Entropy Analysis of Carreau Nanofluid Flow in the Presence of Joule Heating and Viscous Dissipation Past Unsteady Stretching Cylinder

Abstract

ABSTRACT

The study aims to investigate the irreversibilities of a Carreau nanofluid flow over, unsteady stretching cylindrical sheet exposed to radiation, non‐Darcy porous medium, viscous dissipation, joule heating, etc. It provides how energy produced in the nanofluid flow is used efficiently by minimizing the irreversibilities. The governing partial differential equations are transformed into first‐order initial value problems by similarity transformation and linearization. The shooting technique and an open‐source Python programming package are used to solve the initial value problems using the Runge–Kutta sixth‐order, and the numerical approach is validated using published articles. Basic flow profiles and, most importantly, entropy generation are examined using graphs in relation to relevant parameters. Skin friction and the behavior of heat and mass fluxes in response to various parameters are also examined. The results of the study demonstrated that the entropy creation is initiated by an increase in the magnetic and curvature parameters, as well as the Eckert, Brinkman, and porosity parameters. However, when the Forchheimer number increases, entropy generation decreases. An increase in the Eckert number, Prandtl number, and radiation parameter motivates the irreversibility due to heat transfer, whereas as the Weissenberg number rises, the irreversibility of heat transfer falls around the wall. According to the numerical values in the table, growth in Weissenberg number, thermal Biot number, Forchheimer number, curvature parameter, and radiation parameter initiate the magnitude of the rate of heat and mass transfers. In contrast, the rates fell as the Eckert and Prandtl values rose. Analysis of energy conversions and system efficiency can be done using this study, particularly in heat engines, refrigeration systems, and other thermodynamic processes.

Introduction

Entropy is a broad term used metaphorically to describe “disorderness” across a variety of disciplines, such as engineering, information technology, fluid dynamics, thermodynamics, and other natural and social sciences. This study takes into account the entropy that Clausius introduced in the 1850s [1]. It is a purely thermodynamic property that deals with energy loss or energy that cannot be transformed into work because of mass diffusivity, joule heating, radiation of heat, viscous dissipation, etc. [2]. This energy loss is taken as the second law of thermodynamics [3–5], which is not conserved like that of the first law of thermodynamics [6]. Bejan [7] have conducted groundbreaking research on entropy creation and irreversibility minimization. These days, energy efficiency has grown to be a major concern in addition to the quantity of energy generated to complete a certain activity. In this context, the concept of reducing irreversible energy and boosting energy efficiency in various systems, such as heat exchanger design, power plant design, thermal storage, electronic device cooling, etc., has given the due emphasis it deserves [8–11]. The entropy generation of hybrid and trihybrid nanofluid has been investigated by recent articles in different scenarios [12, 13].

Since 1995, when Choi and his colleagues first used the word “nanofluid” [14], studies on the production, properties, applications, and efficacy of nanofluids have been conducted continuously over the past 20 years. In simple language, nanofluids are a stable, homogeneous mixture of metallic nanoparticles and common fluids including water, engine oil, and ethylene glycol [15]. The field receives the attention it merits because of its varied uses in nuclear reactors, heat exchangers, food, and other industries, as well as in the chemical, automotive, medicinal, and electronic sectors [16–19].

The majority of biological and industrial fluids are classified as non-Newtonian fluids. Researchers were able to classify non-Newtonian fluids as Carreau, tangent hyperbolic, Williamson, Casson, Jeffery, etc. [20–22] based on the non-linear shear stress-strain relations. As a result, there is mounting evidence that scientists should look into the thermophysical properties and flow dynamics of different non-Newtonian fluids. Carreau [23] developed the Carreau model by improving the restriction of the power law model, which fails to predict viscosity at the two extreme values of shear rate. Shear thinning and shear thickening are two significant categories introduced by Carreau fluid behavior that are essential to the study of the flow regime [24, 25].

Numerous studies on the entropy analysis of non-Newtonian fluid flow have been conducted since the advent of entropy and various irreversibilities. Numerous studies have taken into account the irreversibilities caused by mass diffusivity, heat transfer, joule heating, and viscous dissipation. Various recommendations have been made to reduce wastage of energy wasted as a result of all of those irreversibilities [26–30]. Mohamed et al. [31] investigated energy degradation in channel design and came to the conclusion that the minimum entropy generations are determined by the optimum inclined angle. Rahila et al. [32] examined the entropy generation of Carreau nanofluid flow on a flat cylinder and found that, in the shear thinning scenario of the Carreau fluid, the temperature difference parameter increases the system's entropy generation. The entropy generation analysis of Carreau nanofluid flows under various conditions has been performed [33–36].

Arrhenius developed an equation that linked temperature, activation energy, and chemical reaction rate [37]. A binary chemical reaction supported by activation energy is one of the driving forces that could accelerate the irreversibility caused by mass diffusivity. Accordingly, scientists have investigated the entropy analysis of non-Newtonian nanofluid flow with chemical reaction supported by Arrhenius activation [38–41].

Flow across a permeable medium exposes mass diffusivity, leading to one of the common irreversibilities. For the purposes of generalization, Forchheimer's model of a porous medium with substantial porosity is employed [42]. The motion of various non-Newtonian nanofluids is influenced negatively by the rise of Forchheimer number [43, 44].

Nowadays, the effective use of the generated energy in any technical or industrial process has received tremendous attention. To reduce the irreversibilities, a deep comprehension of entropy generation is essential. Numerous researchers have studied the entropy analysis of Carreau nanofluid [26, 28–30], but not on unsteady stretching cylindrical sheets with joule heating and viscous dissipation. The Forchheimer porosity model has been applied by researchers [45–47] to the study of basic profiles, but not to the entropy analysis. Following a review of the previously stated literature, the main goal of this study is to use the Forchheimer porosity model to investigate the entropy analysis of Carreau nanofluid flow on an unsteady stretching cylinder in the presence of joule heating and viscous dissipation. Moreover, the heat irreversibility to the total entropy generation is also considered. We believe that the study will improve the design of power plants, heat exchangers, thermal storage, electronic device cooling, etc. Additionally, it will be crucial in igniting more research in the area.

Problem Formulation and Assumptions

The present study examines the entropy generation of five irreversibility conditions of Carreau nanofluid flow with the following assumptions:

• The flow is laminar, incompressible, and exposed to an electrically conducting fluid.
• The flow is two dimensional over unsteady stretching cylindrical sheet with stretching velocity, $$$ {w}_w\left(z,t\right)=\frac{cz}{1- at} $$$, where $$$ a $$$ and $$$ c $$$ are positive constants.
• As seen in Figure 1, a time-dependent magnetic field, $$$ B(t)=\frac{B_0}{\sqrt{1- at}} $$$, is applied radially.
• The flow is subjected to mixed convective boundary conditions with no-slip conditions on a permeable media, which combine the effects of binary chemical reaction, viscous dissipation, and Joule heating.
• The thermal equilibrium between the fluid and the nanoparticles is assumed.
• In the cylindrical coordinates $$$ \left(r,\theta, z\right) $$$, the velocity field of the nanofluid is often expressed as $$$ V=\left(u,0,w\right) $$$, and the transverse magnetic field is $$$ B(t)=\left(\frac{B_0}{\sqrt{1- at}},0,0\right) $$$.
• The action of the Nabla, $$$ \nabla $$$ and Laplacian, $$$ {\nabla}^2 $$$ operators are within the framework of the cylindrical coordinate system.

