Irreversibility scrutinization on EMHD Darcy–Forchheimer slip flow of Carreau hybrid nanofluid through a stretchable surface in porous medium with temperature‐variant properties

The significance of hybrid nanofluids in controlling heat transmission cannot be overemphasized. Therefore, this article scrutinizes the electromagnetized flow of Carreau hybrid nanofluid towards a stretching surface in a Darcy–Forchheimer porous medium with the occurrence of slip conditions. To form the hybrid nanofluid, the amalgamation of silver and alumina nanoparticles (NPs) embedded in water as conventional fluid is assumed. For accurate interception of the rate of heat and mass transport, thermal conductivity and mass diffusion conductance are presumed to be temperature variants. The modeling system of partial differential equations has been translated into a nondimensional form by means of suitable similarity conversions. Then, the subsequent system of ordinary differential equations is handled using overlapping domain decomposition spectral local linearization method to acquire numerical solutions. The choice of the method has been justified through the provision of errors, condition numbers, and computation time. The behavior of distinct fluid parameters on the flow features, quantities of engineering curiosity, and entropy is analyzed. Findings of paramount importance constitute that the superior thermal conductivity, heat transfer efficiency, and low production cost can be achieved through the hybridization of silver and alumina NPs. The role of thermal radiation and temperature‐variant thermal conductivity is to enhance the thermal transport performance of Carreau hybrid nanofluids. The velocity, energy, and mass profiles grow with the utilization of injection effects. The principal aspiration of the second law of thermodynamics (minimizing the rate of entropy generation) can be achieved by considering shear‐thinning Carreau fluid while reducing the porosity parameter and Brinkman number in the existence of velocity slip conditions in the flow system. Outcomes of the current flow model can play a significant role in biomedical, technological, and various manufacturing processes. The approximation of entropy contributes towards power engineering and aeronautical propulsion to anticipate the smartness of the overall system.

Thus far, numerous studies have been conducted on the inspection of flow and heat transfer features using the mono (single) NF model. However, there is no mono NF model that can have sufficient thermophysical properties for maximum efficiency. To try and fill this gap, the concept of adding two or more NPs onto the base fluid to achieve thermophysical properties that are not available in a single NF has been introduced. The resultant fluid formed by the addition of more than one NP onto the conventional fluid is called hybrid NF. 24,25 As an illustration, aluminum oxide (alumina) is extremely stable and demonstrates chemical inertness, whereas it possesses lower thermal conductivity. Alumina NPs are applicable in materials manufacturing industries and biomedicine, where such NPs can be utilized in drug delivery, biosensing, cancer therapy, demolition of microbes, and biomolecular stabilization. To achieve better thermal conductivity in the alumina NF, metallic NPs with high thermal conductivity can be added to the alumina NF. Metallic NPs are the best in heat transfer performance when compared with oxide NPs. However, metal NPs are expensive and not practically usable in mass production. As a result, the utilization of oxide NPs becomes more practical to the economy. The resultant hybrid NF will be a better thermal conductor and have better heat transfer capabilities than the alumina NF but have the lowest production cost than the metal NF. Among the available metal NPs with high thermal conductivity, we have silver NPs. Silver NPs have captured the attention of many researchers owing to their diverse applications in the biomedical field. Furthermore, owing to their wide-spectrum antimicrobial capability, silver NPs have been mostly utilized as sterilizing nanomaterials in consuming and medical products, such as textiles, refrigerator surfaces, and food storage bags. Zagabathuni et al. 26 experimentally investigated heat transfer features of aqueous alumina-silver hybrid NF and established that alumina-silver hybrid NF exhibits greater thermal conductivity increment when compared with mono-typed alumina NF. Limited studies have been reported on the contribution of hybridization of silver and aluminum oxide NPs in flow, heat, and mass transfer analysis through a wide range of geometries. Recently, Abbasi et al. 27 investigated stagnation point flow and heat transfer features in silver-alumina hybrid NF adjacent to a vertical cylinder. Abbasi et al. 27 concluded that thermal rate is enhanced by rising solid volume fraction of NPs. Also, the rate of heat transference was found to be higher in hybrid NF than in regular water, silver NF, and aluminum oxide NF. Gamachu and Ibrahim 28 explored the mixed convective flow of silver-alumina hybrid NF over a rotating disk in the existence of slip and convective boundary constraints. They found that thermal and mass distributions can be controlled using aluminum oxide and silver NPs volume fraction. Also, the 0.03 volume fraction of alumina and silver NPs possesses high heat transfer rate than the 0.01 volume fraction of alumina and silver NPs.
