Significance of Nonsimilar Numerical Simulations in Forced Convection from Stretching Cylinder Subjected to External Magnetized Flow of Sisko Fluid

College of Science, Inner Mongolia University of Technology, Hohhot 010051, China Department of Mathematics, COMSATS University Islamabad, Park Road Chak Shahzad, Islamabad 44000, Pakistan Faculty of Mathematical and Computer Sciences, University of Gezira, Wad Madani 11123, Sudan College of Computer and Information Sciences, Jouf University, Sakaka 72441, Saudi Arabia Department of Mathematics, Faculty of Sciences AlZulfi, Majmaah University, Majmaah 11952, Saudi Arabia Abdus Salam School of Mathematical Sciences, GCU, Lahore 54000, Pakistan


Introduction
e evaluation of non-Newtonian fluids won plenty of attention between the scholars in past because of its large-scale applications in metal spinning, polymer extrusion, and fabrication fields. Since a massive number of non-Newtonian fluids exist, therefore, a bunch of fluid simulations is planned to examine the material characteristics of these fluids. As the power law or generalized Newtonian fluid model is the best relevant model to portend the mindset of non-Newtonian fluids. e procedure of liquid by means of transport in food stuff productions throughout the globe is of fantastic importance. Extremely nutrition handling productions make use of sticky liquids to produce valuable food manufactured goods in majority. Nevertheless, such productions ought to guarantee that involved kit such as pipelines are maintained clean as well as are cleared out for the efficient movement of liquids. e liquids are injected in network which results in friction over the wall up of the tube [1,2]. Movement qualities are the basis of the mass shifts taking place in food stuff productions, e.g., honey, which is utilized in medication and diets due to its biological produce found from nectar, and patients of all age group use these stuffs on regular basis in all seasons. Furthermore, vitamins, high value minerals, and proteins are also components of honey. Consequently, the flow performances of honey at all times have a fascinating subject matter. Honey is also widely utilized as a food preservative and food component as well. e honey found from beehives and scrutinizes has honeycomb, pollen, and other unattractive ingredients. e ejection of these substances is crucial for a generally nice value product alongside with enhancing shelf life. In nutrition industries, honey goes through various procedures, prior to being jam-packed into cans and containers, such as central heating and purification to remove additional waste materials. Furthermore, the humidity substance would be monitored to reduce the risks of fermentation [1][2][3].
When the shear rate becomes very low or high these fluids stop to inspect the flow characteristics, the Sisko fluid model is one of the numerous non-Newtonian fluid models, and it has vast industrial, mechanical, and automobile engineering applications such as lubricating oil, food products, and cementitious slurry. It can be used as an antiwear agent, extreme pressure agent, and corrosion inhibitor and inhibit seizure under high loads and temperatures. In recent years, scholars examined Sisko fluid with different physical hypotheses over various geometries. Sisko [4] worked on greases that depict non-Newtonian behavior in nature having high viscosity with low shear rate and vice versa. Alyiuldiz et al. [5] reviewed the thin film Sisko fluid flow on the top of a moving belt and acquired analytical solutions by employing the homotopy method (HAM). Nadeem and Akbar [6] examined the peristaltic Sisko fluid flow in the uniform inclined duct through. Khan et al. [7] modeled nonlinear equations of incompressible Sisko fluid through a pipe and obtained both approximate and numerical results by using HAM and FDM, respectively. Munir et al. [8] studied the characteristics of 3D, steady flow of Sisko fluid which is triggered due to bi-directional stretching. Malik et al. [9] implored thermal transport of the flow of Sisko fluid across a cylinder. Casson 3D fluid flow subjected to magnetohydrodynamic (MHD) past on top of a permeable surface was numerically viewed by Nadeem et al. [10]. Mustafa and Khan [11] presented MHD drift of Casson nanofluid on top of a nonlinearly expanding cylinder. Farooq et al. [12] inspected the MHD flow of non-Newtonian fluid across a stretching surface. Rashidi et al. [13] inspected numerically the impacts of slip factor, magnetic interaction, and temperature gradient across a rotating disk. Malik et al. [14] analyzed the thermal transport of fluid flow across a cylinder with the consequences of magnetic force. e utilization of numerical procedures in hydraulics networks and various engineering applications can be seen in [15,16].
During the experiments, "the fluids are considered in motion, during motion fluid particles, which transforms some of their kinetic energy (K. E) into thermal heat due to viscosity, and the process is irreversible," and this process is mentioned as viscous dissipation. Brikman [17] investigated viscous dissipation and its properties. He discussed the temperature distribution of the Newtonian fluid in a linear round duct and illuminated the results that were produced in the locked region. Abd El-Aziz [18] measured the outcomes of the viscous dissipation for Sisko fluid flow along with a semiinfinite stretching sheet. Saleem and Nadeem [19] examined the viscous dissipation influences over a vertically rotating cone by applying HAM. Moreover, compilations of publications on viscous dissipation can be found in [20][21][22][23][24][25][26].
Mathematical models of physical procedures in fields of wave dynamics, fluid mechanics, biological kinetics, diffusion, and transportation problems are driven by nonlinear partial differential equations (PDEs). e governing PDEs of a BL fluid flow are transformed into dimensionless PDEs. e considerable mathematical difficulty is associated with the solutions of these BL equations of fluid flow mainly because of their nonlinearity. Several approximate analytical and numerical methods have been established to tackle these equations. Although the local nonsimilar method (LNSM) presented by Sparrow and Yu [27] is more accurate and efficient than other techniques, it is only an approximate method since PDEs are reduced to ODEs after some degree of approximation truncation. Lately, several authors have conducted extensive research on nonsimilar flows primarily using the approach of LNSM [28][29][30][31][32][33][34].
In this study, the magnetized flow of Sisko fluid over a circular cylinder stretching in the axial direction is investigated. e nonsimilar dimensionless system is numerically simulated by employing LNSM up to the second level of truncation via bvp4c. e drawback of local similarity method (LSM) is the governing PDEs that were obtained after the transformation has similar terms. To overcome this drawback, Sparrow et al. [35] used the method of local nonsimilarity. Impacts of the emerging dimensionless parameters, e.g., material parameter, curvature parameter, Prandtl number, and magnetic parameter, are studied on the important numbers in thermal transport analysis such as frictional drag and local Nusselt number and profiles of velocity and temperature through tables and graphs. Practical implications of this study can be found in [36][37][38][39][40][41].

