An implication of magnetic dipole in Carreau Yasuda liquid in ﬂ uenced by engine oil using ternary hybrid nanomaterial

: The aim of this work was to study the enhancement of thermal transportation in Carreau Yasuda liquid passed over a vertical surface in the presence of magnetic dipole. A mixture of tri - hybrid nanoparticles ( ) Al O , MoS , TiO 2 3 3 3 is inserted into the Carreau Yasuda liquid. The transport phenomenon of heat is derived in the presence of heat source/sink contri -bution. The concept boundary layer theory is engaged to derive the mathematical expression for momentum and energy in the form of coupled partial di ﬀ erential equa tions. The derivations are transformed into a set of coupled nonlinear ordinary di ﬀ erential equations ( ODEs ) with the help of suitable similarity transformation. These converted ODEs have been handled numerically via ﬁ nite element method. The grid - independent analysis is established for 300 elements. The impact of numerous involved para meters on temperature and velocity solution is plotted and their contribution is recorded. Temperature pro ﬁ le is inclined versus the higher values of heat generation and viscous dissipation numbers while thermal layers are also increasing the behavior. A vital role of magnetic dipole is examined to raise the production of thermal layers but declination is noticed in ﬂ ow pro ﬁ le.


Introduction
Numerous research works have been carried out on nanomaterials due to their wider applications in medicine, energy system and different industrial mechanisms. The involvement of nanoparticles has also been tested for the treatment of cancer. Researchers used these nanoparticle mixtures in different liquids to study the thermal performance. For instance, Shah et al. [1] worked on ecosystem by studying the inclusion of titanium dioxide particles. Khan et al. [2] examined the contribution of slip effects on Eyring-Powell liquid with heat transport in which graphene particles are mixed. The phenomenon of thin film is further discussed under time-dependent magnetic field. The flow-governing equations have been solved analytically via homotopic solution scheme. They noted the decline in temperature field against Prandtl number. Rehman et al. [3] numerically solved the magnetized non-Newtonian Casson model passed over a radial rotating stretching sheet. They handled the derived boundary layer transformed nonlinear set of differential equations via shooting scheme. They observed the decline in fluid velocity by mounting the values of slip parameter, Casson fluid parameter and magnetic parameter. Alobaid et al. [4] presented the experimental study of carbon-based nanoparticles to examine the degraded soil properties. Ali et al. [5] presented the involvement of thermal radiation and heat generation for the stagnation point flow of viscous liquid past over a stretching cylinder by including the involvement of Brownian motion and thermophoresis. They used shooting scheme to solve the coupled nonlinear ordinary differential equations (ODEs) in unbounded domain. They recorded the decline in velocity against the mounting values of curvature parameter and also the decrease in temperature field is monitored against Prandtl number and radiation parameter. Boarescu et al. [6] [44] studied the performance of nanofluid in Casson material in the presence of Buoyancy-driven convection in parallel hot/cold fins. Ganesh et al. [45] investigated the impacts of second-grade liquid inserting nanoparticles in the occurrence of activation energy in a catalytic surface. An extensive research has been conducted on nanoparticles so far due to their wider applications and utilization. This research is conducted in the presence of magnetic dipole for the inclusion of ternary hybrid nanoparticle mixture in Carreau Yasuda liquid, and numerical computation is performed by using finite element method (FEM) tool in MAPLE18.0 package. The influence of different emerging parameters has been displayed and discussed.