When accounting for the manufacturing process entailing the stretching sheet exercise, the quality of the finished product considerably rests upon the rates of stretching and heat transmission on the sheet. The initial research work regarding convective flow past a stretching sheet was carried out by Crane. 29 Later on, several scholars used the theory of NFs to extend the temperature, concentration, and mass transfer rate. Xia et al. 39 considered to be an oscillatory MHD convection flow of second-grade NF via the Darcy-Forchheimer model in a porous cylindrical medium with thermal radiation and an internal varying heat source. Increment in porosity, Forchheimer, and magnetic field parameters demonstrated a shrinking behavior in the velocity and temperature profiles. Shafiq et al. 40 scrutinized MHD Darcy Forchheimer flow of Casson NF due to a rotating disk. They also reported that the velocity field diminishes with improvement in Forchheimer number and porosity parameter. Rasool et al. 41 used the Darcy-Forchheimer model to inspect flow, heat, and mass transfer features in Jeffery NF past a stretched surface in the presence of the applied magnetic field. It is worth pointing out that factors of porosity and inertial force were found to cause decrement in velocity and mass transfer rate, but resulted in an increment in temperature and concentration distributions. Algehyne et al. 42 scrutinized the importance of Lorentz and inertial forces on radiative Al O 2 3 -H 2 O NF over a curved stretching surface with activation energy, velocity slip, heat, and mass boundary conditions. Rasool et al. 43 investigated electromagnetohydrodynamic (EMHD) Darcy-Forchheimer convection flow over a Riga plate in a porous medium. They concluded that Lorentz and inertial forces strengthen the produced frictional factor. Saeed et al. 44 investigated EMHD flow of couple stress hybrid NF in Darcy-Forchheimer porous medium with variable viscosity. They found that increment in porosity and inertia coefficient provides augmented resistive force for fluid movement, leading to decrement in the flow profile.
Mathematical analysis of entropy production is very essential since it is one of the key factors in fluid mechanics. The existing amount of work that deteriorates through flow irreversibility and brings about chaotic behavior in the system has a direct effect on the performance of thermal devices. For that reason, it is necessary to scrutinize the dynamics characterizing entropy generation to ensure that the efficiency of thermal devices is optimized. Wakif et al. 45 considered irreversibility analysis of dissipative EMHD fluid flow through a moving Riga plate with suction and Joule heating influences. They reported that the incorporation of wall suction augments the rate of heat transmission and thermodynamic irreversibility closer to the Riga surface, and the opposite was true for the magnetic field effect. Gul et al. 46 scrutinized entropy generation, EMHD flow, and heat transfer of couple stress hybrid NF past a stretchable surface. Their findings disclosed that hybrid NFs are the best thermal conductors and can be utilized for blood circulation in different human body parts. Naganthran et al. 47 investigated entropy generation and heat transfer analysis in non-Newtonian Carreau thin hybrid NF film flow past a stretching sheet. The rate of heat transference was found to dwindle when the Carreau fluid displays shear-thickening characteristics. Also, the entropy in the flow system was established to be higher in the hybrid NF than in the case of mono-typed NF.
In view of the aforementioned literature review, no study analyzed the overall volumetric entropy, heat, and mass transmission of non-Newtonian Carreau hybrid NF through Darcy-Forchheimer relation along with the Tiwari and Das NF model under the influence of the combined electric and magnetic field, suction/injection, slip conditions, varying thermal conductivity, and mass diffusivity. Therefore, the main purpose of the current scrutiny is to investigate entropy generation, heat, and mass transfer features in EMHD flow of radiative Carreau hybrid NF past a stretching sheet immersed in Darcy-Forchheimer porous medium with the assumption of temperature-variant properties, suction/injection, and slip conditions at the interface. The composite NF is manufactured by suspension of silver and alumina NPs in pure water as base fluid. These two NPs are mostly used as coolants in medical devices and diverse engineering applications. A set of differential equations is handled using the overlapping domain decomposition spectral local linearization method (SLLM). The other objective of the study is also to account for the superiority of the recently established overlapping domain division spectral collocation method over the usual single-fixed domain spectral collocation method when applied to the Carreau hybrid NF model. This will be achieved through the inspection of errors, condition numbers, and computation time. The behavior of several physical constraints on the flow, heat, and mass transport analysis is disclosed in the form of graphs and tables and then clarified in detail. The computed results are correlated with earlier reported works to demonstrate validity and reliability in the numerical approach utilized. This study is designed to answer the following important research questions: • What are the consequences of the hybridization of silver and alumina NPs on the thermophysical properties of the Carreau hybrid NF model in the existence of Lorenz and inertial forces? • What are the repercussions of the features of electromagnetism, radiative heat flux, suction/ injection, slip factors, varying thermal conductivity, and mass diffusivity on the flow of Carreau hybrid NF? • How is thermal transport capability influenced by thermal radiation, temperature-variant properties, heat generation/absorption, and Joule heating impacts? • What is the impression of activation energy on the motion of NPs along with mass transfer in the considered Carreau hybrid NF model? • How best can entropy minimization be achieved in the flow system to control energy loss?
