Thermal transport and magnetohydrodynamics flow of generalized Newtonian nanofluid with inherent irreversibility between conduit with slip at the walls

This study enlightens the magnetohydrodynamic Jeffery-Hamel flow under an inclined Lorentz force through a non-uniform conduit having slip at walls, which is frequently applied in geothermal applications, electronic cooling devices, and modern energy systems, etc. Therefore, the performance of a two-dimensional purely radial flow inside a converging-diverging channel is explored from the perspective of second law of thermodynamics for Carreau nanofluids. The intersecting walls of conduit are inclined with horizontal plane to construct a converging flow for negative angle $ ({\alpha \lt 0} ) $ (α<0) and a diverging flow for positive angle $ ({\alpha \gt 0} ) $ (α>0). Additionally, second law thermodynamic evaluation offers an effective method for improving thermal performance by reducing entropy production. To accomplish the main objective, rigorous physical theories and assumptions are implemented based on the passive control approach of Buongiorno's model. By applying distinctive modifications, the governing equations are renovated into a system of ordinary differential equations, which are solved numerically by a collocated technique based on finite difference code. Simple shear near the wall influences the flow configurations allow compression in a local flow topology in regions of divergent channel. The temperature profiles increase with sophisticated heat source and Brinkman number. Entropy is minimum and uniform with optimum channel angle and velocity slip.


Gradient operator D/Dt
The material derivative K 0 Chemical reaction rate (sec −1 ) I Identity tensor ∈ Relaxation parameter (s) The second invariant strain tensor n Power-indexed Q 0 Heat generation (Wm −2 ) T The temperature of the fluid (K) T 0 The reference temperature (K) C Fluid concentration (kgm −3 ) C 0 Wall concentration (kgm −3 ) p Pressure (Nm −2 ) c p Specific heat (jkg −1 K −1 ) k f Thermal conductivity (Wm

