Entropy analysis in a mixed convective Carreau nanofluid flow around a wedge: impact of activation energy and sinusoidal magnetic field

Abstract

A wide range of real-world applications have proven the importance of non-Newtonian fluids near a wedge, including the oil and gas industry, the aerospace sector. This study elucidates the dynamics of a Carreau nanofluid around a wedge by combining entropy analysis with periodic magnetohydrodynamics (MHD) and activation energy. The dimensional partial differential equations (PDEs) that describe the fluid flow system undergo non-similar transformations, forming nondimensional PDEs. The numerical solution to these PDEs is obtained by applying quasilinearization followed by the implicit finite difference approach. In the case of n = 0.5 (power-law index), when We (Weissenberg number) improves from 0 to 4, surface friction $\left(R{e}^{1/2}{C}_f\right)$ upsurges by approximately 23% and declines by around 41% in the case of n = 1.5. The mass transport intensity of liquid oxygen is about 18% higher than liquid nitrogen’s. Increasing the wedge angle results in a significant increase in fluid velocity.

KEYWORDS:

• Periodic MHD
• Carreau nanofluid
• activation energy
• mixed convection
• entropy analysis

1. Introduction

Non-Newtonian fluids are integral to the scientific and industrial processes that transfer heat and mass. It is important to note that Newtonian and non-Newtonian fluids’ heat transmission intensities are significantly different. Some examples of non-Newtonian fluids include coolants, dirt, lubricants, blood, ketchup, slurries, and many others. The great majority of fluids used in industrial settings typically exhibit non-Newtonian behaviour. Since non-Newtonian fluids exhibit such a diverse array of characteristics, scientists and engineers have devoted much time and energy to studying them. The production of plastic, dyes, foods, biological fluids, etc, induces such characteristics. Among the several non-Newtonian fluids, the Carreau fluid will be the one we’ll discuss here. Merging the power law and Newtonian fluid models, the Carreau fluid model provides a multi-faceted framework. The equation $\mu \left(\dot{\gamma }\right)={\mu }_{\mathrm{\infty }}+\left({\mu }_0-{\mu }_{\mathrm{\infty }}\right)[1+{\left(\mathrm{\Gamma }\dot{\gamma }\right)}^2{]}^{\frac{n-1}{2}}$ is the constitutive model for the Carreau fluid [1,2]. Here, ${\mu }_0$ is viscosity at zero shear rate and ${\mu }_{\mathrm{\infty }}$ is at infinite shear rate. Also $\mathrm{\Gamma }$ and $n$ are material constant and the power-law index. The Carreau model elucidates the manifestation of shear thickening attributes at elevated shear rates $\left(n>1\right)$, whereas it demonstrates shear thinning attributes at lower shear rates $\left(0<n<1\right)$. Ramesh et al. [3] investigated the radiative Carreau nanofluid flow in a microchannel with magnetic properties. To know the magnetic consequence on the flow of Carreau nanofluid with bioconvection, Tabrez et al. [4] made a numerical study with the help of the bvp4c technique of MATLAB. Results show that the velocity profile improves as the Weissenberg number increases. Irfan et al. [5] also used the bvp4c technique to solve the flow problem of Carreau nanofluid in the presence of Joule heating and activation energy with radiation. They revealed that the temperature field of the Carreau fluid is improved when the thermo-Biot and Dufour components have more significant values.

In the contemporary period characterized by rapid advancements, relying solely on non-Newtonian fluids may prove inadequate in achieving optimal rates of heat transmission. Nanofluids are part and partial of the researchers nowadays. The term “nanofluids” was labelled by Choi and Eastman [6] to elucidate a novel category of heat transfer fluids that consists of suspensions of nanoparticles in a fluid. Nanotechnology-based fluids have thermal properties surpassing their host fluids or conventional particle-fluid suspensions. Buongiorno [7] extended a theoretical framework for analyzing nanofluid flow by incorporating the effects of Brownian motion and thermophoresis. Based on his research, he concluded that the boundary layer’s temperature gradient and thermophoresis might cause nanofluid characteristics to differ significantly. The references [8–11] provide an excellent overview of recent research on this developing concept of nanofluids.

One of the most ubiquitous physical phenomena is magnetohydrodynamics (MHD), which can impact many different physical, chemical, and biological processes. The use of magnetic fields has thus far proven helpful in many sectors, from nuclear fusion research to industrial material manipulation (including crystal growth and separation) and various scientific endeavours. Many types of magnets can generate diverse magnetic fields, such as electromagnets, permanent magnets, hybrid magnets, superconducting magnets, and pulsed magnets. In the recent past, Madhu et al. [12], Krishna et al. [13], Chamkha and Rashad [14], and many others confined their studies to non-periodic MHD, which has many applications in industries. However, this study aims to provide a complete description of a recently identified periodic magnetic field [15–17] produced by the rotation of an infinite number of pairs of magnets. According to preliminary research, water evaporation, protein crystallization, and silver deposition benefit from the periodic magnetic field. It is reasonable to assume that the sample will also generate a periodic force field when exposed to a medium that may cause a periodic magnetic field. Moreover, additional study is crucial to investigate this magnetic field’s potential applications in many sectors. The study by Ahmed et al. [15] investigates the features of a periodic magnetohydrodynamics (MHD) flow through an isothermal oscillating cylinder, considering the influence of temperature-dependent viscosity. Higher magnetic fields indicate more non-smooth curves than lower periodic magnetic fields (M). That is to say, the wavy curves only appear when a periodic magnetic field (M) is applied. The investigation of the periodic magnetohydrodynamics (MHD) effect on a porous stretched sheet has been conducted by Mamun et al. [16] using Casson nanofluid. It has been shown that the periodic magnetic field has considerably enhanced the interaction of nanoparticles in the velocity field.

The notion of activation energy (AE) was initially introduced by the Swedish scientist Svante Arrhenius in 1889. “Activation energy” (AE) refers to the minimal energy a chemical species needs to start a reaction. The AE is contingent upon the specific chemical reaction and can sometimes be zero. Bestman [18] was probably the first to study a binary chemical reaction involving AE. There are several commercial uses for the binary chemical reaction that AE facilitates. Examples are ceramics production, biological systems, thermal lubricant recovery, chemical reaction catalysis, oil reserves, food processing, and insulation. Many authors [19–22] have extensively examined the concept of AE due to its extensive array of potential applications. In their study, Dhlamini et al. [19] investigated the effects of convective boundary conditions and AE on an infinitely long plate. Based on the results, AE considerably positively impacted the concentration profile. Ali et al. [20] investigated AE’s impact on nanofluids’ flow in three dimensions. They concluded that the nanofluid concentration tends to grow with an increase in AE, which is rather astonishing.

Research on a wedge using Carreau nanofluid, considering the influence of activation energy and periodic magnetohydrodynamics, has garnered attention from researchers for its essential applications in aerospace, pharmaceuticals, oil and gas sectors, thermal insulation, dams, and the design of discs for engine gate valves. Due to these significant industrial applications, we have addressed the current issue. Falkner and Skan [23] made a substantial and influential addition to perusing fluid flow around a wedge. Researchers Kumari et al. [24] found that the friction coefficient decreases as the Prandtl number changes, and the Nusselt number rises for combined convection flow in a permeable wedge. Singh et al. [25] investigated the combined convection flows across an inclined wedge with time-dependent characteristics. The self-similarity method to the micropolar fluid flow around a wedge was given by Ishak et al. [26]. The study conducted by Roy and Gorla [27] expanded upon the previous research [26] to include the time-dependent aspect. Their findings indicated that the heat transfer efficiency increased over time.

When developing thermal transport systems, it is essential to maximize energy efficiency. Irreversible losses must happen in these devices as these systems function, generating more entropy and affecting thermal efficiency. The ideal design of a thermal system would have an extremely low entropy. As a result, numerous experts have directed their attention to researching and enhancing the structure of traditional engineering devices utilized for heat removal. Solar energy gathering, geothermal energy transfer, and the cooling of modern electronic gadgets are just a few examples of the many manufacturing processes that benefit from entropy generation (EG) reduction [28]. A plethora of studies, including those by Haq et al. [29], Gangadhar et al. [30], and Ibrahim and Gizewu [31], etc., have yielded valuable findings concerning EG in the systems of horizontal cylinders, stretched surfaces, vertical cylinders, etc.