• What are the advantages of using the overlapping domain decomposition SLLM instead of the single-domain approach when solving the current flow model?

| FORMULATION OF THE PROBLEM
In the current analysis, we account for the flow of Carreau hybrid NF through a stretching surface embedded in Darcy-Forchheimer porous medium. The geometry of the problem is displayed in Figure 1, where the Cartesian coordinate system is adopted in such a way that the x-axis is measured parallel to the stretching sheet in the direction of the fluid flow and the y-axis is considered normal to it. It is presumed that the stretching velocity of the sheet is linear and given by u ax = w in the x-direction, where the positive constant a (>0) designates the stretching rate. The amalgamated impact of Lorentz force and electric current on the stabilization of the fluid has also been considered in the flow regime. The constant magnetic field of intensity B 0 and uniform electric field of strength E 0 are imposed perpendicular to the stretching surface. The induced magnetic field is ignored due to a sufficiently small magnetic Reynolds number. The NF temperature and concentration at the surface of the wall are, respectively, designated by T w and C w , whereas the constant ambient temperature and concentration are signified by T ∞ and C ∞ . In the model, 6% silver NPs (Φ 2 ) are dispersed onto the base fluid (water) to give silver/water mono NF, which is scientifically designated by Ag/H 2 O NF. Then, 10% alumina NPs (Φ 1 ) are suspended onto the Ag/H 2 O NF to yield silver-alumina/water hybrid NF, which is symbolically denoted using Ag-Al O H To control thermal characteristics of the flow system, the consequences of Joule heating, viscous dissipation, exponential heat generation/absorption, and radiative heat flux are embraced. The velocity and thermal slip conditions are also implemented on the boundaries. The mass transfer is characterized by accounting for activation energy in the energy equation. The following equation yields the Cauchy stress tensor of the Carreau fluid model 48 : In Equations (1) and (2), identity tensor, pressure, zero shear rate viscosity, infinity shear rate viscosity, power-law exponent, material time constant, first Rivlin-Erickson tensor, and shear rate, respectively. In most practical instances μ ∞ is chosen as zero. 49 Consequently, the value of μ ∞ in the current scrutiny is taken as zero, which leads to Equation (1) becoming The values of n range between 0 and 1 (i.e., n 0 < < 1), which elucidates shear-thinning/ pseudoplastic liquids, and above 1 (i.e., n > 1), which represents shear-thickening/dilatant liquids. With the above situation and boundary layer approximations, the governing boundary layer equations for the Carreau hybrid NF can be written as 44,46 F I G U R E 1 Flow geometry of the problem and the subjected boundary conditions via the notion of slip theory are Here, u and v are the components of velocity along the xand y-axes, respectively, T and C are respective fluid temperature and concentration, k p is the permeability parameter, F C k = r b p − 1 2 is the inertia factor of porous medium with C b signifying the form of drag coefficient, B 0 and E 0 represent strength of magnetic and electric fields, respectively, Q 0 is the coefficient used for the heat absorption/generation, L 1 and L 2 represent velocity and thermal slip factors, respectively, V w is the mass flux velocity for the surface, k r designates the rate of reaction, E a stands for the activation energy, κ is the Boltzmann constant, m is used for the rate of exponent fitted constant, D μ ρ σ k , , , , hnf hnf hnf hnf hnf , and ρCp ( ) hnf designate respective mass diffusion, dynamic viscosity, density, electrical conductivity, and thermal conductivity and specific heat capacitance of hybrid NF.
The following transformations are used to make the governing equations nondimensional 44,46 : With reference to Table 1 and using the above transformations (9), thermal conductivity and mass diffusivity vary with temperature as 37,50,51 where δ a and δ b are respective variable thermal conductivity and heat conductance parameters ( Table 2). Using the above transformations, the governing equations are written as T A B L E 1 Hybrid nanofluid correlations

Properties Nanofluids Sources
Heat capacity

Base fluid Nanoparticles
Physical characteristics  4 , and H 5 are mathematically communicated as The related boundary condition can be written in the nondimensional form as are the respective dimensionless velocity and thermal slip parameters.

| Quantities of engineering curiosity
The material quantities for drag force, heat, and mass transports are the skin friction C f , local Nusselt number Nu x and local Sherwood number Sh x with their mathematical expressions given by 44 where the surface shear stress, heat, and mass fluxing are perceived as The dimensionless version of these quantities are

| Entropy generation
Entropy generation 45-47 for Carreau hybrid NF over a stretching surface with heat transmission characterized by radiative heat flux, Darcy-Forchheimer porous medium, thermal conductivity fluctuation, viscous dissipation, and electric and magnetic fields is mathematically disclosed as Making use of the similarity variables, the dimensionless form of the entropy generation becomes is the characteristic entropy generation and Br PrEc = is the Brinkman number.

