A numerical frame work of magnetically driven Powell-Eyring nanofluid using single phase model

The current investigation aims to examine heat transfer as well as entropy generation analysis of Powell-Eyring nanofluid moving over a linearly expandable non-uniform medium. The nanofluid is investigated in terms of heat transport properties subjected to a convectively heated slippery surface. The effect of a magnetic field, porous medium, radiative flux, nanoparticle shapes, viscous dissipative flow, heat source, and Joule heating are also included in this analysis. The modeled equations regarding flow phenomenon are presented in the form of partial-differential equations (PDEs). Keller-box technique is utilized to detect the numerical solutions of modeled equations transformed into ordinary-differential equations (ODEs) via suitable similarity conversions. Two different nanofluids, Copper-methanol (Cu-MeOH) as well as Graphene oxide-methanol (GO-MeOH) have been taken for our study. Substantial results in terms of sundry variables against heat, frictional force, Nusselt number, and entropy production are elaborate graphically. This work’s noteworthy conclusion is that the thermal conductivity in Powell-Eyring phenomena steadily increases in contrast to classical liquid. The system’s entropy escalates in the case of volume fraction of nanoparticles, material parameters, and thermal radiation. The shape factor is more significant and it has a very clear effect on entropy rate in the case of GO-MeOH nanofluid than Cu-MeOH nanofluid.


Differentiation concerning to ψ
The first to introduce the theory of the boundary layer was Ludwig Prandtl 1 . A boundary-layer is the tinny region of fluid flow in which flow is influenced by the friction between the solid plate and the liquid. The boundary layer flow has been broadly deliberated in the literature and plays a vital role in fluid dynamics. The investigation of boundary-layer flowing past a horizontal plate had countless manufacturing implementations, such as food manufacturing, glass fibers production, manufacturing of rubber sheets, extrusion, metal spinning, wire drawing, and cooling of massive metallic plates such as an electrolyte [2][3][4] . Makinde and Onyejekwe 5 presented the numerical computations for the boundary-layer flowing model results in the stretching sheet with variable electrical conductivity and variable viscosity using a shooting technique and a sixth-order RK integration algorithm. They concluded that, when the electrical conductivity parameter is increased, convective heat transfer and skin friction coefficient decreases within the boundary surface. Moreover, a rise in the variable viscosity parameter increases viscous force and makes viscous forces dominant over the applied magnetic field. In the use of numerical shootings, Ibrahim and Makinde 6 looked at the boundary-layer movement past a vertical, rotating flat sheet with heating effects and chemical reaction effects from Joule. Heat transmission is the thermal energy transfer from one device to another because of temperature differences. Because of this temperature difference, the heat transmission process takes place between two bodies (or a related body). In many industrial applications such as composite materials manufacturing, geothermal reservoirs, porous solid drying, thermal isolation, oil recovery, and the transport of sub-terrain species, the research into fluid flows and heat transmission produced by stretching media is of great importance. In the above situations, heat transfer and flow assessment are important as the final product efficiency is calculated based on the velocity gradient (skin friction) coefficient and the convective heat shift rate. Elbashbeshy 7 numerically studied viscous fluid and heat transfer flow by assuming the exponentially continuous stretching sheet. In his work, fluid inhabits the distance over an endless horizontal plate, and the nonlinear extending of the plate induces the flow. He implements the numerical technique to solve the modeled equations. The results indicated that the suction parameter could cool the continuous moving stretching surface. The numerical results also showed that the thermal boundary layer's thickening level reduces for larger values of the suction parameter. After that, herein c is a preliminary extending rate. The partial slip, as well as convective conditions, are considered at the boundary. A magnetic field B 0 is utilized in a perpendicular direction to the fluid flow, and an induced magnetic field is neglected in comparison to B . The expressions T w (x, t) = T ∞ + cx as well as T ∞ indicates wall and ambient temperature respectively. For convenience, the sheet has to be fixed at x = 0 and is stretching in the positive x-direction. Moreover, the sheet is considered slippery and is subjected to a temperature gradient. Powell-Eyring nanofluid behaves like shear thickening and is assumed optically thick. The stress tensor expression for the case of Power-Eyring fluid is specified by (see, for example, Powell and Eyring 29 ) where µ nf indicates dynamic viscosity, ∼ β and ς * for material constants. The inside geometry of the physical model is illustrated in Fig. 1.
The controlling modeled formulas (Kumar and Srinivas et al. 47 ) are given by The BCs are bestowed by (for instance 48 )  www.nature.com/scientificreports/ Here, u and v depicts velocities along x as well as y axis, t is the time, T is a temperature of the fluid, µ nf is the dynamical viscidness of the nanofluid, ρ nf is the intensity, σ nf is the electrically conducting. q r is the radiative heat flux, (C p ) nf and κ nf are the specific heat capacitance and the thermal conductance, correspondingly. V w signifies the penetrability of the expanding sheet. The penetrability of nanofluid is signified by k . The expressions regarding thermal conduction and heat transfer coefficient are delineated as k 0 and h f . Table 1 represents physical properties [49][50][51] for the case of Powell-Eyring fluid.
Based on Table 1, φ indicates the nanoparticle volume fraction. Symbols µ f , ρ f and (C p ) f , κ f and σ f are dynamical viscidness, intensity, specific heat capacitance, the thermally and electrically conductive of the basefluid. ρ s , (C p ) s , κ s and σ s are the density, specific heat capacity, the thermal and electrically conducting of the nano-solid particles. Table 2 presents empirical shape factor values in the case of distinguished particle shapes (see for example 52,53 ).
Roseland expression in terms of heat flux (Brewster 54 ) is given by where σ and k * points out Stefan/Boltzmann constant and moreover k * is the absorption coefficient.