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When all of the aforementioned assumptions are taken into account within the conservation laws, the vector form of the flow dynamics can be written as follows [16, 44]: 1 $$$ \kern0.5em \nabla .V=0 $$$ 2 $$$ \kern0.5em \frac{\partial V}{\partial t}+\left(V.\nabla \right)V=\frac{1}{\rho_{nf}}\nabla .{\tau}_1+\frac{1}{\rho_{nf}}J\times B-\frac{\nu_{nf}}{K}V-F\mid V\mid V $$$ 3 $$$ {\displaystyle \begin{array}{ll}\hfill & \frac{\partial T}{\partial t}+\left(V.\nabla \right)T={\alpha}_{nf}{\nabla}^2T+\tau \left[{D}_B\nabla C.\nabla T+\frac{D_T}{T_{\infty }}\left(\nabla T.\nabla T\right)\right]\\ {}\hfill & \kern1.4em +\frac{S:\nabla V}{{\left(\rho {C}_p\right)}_{nf}}+\frac{J.J}{\alpha_{nf}{\left(\rho {C}_p\right)}_{nf}}\\ {}\hfill & \kern1.4em +\left(\frac{\nu_{nf}}{K}+\frac{F}{C_p}|V|\right){\left|V\right|}^2-\frac{\nabla {q}_r}{{\left(\rho {C}_p\right)}_{nf}}+\frac{q^{\prime \prime \prime }}{{\left(\rho {C}_p\right)}_{nf}}\end{array}} $$$ 4 $$$ {\displaystyle \begin{array}{ll}\hfill & \frac{\partial C}{\partial t}+\left(V.\nabla \right)C={D}_B{\nabla}^2C+\frac{D_T}{T_{\infty }}{\nabla}^2T\\ {}\hfill & \kern1em -{K}_r^2\left(C-{C}_{\infty}\right){\left(\frac{T}{T_{\infty }}\right)}^m\mathit{\exp}\left(-\frac{E_a}{\kappa T}\right)\end{array}} $$$ where $$$ {\nu}_{nf}=\frac{\mu_{nf}}{\rho_{nf}} $$$, $$$ {\alpha}_{nf}=\frac{\kappa }{{\left(\rho {C}_p\right)}_{nf}} $$$, $$$ \tau =\frac{{\left(\rho {C}_P\right)}_{np}}{{\left(\rho {C}_p\right)}_{nf}} $$$, $$$ F=\frac{C_b}{r\sqrt{K}} $$$ is the non-uniform inertial coefficient. Moreover, $$$ {\left(\rho {C}_p\right)}_{nf} $$$ and $$$ {\left(\rho {C}_p\right)}_{np} $$$ are the heat capacity of the nanofluid and nanoparticles, respectively.

In the momentum equation, $$$ J\times B={\sigma}_{nf}\left(V\times B\right)\times B={\sigma}_{nf}\left(0,0,-\frac{B_0^2}{1- at}w\right) $$$ is the Lorenz force and $$$ {\tau}_1 $$$ is the Cauchy stress tensor given by: 5 $$$ {\tau}_1=- PI+S $$$ where $$$ P $$$ is the pressure of the nanofluid, $$$ I $$$ is the unit tensor and $$$ S $$$ is the extra stress tensor of Carreau fluid given as: 6 $$$ S=\left[{\mu}_{\infty }+\left({\mu}_0-{\mu}_{\infty}\right){\left[1+{\left(\Gamma \dot{\gamma}\right)}^2\right]}^{\frac{n-1}{2}}\right]{A}_1 $$$ where $$$ {\mu}_0 $$$ and $$$ {\mu}_{\infty } $$$ are the limiting viscosities when the shear rate is at zero and infinity respectively, $$$ \Gamma >0 $$$ is the time constant, $$$ {A}_1=\nabla V+{\left(\nabla V\right)}^T $$$ is the Rivline-Erickson tensor or the velocity gradient tensor and $$$ \dot{\gamma}=\frac{1}{2}\sqrt{tr\left({A}_1^2\right)} $$$. For most practical cases $$$ {\mu}_0>>{\mu}_{\infty } $$$ and hence we can take $$$ {\mu}_{\infty }=0 $$$, which gives, 7 $$$ S={\mu}_0{\left[1+{\left(\Gamma \dot{\gamma}\right)}^2\right]}^{\frac{n-1}{2}}{A}_1 $$$ where 8 $$$ \dot{\gamma}={\left[{\left(\frac{\partial u}{\partial r}\right)}^2+{\left(\frac{\partial w}{\partial z}\right)}^2+{\left(\frac{u}{r}\right)}^2+\frac{1}{2}{\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r}\right)}^2\right]}^{\frac{1}{2}} $$$ The component of the tensor, $$$ S $$$ for the velocity field, $$$ V=\left(u,0,w\right) $$$ is written as: 9 $$$ {\displaystyle \begin{array}{ll}\hfill {S}_{r r}& =2{\mu}_0{\left[1+{\left(\Gamma \dot{\gamma}\right)}^2\right]}^{\frac{n-1}{2}}\frac{\partial u}{\partial r}\\ {}\hfill {S}_{r z}& ={S}_{z r}={\mu}_0{\left[1+{\left(\Gamma \dot{\gamma}\right)}^2\right]}^{\frac{n-1}{2}}\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r}\right),\kern1em {S}_{r\theta}={S}_{\theta r}=0\\ {}\hfill {S}_{z z}& =2{\mu}_0{\left[1+{\left(\Gamma \dot{\gamma}\right)}^2\right]}^{\frac{n-1}{2}}\frac{\partial