Introduction
The Jeffery-Hamel (J-H) flow is a class of flows that was initially recognized by (Jeffery, 1915) and independently by (Hamel, 1917).These flows describe the inflow and outflow of a viscous, incompressible fluid in a wedgeshaped (converging and diverging) channel that is linearly expanding and has a specific angle between the walls.As is customary, the angle between the wedge's walls is given by 2α.Flow through converging and diverging channels (also entitled as inclined channel) is one of the substantial motives in fluid dynamics due to versatile variety of applications in different fields.Due to its significance in numerous domains, including mechanical, industrial, biological, and physical applications, the fluid flow between two inclined plates is an essential and vital component of the physical model.Similar types of flows can be observed in the flow of blood across natural channels, such as arteries and veins (Dinarvand et al., 2022;Ibrahim, 2022;Rathore & Sandeep, 2022a, 2022b).For instance, one may cite the flow through nozzles, diffusers, and reducers that are used in polymer processing procedures (Hooper et al., 1982;Kato & Shibanuma, 1980;Peddieson, 1973).To model the flow of diluted polymer solutions over porous surfaces, convergence/divergence flows have also been employed with considerable effectiveness (Daripa & Paşa, 2004).The cold-drawing process in the polymer sector uses flow through convergent channels as well to enhance the mechanical qualities of goods like plastic sheets and rods (Casas et al., 2006).The convergent and divergent flow of a power-law fluid, whose zero-shear viscosity is neither zero nor infinity for any finite value of the power-law exponent, was explored by Harley et al. (2018).The current contributions on MHD (J-H) flow revealed that (Meher & Patel, 2019) considered such type of flow for non-Newtonian fluid using Eyring-Powell model.They used the Differential Transform Method (DTM) and included thermal influences to put up with an analytical approach.
Then (Onyango et al., 2020) focused on the flow in a divergent/convergent channel with an oblique fluctuating magnetic field.Additionally, they considered how suction and injection would affect the channel walls as they used the collocation method to tackle the problem.The analytical study (Sobamowo et al., 2020) investigated how wall slip and the magnetic field affected Maxwell nanofluid.
Other investigation on non-Newtonian fluids subjected to Jeffery Hamel flow in a convergent/divergent channel has been done by (Asghar et al., 2022;Banerjee et al., 2021;Bég et al., 2022;Garimella et al., 2022;Hashim et al., 2022;James & Roos, 2021;Saifi et al., 2020).Magnetohydrodynamic (MHD) flow has extensive range of applications from nuclear fusion apparatus to magnetohydrodynamic pumps.Convection-diffusion type equations compensate the mathematical model of the MHD equations.The Navier-Stokes equations and Maxwell's equations of electromagnetism via Ohm's law, respectively, determine the fluid velocity and the induced magnetic field.The field of magnetohydrodynamics, or MHD in brief, investigates how magnetized fluids move through channels when a magnetic field is applied.These complications occur in a variety of scenarios, including nuclear reactors, astronomy, meteorology, pumps, accelerators, and measures of blood flow.Hartmann (1937), who investigated the MHD-flow between parallel walls theoretically and practically, founded the field.The research presented here offers key concepts for the creation of a variety of MHD-devices, including MHDpumps, brakes, flow meters, and generators.Magnetic fields are recognized to have an impact on the flow kinematics of various fluid flows, like polymeric additives.In most real situations, we have to deal with the flow of conducting fluid, which behaves differently under the influence of magnetic forces than non-conducting fluid.This makes the flow of electrically conducting non-Newtonian fluid a very essential phenomenon.In these circumstances, the magneto hydrodynamic (MHD) component of the flow must be considered.To better, understand how a transverse uniform magnetic field affects the flow of an electrically conducting fluid between two infinite parallel insulating and stationary plates.A problem on the heat and mass transfer of MHD Jeffrey-Hammel flow in the presence of an angled magnetic field was conducted by (Onyango et al., 2020).The list of authors who also made significant contributions to the research of magnetohydrodynamic fluxes in a converging/diverging channel is found in References (Bhaskar et al., 2022;Gahgah et al., 2020;Hamid et al., 2021;Kimathi et al., 2022;Khan et al., 2022).
Entropy generation (EG) and its consequences point to irreversibility in accordance with the second rule of thermodynamics.Any mechanism that causes a thermal system to lose available work always has some irreversibility.The EG measures this loss of available work.To prevent the loss of the energy output owing to fluid friction, magnetic irreversibility, and irreversible heat transfer, minimizing of entropy formation plays a crucial role in the designing of energy systems.In fact, system irreversibility reduces the maximum achievable performance of the thermal process, which can be explained by the fact that each energy activity results in the destruction of some useful energy.To minimize irreversible losses, many studies have focused on the entropy production in thermal systems.In this regard, Bejan (1980;Bejan, 2013) made a tremendous effort and introduced a magnificent number.The ratio of entropy formation due to fluid friction to thermal irreversibility is referred as the Bejan number Be. (Turkyilmazoglu, 2020) discussed the application of the second law to the subject of thermal transportation in porous metallic channels.Furthermore, the production/generation of entropy was employed in various research to explore the thermophysical properties of micro/mini-channels in order to understand more about the calibre of the available energy (Alrowaili et al., 2022;Chen & Jian, 2022;Guedri et al., 2022;Jing et al., 2019;P. Kumar et al., 2022;Zaman et al., 2022).