Properties Nanofluid
Dynamic viscosity Electrical conductivity

Classical Keller-Box numerical technique
Keller-box method (KBM) 55 is utilized to obtain the numerical solution of modeled equations. This method generally provides fast convergence in contrast to other nonlinear numerical schemes. This scheme provides convergent up to second-order and inherently stable as well. This method assures Von Neumann's stability test in terms of stability analysis. This test sets the criterion for the convergence of the numerical solution to PDEs' real solution using the numerical solution's consistency and stability. The solutions of Eqs. (11)-(12) along with, (13)- (14), are achieved by KBM. The flow chart mechanism of Keller box method is explained below. (see Fig. 2): Stage 1: renovation of ODEs. The ODEs (11)- (14) are stepped down into five first-order ODEs mentioned below www.nature.com/scientificreports/ Stage 2: discretize domain. The discretization of a domain can be done by dividing the domain of the system into small uniform grids to obtain the approximate solution (see Fig. 3). Generally smaller grid provides high accuracy (6).
In this problem, the value of h is fixed to 0.01 . To achieved difference equations the process of central differences has been implemented. Mean averages replace the functions. The ODEs (17)- (22) are transformed into algebraic expressions of nonlinear nature. The replacement of overhead in formulas (23)- (27) and disregard the quadratic and higher bounds of ̟ i j , a linear tri-diagonal system is achieved where The boundary constraints become through the similarity procedure (39) www.nature.com/scientificreports/ Stage 4: the block-tridiagonal matrix. The above formulas (29)-(33) have a tridiagonal block structure.
In a matrix-vector, we write the system accordingly, For j = 1; In matrix formula, That is In matrix formula, In matrix formula,

That is
For j = J; Here R signifies the J × J block-tridiagonal array with each block size of 5 × 5 , whereas, ̟ and p are column vectors of order J × 1 . The LU factorization technique is now useful to discover the solution of ̟.