w}{\partial z}\\ {}\hfill {S}_{\theta \theta}& =2{\mu}_0{\left[1+{\left(\Gamma \dot{\gamma}\right)}^2\right]}^{\frac{n-1}{2}}\frac{u}{r},\kern1em {S}_{z\theta}={S}_{\theta z}=0\end{array}} $$$ In the energy equation, $$$ J.J={\sigma}_{nf}^2\frac{B_0^2}{1- at}{w}^2 $$$, $$$ \nabla {q}_r=\frac{16{\sigma}^{\ast }{T}_{\infty}^3}{3{k}^{\ast }}\left(\frac{\partial^2T}{\partial {r}^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right) $$$, and $$$ {q}^{\prime \prime \prime }={Q}_0\left(T-{T}_{\infty}\right) $$$. Moreover, using the boundary layer approximations and Equations (8) and (9), the viscous dissipation term is simplified as [44]: 10 $$$ {\displaystyle \begin{array}{ll}\hfill S:\nabla V& =\nabla \cdotp \left(S\cdotp V\right)-V\cdotp \left(\nabla \cdotp S\right)\\ {}\hfill & ={\mu}_0{\left[1+{\left(\frac{\Gamma}{\sqrt{2}}\frac{\partial w}{\partial r}\right)}^2\right]}^{\frac{n-1}{2}}{\left(\frac{\partial w}{\partial r}\right)}^2\end{array}} $$$ Using the Buongiorno's flow model [48], adopting the mixed convective heat and mass transfer boundary conditions, and with the aforementioned assumptions and employing the boundary layer approximations, the governing equation of the Carreau nanofluid flow is described as follows [44, 49–51]: 11 $$$ \kern0.5em \frac{\partial w}{\partial z}+\frac{u}{r}+\frac{\partial u}{\partial r}=0 $$$ 12 $$$ {\displaystyle \begin{array}{ll}\hfill & \frac{\partial w}{\partial t}+w\frac{\partial w}{\partial z}+u\frac{\partial w}{\partial r}={\nu}_{nf}{\left[1+{\left(\frac{\Gamma}{\sqrt{2}}\frac{\partial w}{\partial r}\right)}^2\right]}^{\frac{n-1}{2}}\\ {}\hfill & \kern1em \left[\frac{\Gamma^2\left(n-1\right)}{2}{\left(\frac{\partial w}{\partial r}\right)}^2\frac{\partial^2w}{\partial {r}^2}{\left(1+{\left(\frac{\Gamma}{\sqrt{2}}\frac{\partial w}{\partial r}\right)}^2\right)}^{-1}\right.\\ {}\hfill & \kern1em \left.+\frac{1}{r}\frac{\partial w}{\partial r}+\frac{\partial^2w}{\partial {r}^2}\right]-\left(\frac{\sigma_{nf}}{\rho_{nf}\left(1- at\right)}{B}_0^2w+\frac{\nu_{nf}}{K}w+F{w}^2\right)\end{array}} $$$ 13 $$$ {\displaystyle \begin{array}{ll}\hfill & \frac{\partial T}{\partial t}+w\frac{\partial T}{\partial z}+u\frac{\partial T}{\partial r}={\alpha}_{nf}\left(\frac{\partial^2T}{\partial {r}^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)\\ {}\hfill & \kern1.4em +\tau \left[{D}_B\frac{\partial T}{\partial r}\frac{\partial C}{\partial r}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial r}\right)}^2\right]\\ {}\hfill & \kern1.4em +\frac{16{\sigma}^{\ast }{T}_{\infty}^3}{3{\left(\rho {C}_p\right)}_{nf}{k}^{\ast }}\left(\frac{\partial^2T}{\partial {r}^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)\\ {}\hfill & \kern1.4em +\frac{Q_0}{{\left(\rho {C}_P\right)}_{nf}}\left(T-{T}_{\infty}\right)+\frac{\nu_{nf}}{C_p}{\left[1+{\left(\frac{\Gamma}{\sqrt{2}}\frac{\partial w}{\partial r}\right)}^2\right]}^{\frac{n-1}{2}}{\left(\frac{\partial w}{\partial r}\right)}^2\\ {}\hfill & \kern1.4em +\left(\frac{\sigma_{nf}{B}_0^2}{{\left(\rho {C}_p\right)}_{nf}\left(1- at\right)}+\frac{\nu_{nf}}{K{C}_p}\right){w}^2+\frac{F}{C_p}{w}^3\end{array}} $$$ 14 $$$ {\displaystyle \begin{array}{ll}\hfill & \frac{\partial C}{\partial t}+w\frac{\partial C}{\partial z}+u\frac{\partial C}{\partial r}={D}_B\left(\frac{\partial^2C}{\partial {r}^2}+\frac{1}{r}\frac{\partial C}{\partial r}\right)\\ {}\hfill & \kern1em +\frac{D_T}{T_{\infty }}\left(\frac{\partial^2T}{\partial {r}^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right)-{K}_r^2\left(C-{C}_{\infty}\right){\left(\frac{T}{T_{\infty }}\right)}^m\mathit{\exp}\left(-\frac{E_a}{kT}\right)\end{array}} $$$ where $$$ \tau =\frac{{\left(\rho {C}_P\right)}_{np}}{{\left(\rho {C}_p\right)}_{nf}} $$$.

