Dynamics of radiative Williamson hybrid nanofluid with entropy generation: significance in solar aircraft

Sun based energy is the chief source of heat from the sun, and it utilizes in photovoltaic cells, sun-based power plates, photovoltaic lights and sun-based hybrid nanofluids. Specialists are currently exploring the utilization of nanotechnology and sun-based radiation to further develop flight effectiveness. In this analysis, a hybrid nanofluid is moving over an expandable sheet. Analysts are presently exploring the utilization of nanotechnology and sunlight-based radiation to further develop avionics productivity. To explore the heat transfer rate phenomenon, a hybrid nanofluid stream is moving towards a trough having a parabolic type shape and is located inside of solar airplane wings. The expression used to depict the heat transfer phenomenon was sun based thermal radiation. Heat transfer proficiency of airplane wings is evaluated with the inclusion of distinguished effects like viscous dissipation, slanted magnetic field and solar-based thermal radiations. The Williamson hybrid nanofluid past an expandable sheet was read up for entropy generation. The energy and momentum expressions were solved numerically with the utilization of the Keller box approach. The nano solid particles, which are comprised of copper (Cu) and Graphene oxide, are dispersed utilizing SA (Sodium alginate) as an ordinary liquid (GO). A huge number of control factors, for example, temperature, shear stress, velocity, frictional element along with Nusselt number are investigated in detail. Intensification of thermal conduction, viscous dissipation and radiation improve the performance of airplane wings subjected to heat transmission. Hybrid nanofluid performance is much better than the ordinary nanofluid when it comes to heat transmission analysis.

www.nature.com/scientificreports/ of an optimal mixture regarding the concentration of nanoparticles of hybrid nanofluid. Yıldırım and Yurddaş 65 examined the performance of heat transfer regarding solar collector of U-tube form with the employment of hybrid nanofluid (SiO 2-Cu), then made a comparison of it with Cu/water nanofluid performance. According to the results, the addition of SiO 2 nanoparticles into Cu/water nanofluid resulted in improving the heat transfer ability of later along with eliminating the problem of precipitation. Yan et al. 66 employed a non-Newtonian and two-phase model to study the effect of U-shaped tube absorber regarding thermal and hydraulic performance about parabolic solar collectors working with two fluids including hybrid nanofluid. Further, an innovative study representing the heat transmission features of hybrid nanofluids considering various physical situations has been inspected by [67][68][69][70][71][72][73][74] . According to a previous literature survey, a few attempts has been made for analysis of entropy generation using hybrid nanofluids in solar systems. The performance of thermal systems is evaluated through analysis of entropy production. Armaghani et al. 75 analyzed entropy generation of MHD and mixed convection Al 2 O 3-Cu/ water hybrid nanofluid flow in the L-shaped enclosure. The same hybrid nanofluid was utilized by Kashyap et al. 76 for analysis of entropy generation as well as the effect of three different boundary constraints with a two-phase model. Moreover, artificial neural networks were employed by Khosravi et al. 77 for the prediction of entropy generation about hybrid nanofluid which is flowing in microchannel fluid block. Optimization of entropy production, as well as heat transference of non-Newtonian hybrid nanofluid flow, was done by Shahsavar et al. 78 in a concentric annulus. A flattened tube was exploited by Huminic et al. 79 for entropy production analysis considering two-hybrid nanofluids. There is a greater performance of heat transference of MWCNT and Fe 3 O 4 driven water hybrid nanofluid as compared with ND + Fe 3 O 4 /water. However, heat transfer was increased and production of system entropy was reduced when it comes to the base fluid. Similar explorations of entropy generation on the nanofluid with stretched sheet assuming various geometries is administered in [80][81][82][83][84] .
The designation of Fig. 2 is such as to study the effectiveness of solar energy-based aircraft wings. The trough has a parabolic shape called (PTSC) placed inside solar aircraft wings and collects solar thermal energy in the form of solar radiative scattering. Heat transmission on solar aircraft is investigated using hybrid nanofluids and analytical expressions of mathematics in the supplied study, which is said to be the first-ever endeavor. Aviation research will have a significant impact on the hunt for economically costly and alternative fuel supplies. By substituting standard nanofluids with the well-established hybrid nanofluids, the heat transfer rate amplifies. The work's results will be useful for new researchers because they were acquired using all cuttingedge material conditions.
The offered study can cover the gap lying within heat transfer using changing thermal conductivity and temperature as well as radiative Williamson hybrid nanofluid flow on a penetrating stretchy surface. Tiwari and Das model for nanofluids is employed to represent the mathematical flow of the nanofluid. In this work, hybrid nanoparticles of Graphene oxide (GO) together with copper (Cu) are employed as hybrid nanoparticles, with sodium alginate (SA) as the base fluid. The flow influence will be measured using entropy generation analysis. Hybrid nanoparticles are used in the research. The governing modelled equation of the current experimental model (Williamson hybrid nanofluid) will be converted into ordinary differential equations employing suitable similarity variables. The well-established and reliable numerical method labelled as Keller box method will be adopted to handle the system resultant ODEs utilizing the most relevant values of governing distinguished  www.nature.com/scientificreports/ parameters. To better illustrate the numerical outcomes, graphs will be shown. The effects of solar-based thermal radiations, slippage effect at the surface of the sheet due to convection phenomenon, and slippage effect of the moving fluid will be thoroughly investigated.
Aims regarding proposed model. In light of the current investigation, adopting the flow of hybrid nanofluid across a PTSC improves aircraft performance. Many PTSCs are positioned throughout the aircraft. The following are the reasons for this investigation: In the suggested paradigm, PVC cell sheets are substituted with PTSC. Because PTSC is a type of cylindrical form having a bigger surface area than typical PVC sheets, having more ability to collect and store solar energy. Solar aircraft may be manufactured and maintained at a minimal cost, making them economically viable. Scholars are striving to depend on solar-based thermal energy, notably in the sector of airplane manufacturing, as the price of oil constantly rises. According to the results of the experiment, introducing hybrid nanoparticles within the fluid moving across a PTSC improves heat transmission and delivers enormous energy. Thermal conduction and radiation, as well as viscous dissipation processes, are also present. Solar aircraft is environmentally friendly in comparison to other aircraft and does not pollute the atmosphere in any way.