The literature review indicates that, in the presence of AE, no research has been done on the effects of quadratic temperature difference and periodic MHD on the Carreau nanofluid flow over a wedge. Previous studies are restricted to only similarity solutions; herein, we used the non-similarity solutions for the present study. Additionally, the subsequent aspects demonstrate the originality of the current investigation.

• Carreau and Newtonian nanofluid comparison.

• Effect of the power-law index over the flow.

• Wedge angle’s impact on flow characteristics.

• EG analysis.

• Impact of sinusoidal magnetic field on the flow.

2. Mathematical formulation

The work involves the flow of a two-dimensional, incompressible Carreau nanofluid comprising liquid nitrogen over a wedge having a half angle $\phi$. The flow occurs in the x-axis direction, while the y-axis is perpendicular. The y-direction is subjected to an externally applied periodic magnetic field exhibiting sinusoidal periodic behaviour. It appears that there are quadratic changes in temperature between the wedge wall and the fluid. In this case, $T_w$ and $T_{\mathrm{\infty }}$ are meant to represent the wall and fluid temperatures. Figure 1 visually depicts the model utilized in the current investigation. The Buongiorno model is utilized to account for the variations that take place within the nanofluid. The Buongiorno model is used to account for the variations that take place due to the nanofluid. Density difference estimates were calculated using the Boussinesq approximation [32–37]. The liquid nitrogen is also presumed to be evenly diffused throughout the fluid. Based on these presumptions, we obtain the following set of fluid-flow governing equations [1,2,8,19,24]:

Figure 1. Diagrammatic representation of the current model.

Governing Equations: (1) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0,\end{array}$(1) (2) $\begin{array}{rl}& \begin{array}{l}u{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}+v{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}=u_e{\displaystyle \frac{d{u}_e}{\mathrm{dx}}}\\ +\nu {\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}{\left[1+{\mathrm{\Gamma }}^2{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}\right)}^2\right]}^{\frac{n-1}{2}}\\ +\nu \left(n-1\right){\mathrm{\Gamma }}^2{\displaystyle \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}}{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}\right)}^2{\left[1+{\mathrm{\Gamma }}^2{\left({\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}\right)}^2\right]}^{\frac{n-3}{2}}\\ -{\displaystyle \frac{\sigma B_0^2}{\rho }}\left(u-u_e\right){\sin }^2\left({\displaystyle \frac{\mathrm{\pi x}}{L}}\right)\\ -\left({\displaystyle \frac{{\rho }_p-\rho }{\rho }}\right)g\left(\theta -{\theta }_{\mathrm{\infty }}\right)\cos \left({\displaystyle \frac{\mathrm{\pi \alpha }}{2}}\right)\\ +g\left(1-{\theta }_{\mathrm{\infty }}\right)\left[{\text{β}}_1\right(T-T_{\mathrm{\infty }})+{\text{β}}_2{\left(T-T_{\mathrm{\infty }}\right)}^2+\cdots \\ +{\text{β}}_3\left(C-C_{\mathrm{\infty }}\right)+{\text{β}}_4{\left(C-C_{\mathrm{\infty }}\right)}^2+\cdots ]\cos \left({\displaystyle \frac{\mathrm{\pi \alpha }}{2}}\right),\end{array}\}\end{array}$(2) (3) $\begin{array}{rl}& u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}=\frac{\kappa }{\rho C_p}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\\ & +J\left[D_B\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}+\frac{D_T}{T_{\mathrm{\infty }}}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}^2\right]\\ & +\frac{\sigma B_0^2}{\rho C_p}(u-u_e{)}^2{\sin }^2\left(\frac{\mathrm{\pi x}}{L}\right),\end{array}$(3) (4) $\begin{array}{rl}& u\frac{\mathrm{\partial }C}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }C}{\mathrm{\partial }y}=D_s\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{y}^2}-K_r^2\left(C-C_{\mathrm{\infty }}\right){\left(\frac{T}{T_{\mathrm{\infty }}}\right)}^{n_1}{e}^{-\left(\frac{E_a}{\mathrm{\kappa T}}\right)},\end{array}$(4) (5) $\begin{array}{rl}& u\frac{\mathrm{\partial }\theta }{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}=D_B\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{y}^2}+\frac{D_T}{T_{\mathrm{\infty }}}\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}.\end{array}$(5)

