Analysis of entropy production in a bi-convective magnetized and radiative hybrid nanofluid flow using temperature-sensitive base fluid (water) properties

The heat transport characteristics, flow features, and entropy-production of bi-convection buoyancy induced, radiation-assisted hydro-magnetic hybrid nanofluid flow with thermal sink/source effects are inspected in this study. The physical characteristics of hybrid nanofluids (water-hosted) are inherited from the base liquid (water) and none has considered the physical characteristics of base liquid (water) in the study of temperature-sensorial hybrid nanofluid investigations, though the water physical characteristics are not constants in temperature variations. So, the temperature-sensorial attributes of base liquid (water) are taken into account for this hybrid nanofluid (Cu+Al2O3+water\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Cu+{Al}_{2}{O}_{3}+\text{water}$$\end{document}) flow analysis. The mathematical forms of the flow configuration (i.e., the set of coupled, nonlinear PDE form of governing equations) are solved by utilizing the subsequent tasks: (i) congenial transformation; (ii) quasilinearization; (iii) methods of finite differences to form block matrix system, and (iv) Varga’s iterative algorithm. The preciseness of the whole numerical procedure is ensured by restricting the computation to follow strict convergence conditions. Finally, the numerically extracted results representing the impacts of various salient parameters on different profiles (F,G,H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F, G, H$$\end{document}), gradients, and entropy production are exhibited in physical figures for better perception. A few noticeable results are highlighted as: velocity graph shows contrast behaviour under assisting and opposing buoyancy; temperature (G(ξ,η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(\xi ,\eta )$$\end{document}) is dropping for heightening heat source (Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document}) surface friction remarkably declines with the outlying magnetic field (St\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$St$$\end{document}); thermal transport confronts drastic abatement under radiation (R1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}_{1}$$\end{document}), and St\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$St$$\end{document}; the characteristics Reynolds and Brinkman numbers promote entropy. Furthermore, the bounding surface acts as a strong source of SG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S}_{G}$$\end{document}-production. Summarizations are listed at the end to quantify percentage variations.


Governing equations
From Table 1, µ f and (Pr) f can be approximated at different temperatures as 41,42,44 (1) µ f (T) = 1 a 1 + a 2 T , www.nature.com/scientificreports/ where constant coefficients obtained from the curve fitting of thermos-physical data of water at various temperatures are b 1 , b 2 , c 1 and c 2 defined as: The hybrid nanofluid-base liquid correlations for various physical characteristics are given below 46 Here sf = 3 stands for nanoparticles' shape factor ( is the sphericity) (see Table 2) and the other terms φ, µ hnf , ρ hnf , β hnf , C p hnf , k hnf , σ hnf , µ f , ρ f , β f , C p f , k f , σ f , φ s1 , ρ s1 , β s1 , C p s1 , k s1 , σ s1 , φ s2 , ρ s2 , β s1 , C p s2 , k s2 , σ s2 are all given in the Nomenclature. Table 2 shows that the variation in ρ f , C p f with respect to temperature is less than 1% . Combining this fact with the correlations (Eqs. [4][5] can be easily prove that the variation in ρ hnf , C p hnf is also less than 1% . So, from practical point of view, these two physical quantities can be considered as constant (see Table 3).
Consider a 2-D bi-convective (incompressible and steady) water-based hybrid-nanofluid flow for an arbitrarily inclined plate with vertical inclination γ and let the axes x and y are along the surface and perpendicular to it, respectively (see Fig. 1). The convective variation in temperature from the wall to the ambient fluid is deemed moderate ( < 40 °C) and an outer magnetic field normal to x-axis is applied under thermal sink/source and radiation effects. Using Oberbeck-Boussinesq approximation, the equations representing the physical characteristics of the flow become 10,51,52  44,45 .