Nanomaterial is an essential organic fluid used in industry because of its rapid heat exchanger rates.Convective heat exchanger fluids including water or oil do not have the remarkable thermophysical properties that nanofluids do.Numerous investigations exploring the various uses of nanofluids have been done since their discovery.Heat exchangers in welding and milling, heating tubes, nuclear power plants, electronic refrigeration, automotive radiators, and other applications are a few examples.Since of their enhanced thermal properties, nanofluids have captured the interest of many scholars.These fluids are made up of working fluids that contain nanoparticles, which are microscopic pieces such carbides, oxides, metals, and carbon nanotubes.Bionanofluid is another fascinating concept in engineering and medical research.Scientists became interested in nanofluids a few years ago because of their application in the technological and medical fields.For the first time, (Choi & Eastman, 1995) incorporated nonmaterial in fluids to evaluate the thermal conductivity and renewable energy.Researchers are becoming more attracted because of function of its improved thermal conductivity and heat transportation capabilities.But according to (Buongiorno, 2005) nanofluid mechanics, a homogeneous model was able to predict the heat transfer coefficient of nanofluids.The size of the nanoparticle prevents consideration of the effect of dispersion.To address this flaw, Buongiorno created a different model in which he included the seven-slip mechanisms, including Brownian diffusion, inertia, thermophoresis, fluid drainage, and Magnus, and gravity action.He continued by asserting that the two primary slip processes in nanofluids are merely thermophoresis and Brownian diffusion.By considering the assumption that there is zero mass on the channel surface, (Turkyilmazoglu, 2018) used the Buongiorno nanofluid model to estimate the potential for heat transfer.(Waqas et al., 2021) documented the effects of heat transfer and excessive viscosity on the flow of nanofluid through a porous, stretchable cylinder.(Muhammad et al., 2021) evaluated consequences of altered heat and mass oscillations on a three-dimensional Eyring-Powell nanofluid with thermal radiation emission.Moreover, in the recent years the renowned researchers (Baghban et al., 2019;Jeevankumar & Sandeep, 2022;R. N. Kumar et al., 2022;Mahdavi et al., 2020;Qureshi et al., 2023;Raza et al., 2022;Sarada et al., 2022;Shah et al., 2022;Soumya et al., 2022), have deliberated the flow of nanofluids/hybrid nanofluids in various shaped geometries as well.
In many industrial and biological processes, non-Newtonian fluid motion over diverse geometries is often seen.Because they are found in human bodily fluids, industrial chemicals, crude oils, food items, and other things, shear-thinning fluids demand special consideration.It is well known that Navier-Stokes equations are unable to fully capture the characteristics of non-Newtonian liquids due to the complicated interaction between shear and strain rates.Non-Newtonian fluids adapt to the application of shear stress in terms of strain rate significantly, according to experimental observations.Based on this connection, non-Newtonian liquids are separated into shear thinning and thickening liquids.These classifications have raised the importance of non-Newtonian fluids in many technical, technological, and everyday life activities.To illustrate the physical explanation of these fluids, several fluid models are provided.One of these models, the Carreau fluid, is particularly useful for understanding the flow properties of shear thinning and thickening liquids.This model also demonstrates the behaviour of materials that are shear thinning at low and high shear rates.Carreau liquid has a wide range of uses in the production of polymers, capillary electrophoresis, gas turbine engines, crystal development, mud drilling, the manufacture of gels and shampoos, powder technology, and biological applications.The references (Akram et al., 2022;Alsemiry et al., 2022;Bilal & Shah, 2022;Bilal & Shah, 2022;Bilal & Shah, 2022;Song et al., 2022;Ud Din et al., 2022) compile some recent developments addressing the evaluation of Carreau fluid flow in various physical configurations and features.
A glance at the evaluation of literature discloses that no comparative analysis is attempted for Buongiorno's model nanofluid mass, heat transmission attributes, and entropy degradation of Carreau nanofluid across a wedge-shaped (convergent divergent channel) under the influence of an inclined magnetic field.Hence, it is of great importance for the development of highefficiency converging diverging channel and figure out the coupled flow mechanism and thermal-hydraulic performance of Carreau nanofluid, while those inquiries are not described in open database.The revealed magneto laminar flow regime resulting in irreversibility due to heat transfer and flow physics are significant to understood.The effect of an inclined magnetic field strength, friction irreversibility, and mass, heat transport irreversibility leading to entropy generation.The novelty of our work compared to the previous studies dealing with the Jaffrey-Hamel flow and heat, mass transfer of non-Newtonian fluid impinging in the diversity of physical effects, which will allow us to obtain preliminary guidance for optimizing entropy degradation in systems subjected to magnetic field depending on the nature of the fluid and convergent divergent channel opening.The intersecting wall of the channel is horizontal and inclined with respect to the horizontal plane to construct a convergence angle of −5 o for the convergent domain, and a divergent angle of 5 o for the divergent domain.The computations are based on the Navier-Stokes equations solved with a heat, mass transfer mechanism and a modified Buongiorno model.Two distinct heat sources are considered in the thermal transport phenomenon.Entropy accumulation originating from heat and mass flux, viscous dissipation, and joule heating in the flow were also reviewed thoroughly using second law.To accomplish the main objective, rigorous physical theories and assumptions are implemented in this respect to state properly the leading conservation equations based on the passive control approach of Buongiorno's model.By applying distinctive modifications, the governing equations are renovated into a system of ordinary differential equations, which are solved numerically by a Bvp4c procedure for realistic conditions of the geometry.This analysis offers a thermodynamic investigation to predict the various irreversible processes that produce entropy at the local scale throughout the entire channel, such as supersonic flow, two-phase flow in a convergent-divergent nozzle, and solid-propellant rocket engines.