Boundary conditions (BCs): (6) $\begin{array}{r}\begin{array}{l}\mathrm{At}y=0:u=0;v=0;\\ T=T_w;C=C_w;\theta ={\theta }_w;\\ \mathrm{As}y\to \mathrm{\infty }:u\to u_e=u_{\mathrm{\infty }}(\overline{x}{)}^m;\\ T\to T_{\mathrm{\infty }};C\to C_{\mathrm{\infty }};\theta \to {\theta }_{\mathrm{\infty }}.\end{array}\}\end{array}$(6) Non-similar transformations: $\begin{array}{r}\overline{x}=\frac{x}{L};\eta ={\left[\frac{m+1}{2}\frac{u_e}{\mathrm{x\nu }}\right]}^{1/2}y;\end{array}$ $\begin{array}{rl}\psi \left(x,y\right)& ={\left[\frac{2}{m+1}\mathrm{x\nu }{u}_e\right]}^{1/2}f\left(\overline{x},\eta \right);\\ u& =\frac{\mathrm{\partial }\psi }{\mathrm{\partial }y};v=-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }x}\end{array}$ $\begin{array}{rl}u& =u_{e}F;\\ v& =-{\left[\frac{2}{m+1}\mathrm{x\nu }{u}_e\right]}^{1/2}\left\{\frac{m+1}{2x}f+\frac{1}{L}{f}_{\overline{x}}+\frac{m-1}{2x}\mathrm{\eta F}\right\};\end{array}$ $\begin{array}{r}G=\frac{T-T_{\mathrm{\infty }}}{T_w-T_{\mathrm{\infty }}};H=\frac{C-C_{\mathrm{\infty }}}{C_w-C_{\mathrm{\infty }}};S=\frac{\theta -{\theta }_{\mathrm{\infty }}}{{\theta }_w-{\theta }_{\mathrm{\infty }}};\end{array}$Equations (2)–(5) are rewritten as follows after undergoing these non-similar transformations: (7) $\begin{array}{rl}& \begin{array}{l}{\left[1+\left({\displaystyle \frac{m+1}{2}}\right)M_{1}W{e}^2{F}_{\eta }^2\right]}^{\frac{n-3}{2}}\\ \left[1+\left({\displaystyle \frac{m+1}{2}}\right)n{M}_{1}W{e}^2{F}_{\eta }^2\right]F_{\mathrm{\eta \eta }}\\ +\left(f+{\displaystyle \frac{2\overline{x}}{m+1}}{f}_{\overline{x}}\right)F_{\eta }\\ +{\displaystyle \frac{2m}{m+1}}\left(1-F^2\right)-{\displaystyle \frac{2\overline{x}}{m+1}}F{F}_{\overline{x}}\\ +{\displaystyle \frac{2}{m+1}}{M}_2\mathrm{Re}{M}^2{\sin }^2\left(\pi \overline{x}\right)\left(1-F\right)\\ +{\displaystyle \frac{2}{m+1}}{M}_3\mathrm{Ri}\left\{\right(1+{\text{β}}_{t}G)G\\ +\mathrm{Nc}\left(1+{\text{β}}_{c}H\right)H-\mathrm{Nr}S\}=0,\end{array}\}\end{array}$(7) (8) $\begin{array}{rl}& \begin{array}{l}{G}_{\mathrm{\eta \eta }}+\mathrm{Pr}\left(f+{\displaystyle \frac{2\overline{x}}{m+1}}{f}_{\overline{x}}\right)G_{\eta }-{\displaystyle \frac{2\overline{x}}{m+1}}\mathrm{PrF}{G}_{\overline{x}}\\ +\mathrm{Pr}\left(\mathrm{Nb}{H}_{\eta }+\mathrm{Nt}{G}_{\eta }\right)G_{\eta }\\ +{\displaystyle \frac{2\mathrm{Pr}}{m+1}}\mathrm{Ec}\mathrm{Re}{M}^2{M}_4{\sin }^2\left(\pi \overline{x}\right)(F-1{)}^2=0,\end{array}\}\end{array}$(8) (9) $\begin{array}{rl}& H_{\mathrm{\eta \eta }}+\mathrm{Sc}\left(f+\frac{2\overline{x}}{m+1}{f}_{\overline{x}}\right)H_{\eta }-\frac{2\overline{x}}{m+1}\mathrm{Sc}F{H}_{\overline{x}}\\ & -\frac{2}{m+1}\mathrm{Sc}\mathrm{Kc}\mathrm{Re}{M}_{2}H(1+{\mathrm{\Omega }}_{T}G{)}^{n_1}{e}^{-\frac{E}{\left(1+{\mathrm{\Omega }}_{T}G\right)}}=0,\end{array}$(9) (10) $\begin{array}{rl}& S_{\mathrm{\eta \eta }}+\mathrm{Pr}\mathrm{Le}\left(f+\frac{2\overline{x}}{m+1}{f}_{\overline{x}}\right)S_{\eta }-\frac{2\overline{x}}{m+1}\mathrm{Pr}\mathrm{Le}F{S}_{\overline{x}}\\ & +\frac{\mathrm{Nt}}{\mathrm{Nb}}{G}_{\mathrm{\eta \eta }}=0.\end{array}$(10) BCs: (11) $\begin{array}{r}\begin{array}{l}\mathrm{At}\eta =0:F=0;G=1;H=1;S=1;\\ \mathrm{As}\eta \to \mathrm{\infty }:F\to 1;G\to 0;H\to 0;S\to 0.\end{array}\}\end{array}$(11) Where, $M_1=(\overline{x}{)}^{3m-1};M_2=(\overline{x}{)}^{1-m};M_3=(\overline{x}{)}^{1-2m};M_4=(\overline{x}{)}^{1+m};$ $\begin{array}{r}W{e}^2=\frac{{\mathrm{\Gamma }}^2{u}_{\mathrm{\infty }}^3}{\mathrm{\nu L}};\mathrm{Re}=\frac{u_{\mathrm{\infty }}L}{\nu };M^2=\frac{\sigma B_0^2\nu }{\rho u_{\mathrm{\infty }}^2};\end{array}$ $\begin{array}{rl}\mathrm{Ri}& =\frac{g{\text{β}}_1\left(1-{\theta }_{\mathrm{\infty }}\right)\left(T_w-T_{\mathrm{\infty }}\right)\mathrm{Lcos}\left(\mathrm{\pi \alpha }/2\right)}{u_{\mathrm{\infty }}^2};\\ {\text{β}}_t& =\frac{{\text{β}}_2}{{\text{β}}_1}\left(T_w-T_{\mathrm{\infty }}\right);{\text{β}}_c=\frac{{\text{β}}_4}{{\text{β}}_3}\left(C_w-C_{\mathrm{\infty }}\right);\end{array}$ $\begin{array}{rl}\mathrm{Nc}& =\frac{{\text{β}}_3\left(C_w-C_{\mathrm{\infty }}\right)}{{\text{β}}_1\left(T_w-T_{\mathrm{\infty }}\right)};\mathrm{Nr}=\frac{\left({\rho }_p-\rho \right)\left({\theta }_w-{\theta }_{\mathrm{\infty }}\right)}{\rho {\text{β}}_1\left(1-{\theta }_{\mathrm{\infty }}\right)\left(T_w-T_{\mathrm{\infty }}\right)};\\ \mathrm{Pr}& =\frac{\nu }{{\alpha }_m};\end{array}$ $\begin{array}{rl}\mathrm{Nb}& =\frac{J{D}_B\left({\theta }_w-{\theta }_{\mathrm{\infty }}\right)}{\nu {\theta }_{\mathrm{\infty }}};\mathrm{Nt}=\frac{J{D}_T\left(T_w-T_{\mathrm{\infty }}\right)}{\nu T_{\mathrm{\infty }}};\\ \mathrm{Ec}& =\frac{u_{\mathrm{\infty }}^2}{C_p\left(T_w-T_{\mathrm{\infty }}\right)};\end{array}$ $\begin{array}{r}\mathrm{Le}=\frac{\alpha }{D_B};\mathrm{Sc}=\frac{\nu }{D_s};\mathrm{Kc}=\frac{K_r^2\nu }{u_{\mathrm{\infty }}^2};\end{array}$ $\begin{array}{r}{\mathrm{\Omega }}_T=\frac{\left(T_w-T_{\mathrm{\infty }}\right)}{T_{\mathrm{\infty }}}.\end{array}$Let $\xi =(\overline{x}{)}^{\left(1-m\right)/2}$ stand for the dimensionless distance along the wedge $\left(\xi >0\right)$.

So that, $\overline{x}{f}_{\overline{x}}=\overline{x}{f}_{\xi }{\xi }_{\overline{x}}=\left(\frac{1-m}{2}\right)\xi f_{\xi }$ and $\overline{x}{F}_{\overline{x}}=\left(\frac{1-m}{2}\right)\xi F_{\xi }$.

Thus, Equations (7)–(10) can be expressed as follows: (12) $\begin{array}{rl}& \begin{array}{l}{\left[1+\left({\displaystyle \frac{m+1}{2}}\right)M_{5}W{e}^2{F}_{\eta }^2\right]}^{\frac{n-3}{2}}\\ \left[1+\left({\displaystyle \frac{m+1}{2}}\right)n{M}_{5}W{e}^2{F}_{\eta }^2\right]F_{\mathrm{\eta \eta }}+f{F}_{\eta }\\ +\left({\displaystyle \frac{1-m}{1+m}}\right)\xi \left(f_{\xi }{F}_{\eta }-F{F}_{\xi }\right)\\ +{\displaystyle \frac{2m}{m+1}}\left(1-F^2\right)\\ +{\displaystyle \frac{2}{m+1}}{M}_6\mathrm{Re}{M}^2{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)\left(1-F\right)\\ +{\displaystyle \frac{2}{m+1}}{M}_7\mathrm{Ri}\left\{\right(1+{\text{β}}_{t}G)G\\ +\mathrm{Nc}\left(1+{\text{β}}_{c}H\right)H-\mathrm{Nr}S\}=0,\end{array}\}\end{array}$(12) (13) $\begin{array}{rl}& \begin{array}{l}{G}_{\mathrm{\eta \eta }}+\mathrm{Pr}f{G}_{\eta }+\mathrm{Pr}\left({\displaystyle \frac{1-m}{1+m}}\right)\xi \left(f_{\xi }{G}_{\eta }-F{G}_{\xi }\right)\\ +\mathrm{Pr}\left(\mathrm{Nb}{H}_{\eta }+\mathrm{Nt}{G}_{\eta }\right)G_{\eta }\\ +{\displaystyle \frac{2\mathrm{Pr}}{m+1}}\mathrm{Ec}\mathrm{Re}{M}^2{M}_8{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)(F-1{)}^2=0,\end{array}\}\end{array}$(13) (14) $\begin{array}{rl}& H_{\mathrm{\eta \eta }}+\mathrm{Sc}f{H}_{\eta }+\mathrm{Sc}\left(\frac{1-m}{1+m}\right)\xi \left(f_{\xi }{H}_{\eta }-F{H}_{\xi }\right)\\ & -\frac{2}{m+1}\mathrm{Sc}\mathrm{Kc}\mathrm{Re}{M}_{6}H(1+{\mathrm{\Omega }}_{T}G{)}^{n_1}{e}^{-\frac{E}{\left(1+{\mathrm{\Omega }}_{T}G\right)}}=0,\end{array}$(14) (15) $\begin{array}{rl}& S_{\mathrm{\eta \eta }}+\mathrm{Pr}\mathrm{Le}f{S}_{\eta }+\mathrm{Pr}\mathrm{Le}\left(\frac{1-m}{1+m}\right)\xi \left(f_{\xi }{S}_{\eta }-F{S}_{\xi }\right)\\ & +\frac{\mathrm{Nt}}{\mathrm{Nb}}{G}_{\mathrm{\eta \eta }}=0.\end{array}$(15)

Dimensionless BCs: (16) $\begin{array}{r}\begin{array}{l}\mathrm{At}\eta =0:F=0;G=1;\\ H=1;S=1;\\ \mathrm{As}\eta \to \mathrm{\infty }:F\to 1;G\to 0;\\ H\to 0;S\to 0.\end{array}\}\end{array}$(16) Where $M_5={\xi }^{2\left(\frac{3m-1}{1-m}\right)}$; $M_6={\xi }^2$; $M_7={\xi }^{2\left(\frac{1-2m}{1-m}\right)}$; $M_8={\xi }^{2\left(\frac{1+m}{1-m}\right)}$.