Convection Equations for Transport Analysis
Consider a circular cylinder, which is submerged in a stationary fluid. e forced external flow is initiated due to the cylinder expansion in the axial direction with the velocity U(x) � cx, where c is time constant. e flow is steady, incompressible, and axisymmetric. e transverse magnetic field with strength B o is imposed in r-direction. Induced magnetic and electric fields are ignored. Frictional heating due to viscous dissipation is significant because of the high viscosity of the fluid. erefore, the viscous shear stress impact is incorporated in the energy equation.
Problem geometry is indicated in Figure 1.
Using the assumptions stated above, the convection equations are given by [42] 2 Journal of Mathematics under boundary conditions where u − v are the components of fluid velocity along the xaxis and r-direction. e material constants at high shear rate viscosity are a and σ which represent an electrical conductivity, power-law index denoted by n, and b is the consistency index. e magnetic field B o , the fluid density is ρ, α indicates the thermal diffusivity, and C p , T w , and T ∞ are the specific heat, wall, and ambient temperature.

Journal of Mathematics
Nonsimilar boundary conditions are where τ w surface shear stress, q w is the surface flux, and Equations (9) and (10) in dimensionless forms are

Forced Convection Analysis
e behavior of various dimensionless numbers is implored to study the variations in the thermal transport analysis. e significance of the nonsimilar modeling and its simulations are compared, and results are presented in Tables 1 and 2.  Table 1 is the comparison with the existing works of Malik et al. [42] and Rangi and Ahmad [43]. Results of skin friction for varying the curvature parameter values are shown in Table 1. Table 2 reveals the difference between the current results with the published articles of Malik et al. [42] and Akbar et al. [44] for several values of magnetic number. Table 3 describes the impact of increasing c, A, and M on frictional coefficient. It is noticed that friction increases by enhancing numerical values of c, A, and M. Table 4 illustrates that increasing c and Pr increases the rate of thermal transport; however, the impact of increasing Ec is to decline the rate of heat transfer. Figure 2 describes that increase in magnetic force boosts the Lorentz force which resist the fluid flow. us, a rise in the Lorentz force reduces the velocity field.
Curvature parameter behavior against velocity profile is portrayed in Figure 3. e curvature parameter is the ratio of boundary layer thickness to the cylinder's radius. e increasing curvature curtails the cylinder's diameter. e smaller radius implies that the contact surface of the cylinder is less; therefore, it will provide less struggle to fluid flow. As a result, the fluid accelerates, and it is determined that the flow escalates by boosting the curvature parameter. In Figure 4, the property of A on the velocity field is examined. e fluid velocities increase for significant values of the material parameter; physically, it retains because when the material parameter has significant values, viscous effects reduce, so it suggests a reduced amount of resistance to the fluid motion. In Figure 5, effects of the Eckert number are studied. e Eckert number relates the kinetic energy and heat enthalpy difference of the fluid flow. ere is a direct link between the Eckert number and kinetic energy. So, rise in the Eckert number provides an increment in the kinetic energy; as a result, the thermal transport increases. Journal of Mathematics Figure 6 displays the effect of the curvature parameter on thermal distribution. e enlarged curvature parameter contracts the cylinder's radius, so as an outcome, velocity as well as the kinetic energy which boosts the thermal transport enhances. Figure 7 assess the behavior of Prandtl number; it relates fluid viscosity with the thermal conductivity. erefore, it evaluates the relation between momentum and the thermal transport capacity of the fluid. Enhancement in the Prandtl number reduces the thermal dissipation and the temperature distribution falls due to the weak heat transfer rate.
is shows that strengthening the Prandtl number plunges down the temperature distribution.

Conclusion
In this study, the BL flow of Sisko fluid across a stretching cylinder with the impacts of viscous dissipation and the magnetic field is evaluated numerically. e numerical simulations of the nonsimilar model show that the rise in material parameter decays the viscous forces and, in result, enhances the flow rate, and the increment in the curvature parameter lift flow rate, while the augmentation in the magnetic field parameter, enhancing the resistance force which diminishes the velocity profile. Rise in dimensionless curvature and viscous dissipation parameters raises the thermal profile of the fluid. An increment in the Prandtl number decreases the thermal dissipation which declines the temperature distribution because of low heat transfer rate. Rising values of the Eckert number upsurges the temperature distribution profile. Also, increasing estimations of material and curvature parameters enhances the temperature profile. Friction forces extend for greater values of A, M, and c. Increment in c, Pr, and n gives enhancement of the Nusselt number.
Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.