Mathematical formulation
The geometry of interest, which is depicted in Figure 1 as a two-dimensional converging/diverging channel with subtended angle 2α.It is presumable that the channel walls extended in the z-direction to infinity (i.e.perpendicular to the plane).As a result, the problem will be written in terms of plane polar coordinates, with the apex serving as the origin.It is assumed that the fluid itself is an incompressible, electrically conducting, non-Newtonian fluid that abides by the Carreau fluid model's properties as its constitutive equation.The fluid is being applied to a consistent magnetic field of strength B while it moves in the transverse direction.It is thought that the magnetic field has no impact on the fluid physical or rheological characteristics.The flow is further considered symmetric and radial.Due to these presumptions, there is only one non-zero velocity component in polar coordinates v(r, θ), which is solely determined by r and θ in steady-state conditions.The Carreau fluid flow pattern is realized with the assumption of the magnetic field.This physical assumption is supported by the fact that body force is a factor in the momentum equation.The Fourier's heat flux describes the anomalous heat transport of Carreau nanofluid considering the effects of Brownian diffusion and thermophoresis.The heat generating source and non-linear thermal radiation effect is also considered to have a better knowledge of heat transport.The random motion and thermo-migration of nanoparticles on the concentration distribution throughout the channel of nanofluidic is delineated.The concentration equation also accounts for a chemical reaction effects.The appropriate equations of motion in vector forms are the continuity, momentum energy and nanoparticles concentration equation (Hayat et al., 2022;Ketchate et al., 2022;Nath & Murugesan, 2022;Reddy et al., 2022;Wehinger & Scharf, 2022). where is the viscous heat dissipation, K 0 is the first order chemical reaction rate, D B , T the fluid temperature, C the nanofluid concentration, D B is the Brownian diffusion coefficient, D T the thermophoresis diffusion coefficient, T 0 is the channel wall temperature, Q 0 is the heat source/sink term respectively.stands for the source term resulting from the applied magnetic field, also known as the magnetic or Lorentz force.It is established that this force depends on the fluid velocity vector V, the generated electric field E, and the imposed magnetic field B = 0, B 0 r , 0 , is the (constant) electrical conductivity of the fluid, thus (Bhatti et al., 2022;Sadeghy et al., 2007) For this recent model, we adopt that polarizing and applied voltage is zero, i.e., E = 0.After simplifying Eq. ( 5), it becomes (Dolui et al., 2022): ) where σ 1 is the electric conductivity, ( e r , e θ ) is the corresponding radial and tangential direction of the field, J is the current density vector.
In the instance of a Carreau fluid, established the relationship between the shear stress tensor and the shear rate: (Alsemiry et al., 2022;Hassan et al., 2022;Johnston et al., 2004).
where μ 0 is the zero-shear rate viscosity, μ ∞ is the infinite-shear rate viscosity, ∈ the relaxation time, and n is the flow behaviour index.Υ is the shear rate tensor and is the second invariant strain tensor, are given as: where trac is the sum of the elements on the square matrix's ( Υ) 2 major diagonal.
Subject to the following boundary conditions At the channel central line At the wall surface Integrating Eq. ( 9), equation of continuity yields as After implementing Eq. ( 23) into Eq.( 17)-( 20), the above equations can be settled as