Gradients at the wall are given by:

Surface drag coefficient: $\begin{array}{r}{C}_f=\frac{{\tau }_w}{\rho u_e^2},\end{array}$ $\begin{array}{r}\Rightarrow C_f=\frac{\mu \frac{\mathrm{\partial }u}{\mathrm{\partial }y}\{1+{\mathrm{\Gamma }}^2{\left(\mathrm{\partial }u/\mathrm{\partial y}\right)}^2{\}}_{y=0}^{\frac{n-1}{2}}}{\rho u_e^2},\end{array}$ $\begin{array}{rl}\Rightarrow R{e}^{1/2}{C}_f& ={\xi }^{\frac{m+1}{m-1}}{\left(\frac{m+1}{2}\right)}^{1/2}\\ & \times {\left[1+\left(\frac{m+1}{2}\right)M_{5}W{e}^2{F}_{\eta }^2\left(\xi ,0\right)\right]}^{\frac{n-1}{2}}\\ & \times F_{\eta }\left(\xi ,0\right).\end{array}$

Heat transfer strength: $\begin{array}{r}\mathrm{Nu}=\frac{-x{\left(\mathrm{\partial }T/\mathrm{\partial y}\right)}_{y=0}}{\left(T_w-T_{\mathrm{\infty }}\right)},\end{array}$ $\begin{array}{r}\Rightarrow R{e}^{-1/2}\mathrm{Nu}=-{\xi }^{\frac{m+1}{m-1}}{\left(\frac{m+1}{2}\right)}^{1/2}{G}_{\eta }\left(\xi ,0\right).\end{array}$Mass transfer strength of liquid nitrogen: $\begin{array}{r}\mathrm{Sh}=\frac{-x{\left(\mathrm{\partial }C/\mathrm{\partial y}\right)}_{y=0}}{\left(C_w-C_{\mathrm{\infty }}\right)},\end{array}$ $\begin{array}{r}\Rightarrow R{e}^{-1/2}\mathrm{Sh}=-{\xi }^{\frac{m+1}{m-1}}{\left(\frac{m+1}{2}\right)}^{1/2}{H}_{\eta }\left(\xi ,0\right).\end{array}$Mass transfer strength of nanoparticles: $\begin{array}{r}\mathrm{NSh}=\frac{-x{\left(\mathrm{\partial }\theta /\mathrm{\partial y}\right)}_{y=0}}{\left({\theta }_w-{\theta }_{\mathrm{\infty }}\right)},\end{array}$ $\begin{array}{r}\Rightarrow R{e}^{-1/2}\mathrm{NSh}=-{\xi }^{\frac{m+1}{m-1}}{\left(\frac{m+1}{2}\right)}^{1/2}{S}_{\eta }\left(\xi ,0\right).\end{array}$

2.1. Entropy generation (EG)

The following expression describes the volumetric rate at which entropy is generated for an incompressible Carreau nanofluid [1,2]: (17) $\begin{array}{rl}{S}_{\mathrm{gen}}& =\frac{k}{T^2}{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}^2+\frac{\mu }{T}{\left[1+{\mathrm{\Gamma }}^2{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\right]}^{\frac{n-1}{2}}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2\\ & +\frac{\sigma B_0^2}{T_{\mathrm{\infty }}}(u-u_e{)}^2{\sin }^2\left(\pi \overline{x}\right)+\frac{R^{\ast }{D}_s}{C_{\mathrm{\infty }}}{\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }y}\right)}^2\\ & +\frac{R^{\ast }{D}_s}{T_{\mathrm{\infty }}}\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }y}\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)\\ & +\frac{R^{\ast }{D}_{\theta }}{{\theta }_{\mathrm{\infty }}}{\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\right)}^2+\frac{R^{\ast }{D}_{\theta }}{T_{\mathrm{\infty }}}\left(\frac{\mathrm{\partial }\theta }{\mathrm{\partial }y}\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right),\end{array}$(17) Bejan [25] defines the dimensionless EG $S_G\left(\xi ,\eta \right)$ as the quotient of the volumetric EG rate and the characteristic EG rate $S_0=\left({\mathrm{\Omega }}_T^{2}k\right)/x^2$. Thus, with the aid of non-similar transformations and characteristic EG rate, the following expression gives the nondimensional EG: (18) $\begin{array}{rl}{S}_G& =\left(\frac{m+1}{2}\right)\mathrm{Re}\{\frac{1}{\left(1+{\mathrm{\Omega }}_{T}G\right)}[\frac{M_8}{\left(1+{\mathrm{\Omega }}_{T}G\right)}{G}_{\eta }^2\\ & +\frac{\mathrm{Br}}{{\mathrm{\Omega }}_T}{M}_9{\left(1+\frac{m+1}{2}{M}_{5}W{e}^2{F}_{\eta }^2\right)}^{\frac{n-1}{2}}{F}_{\eta }^2]\\ & +\frac{\mathrm{Re}\mathrm{Br}{M}^2}{{\mathrm{\Omega }}_T}\frac{2}{m+1}{M}_8^2(F-1{)}^2{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)\\ & +N_1\frac{{\mathrm{\Omega }}_c}{{\mathrm{\Omega }}_T}{M}_8\left(\frac{{\mathrm{\Omega }}_c}{{\mathrm{\Omega }}_T}{H}_{\eta }+G_{\eta }\right)H_{\eta }\\ & +N_2\frac{{\mathrm{\Omega }}_{\theta }}{{\mathrm{\Omega }}_T}{M}_8\left(\frac{{\mathrm{\Omega }}_{\theta }}{{\mathrm{\Omega }}_T}{S}_{\eta }+G_{\eta }\right)S_{\eta }\}.\end{array}$(18) Where, $M_9={\xi }^{2\left(\frac{3m+1}{1-m}\right)}$; ${\mathrm{\Omega }}_c=\frac{\mathrm{\Delta }C}{C_{\mathrm{\infty }}}$; ${\mathrm{\Omega }}_{\theta }=\frac{\mathrm{\Delta }\theta }{{\theta }_{\mathrm{\infty }}}$; $N_1=\frac{R^{\ast }{D}_s{C}_{\mathrm{\infty }}}{k}$; $N_2=\frac{R^{\ast }{D}_{\theta }{\theta }_{\mathrm{\infty }}}{k}$.