Mathematical analysis
A 2D heat transfer model is carried out in Carreau Yasuda liquid past a stretching surface. A Carreau Yasuda liquid is immersed along with base fluid based on engine oil. Three kinds of nanoparticles (aluminum oxide, MoS 2 and TiO 2 ) are inserted into base liquid as shown in Figure 1. A wall is considered as stretchable to bring motion into fluid particles. A magnetic dipole is considered while the center related to magnetic dipole is placed at the horizontal direction. The flow development is assumed by Figure 2. Transfer of heat energy is considered as absorption and generation into fluid particles. A set of partial differential equations [46,47] is modeled using considerations.  (3) Figure 2 captures the behavior of developed model. It is mentioned that horizontal surface is assumed where a-axis is taken along the horizontal direction and b-axis is considered along the vertical direction. The motion into fluidic particles is produced using movement of wall. Moreover, direction of magnetic dipole is visualized along a-axis due to an implication of magnetic dipole. Boundary conditions [46,47] are Required scalar potential via magnetic field [46,47] is given as Components of magnetic inductions are Using Binomial series and expanding it up to a 2 , Transformations [46,47] are defined as A set of dimensionless ODEs with boundary conditions [46,47] is derived as ( Table 1)   It is noticed that Eqs. (11)- (12) are known as non-Newtonian model in the presence of occurrence of Carreau Yasuda liquid. The present non-Newtonian study is reduced into a case of Newtonian model considering = We 0 and = β 0. The correlations for ternary hybrid nanoparticles [48 and 49] are Drag force coefficient of Carreau Yasuda liquid is formulated as Rate of heat transfer in the presence of tri-hybrid nanoparticles is

Finite element approach
A strong numerical approach based on FEM is implemented to simulate numerical results of ODEs along with boundary conditions. An FEM is used to conduct the solutions of various CDD problems. It has the capacity to handle complex geometries as well as various types of boundary conditions. Six steps of FEMs are discussed below, while six steps are mentioned in Figure 3. An FEM is observed as a good method in view of accuracy analysis, convergence analysis and stability analysis rather than other numerical methods. The following advantages of implementing FEM are as follows.
• Numerous applications of FEM are investigated in computational fluid mechanics problems; • Complex types of geometries are tackled by FEM; • Physical problems based on applied science are developed by FEM; • It has the ability to discretize the derivatives with very ease; • An important role of FEM is to solve various types of boundary conditions; • FEM requires low investment and time rather than other numerical techniques.
Step I: Domain discretization The first step is about domain discretization of problem domain. Domain is broken into small elements of up to 300 elements. Three hundred elements are enough to simulate the solution of current analysis. It is noticed that a system of ODEs is called strong form, whereas weak form is achieved via the residual method.
Step II: Selection of shape function A significant role of shape functions is used to obtain approximation solution of current analysis. Various types of shape functions are used in finite element procedure. In this procedure, linear kind of shape functions is used. The desired form of shape functions is defined as Step III: Weak formulation Eqs. (12)- (14) are known as strong form along with boundary conditions. In this procedure, weak forms are needed to achieve approximation solution. Collection of all terms is placed on one side and integrated over 300 elements. The desired residuals of present problems are derived as Step IV: Finite element formulation In this step, stiffness elements are obtained from current problem. Finally, global stiffness matrices are achieved over each element. The stiffness elements are derived as Step V: Assembly process Assembly process is an integral part of FEM. Stiffness matrices are formulated using concept of assembly approach.
Step VI: Solution of algebraic equations Finally, a system of linear algebraic equations is numerically solved within computational tolerance ( − 10 5 ). The stopping condition is listed below. Flow chart of finite element procedure is given in Figure 4. Furthermore, validation of numerical results in terms of Nusselt number is shown in Table 3. Moreover, programming of FEM is designed on MAPLE 18. Homemade code regarding FEM is developed using MAPLE 18, whereas this code is tested with already published studies.

Mesh-free study
The convergence of problem is investigated through investigation of mesh-free. It is noticed that the present problem becomes grid independent by observing 300 elements. The outcomes of velocity and temperature profiles against 30-300 elements are recorded in Table 2. The convergence of problem is achieved by observing 300 elements. It is included that numerical as well as graphical study is simulated via 300 elements.

Results and discussion
A developing model is analyzed inserting heat generation and heat absorption phenomena in Carreau Yasuda martial past a stretching surface. A role of magnetic dipole is implemented toward stretching surface. A viscous dissipation effect is added into heat energy. Such complextype model is handled with the help of FEM. Graphical simulations and tables are tabulated, whereas graphical discussion of heat energy and velocity fields versus physical parameters is listed below. Here, base fluid is considered as engine oil in ternary hybrid nanofluid. Numerical value of Prandtl fluid [50] is taken as = Pr 6,450.