| SOLUTION OF THE PROBLEM
The flow of the non-Newtonian Carreau fluid model results in nonlinear complex mathematical models due to the non-Newtonian nature. Such models are associated with complex flow equations that are very challenging to solve and obtain accurate solutions. In other words, such flow problems are described by strongly coupled and highly nonlinear differential equations, which yields to difficulty or even impossible to find closed-form solutions. Consequently, we have to rely on approximate solutions via numerical methods. Spectral methods have been confirmed in literature to be superior in giving the most accurate and convergent solutions with great efficiency than other traditional techniques (Keller box scheme, finite element methods, RK4-method, finite difference approach, etc.). Spectral methods can be modified to handle problems with nonlinear behavior resulting from severe deformations and materials that are nonlinear, and those problems in complex geometries and diverse kinds of boundary conditions. For single-domain spectral collocation methods, such accuracy and efficiency are only limited to smaller computational domains. Also, using many grid points fails to improve the accuracy, but instead it makes the situation worse.
Recently, researchers 14,53-57 tried to circumvent these limitations by implementing the overlapping grid idea with spectral collocation methods to ensure accuracy, stability, and computational efficiency are maintained when solving differential equations defined on large computational domains. These techniques are very powerful methods for the numerical computation of solutions to nonlinear boundary value problems. They consist of the following major steps, namely, linearization, domain decomposition and discretization, domain transformation, interpolation, and spectral collocation. Their main attractive features include simplicity in solving boundary value problems defined on semiinfinite domains, computational effectiveness, rapid convergence, superior accuracy, and stability, which are all achieved through the utilization of less number of grid points. The promising benefits of these methods appeal to more work to be done on generalizing and testing their robustness in various classes of complex differential equations with strong nonlinearity and coupling. The approach used to solve the current coupled nonlinear complex heat and mass transfer problem involves blending the concepts of domain decomposition via overlapping grid technique and univariate Lagrange interpolation in conjunction with spectral collocation-based local linearization method (LLM). The linearization is based on employing the truncated Taylor-series approximations to simplify nonlinear terms of nonlinear differential equations. In the LLM approach, one nonlinear function and its derivatives are linearized at a time in a sequential manner, starting with the first equation. The other functions and their derivatives are assumed to be known. The updated solution is then used in the subsequent equations. The process is repeated until the last equation. Thus, the linearized form of Equations (11)-(13) becomes where The first step in the implementation of the method involves truncating the semibounded domain [0, ) ∞ on which the governing equations are defined into a convenient interval η [0, ] max , where η max is a finite value that is selected to allow the method to be applicable at ∞. Next, the solution domain η [0, ] max  ∈ is decomposed into P subdomains by overlapping 1 grid point as (see Figure 2) where M + 1 η is the number of Chebyshev-Gauss-Lobatto (C-G-L) collocation points that are used in the numerical discretization of each subdomain. For easy implementation of the method, two restrictions are necessary. First, the number of collocation points in each subdomain must be the same. According to Yang et al., 58 the method becomes very complicated if the number of collocation points is varied in each subdomain. Second, the length of each subdomain must fixed if linear transformation is used, which appears in many studies 53-56 as follows: The spectral method is implemented in the interval [−1, 1], so it is necessary to transform the computational domain into [−1, 1] through the linear mapping Consequently, the physical coordinate η η η , within each subdomain. In the process of searching for the approximate solution, the solution that we require can be expressed as an interpolation polynomial in the Lagrangian form. Thus, within each subinterval, the solution of f η ( ) is approximated as where L η ( ) ι signifies the Lagrange basis polynomials. The crucial aspect in the implementation of the spectral method is the establishment of the differentiation matrix which is utilized in an approximation of derivatives of the unknown variables in each subinterval. Thus, the derivative of f η ( ) within each subinterval is given by the following matrix-vector product at the collocation points η (ˆ) In the overlapping grid approach, solutions are attained concurrently throughout all the subdomains. Since the grid points that overlap become common, the rows that correspond to those repeated grid points are discarded when assembling the matrix D. Thus, the differentiation matrix D demonstrates sparseness arrangement as follows: 1,1 ,0 , −1 , where the unoccupied entries contain zero elements. Higher-order derivatives are computed as powers of D as follows: where the vector function As a matrix system of dimension ( + 1) × ( + 1)   , Equation (31) become The matrix systems for Equations (32) and (33) are similar to Equation (34). The boundary conditions are also evaluated at the collocation points and the discrete forms of boundary restrictions are inserted into the matrix systems. The approximate solutions at the collocation points are acquired by solving the matrix systems. The functions taken as guesses for initiating the iteration procedure are which are favored because they are in complement with the entire boundary constraints.