Since the medium is preamble, there will be suction/injection of fluid on the surface. Moreover, a net zero mass and heat flux (mixed convective) is also assumed on the surface because of the injection and suction effects. Thus, the boundary conditions of the flow dynamics are as follows [50, 51]: 15 $$$ {\displaystyle \begin{array}{ll}\hfill & At\kern1em r=R,\kern1em w={w}_w\left(z,t\right)=\frac{cz}{1- at},u=-{u}_w(t)=-\frac{u_0}{\sqrt{1- at}}\\ {}\hfill & \kern1em -{k}_w\frac{\partial T}{\partial r}={h}_w\left({T}_w-T\right),\kern1em {D}_B\frac{\partial C}{\partial r}+\frac{D_T}{T_{\infty }}\frac{\partial T}{\partial r}=0\end{array}} $$$ 16 $$$ \kern0.5em As\kern1em r\to \infty, \kern1em w\to 0,\kern1em T\to {T}_{\infty },\kern1em C\to {C}_{\infty } $$$ The governing Equations (12)–(14) with the boundary conditions (15) and (16) are transformed into a system of non-linear ODEs using the similarity variables as follows [43, 49]: 17 $$$ {\displaystyle \begin{array}{ll}\hfill \eta & =\frac{r^2-{R}^2}{2R}\sqrt{\frac{c}{\nu_{nf}\left(1- at\right)}},\kern1em \psi =\sqrt{\frac{c{\nu}_{nf}}{1- at}} zRf\left(\eta \right),\\ {}\hfill \theta & =\frac{T-{T}_{\infty }}{T_w-{T}_{\infty }},\kern1em \phi =\frac{C-{C}_{\infty }}{C_w-{C}_{\infty }}\end{array}} $$$ The stream function, $$$ \psi =\sqrt{\frac{c{\nu}_{nf}}{1- at}} zRf\left(\eta \right) $$$ with the component velocities defined below satisfies (11): 18 $$$ w\kern0.5em =\frac{1}{r}\frac{\partial \psi }{\partial r}=\frac{cz}{1- at}{f}^{\prime}\left(\eta \right) $$$ 19 $$$ u\kern0.5em =-\frac{1}{r}\frac{\partial \psi }{\partial z}=-\frac{1}{r}\sqrt{\frac{c{\nu}_{nf}}{1- at}} Rf\left(\eta \right) $$$ Using (18) and (19), the governing Equations (12)–(14) are transformed into a couple of dimensionless equations: 20 $$$ {\displaystyle \begin{array}{ll}\hfill & {\left(1+\frac{We^2}{4}{f^{\prime \prime}}^2\right)}^{\frac{n-1}{2}}\left[{f}^{\prime \prime \prime }+\frac{2\gamma }{1+2\eta \gamma}{f}^{\prime \prime}\right.\\ {}\hfill & \kern1em \left.+\left(n-1\right){\left(1+\frac{We^2}{4}{f^{\prime \prime}}^2\right)}^{-1}\frac{We^2}{4}{f^{\prime \prime}}^2\left({f}^{\prime \prime \prime }+\frac{\gamma }{1+2\eta \gamma}{f}^{\prime \prime}\right)\right]\\ {}\hfill & \kern1em -\frac{1}{1+2\eta \gamma}\left[\frac{A\eta}{2}{f}^{\prime \prime }+\left(1+ Fr\right){f^{\prime}}^2-f{f}^{\prime \prime }+\left(A+M+\lambda \right){f}^{\prime}\right]=0\end{array}} $$$ 21 $$$ {\displaystyle \begin{array}{ll}\hfill & \left(1+2\eta \gamma \right)\left[\frac{1+\frac{4}{3} Rd}{\mathit{\Pr}}{\theta}^{\prime \prime }+ Nt{\theta^{\prime}}^2+ Nb{\theta}^{\prime }{\phi}^{\prime}\right.\\ {}\hfill & \kern1em \left.+ Ec{\left(1+\frac{We^2}{4}{f^{\prime \prime}}^2\right)}^{\frac{n-1}{2}}{f^{\prime \prime}}^2\right]+ Q\theta + Ec\left(M+\lambda \right){f^{\prime}}^2\\ {}\hfill & \kern1em + Ec Fr{f^{\prime}}^3+\left[\frac{2\gamma \left(1+\frac{4}{3} Rd\right)}{\mathit{\Pr}}-\frac{A\eta}{2}+f\right]{\theta}^{\prime }=0\end{array}} $$$ 22 $$$ {\displaystyle \begin{array}{ll}\hfill & \frac{1+2\eta \gamma}{Sc}\left[{\phi}^{\prime \prime }+\frac{Nt}{Nb}{\theta}^{\prime \prime}\right]+\frac{2\gamma Nt}{Sc Nb}{\theta}^{\prime }+\left(\frac{2\gamma }{Sc}-\frac{A\eta}{2}+f\right){\phi}^{\prime}\\ {}\hfill & \kern1em -\beta \phi {\left(1+\delta \theta \right)}^m\mathit{\exp}\left(-\frac{E}{\delta \theta +1}\right)=0\end{array}} $$$ with the transformed boundary conditions 23 $$$ {\displaystyle \begin{array}{ll}\hfill f\left(\eta \right)& ={S}_1,\kern1em {f}^{\prime}\left(\eta \right)=1,\kern1em {\theta}^{\prime}\left(\eta \right)=-B{i}_1\left(1-\theta \left(\eta \right)\right)\\ {}\hfill {\phi}^{\prime}\left(\eta \right)& =\frac{Nt}{Nb}B{i}_1\left(1-\theta \left(\eta \right)\right)\kern1em \mathrm{at}\kern1em \eta =0\end{array}} $$$ 24 $$$ {f}^{\prime}\left(\eta \right)\kern0.5em \to 0,\kern1em \theta \left(\eta \right)\to 0,\kern1em \phi \left(\eta \right)\to 0\kern2em \mathrm{as}\kern1em \eta \to \infty $$$ where $$$ {\displaystyle \begin{array}{ll}\hfill We& =\frac{czr\Gamma}{R\left(1- at\right)}\sqrt{\frac{2c}{\nu_{nf}\left(1- at\right)}},\kern1em A=\frac{a}{c},\\ {}\hfill \gamma & =\sqrt{\frac{\nu_{nf}\left(1- at\right)}{R^2c}},\kern1em M=\frac{\sigma {B}_0^2\left(1- at\right)}{\rho_{nf}c},\\ {}\hfill Nt& =\frac{\tau {D}_T\left({T}_w-{T}_{\infty}\right)}{\nu_{nf}{T}_{\infty }},\kern1em Nb=\frac{\tau {D}_B\left({C}_w-{C}_{\infty}\right)}{\nu_{nf}},\\ {}\hfill \mathit{\Pr}& =\frac{\nu_{nf}}{\alpha_{nf}},\kern1em Rd=\frac{4{\sigma}^{\ast }{T}_{\infty}^3}{\kappa_{nf}{k}^{\ast }},\kern1em \\ {}\hfill \lambda & =\frac{\nu_{nf}\left(1- at\right)}{Kc},Q=\frac{Q_0\left(1- at\right)}{c{\left(\rho {C}_p\right)}_{nf}},\\ {}\hfill Sc& =\frac{\nu_{nf}}{D_B},\kern1em E=\frac{E_a}{k{T}_{\infty }},\kern1em {S}_1=\frac{v_0}{\sqrt{c{\nu}_{nf}}},\kern1em Fr=\frac{C_b}{\sqrt{K}},\\ {}\hfill Ec& =\frac{c^2{z}^2}{C_p\left({T}_w-{T}_{\infty}\right){\left(1- at\right)}^2},\kern1em B{i}_1=\frac{h_w}{k_w}\sqrt{\frac{\nu_{nf}\left(1- at\right)}{c}},\\ {}\hfill \beta & =\frac{K_r^2\left(1- at\right)}{c},\kern1em \delta =\frac{T_w-{T}_{\infty }}{T_{\infty }}\end{array}} $$$