Skin friction (C f ) and Nusselt number (Nu x )
The expression regarding (C f ) and (Nu x ) are bestowed by (See for example Khan et al. 56 ) here τ w and q w represents stress as well as heat flux at the wall bestowed by Using similarity transformations (12), above (68) R̟ = p,

Entropy generation analysis
Generally speaking, MHD and porous media amplify entropy. Jamshed and Aziz 39 defined entropy generation expression mentioned below.
The first term depicts the transfer of heat irreversibility. The second term in the entropy expression indicates fluid friction and MHD as well as porous media effects are given at the end, respectively. The dimension-less expression regarding entropy generation NG is manifested by (for instance: [57][58][59] Using similarity transformations (12) Code validity. The authenticity of the proposed technique was scrutinized by taking comparison with already available literature [60][61][62][63] . Table 3 shows a strong agreement with our proposed numerical scheme. It is found that the present numerical solution is accurate up to 5 significant figures. Hence, outcomes are reliable and numerically authentic.

Numerical consequences and discussion
This section is devoted to studying the influence of sundry parameters like ω , K , , M , φ , , Nr , Bi , Ec , Q , S , Re , Br and m on velocity, temperature, and entropy generation in terms of Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 and 27 in the case of Cu-MeOH and GO-MeOH nanofluids. Table 5 displays the distinguished physical quantities against surface drag factor as well as temperature field gradient.  Table 4.  Fig. 4. This trend assures our numerical scheme's authenticity. Moreover, a positive variation in depreciates the fluid velocity and shows a decrement in fluid motion. Incremental change in fluid viscosity depreciates yield stress. In the case of distinguished nanofluids, (when = 0.2 ) the momentum boundary layer of GO-MeOH nanofluid is heavier than the Cu-MeOH nanofluid. The upsurge in nanofluid temperature is seen in Fig. 5 for magnification in . Heat transport rate falls in the fluid stream since an incremental change in elasticity stress. Figure 6 shows that magnification in improves overall system entropy. It is expected that a magnification upsurges fluid viscosity which ultimately retards the fluid motion and enlarges the temperature of the fluid and entropy phenomenon. Entropy amplifies due to a decrement in the heat transfer rate. Increment change in depreciates available energy amount.