| RESULTS AND DISCUSSION
In this segment of the article, the behavior of various flow constants on velocity, thermal, and concentration distributions, coefficients of skin friction, heat, and mass transport together with entropy generation rate is provided and discussed in detail. The default values of parameters are chosen to be fixed throughout the study as   61 Wahid et al., 62 Wang, 63 Khan and Pop, 64 Devi and Devi, 25 Waini et al., 65 and Yahya et al. 66 The comparison is provided in Tables 3 and 4, where clear consistency has been noted. Thus, the accuracy of the method and the considered model is approved. Table 5 is provided to assess the outcomes of a number of collocation points per subinterval on the accuracy and efficiency of the numerical scheme. It is clearly evident in Table 5 that there is a remarkable improvement with the use of multiple domains than considering onefixed domain. It is very much important to establish how best can the number of collocation points be distributed among the respective subintervals to ensure good accuracy and computational effectiveness. Table 5 illustrates that quick computation of accurate and stable solutions relies on the smallest number of collocation points per subinterval. Basically, the accuracy, stability, and computational efficacy of the numerical scheme are improved by minimizing the number of collocation points while maximizing the number of subintervals used in the numerical computation. This observation is supported by decreasing errors, condition numbers, and calculation time with minimal grid points per subdomain. The best results are noted when two grid points per subdomain are used with 60 subdomains. These best results are accredited to the use of less grid points ( = 61  ) in the entire calculation domain. Figure 3 conveys the position and number of nonzero components in the coefficient matrix C. The figure indicates that when there is no partitioning of the main domain, the coefficient matrix is full of nonzero entries, whereas in the partitioned domain with many subintervals, there is a dominance of zero elements in the coefficient matrix. This implies that partitioning the main domain introduces sparsity in the coefficient matrix. Since the coefficient matrix in the overlapping domain decomposition approach contains mostly zero values, then performing operations across this matrix can take short time as computation will involve multiplying zero values together. This is clearly confirmed in Table 5, where the execution time is less in the multidomain approach than in the one-domain approach. The computational time reduces significantly with increasing P since additional subdomains reduce the size of the differentiation matrices and implies less nonzero entries in the coefficient matrix as seen in Figure 3. Calculating with these reduced size differentiation matrices contributes towards the attainment of solutions with reduced round-off errors, but best spectral accuracy and stability.   for multidomain approach ( Figure 4B) justify the attainment of accurate solutions after very few iterations and expected the same total number of collocation points. It is noticeable that there is a significant accuracy improvement with the method implemented on multiple domains. Figures 5-14 disclose the velocity, energy, and mass profiles for regular fluid, single NF, hybrid NF, and distinct fluid parameters in both cases of suction (γ < 0) and injection (γ > 0). In all figures, the velocity, temperature, and concentration profiles are seen to be higher in the case of injection compared with suction fluid. This implies that momentum, thermal, and solutal boundary layer thicknesses improve with the injection fluid. The suction of the fluid leads to deceleration in the speed of moving particles owing to mass transfer at the suction of the wall, whereas heat and mass contents in the particles are dominant compared with the case of injection. These results imply that the suction/injection process can be an efficient tool for controlling flow dynamics, rates of heat, and mass transport.