3. Numerical method

Equations (12) through (15) demonstrate a highly nonlinear characteristic of the system. The linearized forms of Equations (12)–(15) are provided below, following the application of the Quasilinearization method [38–42]. (19) $\begin{array}{rl}& F_{\mathrm{\eta \eta }}^{i+1}+A_1^i{F}_{\eta }^{i+1}+A_2^i{F}_{\xi }^{i+1}+A_3^i{F}^{i+1}+A_4^i{G}^{i+1}+A_5^i{H}^{i+1}\\ & +A_6^i{S}^{i+1}=A_7^i,\end{array}$(19) (20) $\begin{array}{rl}& G_{\mathrm{\eta \eta }}^{i+1}+B_1^i{G}_{\eta }^{i+1}+B_2^i{G}_{\xi }^{i+1}+B_3^i{F}^{i+1}+B_4^i{H}_{\eta }^{i+1}=B_5^i,\end{array}$(20) (21) $\begin{array}{rl}& H_{\mathrm{\eta \eta }}^{i+1}+C_1^i{H}_{\eta }^{i+1}+C_2^i{H}_{\xi }^{i+1}+C_3^i{H}^{i+1}+C_4^i{F}^{i+1}\\ & +C_5^i{G}^{i+1}=C_6^i,\end{array}$(21) (22) $\begin{array}{rl}& S_{\mathrm{\eta \eta }}^{i+1}+D_1^i{S}_{\eta }^{i+1}+D_2^i{S}_{\xi }^{i+1}+D_3^i{F}^{i+1}+D_4^i{G}_{\mathrm{\eta \eta }}^{i+1}=D_5^i,\end{array}$(22)

Boundary Conditions: (23) $\begin{array}{r}\begin{array}{l}{F}^{i+1}\left(\xi ,0,\tau \right)=0,G^{i+1}\left(\xi ,0,\tau \right)=1,\\ H^{i+1}\left(\xi ,0,\tau \right)=1,S^{i+1}\left(\xi ,0,\tau \right)=1,\text{at}\eta =0,\\ F^{i+1}\left(\xi ,{\eta }_{\mathrm{\infty }},\tau \right)=1,G^{i+1}\left(\xi ,{\eta }_{\mathrm{\infty }},\tau \right)=0,\\ H^{i+1}\left(\xi ,{\eta }_{\mathrm{\infty }},\tau \right)=0,S^{i+1}\left(\xi ,{\eta }_{\mathrm{\infty }},\tau \right)=0,\text{at}\eta ={\eta }_{\mathrm{\infty }}.\end{array}\}\end{array}$(23) The coefficients of the functions having iterative positions i + 1 and i are, respectively, unknown and known.

In order to get the numerical solutions, Equations (19) – (22) are attempted to be solved using the implicit finite difference technique [43–46]. The investigation used a central difference along $\eta$ and a backward difference along $\xi$. Each of the step lengths $\mathrm{\Delta \xi }$ and $\mathrm{\Delta \eta }$, is set to have a value of 0.01 throughout. The numerical computations and graphical displays of results were accomplished with the help of the MATLAB programme. When the largest absolute difference between two consecutive iterations is smaller than 10⁻¹⁰, the solution is “converged” i.e. $\begin{array}{rl}& \mathrm{Max}\{|(F_{\eta }{)}_w^{i+1}-(F_{\eta }{)}_w^i|,|(G_{\eta }{)}_w^{i+1}-(G_{\eta }{)}_w^i|,|(H_{\eta }{)}_w^{i+1}\\ & -(H_{\eta }{)}_w^i|,|(S_{\eta }{)}_w^{i+1}-(S_{\eta }{)}_w^i|\}\le {10}^{-10}.\end{array}$The coefficients of Equations (19–22) are given by: $\begin{array}{rl}{A}_1^i& =\left[f+\left(\frac{1-m}{1+m}\right)\xi f_{\xi }\right]\mathrm{dff}\\ & +\left(\frac{2}{1+m}\right)\mathrm{df}\left[m\right(1-F^2)-{\displaystyle \frac{\left(1-m\right)}{2}}\mathrm{\xi F}{F}_{\xi }\\ & -M_6\mathrm{Re}{M}^2{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)\left(F-1\right)\\ & ++M_7\mathrm{Ri}\left\{\left(1+{\text{β}}_{t}G\right)G+\mathrm{Nc}\left(1+{\text{β}}_{c}H\right)H-\mathrm{Nr}S\right\}];\end{array}$ $\begin{array}{rl}{A}_2^i& =-\left(\frac{1-m}{1+m}\right)\frac{1}{Q}\mathrm{\xi F};\\ A_3^i& =\frac{-2}{Q\left(m+1\right)}\{\frac{\left(1-m\right)}{2}\xi F_{\xi }+2\mathrm{mF}\\ & +M_6\mathrm{Re}{M}^2{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)\phantom{\frac{\left(1-m\right)}{2}}\};\end{array}$ $\begin{array}{rl}{A}_3^i& =\frac{-2}{Q\left(m+1\right)}\{\frac{\left(1-m\right)}{2}\xi F_{\xi }+2\mathrm{mF}\\ & +M_6\mathrm{Re}{M}^2{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)\};\end{array}$ $\begin{array}{rl}{A}_4^i& =\frac{2}{Q\left(m+1\right)}{M}_7\mathrm{Ri}\left(1+2{\text{β}}_{t}G\right);\\ A_5^i& =\frac{2}{Q\left(m+1\right)}{M}_7\mathrm{Ri}\mathrm{Nc}\left(1+2{\text{β}}_{c}H\right);\end{array}$ $\begin{array}{r}{A}_6^i=\frac{-2}{Q\left(m+1\right)}{M}_7\mathrm{Ri}\mathrm{Nr};\end{array}$ $\begin{array}{rl}{A}_7^i& =A_1^i{F}_{\eta }+A_2^i{F}_{\xi }\\ & -\frac{1}{Q}\left\{\frac{2}{m+1}\left[\begin{array}{l}{\displaystyle \frac{\left(1-m\right)}{2}}\xi f_{\xi }{F}_{\eta }+m\left(1+F^2\right)\\ +M_6\mathrm{Re}{M}^2{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)\\ -M_7\mathrm{Ri}\left({\text{β}}_t{G}^2+\mathrm{Nc}{\text{β}}_c{H}^2\right)\end{array}\right]\right\};\end{array}$ $\begin{array}{rl}\mathrm{dff}& =\frac{Q_1^{\frac{1-n}{2}}\left\{Q_2\left[\left(3-n\right)P{F}_{\eta }^2+Q_1\right]-2\mathrm{nP}{Q}_1{F}_{\eta }^2\right\}}{Q_2^2};\\ \mathrm{df}& =\frac{Q_1^{\frac{1-n}{2}}P{F}_{\eta }\left\{\left(3-n\right)Q_2-2n{Q}_1\right\}}{Q_2^2};\end{array}$ $\begin{array}{rl}P& =\left(\frac{m+1}{2}\right)M_{5}W{e}^2;Q_1=\left[1+P{F}_{\eta }^2\right];\\ Q_2& =\left[1+\mathrm{nP}{F}_{\eta }^2\right];Q=Q_1^{\frac{n-3}{2}}{Q}_2;\end{array}$ $\begin{array}{rl}{B}_1^i& =\mathrm{Pr}\left\{f+\left(\frac{1-m}{1+m}\right)\xi f_{\xi }+\mathrm{Nb}{H}_{\eta }+2\mathrm{Nt}{G}_{\eta }\right\};\\ B_2^i& =-\left(\frac{1-m}{1+m}\right)\mathrm{\xi Pr}F;\end{array}$ $\begin{array}{rl}{B}_3^i& =\frac{-\mathrm{Pr}}{m+1}\left\{\right(1-m)\xi G_{\xi }-4\mathrm{Ec}\mathrm{Re}{M}^2{M}_8{\sin }^2\\ & \times \left(\pi {\xi }^{\frac{2}{1-m}}\right)\left(F-1\right)\};\\ B_4^i& =\mathrm{Pr}\mathrm{Nb}{G}_{\eta };\end{array}$ $\begin{array}{rl}{B}_5^i& =B_2^i{G}_{\xi }+B_4^i{H}_{\eta }+\mathrm{Pr}\mathrm{Nt}{G}_{\eta }^2\\ & +\frac{2\mathrm{Pr}}{m+1}\mathrm{Ec}\mathrm{Re}{M}^2{M}_8{\sin }^2\left(\pi {\xi }^{\frac{2}{1-m}}\right)\left(F^2-1\right);\end{array}$ $\begin{array}{rl}{C}_1^i& =\mathrm{Sc}\left\{f+\left(\frac{1-m}{1+m}\right)\xi f_{\xi }\right\};\\ C_2^i& =-\mathrm{Sc}\left(\frac{1-m}{1+m}\right)\mathrm{\xi F};\end{array}$ $\begin{array}{rl}{C}_3^i& =-\frac{2}{m+1}\mathrm{Kc}\mathrm{Sc}\mathrm{Re}{M}_6(1+{\mathrm{\Omega }}_{T}G{)}^{n_1}{e}^{\frac{-E}{\left(1+{\mathrm{\Omega }}_{T}G\right)}};\\ C_4^i& =-\mathrm{Sc}\left(\frac{1-m}{1+m}\right)\xi H_{\xi };\end{array}$ $\begin{array}{rl}{C}_5^i& =C_3^i\frac{{\mathrm{\Omega }}_T}{\left(1+{\mathrm{\Omega }}_{T}G\right)}\left\{n_1+\frac{E}{\left(1+{\mathrm{\Omega }}_{T}G\right)}\right\};\\ C_6^i& =C_2^i{H}_{\xi }+C_5^{i}G;\end{array}$ $\begin{array}{rl}{D}_1^i=& \mathrm{Pr}\mathrm{Le}\left\{f+\left(\frac{1-m}{1+m}\right)\xi f_{\xi }\right\};\\ D_2^i& =-\mathrm{Pr}\mathrm{Le}\left(\frac{1-m}{1+m}\right)\mathrm{\xi F};\end{array}$ $\begin{array}{r}{D}_3^i=-\mathrm{Pr}\mathrm{Le}\left(\frac{1-m}{1+m}\right)\xi S_{\xi };D_4^i=\frac{\mathrm{Nt}}{\mathrm{Nb}};D_5^i=D_3^{i}F.\end{array}$