Properties Copper Alumina
with The non-dimensional parameters buoyancy ( ), Reynolds number(Re ), Grashof number(Gr ), Stuart number(St ), radiation(R 1 ), heat source(Q ) are defined, respectively, as follows: All the other constants and coefficients are prescribed below: www.nature.com/scientificreports/ Salient gradients. Friction ( C f x ).

Generation of entropy
The EG model for MHD hybrid nanofluid can be written as 53 : The first brace (HTI) includes the terms representing irreversibility for heat transfer, terms inside second brace (FFI) conveys the irreversibility for fluid friction. The characteristics entropy rate S 0 = is utilized to get the dimensionless form ( S G ) of total entropy ( S gen ) i.e., S G = S gen Here the notations = T T ∞ , and Br = U 2 µ f k f �T stand for temperature ratio and Brinkman number, respectively. The comparative study of relative irreversibility sources can be accomplished with Bejan number (Be). Mathematically, it is defined by

Numerical method and validation
The set of coupled non-linear Eqs. (14)(15) has been made linear by employing the quasilinearization technique and the equations turned into with the boundary constraints Here the system (17-18) is linear for iterative indices (k + 1) as superscripts with the coefficients: Irreversiblity due to heat transfer total local entropy . for fixed m, where N is the number intervals of this mesh system and the vectors, coefficient matrices are: where the entries of A n , B n , C n , and D n are defined as:  www.nature.com/scientificreports/ W 1 and W N+1 at the boundaries (at η = 0 and η = η ∞ ) become: Hereafter, Varga's algorithm 34 , as defined below, is used to solve Eqs. (20) with constraints given by Eq. (21).
where E n = {B n − A n E n−1 } −1 C n ; The numerical solutions are reached under the strict convergence criterion and compared in Table 4 with previously published works [54][55][56] and found in a friendly match-up (see Table 4).