It is noticeable in Figure 5A,C that low velocity and concentration correspond to hybrid NF, whereas high velocity corresponds to conventional fluid (water). The addition of single NPs onto the regular fluid dwindles momentum and solutal boundary layer thickness, thus decelerates the fluid motion. This phenomenon is due to the enhancement of viscosity when the fluid becomes concentrated with NPs. The resistance to movement of fluid particles becomes severe with the suspension of more than one NP (hybrid NF) since the fluid will be more concentrated with NPs. On the other hand, it is noticeable in Figure 5B that the base fluid has the lowest temperature distribution of both single NF and hybrid NF. However, when silver NPs (mono-typed NF) are dispersed onto the base fluid, the temperature distribution improves. This is because the existence of silver NPs contributes towards the dissipation of heat energy. When silver and alumina NPs (hybrid NF) are orderly suspended onto the base fluid, extra radiation occurs, which promotes a significant increment of thermal boundary layer thickness. The hybrid NF temperature is expected to be higher than that of conventional fluid and monotyped NF due to the fact that hybrid NF constitutes more NPs. These findings indicate that velocity, thermal, and mass distributions can be controlled using alumina and silver NPs. Such a conclusion was also made by Gamachu and Ibrahim. 28 Also hybrid NFs including the Ag-Al O H 2 3 2 ∕ O have been confirmed as an extraordinary replacement capability in medium temperature utilization such as solar collectors and heat exchangers structure as improved heat transfer fluids.  Figure 6A that intensification in Φ 1 leads to a decrement in velocity profiles. As explained earlier, the reduction in velocity profiles is due to the fact that viscosity augments with an increment in Φ 1 , leading to remarkable opposition to the fluid movement. From Figure 6B, an improvement in temperature profiles is evident with augmentation in Φ 1 . This rise is due to the improvement of thermal conductivity and density of solid NPs with escalating Φ 1 , which boosts the entire thermal conductivity and density of the hybrid NF. Physically, alumina NPs possess high thermal conductivity such that the addition of extra alumina NPs provides more energy that raises the temperature and thickens the thermal boundary layer, which in turn leads to heat relocation. The enhanced NP concentration from rising Φ 1 is what causes a reduction in the concentration field as seen in Figure 6C. Figure 7 is plotted to inspect the outcome of velocity slip (β f ) and porosity (Λ) parameters on the velocity of hybrid NF. It is portrayed in Figure 7A that the no-slip condition (β = 0 f ) has a higher velocity compared with the case of the velocity slip condition of the first order (β > 0 f ). Also, the velocity profiles and momentum boundary layer thickness are evident to depreciate with escalating β f . This happens due to the fact that augmentation in lubrication and slippery at  Figure 7B demonstrates that the velocity field dwindles with escalating Λ. This situation is justified by the physics that rising values of Λ lead to intensification of void spaces in the medium, which in turn provides extra resistive force to the movement of fluid particles. In this procedure, the flow speed of the fluid is minimized. These results are in satisfactory accordance with those outlined in. [38][39][40]43,44 Figure 8A illustrates that improvement in the strength of the electric field leads to advancement in the velocity profiles. This is because the addition of the electric field term in the momentum equation augments the momentum boundary layer. With the inclusion of the electric field, there is an opposition in the resistance to the movement of fluid particles. It is unveiled in Figure 8B that increasing values of Darcy-Forchheimer number (F c ) contribute towards the production of resistive force to the movement of the fluid. This causes moderation in the motion of fluid and decrement in the velocity field. Findings from Figure 8 are in coincidence with outcomes disclosed by Saeed et al. 44 and Gul et al. 46 Figure 9 discloses the impact of varying the Weissenburg number We ( ) on the velocity field of hybrid NF for both cases of shear-thinning (n < 1) and shear-thickening (n > 1) Carreau fluid. In both cases of suction and injection, the velocity profiles drop in the shear-thinning fluid. Physically, We signifies the correlation between relaxation time and the exercise by which time augments the viscosity of the fluid. An increment in We signifies enhanced relaxation time, which means that the reaction of Carreau fluid with external forces uses much more time. Thus, dominance in the repercussions of viscosity brings about resistance within the fluid particles, which leads to a reduction in hybrid NF velocity. The opposite scenario takes place in the shear-thickening Carreau fluid.
It is shown in Figure 10A that an increment in the thermal slip parameter (β T ) boosts the thermal field in injection fluid while reducing the temperature distribution of the suction fluid. The rise in thermal gradient and thermal boundary layer thickness is owed to the fact that greater values of β T lead to a decrement in resistance of fluid motion at the surface. In the case of suction, minimal heat is transported from the surface to the fluid as the strength of the thermal slip condition magnifies, leading to a decrement in hybrid NF temperature. The opposite is true for the injection fluid. Figure 10B discloses that the thermal field improves with rising porosity parameter. From the physical perspective, the inclusion of porosity term augments kinetic viscosity and the extent of the stretchable surface area. The utilization of a porous medium provides enough space for more entry of heat onto the flow of the Carreau fluid. Thus, as heat augments, energy profiles turn to be impacted and start to be enhanced. Figure 11 is provided to convey the behavior of radiation Rd ( ) and heat generation/ absorption (Q) parameters on energy profiles of hybrid NF. A major boost in thermal radiation