3.1. Validation of results

In order to assess the effectiveness of our approach, we conducted a comparison between the results of Mabood et al. [47] and Ishak et al. [48] with the current observations of $R{e}^{1/2}{C}_f$ (as shown in Table 1) for the scenario involving a Newtonian case without periodic magnetic field, mixed convection and nanofluid when $\xi =0$. Mabood et al. [47] and Ishak et al. [48] solved the equations reduced by similarity transformations numerically with the aid of Runge–Kutta-Fehlberg with shooting technique and implicit finite difference Keller box method, respectively. In the present analysis, we used the Quasilinearization technique in combination with the implicit finite difference scheme. The results of [47,48] strongly corroborate the current findings.

Table 1. Assessment of $R{e}^{-1/2}\mathrm{Nu}$ and $R{e}^{1/2}{C}_f$ for the Newtonian case without periodic magnetic field, mixed convection and nanofluid when $\xi =0$.

	$R{e}^{1/2}{C}_f$
m	Mabood et al. [47]	Ishak et al. [48]	Present results
0.0	0.469599988	0.469604	0.46974968
0.0141	0.504614318	0.504618	0.50461538
0.0909	0.654993688	0.654983	0.65497512
0.1429	0.731998540	0.732003	0.73200918
0.2	0.802125593	0.802131	0.80212253

4. Results and discussion

This section outlines the key findings derived from our investigation according to various parameters. The wedge is transformed into a flat horizontal plate when the angle is 0⁰ (m = 0) and a vertical plate when it is 90⁰ (m = 0.3333). Therefore, it is crucial to investigate the wedge for values of m ranging from 0 to 0.2. These values correspond to $\phi =5^0$ (m = 0.0141), $\phi ={30}^0$ (m = 0.0909), $\phi ={45}^0$ (m = 0.1429), and $\phi ={60}^0$ (m = 0.2000). To this end, we fixed m in our analysis at 0.0909 $\left(\phi ={30}^0\right)$. A few consistent values are maintained for the parameters, such as Re = 10, Pr = 7, Le = 10. As an added consideration, a practical value of the Schmidt number is used with liquid nitrogen (Sc = 240) at 27°C [49].

4.1. Impact of the power-law index (n) and Weissenberg number (We)

This section presents the results of the impact of the Carreau nanofluid. The Carreau model exhibits a bimodal behaviour of the fluid, with shear thickening characteristics at high shear rates $\left(n>1\right)$ and shear thinning characteristics at low shear rates $\left(0<n<1\right)$. Figures 2–5 analyze the profiles of dimensionless velocity $\left(F\left(\xi ,\eta \right)\right)$, temperature $\left(G\left(\xi ,\eta \right)\right)$, species concentration $\left(H\left(\xi ,\eta \right)\right)$, and nanoparticle concentration $\left(S\left(\xi ,\eta \right)\right)$ for different values of Weissenberg number (We) and power-law index (n). In this case, the usual Newtonian fluid is represented by $\mathrm{We}=0$, but the Carreau fluid is represented by $\mathrm{We}>0$. We are comparing the Newtonian base fluid and the non-Newtonian Carreau fluid in the scenarios of shear thinning (n = 0.5) and shear thickening (n = 1.5). Growing values of We from 0 to 4 increase $F\left(\xi ,\eta \right)$ for n = 0.5, while for n = 1.5, the magnitude of $F\left(\xi ,\eta \right)$ decreases for the same variations of We, which is depicted in Figure 2. Also, as evident from Figures 3–5, the contrasting results are observed for the $G\left(\xi ,\eta \right)$, $H\left(\xi ,\eta \right)$ and $S\left(\xi ,\eta \right)$. This is because n = 0.5 indicates a shear-thinning liquid, meaning the fluid’s viscosity is very low. On the other hand, when n = 1.5, it corresponds to a shear-thickening liquid, indicating that the fluid’s viscosity is larger.

Figure 2. Impact of n and We over $F\left(\xi ,\eta \right)$.

Figure 3. Impact of n and We over $G\left(\xi ,\eta \right)$.

Figure 4. Impression of n and We over $H\left(\xi ,\eta \right)$.

Figure 5. Impression of n and We over $S\left(\xi ,\eta \right)$.

Further, the impact of We and n on the skin friction $\left(R{e}^{1/2}{C}_f\right)$, energy transfer $\left(R{e}^{-1/2}\mathrm{Nu}\right)$, mass transfer strength $\left(R{e}^{-1/2}\mathrm{Sh}\right)$, and nanoparticle mass transfer strengths $\left(R{e}^{-1/2}\mathrm{NSh}\right)$ are pictured in Figures 6–9. A periodic magnetic field causes sinusoidal fluctuations in the gradients due to the presence of sine function, as seen through Figures 6, 7, and 9. Nevertheless, according to Figure 8, it is evident that the periodic magnetic field has significantly less impact on the pace at which liquid nitrogen undergoes mass transfer. The gradients yield contrasting outcomes compared to the profiles. For instance, in the case of n = 0.5, when We improves from 0 to 4, $R{e}^{1/2}{C}_f$ upsurges by approximately about 23% and declines by about 41% in the case of n = 1.5 at $\xi =1$. Furthermore, a fall in the magnitudes of $R{e}^{-1/2}\mathrm{Nu}$, $R{e}^{-1/2}\mathrm{Sh}$ and $R{e}^{-1/2}\mathrm{NSh}$ about 19%, 6% and 16%, respectively, arises when We enhance from 0 to 4 at $\xi =1$, $\xi =0.5$ and $\xi =1$ in case of n = 0.5. Again, this is because when n = 0.5, it indicates a shear-thinning liquid, meaning that the fluid’s viscosity is very low. On the other hand, when n = 1.5, it corresponds to a shear-thickening liquid, indicating that the fluid’s viscosity is larger. At the same time, we observe increments of $R{e}^{-1/2}\mathrm{Nu}$, $R{e}^{-1/2}\mathrm{Sh}$ and $R{e}^{-1/2}\mathrm{NSh}$ about 10%, 17%, and 34%, respectively at $\xi =1$, $\xi =0.5$ and $\xi =1$ in case of n = 1.5 when We rises from 0 to 4.