Formulations regarding flow model
The moving horizontal plate with irregular expanding velocity and isolated surface temperature are characterised as follows in the mathematical flow equations 85 : where b and b * are original expanding rate and temperature variation, recpectively. Y = w and Y = ∞ represent the temperature of surface and surrounds respectively. The plate is supposed to be slippery and temperature change is imperiled to the surface. Further, the hybrid nanofluid is expressed at 1 st with the addition of Copper (Cu) solid nano-particles in SA-based liquid at a interaction volume fraction ( φ β ) and its value was fixed at 0.09 while testing. Graphene oxide GO nanomolecules have been improved in mixture to get hybrid nanofluid having concentrated size ( φ ).
Suppositions and terms of model. Following are the principals as well as conditions applicable to the flow model: • 2-D flow having laminar and time-dependent features.
• Flow with thermal radiative features.
The tensor of stress in Williamson type is specified as 86 where where τ ij , µ 0 , µ ∞ , ζ > 0 and A β represent the additional stress-tensor, limited viscidness when shear rate is zero, limited viscidness when shear rate is infinite and the 1st Rivlin-Erickson tensor respectively. γ is: Herein, we presumed µ ∞ = 0 and γ < 1 . Thus formula (3) can be inscribed as Which can discribed as follows by used binomial-expansion Inside PTSC, geometry of flow model is presented in Fig. 3 as:  88 reported the associated boundary constraints: Fluid velocity in vector form is well-defined as ← G = G 1 x, y, t , G 2 x, y, t , 0 . Time is denoted by t , Y = represent temperature of the fluid. Here N w signifies the slip length. V w is representing the porosity of the encompassing plate while k 0 indicates the porousness of material.
The expressions of Table 1 summarize WNF variables of the material 89 .

Characteristics Nanoliquid
Dynamic viscosity (µ) www.nature.com/scientificreports/ φ is representing size coefficien of nano solid-particle. µ f , ρ f , (C p ) f and κ f are dynamic viscidness, intensity, operational heat capacitance as well as thermal conductance of standard fluid correspondingly. Extra factors ρ s , (C p ) s and κ s are intensity, effective heat capacity and thermal conductivity of nanomolecules, respectively.
Content of WHNF variants is described in Table 2 90,91 .
The thermo-physical material features of sodium alginate, copper and graphene oxide used in the numerical computation during research study have been reported in Table 3 92,93 .
Ensuing the research study of Brewster 94 , for an optically thick nanofluid, the radiation heat flux q r is written employing Rosseland approximation as: where σ * designates the Stefan-Boltzmann constant and k * shows the absorption coefficient.