Entropy Analysis

For an MHD Carreau nanofluid flow with thermal radiation, Joule heating, and viscous dissipation, the volumetric entropy generation, $$$ {E}_G $$$, for the boundary layer flow of heat and mass transfer is provided by [52]: 25 $$$ {\displaystyle \begin{array}{ll}\hfill {E}_G& =\frac{k_{nf}}{T_{\infty}^2}\left[1+\frac{16{\sigma}^{\ast }{T}_{\infty}^3}{3{k}^{\ast }{k}_{nf}}\right]{\left(\frac{\partial T}{\partial r}\right)}^2\\ {}\hfill & \kern1em +\frac{\mu_{nf}}{T_{\infty }}{\left[1+{\left(\frac{\Gamma}{\sqrt{2}}\frac{\partial w}{\partial r}\right)}^2\right]}^{\frac{n-1}{2}}{\left(\frac{\partial w}{\partial r}\right)}^2\\ {}\hfill & \kern1em +\left(\frac{\sigma_{nf}{B}_0^2}{T_{\infty}\left(1- at\right)}+\frac{\mu_{nf}}{K{T}_{\infty }}\right){w}^2\\ {}\hfill & \kern1em +\frac{\rho_{nf}F}{T_{\infty }}{w}^3+\frac{R{D}_m}{C_{\infty }}{\left(\frac{\partial C}{\partial r}\right)}^2+\frac{R{D}_m}{T_{\infty }}\frac{\partial C}{\partial r}\frac{\partial T}{\partial r}\end{array}} $$$ The characteristic entropy transfer rate, $$$ {E}_0 $$$ is defined as 26 $$$ {E}_0=\frac{k_{nf}c\left({T}_w-{T}_{\infty}\right)}{\nu_{nf}{T}_{\infty}\left(1- at\right)} $$$ The non-dimensional volumetric entropy generation rate, $$$ {N}_d $$$ is defined as the ratio of $$$ {E}_G $$$ to $$$ {E}_0 $$$ [52]. That is: 27 $$$ {\displaystyle \begin{array}{ll}\hfill {N}_d=\frac{E_G}{E_0}& =\left(1+2\eta \gamma \right)\left[\left(1+\frac{4}{3} Rd\right)\delta {\theta^{\prime}}^2\right.\\ {}\hfill & \kern1em \left.+ Br{\left(1+\frac{We^2}{4}{f^{\prime \prime}}^2\right)}^{\frac{n-1}{2}}{f^{\prime \prime}}^2+{\lambda}_1\left(\frac{\delta_1}{\delta }{\phi^{\prime}}^2+{\theta}^{\prime }{\phi}^{\prime}\right)\right]\\ {}\hfill & \kern1em + Br\left(M+\lambda \right){f^{\prime}}^2+ Br Fr{f^{\prime}}^3\end{array}} $$$ where $$$ {\displaystyle \begin{array}{ll}\hfill Br& =\frac{\mu_{nf}{c}^2{z}^2}{k_{nf}\left({T}_w-{T}_{\infty}\right){\left(1- at\right)}^2},\kern1em {\lambda}_1=\frac{RDm\left({C}_w-{C}_{\infty}\right)}{k_{nf}},\\ {}\hfill {\delta}_1& =\frac{C_w-{C}_{\infty }}{C_{\infty }}\end{array}} $$$ In this paper, the entropy generation due to five irreversibility conditions is considered, which include entropy generation due to heat transfer, $$$ {H}_t $$$, viscous dissipation, $$$ {V}_d $$$, mass diffusivity, $$$ {M}_d $$$, Lorentz force, $$$ {L}_f $$$ and inertia forces, $$$ {I}_f $$$. Hence, 28 $$$ {N}_d={H}_t+{V}_d+{M}_d+{L}_f+{I}_f $$$ where $$$ {\displaystyle \begin{array}{ll}\hfill {H}_t& =\left(1+2\eta \gamma \right)\left(1+\frac{4}{3} Rd\right)\delta {\theta^{\prime}}^2,\\ {}\hfill {V}_d& =\left(1+2\eta \gamma \right) Br{\left(1+\frac{We^2}{4}{f^{\prime \prime}}^2\right)}^{\frac{n-1}{2}}{f^{\prime \prime}}^2,\\ {}\hfill {M}_d& =\left(1+2\eta \gamma \right){\lambda}_1\left(\frac{\delta_1}{\delta }{\phi^{\prime}}^2+{\theta}^{\prime }{\phi}^{\prime}\right),\kern1em {L}_f= Br M{f^{\prime}}^2,\\ {}\hfill {I}_f& = Br\left(\lambda {f^{\prime}}^2+ Fr{f^{\prime}}^3\right)\end{array}} $$$ The Bejan number, $$$ Be $$$ is the proportion of heat irreversibility to the total entropy generation [3, 53] expressed as: 29 $$$ {\displaystyle \begin{array}{ll}\hfill Be& =\frac{H_t}{N_d}\\ {}\hfill & =\frac{\left(1+\frac{4}{3} Rd\right)\delta {\theta^{\prime}}^2}{\begin{array}{c}\left(1+\frac{4}{3} Rd\right)\delta {\theta^{\prime}}^2+ Br{\left(1+\frac{W{e}^2}{4}{f^{\prime \prime}}^2\right)}^{\frac{n-1}{2}}{f^{\prime \prime}}^2\\ {}+{\lambda}_1\left(\frac{\delta_1}{\delta }{\phi^{\prime}}^2+{\theta}^{\prime }{\phi}^{\prime}\right)+\frac{Br}{1+2\eta \gamma}\left(M+\lambda + Fr{f}^{\prime}\right){f^{\prime}}^2\end{array}}\end{array}} $$$ The study also includes the three physical quantities that are specified below for engineering and other sciences: the skin friction coefficient $$$ {C}_f $$$, the local Sherwood number $$$ S{h}_z $$$, and the local Nusselt number $$$ N{u}_z $$$. 30 $$$ {\displaystyle \begin{array}{ll}\hfill {C}_f& =\frac{\tau_w}{\rho_{nf}{w}_w^2},\kern2em N{u}_z=\frac{z{q}_w}{\kappa_{nf}\left({T}_w-{T}_{\infty}\right)}\\ {}\hfill S{h}_z& =\frac{z{q}_m}{D_B\left({C}_w-{C}_{\infty}\right)}\end{array}} $$$ where the skin friction $$$ {\tau}_w $$$, wall heat flux $$$ {q}_w $$$ and wall mass flux $$$ {q}_m $$$ are given by: 31 $$$ {\tau}_w\kern0.5em ={\left({\mu}_{nf}\frac{\partial w}{\partial r}{\left[1+{\left(\frac{\Gamma}{\sqrt{2}}\frac{\partial w}{\partial r}\right)}^2\right]}^{\frac{n-1}{2}}\right)}_{r=R} $$$ 32 $$$ {q}_w\kern0.5em =-\left({\kappa}_{nf}+\frac{16{\sigma}^{\ast }{T}_{\infty}^3}{3{k}^{\ast }}\right){\left(\frac{\partial T}{\partial r}\right)}_{r=R} $$$ 33 $$$ {q}_m\kern0.5em =-{D}_B{\left(\frac{\partial C}{\partial r}\right)}_{r=R} $$$ Using (31)–(33) and the local Reynolds number $$$ R{e}_z=\frac{w_wz}{\nu_{nf}} $$$, we get: 34 $$$ \kern0.5em {C}_fR{e}_z^{\frac{1}{2}}={f}^{\prime \prime }(0){\left(1+\frac{W{e}^2}{4}{\left({f}^{\prime \prime }(0)\right)}^2\right)}^{\frac{n-1}{2}} $$$ 35 $$$ \kern0.5em N{u}_zR{e}_z^{-\frac{1}{2}}=-\left(1+\frac{4}{3} Rd\right){\theta}^{\prime }(0) $$$ 36 $$$ \kern0.5em S{h}_zR{e}_z^{-\frac{1}{2}}=-{\phi}^{\prime }(0) $$$

Numerical Solution

It is difficult to find a closed-form solution for Equations (20)–(22) since they are substantially non-linear higher-order boundary value problems. As shown below, in order to implement the sixth-order Runge–Kutta (RK6) using the shooting technique, the boundary value problems must be converted to a system of first-order initial value problems (IVPs). Then, using the shooting and linearization strategies, the system of IVPs is solved by systematically estimating $$$ {\zeta}_1 $$$, $$$ {\zeta}_2 $$$, and $$$ {\zeta}_3 $$$ (45) until the boundary conditions are satisfied. Table data are evaluated and graphs are simulated using an open-source Python programming tool. The numerical solutions are found using an error of tolerance $$$ 1{0}^{-7} $$$ and a step size of 0.1.