Pr
Ref. 60 Ref. 61 Ref. 62 Ref. 63  www.nature.com/scientificreports/      Fig. 8 it is noted that the M is inversely related to fluid density, so a positive variation in M amplifies the boundary layer's temperature. Table 5 validate that the Nusselt number lessens but the drag coefficient amplifies as a result of positive change in M . Figure 9 displayed the fact the overall entropy booms owing to an incremental change in a magnetic field. A magnification in magnetic field strength urging the fluid's speed to slow down and produces more heat, which furthermore elevates the entropy phenomenon.
Nanoparticle concentration size ( φ ) Impact. Figures 10 and 11 reflect the impact of φ = 0.1, 0.15, 0.2 on fluid velocity along with temperature field as well. It is noteworthy that a positive variation in φ makes the fluid dense to flow over the surface which lessens the fluid velocity and thickness of the momentum boundary layer as well. It is observed that the addition of nanoparticles in base fluid amplifies heat transfer rate and thermal conduction phenomenon. As a result temperature of the fluid and thickness of the thermal-based boundary layer have been improved tremendously as depicted in Fig. 11. The velocity as well as and temperature in terms of φ is portrayed in Table 5. Figure 12 showed that a positive change in entropy as a result of magnification in φ .   Fig. 13. This phenomenon happens because it reduces the stretching effect retards fluid velocity. Figure 14 shows that is inversely related to temperature. Amplification in diminishes heat transfer but elevates temperature at the boundary. Overall entropy booms owing to an amplification in the temperature because the slip phenomenon depreciates the friction effect which ultimately magnifies the temperature as well as entropy as displayed in Fig. 15.
Biot number ( Bi ), and Eckert number ( Ec ) impact. Figures 16,17,18 and 19 are planned to reflect the influence of Bi and Ec on temperature as well as entropy fields likewise. Convection in terms of heat transfer from the boundary towards the fluid is getting better and better owing to an amplification in the values of Bi . The temperature as well as thickness in terms of a thermal layer at the boundary booms as a result of enrichment in Bi (Fig. 16). No significant change is reported in the case of the velocity field in the case of a positive variation in Bi . Figure 18 sketches the change in temperature field for the case of the diverse values of Ec = 0.2, 0.4, 0.6 . Greater Ec , a ratio of kinetic energy to enthalpy difference. Molecules collide more randomly as a result of an increment in Ec because the kinetic energy amplifies the molecules' friction and internal heat generation capacity which  www.nature.com/scientificreports/ elevates the heat transfer phenomenon in a temperature field as portrayed in Fig. 18. Figures 17 and 19 it is quite evident that a substantial amplification in parameters Bi as well as Ec provides substantial heat to the fluid which upsurges temperature phenomenon and entropy phenomenon as well. In the case of χ = 0.3 , the entropy showed a cross-over point. Entropy amplifies and declined before as ell as after that point.
Thermal radiative ( Nr ) and heat source parameters ( Q ) impact. Figure 20 shows the impact of Nr on the temperature distribution field for various values of Nr = 0.3, 0.5, 0.8 . Thermal radiation is used where a large temperature difference is required like combustion reactions, nuclear fusions, ceramic productions, etc. In the presence of Nr temperature as well heat transfer phenomenon escalates by the virtue of amplification in Nr .
It is quite interesting that more is generated inside the fluid on the behalf of augmentation in the heat source Q which ultimately makes a pathway for magnification in the temperature field as sketched in Fig. 22. Figures 21  and 23 exhibit the influence of Nr as well as Q parameters on entropy profile. It is quite clear that in the case of χ = 0.3 , the entropy outline depicts incompatible facts. Entropy of the system amplifies before that point while depreciates after that point. Physically, the crossover point is a sign for effective modification of the thermal system. We can say that χ = 0.3 has situated nearby the sheet and the entropy always upsurges close to the boundary sheet and depreciates in the case of away from the surface.    Table 2. From Fig. 24 it is observed that nanofluid temperature upsurges by the virtue of an improvement in m . Temperature of the fluid is getting lower at m = 3 spherical-shaped type nanoparticles. The sphere occupies a large superficial zone and booms heat transmission rate from the sheet surface towards fluid inside. It is noted from Fig. 25, entropy m escalates. The system's entropy is getting lower and lower for sphere structure shape-particles as the heat trick inside the scheme is getting smallest. It is noteworthy that the inertial forces topple viscous forces in the case of magnification in Re which furthermore enhances the overall entropy of the thermal system shown in Fig. 26. Figure 27 sketches the Br effect on entropy. In the case of augmentation in Br , heat dissipates more quickly as compared to the conduction phenomenon at the surface, which moreover amplifies entropy of the system. The results are quite reliable in comparison with Abbas et al. 66 , who reported similar results.

Conclusions
Computational surveys of boundary-layer flow for Cu and GO methanol-based nanofluids were achieved over a permeable elongating surface. This research considered MHD, porous medium, viscous dissipative, thermal radiative, Joule-heating, and particle shapes with Keller box methods help. Significance of the effects of different dimensionless parameters against velocity, Temperature, and entropy profiles are displayed in terms of figures.
The C f as well as Nu x for diverse amounts of sundry factors are portrayed in the form of a table. Some pertinent concluding observations from the present study are enumerated underneath.
1. Velocity profile owing to amplification in and φ.  www.nature.com/scientificreports/ 2. Temperature profile increased the function of parameters , K , M , φ , Nr , Bi , Q and Ec whereas reduced parameters S > 0. 3. Amplification in nanoparticles concentration φ guides an improvement in temperature and thermal boundary layer thickness. 4. The GO-MeOH nanofluid is better in terms of thermal conduction instead of Cu-MeOH nanofluid. 5. The heat transport rate is more significant for the lesser number of shape factors. 6. Overall systems entropy depreciates by the virtue of magnification in slip parameter. 7. Lamina-shaped particles deliver more heat at the boundary layer, while the temperature is getting lower for the case of spherical-shaped nanoparticles.