Graphical simulations of velocity field
A variation in β, We and m is observed against velocity curves inserting ternary hybrid nanomaterials, whereas these simulations are noticed by Figures 5-7. Figure 5 is prepared to notice variation in velocity curves versus the implication of β. A role of β appeared because of magnetic dipole, while a magnetic dipole is applied at the surface of wall. It is noticed that a magnetic dipole attracts fluid particles at the surface of wall and this attraction of fluid particles toward magnetic dipole creates frictional force among particles and layers. So, this attraction force is the reason for slow down velocity of fluid particles. Therefore, it is included that velocity curves have decreasing function against implication of β. This graph is studied for a case without dipole and presence of magnetic dipole. It is investigated that ferrohydrodynamic interaction number is a dimensionless parameter. The viscosity of fluid is enhanced when ferrohydrodynamic interaction parameter is increased. Physically, viscous force is produced into motion via fluid particles. Moreover, thermal layer thickness is reduced using argument values of β. An influence of We on velocity curves using ternary hybrid nanoparticles is carried out by Figure 6. Physically, a ratio among viscous force and elastic force makes a Weissenberg number. It is visualized that an increment in We results in increment in viscosity of fluid particles. Hence, fluid becomes significantly vicious when We is increased. Moreover, layers    Figure 9 shows the comparison of tri-hybrid nanoparticles, fluid, nanofluid and hybrid nanomaterials. Increase is investigated into heat energy when β is increased. Appearance of β is occurred using the strength of magnetic dipole. A magnetic dipole is used to slow down velocity in particles. It is predicted that higher values of β lead to enhanced thermal energy in fluid particles. This effect occurred due to interaction of magnetic field and nanoparticles. So, a frictional heating phenomenon is enhanced in fluid particles because of interaction of magnetic field in fluid particles. Thickness regarding thermal layers is declined versus argument numerical values of β. Physically, it is a dimensionless parameter which is based on the strength of magnetic dipole. Hence, fluid particles absorbed more heat energy when β is increased. Figure 8 is developed to characterize thermal energy among fluid layers, hybrid nanoparticles layers, nanofluid layers and tri-hybrid nanofluid layers. Figure 8 is most significant visualization among layers using hybrid nanoparticles, nanofluid and tri-hybrid nanofluid. It is concluded that tri-hybrid nanoparticles (mixture of TiO , Al O 2 2 3 and SiO 2 in engine oil) are observed to be most significant among fluid layers for the development of more heat energy rather than heat energy is manufactured for nanofluid, fluid and hybrid nanofluid. Hence, maximum amount of heat energy is achieved for the case of tri-hybrid nanomaterials. Figure 10 exhibits an effect of heat generation parameter on temperature field. Maximum production in thermal energy is generated when external heat source is implemented at the surface of wall. Physically, it happened due to the occurrence of external heat source. It is noticed that two kinds of heat phenomena occurred based on heat absorption and heat generation. Heat absorption is based on < H 0 t , whereas heat generation is based on > H 0 t . For both cases, heat energy is augmented by implanted higher values of H t because external heat source is placed at wall. Basically, viscous dissipation number is observed as dimensionless parameter based on viscous dissipation. In energy equation, viscous dissipation parameter appeared in viscous   Figure 11. Heat energy is boosted against implication of viscous dissipation.

Final outcomes
Mathematical model of Carreau Yasuda liquid is developed in the presence of magnetic dipole via stretching surface. Ternary hybrid nano-structures are used to visualize the thermal performance under heat source sink. A finite element scheme is implemented to conduct numerical consequences via flow and temperature profiles. Key remarks are summarized as follows: • Three hundred elements are ensured for visualization of convergence simulations; • It is noticed that ternary hybrid nanomaterials are observed as a significant source to conduct maximum inclination into thermal energy rather than hybrid nanostructures and nanofluid; • Temperature profile is inclined versus the higher values of heat generation and viscous dissipation numbers while thermal layers are also increasing the behavior; • A vital role of magnetic dipole is examined to raise the production of thermal layers but declination is noticed in flow profile; • Flow rate and heat transfer rate are declined versus argument numerical values of viscous dissipation parameter but opposite behavior on heat transfer rate and flow is studied versus power law number; • Heat transfer rate is boosted against higher impact of viscous dissipation.