To change the non-linear higher-order boundary value problems into a system of first-order IVPs, we have used the following relations. 37 $$$ {\displaystyle \begin{array}{ll}\hfill & \mathrm{Let}\kern1em {f}_1=f,\kern1em {f}_2={f}_1^{\prime },\kern1em {f}_3={f}_2^{\prime}\\ {}\hfill & {f}_4=\theta, \kern1em {f}_5={\theta}^{\prime },\kern1em {f}_6=\phi, \kern1em {f}_7={\phi}^{\prime}\end{array}} $$$ Now using the relation in Equation (37), the transformed Equations (20)–(22) linearized in to the system of IVPs as follows: 38 $$$ {f}_1^{\prime}\kern0.5em ={f}_2 $$$ 39 $$$ {f}_2^{\prime}\kern0.5em ={f}_3 $$$ 40 $$$ {\displaystyle \begin{array}{ll}\hfill {f}_3^{\prime }& =-\frac{1}{\left(1+2\gamma \eta \right)\left(1+\left(n-1\right)\frac{W{e}^2}{4}{f}_3^2{\left(1+\frac{W{e}^2}{4}{f}_3^2\right)}^{-1}\right)}\\ {}\hfill & \kern1em \left[2\gamma {f}_3+\left(n-1\right){\left(1+\frac{W{e}^2}{4}{f}_3^2\right)}^{-1}\frac{W{e}^2}{4}\gamma {f}_3^3\right]\\ {}\hfill & \kern1em +\frac{1}{\begin{array}{c}\left(1+2\eta \gamma \right)\left[{\left(1+\frac{W{e}^2}{4}{f}_3^2\right)}^{\frac{n-1}{2}}\right.\\ {}\left.+\left(n-1\right)\frac{W{e}^2}{4}{f}_3^2{\left(1+\frac{W{e}^2}{4}{f}_3^2\right)}^{\frac{n-3}{2}}\right]\end{array}}\\ {}\hfill & \kern1em \left[\frac{A\eta}{2}{f}_3-{f}_1{f}_3+\left(A+M+\lambda \right){f}_2+\left(1+ Fr\right){f}_2^2\right]\end{array}} $$$ 41 $$$ {f}_4^{\prime}\kern0.5em ={f}_5 $$$ 42 $$$ {\displaystyle \begin{array}{ll}\hfill {f}_5^{\prime }& =-\frac{\mathit{\Pr}}{1+\frac{4}{3} Rd}\left[ Nt{f}_5^2+ Nb{f}_5{f}_7+ Ec{\left(1+\frac{W{e}^2}{4}{f}_3^2\right)}^{\frac{n-1}{2}}{f}_3^2\right]\\ {}\hfill & \kern1em -\frac{2\gamma }{1+2\eta \gamma}{f}_5-\frac{\mathit{\Pr}}{\left(1+2\eta \gamma \right)\left(1+\frac{4}{3} Rd\right)}\\ {}\hfill & \kern1em \left[-\frac{A\eta}{2}{f}_5+{f}_1{f}_5+Q{f}_4+ Ec\left(M+\lambda \right){f}_2^2+ Ec Fr{f}_2^3\right]\end{array}} $$$ 43 $$$ {f}_6^{\prime}\kern0.5em ={f}_7 $$$ 44 $$$ {\displaystyle \begin{array}{ll}\hfill {f}_7^{\prime }& =\frac{PrNt}{Nb\left(1+\frac{4}{3} Rd\right)}\left[ Nt{f}_5^2+ Nb{f}_5{f}_7\right.\\ {}\hfill & \kern1em \left.+ Ec{\left(1+\frac{W{e}^2}{4}{f}_3^2\right)}^{\frac{n-1}{2}}{f}_3^2\right]\\ {}\hfill & \kern1em +\frac{NtPr}{Nb\left(1+2\eta \gamma \right)\left(1+\frac{4}{3} Rd\right)}\\ {}\hfill & \kern1em \left[-\frac{A\eta}{2}{f}_5+{f}_1{f}_5+Q{f}_4+ Ec\left(M+\lambda \right){f}_2^2+ Ec Fr{f}_2^3\right]\\ {}\hfill & \kern1em +\frac{Sc}{1+2\eta \gamma}\left[\left(\frac{A\eta}{2}-\frac{2\gamma }{Sc}\right){f}_7\right.\\ {}\hfill & \kern1em \left.-{f}_1{f}_7+\beta {f}_6{\left(\delta {f}_4+1\right)}^m\mathit{\exp}\left(\frac{-E}{\delta {f}_4+1}\right)\right]\end{array}} $$$ with the corresponding initial conditions: 45 $$$ {\displaystyle \begin{array}{ll}\hfill {f}_1(0)& ={S}_1,\kern1em {f}_2(0)=1,\kern1em {f}_3(0)={\zeta}_1,\kern1em {f}_4(0)={\zeta}_2\\ {}\hfill {f}_5(0)& =-B{i}_1\left(1-{\zeta}_2\right)\\ {}\hfill {f}_6(0)& ={\zeta}_3,\kern1em {f}_7(0)=\frac{Nt}{Nb}B{i}_1\left(1-{\zeta}_2\right)\end{array}} $$$ where $$$ {\zeta}_1,{\zeta}_2 $$$ and $$$ {\zeta}_3 $$$ are systematically guessed using the boundary conditions $$$ {f}_2\left(\eta \right)\to 0,{f}_4\left(\eta \right)\to 0 $$$ and $$$ {f}_6\left(\eta \right)\to 0 $$$ as $$$ \eta \to \infty $$$.

As shown in Table 1, we have compared our method and the program we implemented on common parameters and assumptions, making the remaining zero, with previously published works to validate our results.

TABLE 1 Comparison on $$$ -{\theta}^{\prime }(0) $$$ for $$$ n=1 $$$.

$$$ \mathit{\Pr} $$$	Makinde et al. [54]	Abolbashari et al. [55]	Obalalu et al. [40]	Present result
0.2	0.1691	0.1691	0.1734	0.1721
0.7	0.4539	0.4539	0.4539	0.4547
2.0	0.9114	0.9114	0.9114	0.9141
7.0	1.8954	1.8954	—	1.9021

Result and Discussions

The sixth-order Runge–Kutta with the shooting technique has been implemented, and an open-source Python programming package is used to generate values in the tables for the rate of momentum, heat, and mass fluxes as well as to simulate graphs for entropy generations due to various irreversibilities along with the fundamental flow profiles (velocity, temperature, and concentration). Unless and otherwise mentioned, $$$ We=0.4,\kern1em \gamma =0.8,\kern1em Fr=\lambda =E=A={S}_1=m=0.5,\kern.5em Ec=Q=\delta =0.1,\kern.7em M=0.6, Rd= Bi1= Nt=0.2,\kern1em Nb= Nt=\beta =0.3 $$$ and $$$ \mathit{\Pr}= Sc=1 $$$ have been used to simulate graphs and generate tables as default values. The behavior of the graphs remains unchanged for both cases (shear thinning and thickening) of the Carreau nanofluid, only the thinner boundary layer structure becomes more thinner and the thicker ones do the same. Consequently, $$$ n=1.5 $$$ is set as the default value.

Effects of Pertinent Parameters on Velocity Profile

For Figures 2–8, the effect of a few relevant parameters on the Carreau nanofluid's velocity is examined. For a variety of reasons, the selected relevant parameters typically have a detrimental effect on the nanofluid's mobility.

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The stretching rate $$$ c $$$ has an inverse relationship with the unsteadiness parameter and a direct relationship with the flow velocity. Therefore, as seen in Figure 2, as the unsteadiness parameter increases, the mobility of the nanofluid decreases. As illustrated in Figure 3, an increase in the porosity parameter causes a resistance inertia force to the nanofluid flow, which lowers its velocity. As displayed in Figure 4, an increase in the relaxation time causes the nanofluid's viscosity to grow, which in turn affects the fluid's motion negatively by raising the Weissenberg number. Figure 5 illustrates how the velocity drops as the power law index parameter increases. As shown in Figure 6, an additional nanofluid is introduced into the system through suction (for $$$ {S}_1>0 $$$) in a permeable wall, increasing the nanofluid's viscosity and potentially creating a resistance to the nanofluid motion. However, as seen in Figure 7, the injected (for $$$ {S}_1<0 $$$) nanofluid accelerates the mobility of the nanofluid that remained in the system by removing a portion of it via the permeable wall. A cylinder's radius and curvature have an inverse relationship. Therefore, a cylinder's radius decreases as its curvature increases, resulting in a lower surface area. This results in less contact with the wall and a large portion of the fluid will not be subject to wall friction, which eventually improves the motion of the nanofluid as depicted in Figure 8

Effects of Pertinent Parameters on Temperature Profile

The next six simulations signify how the temperature behaves with a rise in some pertinent parameters. The Carreau nanofluid within the boundary layer will experience heat gain or loss as long as there is a temperature difference between the wall and the nanofluid. A cooler wall might be utilized as a sink ($$$ Q<0 $$$), and the nanofluid would lose some heat to the wall. This would result in a decrease in temperature and a thinner thermal boundary layer, as shown in Figure 9. A wall that is substantially hotter may leak heat (as a source, $$$ Q>0 $$$) into the nanofluid, increasing its temperature as shown in Figure 10. The temperature of the nanofluid rises as the thermal Biot number rises, as Figure 11 illustrates. The dissipation (energy generated by the friction force between nanofluid layers and/or collision among nanoparticles) is dependent on the Eckert number, sometimes referred to as the dissipation parameter. As seen in Figure 12, the energy generated by dissipation boosts the temperature of the nanofluid by introducing heat energy into the system. As depicted in Figure 13, the decreased temperature is the result of the additional nanofluid (suction) cooling via a permeable medium. Figure 14 depicts the rise in nanofluid's temperature when a nanofluid is injected through a permeable material.