Results and discussion
The investigation of bi-convective MHD flow in light of temperature-sensorial water properties with radiation, thermal suction/injection effects is accomplished in this manuscript considering Cu + Al 2 O 3 /water hybrid nanofluid as working fluid. The acquired outcomes are featured out graphically to analyse the flow features, transport characteristics and energy distribution in comprehensive approach.
Velocity. Figure 2 is plotted to display the variable behaviour of the flow intensity ( F(ξ , η) ) against the buoyancy force . It may be noted that F(ξ , η) increases with , sometimes overshoot occurs. In the physical aspect, assisting buoyancy force always surpluses pressure gradient in flow and enhances flow intensity. As numerical supporting evidence, it is seen for = 1 and = 2 at ξ = 0.5, η = 1.40 that the velocity overshoots are 15% and 33% , respectively. In contrast, F(ξ , η) decreases for < 0 , and in this case, for = −1.0 backflow is recorded within the region 0.0 < η ≤ 0.85, ξ = 0.5.
Gradients. Skin friction. The variation characteristics of friction coefficient (  Table 4. Comparison of current results with available works [54][55][56] in literature for the case of steady-state with ǫ = 0, φ = 0, γ = 0, = 0, B 0 = 0, q r = 0, Q 0 = 0 for −G η (0) at η = 0. ) in combination with nanoparticles' shape effects are portrayed in Fig. 5. The results indicate that the outlying force field ( St ) has a destructive impact on Nu x √ Re , and among all the considered shapes, spherical-shaped nanoparticles affect most. In particular at ξ = 0.5, the decrement in sphericity (i.e., increment in sf = 3 ) from 1.0 to 0.36 enhances Nu x √ Re almost by 7%. Figure 6 depicts the effects of thermal radiation ( R 1 ) on local thermal transport coefficient ( Nu x √ Re ) and it is clearly visible in the graph that Nu x √ Re is a decreasing function of R 1 . Basically, the increasing magnitude of R 1 directly enhances systems' temperature, and the fluid in BL tries to become thermally equipoise. Hence temperature gradient gets reduced, which results in less thermal transport. At the instant ξ = 1.0 , reduction in Nu x √ Re is 35% for imposing R 1 of strength 1.0.  www.nature.com/scientificreports/ Entropy production and Bejan lines. Figures 7, 8, 9, 10, 11 and 12 illuminate the contributions of different salient parameters on the productions of irreversible heats (entropy production S G ) and their respective shares on gross entropy. Figure 7 indicates that the rate of S G -production increases with Re , but Re 's contribution on S G is immensely high at the surface proximity. Physically, augmentation of Re increases the entropy generation S G due to fluid friction and heat transport (via inertia). For higher Re , fluid inertia augments thermal transport, i.e., HTI takes over the other irreversibility sources. In contrast for lower Re , as viscous force is high, FFI dominates the total S G close to the wall. Thus, the friction force gets mitigated within the boundary layer and HTI takes over the dominant place. Hence, Bejan lines for lower Re intersect the lines for higher Re within the boundary layer. Moreover, all the Bejan lines converge to zero at the boundary layer edge since HTI gradually reduces to zero at the edge of the boundary layer. It is also noticed that the surface plays a high intense S G -production source and is evidenced by the following specific calculation: at η = 0.0 , S G elevates by 46% for varying Re in 10 − 12.5 while the change is only 20% at η = 1.5 for the same variation of Re.  www.nature.com/scientificreports/ Figures 9 and 10 manifested the S G -production and Bejan line regarding different magnitudes of viscous heating ( Br ). As exhibited in Fig. 12, higher Br boosts S G at the wall's proximity but discloses an opposite trend away from the surface. Lifting up the Br value causes added viscous force to the fluid and enhances frictional heating. This frictional heating turns up excessive S G -production. This fact is also evidenced in Fig. 10, which shows lifted down Bejan lines for higher Br , which physically represents that most S G -productions are due to frictional heating (FFI), the associated entropy produced in other modes (i.e., HTI and DI) are comparatively less. Analysing the result data, 32% enhancement in S G is noticed for changing Br from 0.01 to 0.2 at η = 0.5. Figures 11 and 12 demonstrate how S G and Be get affected under the forces of buoyancy ( ). As one can point out from Fig. 11 that S G shows a growing trend for the increase of . The earlier discussions proclaimed that larger pushes the fluids to move faster generates excessive friction at the wall and hence the irreversibility enhances (via FFI, as shown in Fig. 12). Since the buoyancy effect is induced by the thermal imbalance between the wall and neighbouring fluids, the effect of is predominantly noticeable at the wall proximity. Hence, the irreversibilities due to variation vanish at the boundary layer edge and all S G -lines converge at the edge of the boundary layer.  www.nature.com/scientificreports/

Conclusions
This paper performs an analysis on a hybrid nano-liquid flow for an inclined surface under various realistic and practical physical situations by considering the basic temperature-sensorial inheriting characteristics (thermosphysical) of base fluid water. The bearings of flow features, thermal transport characteristics, and EG of magnetized bi-convective hybrid nano-liquid flow with nanoparticles' sphericity, radiation and thermal source/sink effects are studied in this investigation. The immensely nonlinear PDEs are changed into suitable form and then into linear form utilizing compatible transformation and quasilinearization techniques, respectively. Hereafter, implicit difference methods changed the resulting equations into a matrix system which was further solved by Vargas' block matrix iterative method. The acquired results of this study are manifested in graphs and discussed in details. The concluding remarks from the investigated results are summarized and expressed with numerical percentile calculations as observed in this specific study:  www.nature.com/scientificreports/ i. The trend of F(ξ , η)-profiles shows increment for assisting ( > 0 ) and decrement for opposing buoyancy ( < 0 ). In particular, for = 2 , almost 33% overshoot is observed when at η = 1.40, ξ = 0.5 but in contradiction almost 25% backflow is noticed at η = 0.4, ξ = 0.5 when = −1.0. ii. Temperature-profile ( G(ξ , η) ) rising along with the heat source strength Q.