Figure 6. Impression of n and We over $R{e}^{1/2}{C}_f$.

Figure 7. Impact of n and We over $R{e}^{-1/2}\mathrm{Nu}$.

Figure 8. Impact of n and We over $R{e}^{-1/2}\mathrm{Sh}$.

Figure 9. Impact of n and We over $R{e}^{-1/2}\mathrm{NSh}$.

4.2. Velocity and surface drag coefficients

One way to make heat transmission more efficient is to account for quadratic convection. This effect can manifest when there is a significant temperature differential between the wall and the fluid around it. Figures 10 and 11 illustrate the velocity $\left(F\left(\xi ,\eta \right)\right)$ and surface drag $\left(R{e}^{1/2}{C}_f\right)$ changes for several values of the Richardson attribute Ri and the quadratic convective attribute ${\text{β}}_t$. The Ri exhibits a favourable pressure gradient nature, growing the velocity and surface drag. This effect is observed when Ri values are enhanced, regardless of the presence $\left({\text{β}}_t=1\right)$ or absence $\left({\text{β}}_t=0\right)$ of quadratic convection. We have looked at two different ranges for Ri in this discussion. When Ri is negative, the buoyancy acts against the flow, and when it’s positive, it assists the flow. However, a dual nature impact of the Ri on $F\left(\xi ,\eta \right)$ and $R{e}^{1/2}{C}_f$ is seen from Figures 10 and 11. i.e. When Ri is negative, the flow’s velocity and drag are declined; when it is positive, both are enhanced. More precisely, at $\xi =1$ and $\mathrm{Ri}=10$, a rise of about 53% in the $R{e}^{1/2}{C}_f$ is noted for the quadratic convection instead of its absence. This phenomenon can be ascribed to the disparity in the temperature between the wedge’s surface and the surrounding fluid.

Figure 10. Impression of Ri and ${\text{β}}_t$ over $F\left(\xi ,\eta \right)$.

Figure 11. Impression of Ri and ${\text{β}}_t$ over $R{e}^{1/2}{C}_f$.

Figures 12 and 13 provide a comprehensive analysis of the effects of a periodic magnetic field and the buoyancy ratio of a nanoparticle. This model accounts for both the presence or absence of periodic magnetohydrodynamics (MHD) events ($M=0$ and $M\ne 0$ respectively) and the ratio of nanoparticle buoyancy ($\mathrm{Nr}=0$ and $\mathrm{Nr}=1$ respectively). It can be seen in Figures 12 and 13, the flow’s velocity drops drastically, and the surface drag grows due to the Lorentz force induced by the imposed periodic MHD. Mainly, $R{e}^{1/2}{C}_f$ surges approximately by 52% when M becomes 3 from 0 at $\xi =1$ and $\mathrm{Nr}=1$. Moreover, nanoparticle buoyancy demonstrates a simultaneous reduction in velocity and surface drag. Specifically, a surge in Nr leads to a diminution of around 23% in surface drag at $\xi =1$ and $M=3$. This phenomenon occurs due to the disparity in the concentration between the wall and the surrounding fluid. Also, the wedge angle impacts on the fluid’s velocity is described in Figure 14. The wedge angle significantly influences the fluid flow surrounding it. The wedge angle $\phi$ and m have a direct and unique relationship. Where, $m=0.0909\left(\phi ={30}^0\right)$, $m=0.1429\left(\phi ={45}^0\right)$, $m=0.2\left(\phi ={60}^0\right)$. Increasing the wedge angle results in a dramatic growth in $F\left(\xi ,\eta \right)$ (see Figure 14). In physical terms, the fluid’s motion becomes more concentrated on the surface of the wedge as the angle upsurges.

Figure 12. Impression of M and Nr over $F\left(\xi ,\eta \right)$.

Figure 13. Impression of M and Nr over $R{e}^{1/2}{C}_f$.

Figure 14. Impact of Wedge angle m over $R{e}^{1/2}{C}_f$.

4.3. Temperature and heat transfer strength

This paragraph offers a concise elucidation of the impacts of nanofluid employing the Buongiorno model. This model demonstrates the seven slip mechanisms that affect the movement of nanofluids resulting from including nanoparticles in the fluid. Although all seven are essential for heat transport, thermophoresis, and Brownian diffusion are the most important. So, the mobility of nanoparticles resulting from Brownian motion and thermophoresis is depicted in Figure 15, which illustrates the temperature profile $G\left(\xi ,\eta \right)$. The absence of thermophoresis $\left(\mathrm{Nt}=0\right)$ has a lesser temperature than its presence $\left(\mathrm{Nt}=1\right)$. This is because Nt defines how nanoparticles travel along a temperature gradient and how far they can go into a fluid. Figure 15 provides evidence that higher values of the Brownian diffusion attribute Nb increase the fluid temperature. The rise in Nb, associated with the random mobility of the nanoparticles, leads to an elevation in the fluid temperature. Additionally, Figure 16 is depicted to assess the influence of periodic magnetic field (M) and negative and positive Eckert numbers (Ec). When Ec is negative, increasing M results in a more robust energy transfer, and vice versa when Ec is positive. The reason is that the fluid’s temperature drops when an Ec value is negative, whereas the fluid’s temperature rises when an Ec value is positive.

Figure 15. Impression of Nb and Nt over $G\left(\xi ,\eta \right)$.

Figure 16. Impression of Ec and M over $R{e}^{-1/2}\mathrm{Nu}$.

4.4. Impact of activation energy (AE) and Schmidt number

The AE is the least energy required to set atoms or molecules in motion to continue a chemical process. This subsection sheds information on the impression of activation energy on the flow characteristics. Figures 17 and 18 demonstrate the concentration profile $H\left(\xi ,\eta \right)$ and mass transfer $R{e}^{-1/2}\mathrm{Sh}$ of liquid nitrogen (Sc = 240), which vary according to the activation energy E, chemical reaction property Kc, and Schmidt number Sc. For comparison, we have chosen to examine liquid oxygen (Sc = 340) in our study. When molecules or atoms undergo the activation energy, they receive the minimum energy necessary to start a chemical process. This prodigy enables us to examine the dominance of mass transfer by employing Arrhenius activation energy to the binary chemical reaction. When we surge the AE for a chemical reaction, the $H\left(\xi ,\eta \right)$ rises while its mass transfer falls. The inverse outcomes can be observed for the growing chemical reaction characteristic Kc. More damaging chemical reactions occur with higher levels of the chemical reaction parameter, which explains this trend. For instance, as Kc increases from 0.5 to 1.5, the $R{e}^{-1/2}\mathrm{Sh}$ is improved by almost 71% at $\xi =1$, $E=2$ and $\mathrm{Sc}=240$. Furthermore, it can be observed from Figures 17 and 18 that as the Sc steps up from 240 to 340, the concentration profile decreases, and the intensity of mass transfer boosts. The Sc (Schmidt number) differs inversely with the mass diffusivity, consequently affecting the outcomes. Specifically, liquid oxygen’s intensity of mass transport is around 18% more than liquid nitrogen’s at $\xi =1$, $E=2$ and $\mathrm{Kc}=1.5$. This is because the Schmidt number is inversely proportional to the mass diffusivity.

Figure 17. Impact of E and Kc over $H\left(\xi ,\eta \right)$.

Figure 18. Impact of E and Kc over $R{e}^{-1/2}\mathrm{Sh}$.

Figure 19. Impression of Nr and Le over $S\left(\xi ,\eta \right)$.

Figure 20. Impression of Nr and Le on nanoparticle’s mass transfer rate.