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Effects of Pertinent Parameters on Concentration Profile

The concentration behavior was determined by the relevant parameters in the next eight simulations. The mixed convective boundary condition we employed causes the boundary layer structures to seem inverted, with all of them lying below the $$$ \eta $$$-axis.

According to the findings depicted in Figures 15 and 16, an increase in the destructive ($$$ \beta >0 $$$) and constructive ($$$ \beta <0 $$$) chemical reaction causes the concentration of nanoparticles to disperse and accumulate, resulting in thinner and thicker boundary layer structures, respectively. With an increase in the activation energy parameter, Figure 17 shows a thicker concentration boundary layer. This may occur as a result of the nanoparticles' increased activation energy, which facilitates chemical reactions and creates a stable mixture around the wall. The temperature gradient in the system generates the thermophoretic force. As demonstrated in Figure 18, the force causes hot nanoparticles to move from the wall into the flow regime, widening the boundary layer structure for higher thermophoresis parameter values. As the Brownian motion parameter increases, a thinner concentration boundary layer structure is seen in Figure 19. This might be as the result of random nanoparticle collisions that produce kinetic energy which scatters comparatively stable nanoparticles along the wall. The inverse relationship between the Schmidt number and molecular diffusivity leads to a smaller boundary layer structure with high Schmidt number values, as seen in Figure 20. Increasing the suction and injection parameters results in thinner and thicker boundary layer structures, respectively, as Figures 21 and 22 illustrate. This might be as the result of the preamble wall gaining and losing nanoparticles through injection and suction, respectively.

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Entropy Analysis of the Flow

The effects of pertinent factors on the Carreau nanofluid flow's entropy analysis are shown in the following ten simulations. Different factors contribute to the increase in entropy generation surrounding the wall, as shown in Figures 23–26. The presence of significant friction between viscous fluid layers is indicated by an increase in the Eckert number. This friction generates waste heat that may contribute to an increase in entropy generation. Entropy generation rises as a result of heat transfer caused by the Lorentz force created by the magnetic field acting against the motion of the nanofluid. When the Brinkman number rises, more energy is lost as a result of viscous friction, which drives the creation of entropy. Increasing the cylinder's curvature will reduce the flow regime's surface area, which could increase entropy generation due to the accumulation of nanofluid layers. The entropy generation is affected differently by the two porosity parameters, as Figures 27 and 28 demonstrate. The Darcy-Forchheimer porosity model can be used to lower the entropy generation of the nanofluid, which is increased by the friction caused by the inertia force created. Analyzing and developing a system to lessen the energy lost, which is regarded as irreversible energy, is the goal of the second law of thermodynamics. Therefore, lowering the cylinder's curvature, viscous friction, and applying the Darcy-Forchheimer porosity model to the permeable wall can all help to limit the creation of entropy.

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The percentage of heat irreversibility to total entropy generation is defined as the Bejan number [3]. A rise in the Eckert number, Prandtl number, and radiation parameter initiates the irreversibility of heat, as seen in Figures 30–32. However, Figure 29 shows that the heat irreversibility decreases along the wall as the Weissenberg number increases. Small values of the Eckert number, Prandtl number, and radiation parameter might thereby reduce the irreversibility caused by heat transfer.

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Dynamism of Skin Friction, Heat, and Mass Transfer Rates on the Wall

The dynamics of the flow on the extending cylinder wall are examined in this section. As a result, the coefficient of skin friction, local Nusselt, and Sherwood numbers are examined in the following two tables.

Table 2 shows that the coefficient of skin friction is demotivated when the injection parameter ($$$ {S}_1<0 $$$) and Forchheimer number increase, whereas it is motivated when the Weissenberg number, magnetic, curvature, unsteadiness, power law index, and suction ($$$ {S}_1>0 $$$) parameters increase. According to Table 3, an increase in Weissenberg, Biot thermal, and Forchheimer numbers, as well as curvature and radiation parameters, initiate the magnitude of the rate of heat and mass transfers. As the Eckert and Prandtl numbers increase, the rates decrease. An increase in the thermophoresis parameter demotivates and motivates the rates of mass and heat transfers, respectively. While the rate of mass transfer decreases, the rate of heat transmission remains unaffected by the increase in the Brownian motion parameter.

TABLE 2 Impacts of some parameters on coefficient of skin friction.

$$$ We $$$	$$$ M $$$	$$$ \gamma $$$	$$$ Fr $$$	$$$ {S}_1\left(+ ve\right) $$$	$$$ {S}_1\left(- ve\right) $$$	$$$ A $$$	$$$ n $$$	$$$ -{f}^{\prime \prime }(0) $$$
0.2								2.466974
0.4								2.539640
0.6								2.657475
	0.2							2.408273
	0.5							2.507507
	0.8							2.602616
		0.5						2.409360
		0.7						2.496533
		1.0						2.625060
			0.3					2.585222
			0.6					2.518463
			0.9					2.460325
				0.2				2.368820
				0.4				2.481190
				0.6				2.599645
					−0.1			2.211083
					−0.15			2.186002
					−0.2			2.161257
						0.2		2.468242
						0.4		2.516028
						0.6		2.563070
							0.6	2.363767
							1.0	2.443353
							1.2	2.632781

TABLE 3 Impact of some parameters on the rate of heat transfer.

$$$ We $$$	$$$ Fr $$$	$$$ \gamma $$$	$$$ B{i}_l $$$	$$$ Rd $$$	$$$ Ec $$$	$$$ Nt $$$	$$$ Nb $$$	$$$ \mathit{\Pr} $$$	$$$ N{u}_zR{e}_z^{-\frac{1}{2}} $$$	$$$ -S{h}_zR{e}_z^{-\frac{1}{2}} $$$
0.2									0.110331	0.087103
0.4									0.130199	0.102789
0.6									0.138465	0.109314
	0.3								0.108557	0.085703
	0.6								0.137844	0.108824
	0.9								0.153348	0.121064
		0.5							0.110484	0.087224
		0.7							0.124692	0.098441
		1.0							0.139161	0.109864
			0.2						0.130199	0.102789
			0.4						0.219587	0.173358
			0.6						0.284617	0.224697
				0.2					0.130199	0.102789
				0.4					0.171059	0.111560
				0.6					0.212159	0.117866
					0.1				0.130199	0.102789
					0.2				0.048662	0.038417
					0.3				−0.036829	−0.029076
						0.2			0.131349	0.069131
						0.5			0.127823	0.168188
						0.9			0.122741	0.290703
							0.1		0.130199	0.308366
							0.4		0.130199	0.077091
							0.7		0.130199	0.044052
								1.0	0.130199	0.102789
								1.5	0.099453	0.078515
								2.0	0.069646	0.054984