Features Hybrid nanofluid
Thermal conductivity (κ) Table 3. Thermo-physical characteristics.  (2) is accurate. The representation ′ is demonstrating derivatives with respect to χ . The default values of emerged flow parameters alongwith their expressions are mentioned in Table 4.
Drag force C f along with local Nusselt number (Nu x ) is indicating the potential awareness which is controlling flow and is provided in detail as 87 here τ w and q w are signifying thermal flux which is Following expressions are attained when dimensionless transformations (15) are employed where C f embodies the drag force coefficient. Re x = u w x ν f is the local Reynold's number according to the elongated velocity u w (x).

Keller-box technique
As the convergence of the Keller-box method (KBM) 95 can be obtained rapidly, solutions regarding governing equations of the model are obtained by employing it (Fig. 4). Localized solutions of (16)-(17) with constraints (18) can be gained with the help of KBM. The steps of KBM are specified as below: Stage 1: ODEs modification. At first, governed ODEs are moldified to 1st-order ODEs (16)-(18)  www.nature.com/scientificreports/ Stage 2: domain discretization. The estimated solution can be computed when the domain procedure is discretized. Normally, discretization helps the field to divide into equal sizes of the grid (Fig. 5). High estimations result in less grid with the help of the computational outcomes.
The j indicates the position of the coordinates that are used in h-spacing in horizontal direction. As there is no initial estimation, so obtained solution is unsertain. So = 0 and = ∞ are initial values to guess, as temperature variations, speed, temperatures and entropy outlines are important to find out. Estimated solutions are obtained when resulting outcomes are found, provided that they satisfy the boundary conditions regarding the problem. According to observation, when many initial estimations are being done, final results may seem to be equal. However, varied approximations are conducted when time and iterations are taken.
Centre difference formulae are employed to obtain differences formulas. Replacements are done for average functions. The 1st order ODEs (25)- (29) are converted into the next series of nonlinear algebraic expressions.  Following linear equation is obtained as we substituted the formula into Eqs. (31)(32)(33)(34)(35) and ignored the higher orders of i j from 2 to above.

The boundary conditions become
The method is complete when boundary conditions are satisfied even for the whole set of iterations. So initial estimation is employed to obtain the actual values of every iteration.
For j = 1; In array structure,

That is
For j = 2; (a 1 ) 2 �f 2 + (a 2 ) 2 �f 1 + (a 3 ) 2 �z 12 + (a 4 ) 2 �z 11 + (a 5 ) 2 �z 22 + (a 6 ) 2 �z 21 In array arrangement, That is where R is shown as J × J tridiagonal-block matrix for all 5 × 5 bulk-size, meanwhile together with p are vector columns of order J × 1 . Solution of is obtained by factorizing LU. Factorization can be done only if matrix R is non-singular. However,p vector is obtained when R = p works on a vector, that is directed to tridiagonal block matrix R. Next, factorization of bulk tridiagonal matirx R is devided into upper as well as lower triangular matrices. In this way, R = LU is written as LU = p . Assume that U = y results in Ly = p . So y can be obtained. solution for is obtained with putting solution of y again into U = y. In triangular matrices, replacement is done for further work.

Verification of code
Verification is done for acquired results with the help of comparison with available literature 96,97 . Comparison of consistencies available in studies is summarized in Table 5. However, highly accurate outcomes about the present analysis are obtained.

Second law of thermodynamics: entropy generation
Entropy formation for considered model is 98 : Entropy analysis has following dimension-less expression: By formula (15), the non-dimensional entropy formula is: Here R e is representing Reynolds number, Brinkmann number is B r and symbolizes the dimensionless temperature gradient.