implies that extra heat is introduced into the system. For the case of injection in Figure 11A, the thermal field is seen to decay near the surface and then boosted away from the surface with escalation values of Rd. However, for the suction fluid in Figure 11A, thermal distribution enhances throughout the flow regime with increasing values of Rd. The rise in thermal gradient is due to the fact that intensification of Rd transmits more heat into the fluid and bring about augmentation in thermal boundary layer thickness. As thermal radiation improves near the surface of the injection fluid, additional energy in form of heat is being dismissed from the body of the fluid to nearby space, thus hybrid NF temperature diminished remarkably. It is evident in Figure 11B that the temperature profiles of hybrid NF increase from the case of heat absorption to the instance of heat generation. An increment in values of Q means extra heat energy is administered to the area of the boundary layer, which is responsible for accelerating the movement of fluid particles and enhancement of thermal boundary layer thickness. Figure 12A demonstrates that in the case of injection, the thermal field minimizes near the surface while maximizing away from the surface with the consideration of temperature-variant thermal conductivity. For the case of injection, thermal distribution improves throughout the flow regime with the presumption of varying thermal conductivity. Increment of thermal conductivity parameter (δ a ) leads to higher thermal conductivity, thus additional heat is transmitted from the plate to the fluid. This in turn results in the advancement of hybrid NF temperature with corresponding thermal boundary layer thickness. This discovery implies that to improve heat transfer attributes in diverse thermal extrusion processes, the involvement of temperature-variant thermal conductivity can be of paramount importance. Figure 12B elucidates that thermal distribution improves with increasing values of the Eckert number (Ec). The higher values of Ec signify more kinetic energy for NPs which behave as an operating force for greater heat transference rate. The frictional heating emerges at the surface producing escalation in the hybrid NF temperature. Figure 13A reveals that mass distribution strengthens with escalating variable heat conductance parameter δ ( ) b within the mass boundary. This is because rising δ b contributes towards the improvement of mass diffusivity of the fluid particles, which in turn enhances the mass boundary layer thickness. It is unveiled in Figure 13B that mass profile and relevant boundary layer thickness augment with increment in the porosity parameter (Λ). This phenomenon emerges because escalating viscosity slows down the fluid motion, thus expedition of the concentration can gradually intensify.
The disparity of chemical reaction (K c ) and activation energy (E 1 ) parameters across mass distribution is confirmed in Figure 14. It is noticeable in Figure 14A that mass profiles are diminishing against mounting values of K c . Such a situation is made possible by the fact that strong chemical reaction results in a negative outcome that contributes towards the decaying of reactant species. It is witnessed in Figure 14B that amplification in E 1 brings about enhancement in mass profiles. This situation arises because mounting values of E 1 cause a dwindling in the coefficient K exp(− ) Physically, the rate of reaction is weakened by excessive activation energy, leading to resistance in the process of a chemical reaction.
The role of β f and Λ on entropy generation profiles is depicted in Figure 15A. It is confirmed in the figure that entropy generation diminishes with escalating values of β f . Such decrement in entropy generation rate signifies cooling down of the system. When β f is increased, temperature gradients within the fluid boundary layer dwindle while maintaining the fluid friction. This results in depreciation in entropy produced due to commitment to heat transmission. It is expected that if entropy inside the boundary layer diminishes, it should augment by the same magnitude outside the boundary layer. From Figure 15B, the entropy generation profiles are elevated with escalating values of Λ. This is because higher values of Λ promote fluid flow resistance, thus enhancing the entire entropy production rate.  Figure 16A is plotted to highlight the impact of Weissenburg number We ( ) on the entropy generation profiles for shear-thinning and shear-thickening Carreau fluid. Entropy generation is evident to be lower for shear-thinning Carreau fluid than shear-thickening Carreau fluid. The condition of the Carreau liquid conveying shear-thinning characteristic and then depicting shear-thickening attribute produces more energy in form of heat onto the system. The rate of entropy is noted to be maximized in shear-thickening feature with rising values of We. Physically, more heat is lost during the movement of particles because of an increment in relaxation time. The reverse behavior of We on the rate of entropy generation is witnessed in shear-thinning Carreau fluid. The contribution of the Brinkman number (Br) to entropy production is unveiled in Figure 16B. It is noticeable in the figure that entropy generation is boosted by intensifying values of Br. The Brinkman number signifies the ratio of heat produced by fluid friction to heat transference through the thermal conduction process. Thus, augmenting values of Br yield higher temperatures since heat production through viscous dissipation is not rapidly transported by conduction. As a result, temperature variation among the system and neighboring is augmented resulting in more heat transmission, which by that means enhances entropy production. Since the Brinkman number along with thermal radiation are contributors to Joule heating that administers excessive heat into the system, their intensification is capable of causing chaotic behavior in the entire system. Table 6 is provided to unveil the behavior of quantities of engineering curiosity when values of NP volume fraction (Φ 1 and Φ 2 ) are varied. In Table 6, the third