Conclusions

The main focus of this communication is the entropy analysis of Carreau nanofluid flow with an unsteady stretching cylindrical sheet over a non-Darcy porous medium. Utilizing the conservation laws, the flow dynamics were formulated. The similarity transformation and the linearization approach are used to convert the non-linear PDEs into a system of IVPs. An open-source Python programming package and the $$$ RK6 $$$ is used in conjunction with the shooting method. The Prime findings are summarized as follows:

• An increase in $$$ A $$$, $$$ {S}_1\ge 0 $$$, $$$ \lambda $$$, $$$ We $$$, and $$$ n $$$ depresses the motion of the nanofluid, while an increase in $$$ \gamma $$$ and $$$ {S}_1<0 $$$ motivates it.
• A rise in $$$ {S}_1<0 $$$, $$$ Ec $$$, $$$ Q\ge 0 $$$, and $$$ B{i}_1 $$$ motivates the temperature of the nanofluid, while a rise in $$$ {S}_1\ge 0 $$$ and $$$ Q<0 $$$ demotivates it.
• The structure of the concentration boundary layers becomes thinner for bigger values of $$$ {S}_1\ge 0 $$$, $$$ Sc $$$, $$$ Nb $$$, and $$$ \beta \ge 0 $$$ and thicker for an increase in $$$ \beta <0 $$$, $$$ E $$$, $$$ Nt $$$, and $$$ {S}_1<0 $$$.
• The entropy generation is motivated when $$$ Ec $$$, $$$ Br $$$, $$$ \lambda $$$, $$$ M $$$, and $$$ \gamma $$$ increase. However, the generation is declining as $$$ Fr $$$ increases. Thus, lowering the cylinder's curvature, viscous friction, and applying the Darcy-Forchheimer porosity model to the permeable wall can all help to limit the creation of entropy.
• An increase in $$$ Ec $$$, $$$ \mathit{\Pr} $$$, and $$$ Rd $$$ initiates the irreversibility of heat. However, when $$$ We $$$ rises, the irreversibility of heat diminishes along the wall. Thus, small values of the radiation parameter, Prandtl number, and Eckert number may lessen the irreversibility brought on by heat transfer.
• An increase in $$$ We $$$, $$$ B{i}_1 $$$, $$$ Fr $$$, $$$ \gamma $$$, and $$$ Rd $$$ initiates the magnitude of the rate of heat and mass exchanges. Reduced rates were observed as $$$ Ec $$$ and $$$ \mathit{\Pr} $$$ rose.

Recommendations

The study has been carried out theoretically, with no experimental settings which is considered to be the limitation of this study. Thus, we propose the following potential future works in this regard.

• Some biological and industrial fluids can be modeled by using Carreau's model and specific nanoparticles can be considered so that the thermophysical characteristics of real nanofluids (biological and industrial) can be studied.
• The entropy analysis of other non-Newtonian nanofluids can be studied with experimental settings.

Nomenclature

• A
• Unsteady parameter
• B₀
• Magnetic field strength
• Br
• Brinkman number
• Bi₁
• Thermal Biot number
• C
• Concentration
• C_f
• Skin friction coefficient
• C_p
• Specific heat
• C_s
• Concentration susceptibility
• C_w
• Concentration at the wall
• C_∞
• Concentration at the free stream
• D_B
• Brownian diffusion coefficient
• D_m
• Molecular diffusivity of the species concentration
• D_T
• Thermal diffusion coefficient
• E
• Activation energy parameter
• Ea
• Activation energy
• Ec
• Eckert number
• f
• Dimensionless stream function
• h_w
• Wall heat transfer
• $$$ {\kappa}_{nf} $$$
• Thermal conductivity of the nanofluid
• K
• Permeability of porous medium
• k∗
• Rosseland mean absorption coefficient
• k_m
• Wall mass transfer coefficient
• $$$ {K}_r^2 $$$
• Binary chemical reaction parameter
• m
• Fitted rate constant
• M
• Magnetic parameter
• Nb
• Brownian motion parameter
• Nt
• Thermophoresis parameter
• Nu_z
• Local Nusselt number
• Pr
• Prandtl number
• Q
• Heat generation/absorption
• Q₀
• Coefficient of heat source
• q_r
• Radiative heat flux
• R
• Molar gas constant or Constant in entropy generation
• Rd
• Radiation parameter
• Re_z
• Local Reynolds number
• RK
• Runge Kutta
• S₁
• Suction/injection parameter
• Sc
• Schmidt number
• Sh_z
• Local Sherwood number

• T
• Fluid temperature
• T _w
• Temperature at the wall
• T _∞
• Ambient fluid temperature
• w_w
• Stretching velocity
• We
• Weissenberg number
• u,  w
• Velocity components in the $$$ r- $$$ and $$$ z- $$$ axes

Greek Symbols

• $$$ {\sigma}_{nf} $$$
• Thermal conductivity of nanofluid
• $$$ \theta $$$
• Dimensionless temperature
• $$$ \phi $$$
• Dimensionless concentration
• $$$ \sigma $$$
• Electrical conductivity
• $$$ {\sigma}^{\ast } $$$
• Stefan-Boltzmann constant
• $$$ \psi $$$
• Stream function
• $$$ \eta $$$
• Similarity variable
• $$$ \beta $$$
• Chemical reaction parameter
• $$$ {\beta}_C $$$
• Coefficients of volumetric concentration
• $$$ {\beta}_T $$$
• Coefficients of thermal expansion
• $$$ \delta $$$
• Temperature difference parameter
• $$$ {\delta}_1 $$$
• Concentration difference parameter
• $$$ \gamma $$$
• Curvature parameter
• $$$ \Gamma $$$
• Time dependent material constant
• $$$ \lambda $$$
• Porosity parameter
• $$$ {\lambda}_1 $$$
• Diffusion parameter
• $$$ {\nu}_{nf} $$$
• Kinematic viscosity of nanofluid
• $$$ {\rho}_f $$$
• Density of the base fluid
• $$$ {\rho}_p $$$
• Density of the nanoparticles
• $$$ {\rho}_{nf} $$$
• Density of the nanofluid
• $$$ \tau $$$
• Ratio of heat capacities
• $$$ {\left(\rho {C}_p\right)}_f $$$
• Heat capacity of the base fluid
• $$$ {\left(\rho {C}_p\right)}_p $$$
• Heat capacity of the nanoparticles
• $$$ {\left(\rho {C}_p\right)}_{nf} $$$
• Heat capacity of the nanofluid

Author Contributions

Eshetu Haile Gorfie: conceptualization; writing – review and editing; formal analysis; validation; investigation. Gizachew Bayou Zegeye: conceptualization; writing – original draft; formal analysis. Gurju Awgichew Zergaw: methodology; writing – review and editing.

Conflicts of Interest

The authors declare no conflicts of interest.

Peer Review

The peer review history for this article is available at .

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

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