4.5. Nanoparticle concentration and their mass transport strength

Lewis numbers quantify heat diffusion (conduction) relative importance compared to mass diffusion (convection). Another way to understand the connection between nanoparticles and buoyancy is by referring to the buoyancy ratio attribute Nr. Increasing the Nr enhances the intensity of the nanoparticle’s profile $S\left(\xi ,\eta \right)$, but it diminishes the nanoparticles’ ability to transmit mass $\left(R{e}^{-1/2}\mathrm{NSh}\right)$ (Figures 19 and 20). This phenomenon occurs due to the disparity in the concentration between the wall and the surrounding fluid. Further, the enhancing Le value lowers $S\left(\xi ,\eta \right)$ and elevates the intensity of $R{e}^{-1/2}\mathrm{NSh}$. Due to the inverse relationship between Le and mass diffusivity, we noticed a diminution in the magnitude of $S\left(\xi ,\eta \right)$ and a growth in $R{e}^{-1/2}\mathrm{NSh}$. Specifically, $R{e}^{-1/2}\mathrm{NSh}$ is approximately 26% more intensified for the more significant Le value at $\xi =1$ and $\mathrm{Nr}=1$.

4.6. Entropy generation

The focal point of this writing lies in the examination of entropy generation $S_G\left(\xi ,\eta \right)$ (EG). A thorough examination of the entropy generated by an engineering machine is of paramount relevance. For the simple reason that entropy is the quantity of energy that does not contribute to the production of any mechanically beneficial activity. This is well demonstrated in Figures 21–25. The effects of the power-law index (n) and Weissenberg number We on $S_G\left(\xi ,\eta \right)$ is depicted in Figure 21. For the dilatant fluid (n = 1.5), the $S_G\left(\xi ,\eta \right)$ is noted to be reduced near the wedge’s wall for rising We characteristics and pseudoplastic fluid (n = 0.5), the $S_G\left(\xi ,\eta \right)$ enriched for the equivalent changes in We. This is because when n = 0.5, it signifies a shear-thinning liquid, implying that the fluid’s viscosity is extremely low. Conversely, when the value of n is 1.5, the fluid exhibits characteristics similar to those of a shear-thickening liquid, suggesting that its viscosity is higher. The effects of Ri, ${\text{β}}_t$, Le, and Sc on the EG are shown in Figures 22 and 23. The positive pressure gradient characteristic of Ri, along with the more significant temperature difference caused by ${\text{β}}_t$, leads to the increased magnitude of $S_G\left(\xi ,\eta \right)$. Additionally, diffusion of nanoparticles and liquid nitrogen produces a larger magnitude of $S_G\left(\xi ,\eta \right)$.

Figure 21. Impression of n and We over EG.

Figure 22. Impact of Ri and ${\text{β}}_t$ over EG.

Figure 23. Impact of Le and Sc over EG.

Figure 24. Impact of Ec and M on entropy generation.

Figure 25. Impact of N₁ and N₂ on entropy generation when $N_i=0.5,i=1,2$.

In addition, Figure 24 illustrates the evaluation of the impact of periodic MHD, negative and positive Eckert numbers (Ec). With a negative Ec, $S_G\left(\xi ,\eta \right)$ is reduced as a result of more robust energy transfer when M is increased, and a positive Ec causes the reverse effect. The reason is that a positive Ec number causes the fluid to warm up, whereas a negative one causes it to cool down. A novel result on the $S_G\left(\xi ,\eta \right)$ due to the diffusing components is portrayed in Figure 25. Here, N₁ stands for liquid nitrogen diffusion and N₂ for nanoparticle diffusion. If $N_1=0$ and $N_2=0$, it represents there is no diffusion, if $N_1\ne 0$ and $N_2=0$, only diffusion of liquid nitrogen occurs and if both $N_1\ne 0$ and $N_2\ne 0$, both liquid nitrogen and nanoparticles are diffused. It can be viewed from Figure 25 that more EG is produced for $N_1\ne 0$ and $N_2\ne 0$ compared to $N_1\ne 0$, $N_2=0$ and $N_1=0$, $N_2=0$ cases. In order to achieve a more efficient energy system, it can be concluded to utilize a Carreau fluid with a dilatant nature (i.e. n > 1) and higher Weissenberg numbers combined with lower Eckert values.

5. Conclusions

This work aimed to investigate the impact of the flow of a periodic magnetized Carreau nanofluid across a wedge surface on entropy formation with liquid nitrogen diffusion and activation energy. According to our research, the conclusions can be summarized as follows:

• The angle of the wedge significantly influences the fluid flow surrounding it. Increasing the wedge angle results in a dramatic increase in fluid velocity.

• The velocity $F\left(\xi ,\eta \right)$ becomes larger as We rises from 0 to 4 for n = 0.5, but shrinks for n = 1.5 for the same We changes. The surface drag produces divergent results in comparison to the velocity.

• The application of a periodic MHD causes sinusoidal fluctuations in the gradients. Also, the periodic magnetic field does not impact the pace at which liquid nitrogen undergoes mass transfer.

• The flow’s velocity $F\left(\xi ,\eta \right)$ drops drastically, and the surface drag grows owing to the Lorentz force induced by the imposed periodic MHD. Particularly, $R{e}^{1/2}{C}_f$ surges approximately by 52% when M becomes 3 from 0 at $\xi =1$ and $\mathrm{Nr}=1$.

• Enhancement in We from 0 to 4, upsurges $R{e}^{1/2}{C}_f$ by approximately about 23% in case of n = 0.5, and declines by around 41% in the case of n = 1.5 at $\xi =1$.

• Liquid oxygen’s intensity of mass transport is around 18% more than liquid nitrogen’s at $\xi =1$, $E=2$ and $\mathrm{Kc}=1.5$.

• When we surge the activation energy E, for a chemical reaction, the $H\left(\xi ,\eta \right)$ rises while its mass transfer falls.

• The $S_G\left(\xi ,\eta \right)$ is noted to be reduced near the wedge’s wall for rising We characteristics for the dilatant fluid (n = 1.5), and for pseudoplastic fluid (n = 0.5), the $S_G\left(\xi ,\eta \right)$ enriched for the equivalent changes in We.

• To achieve a more efficient energy-consumption system, it can be concluded to utilize a Carreau fluid with a dilatant nature (i.e. n > 1) and higher Weissenberg numbers combined with lower Eckert values.

Nomenclature

H	=	nondimensional concentration (liquid nitrogen)
We	=	Weissenberg number
G	=	nondimensional temperature
Re	=	Reynolds number
T	=	dimensional temperature (K)
Pr	=	Prandtl number
S	=	nondimensional nanoparticles’ concentration
Nb	=	Brownian diffusion characteristic
Ri	=	Richardson number
Nt	=	thermophoresis attribute
G	=	gravitational acceleration (ms−2)
$C$	=	liquid nitrogen concentration (mol/L)
Kc	=	chemical reaction attribute
f	=	nondimensional stream function
Le	=	Lewis number
Sc	=	Schmidt number
u, v	=	velocity components (ms−1)
M	=	magnetic parameter
E	=	dimensionless activation energy
$u_{\mathrm{\infty }}$	=	mainstream velocity constant (ms−1)
$T_{\mathrm{\infty }}$	=	mainstream temperature (K)
$T_w$	=	wall temperature (K)
F	=	nondimensional velocity
$R^{\ast }$	=	gas constant (Jmol−1K−1)
Ec	=	Eckert number

Greek signs

${\text{β}}_1,{\text{β}}_2$	=	thermal concentration expansion coefficient
${\text{β}}_3,{\text{β}}_4$	=	concentration expansion coefficient
${\text{β}}_t,{\text{β}}_c$	=	quadratic convection parameter for temperature and concentration
$\sigma$	=	electrical conductivity (Sm−1)
${\mathrm{\Omega }}_{\theta }$	=	nanoparticle density difference ratio
$\psi$	=	stream function (m2s−1)
${\mathrm{\Omega }}_T$	=	temperature difference ratio
$\phi$	=	half angle of the wedge (rd)
$\theta$	=	nanoparticles concentration (mol/L)
$\nu$	=	kinematic viscosity (m2s−1)
$\xi ,\eta$	=	transformed variables

Abbreviations

AE	=	Activation Energy
EG	=	Entropy Generation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the DST/INSPIRE Fellowship/2019/IF190225, awarded to the second author by the Department of Science and Technology, Ministry of Science and Technology, India, Innovation in Science Pursuit for Inspired Research (INSPIRE), New Delhi.