathered and focus to resolve this major problem. Finally, they developed a new sort of fluids known as "Nanofluids" [1,2]. These fluids are the composition of nanosized metallic particles and the base liquids. The base liquids and the metallic particles are thermally in equilibrium. These fluids became very productive for industrialists and engineers.
Due to the remarkable thermal performance rate, these fluids attained much popularity among the researcher's and engineers in a very short period. The applications of these fluids found in medical sciences, chemical and mechanical engineering, in the detection of cancer cells, aerodynamics, home appliances, paint industry, electronics, computer chips, manufacturing of air crafts parts, various industrial productions and in many more. In view of these applications, the researchers and industrialists focused on the analysis of nanofluids under various flow conditions.
The investigation of heat transfer in the nanofluids over a stretching cylinder is significant from the industrial point of view. In 2017, Pandey et al. [3] reported the heat transfer investigation in the nanofluid over a stretching cylinder. They studied the heat transfer in the additives of water and Cu nanoparticles by considering the slip effects and thermal radiations. They revealed that the temperature of the nanofluid rises due to high volume fraction and the momentum boundary layer region decreases. In 2017, Salahuddin et al. [4] presented the heat transfer investigation in Carreau nanofluid over a stretchable cylinder. They considered the effects of generalized slip flow conditions over the cylinder surface. Further, the finite-difference algorithm is implemented for the mathematical analysis and studied the behavior of the velocity and heat transfer by varying the particular flow parameters.
The analysis of the heat transport and the velocity for different nanofluids in the presence of Lorentz forces over a stretchable cylinder was investigated in [5]. A novel thermal analysis in the nano bioconvection model over a stretchable sheet was presented by Zheng et al. [6]. They studied the effects of thermal radiations, velocity and thermal jump conditions in the governing model. They presented a comprehensive discussion on the behavior of the velocity and the temperature against various flow parameters. The temperature behavior of water suspended by Cu nanomaterial over a stretching/shrinking cylinder was reported by Sulochana et al. [7] in 2016. They studied the flow characteristics for different values of the flow parameters. Bakar et al. [8] discussed the stability analysis in the nanofluid over a shrinking/stretching cylinder. They discussed the influence of the suction parameter in the flow field. The flow of nanofluid by considering the influence of corner radius over a square cylinder was presented in [9].
The analysis of the heat and mass transfer in the presence of porosity parameter over a horizontal cylinder was reported by Reddy et al. [10]. A comprehensive discussion was provided on the flow behavior against the particular flow parameters. The influence of uniform suction and injection on the heat and mass transfer were described in [11]. Recently, Mohyud-Din et al. [12] reported the temperature analysis in a hybrid colloidal model by incorporating the effects of Cattaneo Christov model in the energy equation. They considered the flow of hybrid host liquid (H 2 O-C 2 H 6 O 2 ) saturated by hybrid tiny particles of GO-MoS 2 . From the analysis, they concluded that the particular hybrid nanoliquid have a high thermal transport rate and reliable for the industrial and engineering applications. The analysis of gyrotactic microorganism's boundary layer flow was presented by Aziz et al. [13]. They tackled the model numerically and presented the results for the flow fields. Furthermore, the significant studies related to the flow fields and relevant analysis was reported in [14][15][16][17][18][19][20][21][22][23].
The thermal performance in Al 2 O 3 and γAl 2 O 3 based hybrid suspensions over a cylinder is significant from the industrial and engineering point of view. It is investigated that the thermal transport in Al 2 O 3 -H 2 O and γAl 2 O 3 -H 2 O colloidal suspensions was not reported so far. Therefore, the analysis is presented to fill this significant research gap. The two nanofluid models are considered over a cylinder and treated numerically. Then, graphical results for the velocity and the temperature profiles are plotted by altering the particular parameters and discussed in detail. Finally, the major findings of the analysis are incorporated in the last section.

Statement and Geometry
The flow of Al 2 O 3 -H 2 O and γAl 2 O 3 -H 2 O nanoliquids over a cylinder with uniform suction/blowing is considered. It is assumed that the nano particles dispersed continuously in the host liquid and there is no slip between them. The study is conducted in cylindrical coordinates r; '; z ð Þ and corresponding velocity components are u; v; w ð Þ: Further, length of the cylinder is infinite and a denotes the cylinders radius. The flow is an inward direction which forms the stagnation circle with r ¼ a and z ¼ 0. Fig. 1 presents the flow configuration. The axisymmetric nanofluid model which governs the flow over a cylinder is described as [24]: The associated flow conditions described by the following equations: r ! 1;T ¼T 1 ; The associated invertible transformations that reduced the nanofluid model into a self-similar form are described in the Eqs. (9)-(13):

Adopted Models for Nanofluids Thermal Performance
The following nanofluid models are adopted for thermal performance in Al 2 O 3 -H 2 O and γAl 2 O 3 -H 2 O nanofluids [25]: After using the invertible transformations labeled in Eqs. (9)-(13), the associated flow conditions labeled in Eqs. (5)-(8) and the thermal performance models labeled in Eqs. (14)- (19) in the governing model labeled in Eqs. (1)-(4), the following dimensionless form of the models are obtained:

Al 2 O 3 -H 2 O Nanofluid Model
By using the effective nanofluid models, the following nonlinear model is attained for Al 2 O 3 -H 2 O:

cAl 2 O 3 -H 2 O Nanofluid Model
By using the effective models labeled in Eqs. (14)- (19), the following model is obtained for The associated flow conditions at the surface and far from the cylinder surface are labeled in the following equations:

Quantity for Thermal Performance
The local heat transfer is significant from an engineering, technological, and industrial point of view. The mathematical relation describing the quantity is labeled in the following equations: Nu ¼ ah By using the values from Eq. (28) in Eq. (29), the following self-similar formulas are obtained for the local heat transfer coefficient: It is not easy to tackle such models for exact solutions. Therefore, we treated the models numerically. For said purpose, Runge-Kutta algorithm is adopted with the coupling of shooting technique. The algorithm is applied to the Initial Value Problem (IVP). For this, firstly, we transformed the models into IVP. The following transformations are introduced according to the models: By using the transformations described in Eqs. (32) and (33) in the models given in (20)- (23), the following systems are obtained: The reduced models are then solved by using aforementioned techniques and plotted the results for the flow regimes by varying the particular parameters embedded in the models.

Physical Interpretation of the Results
The behavior of the velocity and heat transfer by changing the particular flow parameters are decorated in this section over the feasible domain.

The Velocity against Preeminent Parameters
The  Fig. 2b. Although, the velocity rises for suction of the fluid from the cylinder surface and these changes are quite slow in comparison with injecting case. Physically, due to suction, more fluid particles get stuck to the surface of the cylinder that opposes the fluid motion. Therefore, the fluid particles move slowly.
In the analysis of nanofluids, the volumetric fraction f is a significant parameter that effectively alters the fluid characteristics.  It is detected that for γAl 2 O 3 -H 2 O nanofluid, the velocity decrement is very rapid due to high dynamic viscosity. Physically, due to high dynamic viscosity and injection of the fluid from the cylinder surface, rapid decrement in the velocity is occurred. Almost inconsequential decrement in the velocities is detected for both injection and suction of the fluid from the cylinder surface for γAl 2 O 3 -H 2 O nanofluid.
The velocity trends for variable injection and suction values are decorated in Figs. 4a and 4b, respectively. The velocity for both colloidal suspensions increases significantly near the cylinder surface. Further, it is examined that the velocity of Al 2 O 3 -H 2 O prevailed for both suction and injection of the fluid from the cylinder surface. Physically, for injection of the fluid, the velocity rises rapidly because injecting fluid disturbed the fluid particles therefore, the particles move rapidly.  A pertinent parameter known as the volumetric fraction f of the tiny particles is significant in the thermal analysis of the nanofluids. The trends against f are decorated in Fig. 6 for both injection and suction, respectively. It can be seen that the thermal performance of the nanofluids enhances due to increasing volumetric fraction of the tiny particles. From the analysis, it is examined that the thermal performance of γAl 2 O 3 -H 2 O nanofluid dominated.

Local Thermal Performance Rate
The local thermal performance rate (local heat transport) against multiple values of the volumetric fraction f, Reynolds number, injection and suction of the fluid are described in Tab. 1. From the analysis, it is observed that the local thermal performance rate of the nanofluids rises against the high-volume fraction of the tiny particles. However, in the current scenario, the thermal performance rate of Al 2 O 3 -H 2 O is higher than γAl 2 O 3 -H 2 O. For injecting fluid, thermal performance rises abruptly in comparison  with suction case. Similarly, it is examined that for higher Reynolds number, the local heat transfer rase for both sorts of nanofluids under the injection and suction effects.

Streamlines and Isotherms
The preeminent flow parameters that appeared in the flow models are significantly change the flow patterns and isotherms over the domain of interest. Fig. 7 describes the flow pattern against varying Reynolds number. It is investigated that for higher Reynolds values, the streamlines pattern expanded around the cylinder surface. Similarly, the streamlines pattern for multiple injecting fluid parameters decorated in Fig. 8. Figs. 9 and 10 portraying the isotherms profiles for multiple Reynolds range at constant injection of the fluid.

Conclusions
The thermal performance rate and the behavior of flow regimes for Al 2 O 3 -H 2 O and γAl 2 O 3 -H 2 O are conducted over a cylinder. The effective nanofluid characteristics are implemented and obtained the dimensionless flow models. The invertible transforms are used for nondimensionalization of the problem. The impacts of the flow quantities on the fluid velocity, temperature, and local thermal performance rate are plotted and explained in detail. It is observed that the velocity of Al 2 O 3 -H 2 O rises promptly it prevailed throughout the analysis. The volumetric fraction of the tiny particles resists the nanofluids motion. it is perceived that the temperature enhances against the volumetric fraction. It is also noticed that the thermal performance rate of both the nanofluids increases. Therefore, these nanofluids are reliable for the industrial, engineering and technological purposes, where high thermal performance rate is required to accomplish the production processes.