Results and discussion
The features of two kinds of sodium alginate-based nanofluids e.g., Cu-SA and GO-Cu/SA are numerically scrutinized by employing the Keller-Box Technique. To authenticate our computational technique, we equate our simulated consequences with the analytical ones, for the limiting case of classical Newtonian flow in the present geometry. The upshots of emerging flow parameters are enumerated via different graphical and tabulated numerical findings. Figures 6 and 7 correspondingly evident sudden increase in entropy rate as well as thermal status with respect to induced temperature in domain of flow employing radiation process represented by factor of thermal radiation N r . Furthermore, there is relatively little effect of radiation on fluctuations in entropy, which may be due to a strong influence on flow constraints Fig. 7. In this regard, Cu-SA nanofluid has a greater potential than GO-Cu/SA hybrid nanofluid. For hybrid GO-Cu/SA as well as Cu-SA fluid combo of nanofluids, variations in thermal as well as entropy formation regarding Eckert numbers ( E c ) is represented in Figs. 8 and 9 respectively. Eckert number presents thermal fluctuation as well as entropy in both conditions. Internal friction of a fluid, when it is mixed with a temperature of the surface, the thermal condition regarding fluids improves. Effect on  www.nature.com/scientificreports/ the temperature profile of Biot Number B i is portrayed in Fig. 10. The graph shows that the rising attitude of Biot Number B i assessed the temperature profiles. For small entities of B i , thin thermal strips are related i.e., usually, there are uniform temperatures in the body (nano polymer surface). Generally, Biot number too greater than 1 indicates thermally thick situations whenever temperature non-uniformity is obtained. Graphical behviour of N G aginst progressive values of Biot number B i in Fig. 11 explores that it is insensitive (gradual increase) to vary at   www.nature.com/scientificreports/ the surface as compared to away from it. i.e. less enhancement is observed near the stretching walls. Away from the surface, when deceleration in entropy generation is substituted with increasing of Biot number. Figures 12, 13, and 14 clarify the viscidness-depended impacts over Williamson parameter . More than the flowing, thermal and entropy establishment of Cu-SA WNF together with hybridized form of GO-Cu/SA. Figure 12 shows flowing nature of together liquids for differing values of . In both cases, the material parameter    Fig. 14. Surprisingly, the conventional NF has more entropy production distant from -the wall. Nanoparticle volumetric fraction parameter φ effect toward velocity is shown in Fig. 15. As φ intensified, the speed of the fluid flow is lessened. This occurrence happens due to fluid viscosity rising with growing nanofluid concentration, friction escalations as well. The hybrid nanofluid has an advanced velocity than conventional nanofluid as φ augmented. It is found that the temperature of the system amplified along with φ in Fig. 16. It is value mentioning that fluid velocity is critical for heat transmission. The movement of the particles in the fluid slowed down will cause the heat to accumulate in the system. Hence, the temperature of the system doubles up Fig. 16. It is also expected that the system's entropy will amplify due to heat accumulation-this evidence is illustrated in Fig. 17. The results in Fig. 17 demonstrate that the nanoparticle volume fraction substantially influences the produced entropy. On the other hand, hybrid nanofluid generates less entropy than nanofluid, as seen in Fig. 17. This finding implies that hybrid nanofluids can better control the entropy system than nanofluids. Figures 18, 19, and 20 presented the consequences due to improved slip conditions on flow nature, thermal aspects as well as entropy formations correspondingly. In Williamson fluid combinations, the flow situations mainly focused around the viscous behaviour. Along with it, the slip conditions becomes more crucial in all fluids facets such as velocity variations, thermal distributions as well as entropy generations. It can be noted that the viscous nature of the Sutterby fluid along with increased slip flow conditions, creates tougher situations for fluidity and makes it reducing further for single suspended nanofluid than that of hybrid suspended Williamson nanofluid. This flow hierarchy reflects in thermal distributions like Cu-SA nanofluid holds a greater thermal state as compared to GO-Cu/SA hybrid nanofluid (Fig. 19). A descending trend can be evident in entropy formation for larger amounts of slip parameters due to slip flow acting opposite to entropy formation across the domain       www.nature.com/scientificreports/ (Fig. 20). The numerical observations of skin friction coefficients and wall temperature gradients against related flow parameters have been presented in Table 6.

Final outcomes
In the current investigation, distinguished effects are investigated on hybrid nanofluid flow past a PTSC in solar-powered airplane wings. The present study is driven by the need to improve the phenomenon of solar energy, which will then be used in solar aviation for a variety of applications and a rise in aircraft perseverance. For this objective, Williamson hybrid nanofluid is considered. Tables along with plots are completely inspected and displayed for various parametrical effects: slated magnetic field effect, viscosity-based dissipation as well as the thermal liquid on PTSC together with a solar energy-based airplane. Coming up next are some significant outcomes from the current examination. Williamson hybrid nanofluid (GO-Cu/SA) is observed to be a better thermal conductor than ordinary Williamson nanofluid (Cu-SA). Velocity is diminished with a swelling impact of , φ , and φ hnf . Higher concentration of nanoparticles resulted in an increasing heat transfer rate. System entropy is enhanced with increasing values of material factor, Reynolds number R e , thermal radiation factor N r , Brinkman number B r along with nanoparticle volumetric concentration parameter hnf while reduction is observed with the rise in parametric values of material as well as velocity slip.
Future guidance. Outcomes of the current study can help in future improvements where the heat effect of the heating system may be assessed by taking into account different non-Newtonian hybrid nanofluids (i.e., Carreau, second-grade, Casson, Maxwell, micropolar nanofluids, etc.). Additionally, impacts of porosity, as well as viscosity that is dependent on temperature together with magneto-slip flow, can be represented by expanding technique.

Data availability
All data generated or analysed during this research investigation are included in this research article.