row corresponds to regular fluid, the fourth to eighth rows represent the case of single-typed NFs and the 9th-12th rows signify the case of hybrid NF. It is observable that the coefficient of skin friction and local Sherwood number are minimized by the suspension of either single-typed silver or alumina NPs on the conventional fluid. However, the local Nusselt number is minimized by adding single-typed alumina NPs while maximized by dispersion of single-typed silver NPs onto the base fluid. A similar trend also takes place when values of NP volume fraction are, respectively, ∕ O NF with low thermal conductivity enhances the local Nusselt number. The local Nusselt number is reduced for rising values of Φ 1 and fixing Φ 2 , but elevates for growing values of Φ 2 and fixing Φ 1 in the hybrid NF. On the basis of the aforementioned findings, effectiveness in thermal conductivity and heat transfer abilities can be maintained by the hybridization of silver and alumina NPs while minimizing production cost. Also, for advanced thermal performance, silver NPs are recommended for use in the formation of a hybrid NF due to their significant contribution to optimized heat transfer. Table 7 provides numerical values of skin friction coefficient for diverse values of M We F , ,Λ, c , and E e for the shear-thinning/thickening fluid and no-slip/slip conditions. It is disclosed in the table that the skin friction coefficient is higher in the shear-thinning fluid than in the shear-thickening Carreau fluid. The skin friction coefficient is also witnessed to be enhanced by the inclusion of the velocity slip condition on the flow system. The wall shear stress improves with escalating We and E e , whereas dwindles with mounting values of M, Λ, and F c . The impact of We Rd δ , Λ, , a , and β T on the thermal rate is unveiled in Table 8 for the case of heat absorption and heat generation. It is clear that the local Nusselt number is significantly lower in the case of heat generation than in the instance of heat absorption. This is justified by the fact that a thermal source can administer excess heat to the boundary-layer area and boosts the thermal boundary layer thickness, thus dwindling the rate of heat transference from the surface to the fluid. Thermal rate is enhanced by increasing values of We Rd , , and δ a , but minimized by the use of porous medium and thermal slip conditions in the flow system. The contribution of We δ E , Λ, , b 1 , and Sc on the local Sherwood number for the first-order destructive K ( > 0) c and generative (K < 0 c ) chemical reaction is provided in Table 9. The rate of mass transfer is evident to be substantially higher in the case of a destructive chemical reaction K ( = 0.5) c than that in the situation of a generative chemical reaction K ( = −0.5) c . The local Sherwood number upsurge for rising Schimdit number and activation energy, whereas drops for mounting values of We, Λ, and δ b .

| CONCLUSION
In this study article, entropy generation and EMHD flow of Carreau hybrid NF towards a stretching surface in a Darcy-Forchheimer porous medium are scrutinized in the presence of suction/injection and partial slip conditions. The theoretical study also accounts for the contribution of heat generation/absorption, radiative heat flux, variation in thermal conductivity, and mass diffusivity. The study elucidated that the application of this Carreau Ag-Al O H 2 3 2 ∕ O hybrid NF along with the incorporated parameters could be among contributing factors in achieving thermal improvement in engineering and industrial solicitations. However, it is necessary to control the amount of these parameters to ensure that the needed heat transport performance is achieved, as these parameters yield various effects that may offset each other. Residual and solution errors, condition numbers, and calculation time are provided to support the argument regarding the choice of the used numerical approach. Numerical results unveiled that there is a substantial improvement with the use of multiple domains than considering one-fixed domain. Also, rapid computation of accurate and stable solutions relies on the smallest number of collocation points per subinterval. The additional key results are outlined as follows: • The velocity, energy, and mass profiles are significantly enhanced in the case of injection.
• The hybrid NF velocity upsurges with the inclusion of an electric field while diminishes with the utilization of Darcy-Forchheimer porous medium and slip conditions. • The rate of entropy generation is minimized by shear-thinning Carreau fluid and velocity slip conditions, whereas maximized by rising values of porosity parameter, Brinkman number, and Weissenburg number for shear-thickening Carreau fluid. • The wall shear stress is enhanced by the consideration of electric field, velocity slip conditions, and shear-thinning Carreau fluid. The opposite is true for the inclusion of magnetic field, porous medium, and shear-thickening Carreau fluid. • The contribution of thermal radiation and temperature-variant thermal conductivity is to enhance the temperature and heat transmission rate of hybrid NF. However, the growth of heat source/sink parameter augments temperature but diminishes the rate of heat transport.
• The mass profiles improve with variable heat conductance, porosity, and activation energy parameters, but dwindles with destructive chemical reactions. On the other hand, mass transfer gradients accelerate with stronger destructive chemical reactions and higher activation energy. • To maintain effectiveness in thermal conductivity and heat transfer capabilities while minimizing production cost, hybridization of silver and economic alumina NPs can be useful.
The blending of silver and alumina NPs can also be effective when used as coolants in diverse technological and medical devices. The present theoretical work can be helpful in understanding why thermal systems are sometimes inefficient. Also, the study can assist in discovering suitable means for entropy generation minimization to ensure that the loss of fruitful and rare energy resources is alleviated. Rd thermal radiation parameter