Engineering quantities
The flow, heat, and mass transfer rate properties of a nanofluid interacting with a boundary are quantified using physical parameters such as skin friction coefficient (C f ), Nusselt number (Nu), and Sherwood number (Sh).
The examples above are provided as the generic versions of several physical quantities. where

Entropy generation equation
Any fluid flow system with a heat transfer mechanism has irreversibility, which are determined by the entropy analysis.Any system's inevitability depletes our usable energy.Establishing the range of values for the parameters that control any genuine process is our primary goal to maximize the available energy.Entropy gauges the degree of disorder in a system.Process reversibility is a prerequisite for entropy formation.Entropy increases with time for non-reversible reactions in physical system.It is crucial to research and comprehend the effects of these nanoparticles on entropy formation in actual systems as the use of nanofluids and nanoparticles in engineering and medical applications increases.(Buongiorno, 2005; Hashemi-Tilehnoee & Palomo del Barrio, 2022;Mandal et al., 2022): The first term N H,G measures irreversibility for the sake of heat transfer, the second term N M,G signify the entropy generation of local mass transfer due to the presence of nanoparticles, the third term N F,G is because of fluid friction and the fourth term N J,G is due to magnetic interaction of the nanofluid.
In above Eq.( 30), ∇T and ∇C is the heat, mass flux, C a = C+C 0 2 , T a = T+T 0 2 is the mean concentration and temperature between the core region and channel wall respectively, J is the current density, q is the electric charge density, V is the velocity vector, E is the electric field intensity, R is the molar gas constant (8.31446261815324J Invoking the assumption that the flow regime is purely radial, the local entropy generation Equation ( 30) can be settled as
In order to highlight the dominance of entropy formation in the context of heat transport and fluid friction, we now define Bejan number (Bejan, 1979) as The thermophoresis parameter.
Skin friction coefficients along (r, θ)-direction, local Nusselt number and Sherwood number are the measures of industrial interest designated in the non-dimensional form as:

Solution methodology
The Bvp4c technique via MATLAB computational software is employed in the numerical scheme to mathematically interact the consequential framework of the innovative non-linear transfigured problem subject through appropriate boundary constraints.Using MAT-LAB bvp4c feature, the mathematical modelled problem was resolved.Particularly for higher order nonlinear boundary value problems, the bvp4c approach in MATLAB can address a variety of complex and challenging situations.To handle systems of nonlinear equations, the method is built on an iteration framework.The Bvp4c employs a finite difference formulation to execute the three-stage Lobatto IIIa formula and provides a continuously driven solution that is uniformly precise in fourth order in the domain x ∈ [a, b].The residual error is lower, and the numerical method is more stable.Additionally, there are no complex discretization in such numerical approximations.To simulate the problem at hand, the bvp4c solver is built-in collocation method.
To solve the system of Eqs. ( 43), ( 45) and ( 47), with the missing slopes t 1 = f 2 (1), and t 2 = (1) are t 2 = (1) are appropriately selected.Newton's method is used to calculate these slopes precise values iteratively until the far field criteria are met with error tolerance 10 −5 .Calculations are done for ζ max = 20, a range of parameters that satisfy the requirements of the far field circumstances.It is establish that the maximum residual is fewer than 10 −3 in all the considered cases.

Model certification
The authenticity of computational methodology was determined by comparison the current technique's flow field (Table 1), wall dragg force (Table 2) and thermal transmitting rate (Table 3) to the verified outcomes of prior studies are providing the summary of the coherence of results regarding previous techniques with standing results.According, to the established tables, the comparsion of the current model presenting highly accurate results.The Equation ( 34) are identical to Newtonian fluid (Rana et al., 2019;Turkyilmazoglu, 2014) by setting We = 0, or n = 1, M = 0.

Numerical outcomes and debate
This section describes the outcomes attained.The investigation's findings were achieved by visually illustrating the relationship between hydrothermal characteristics (velocity, temperature, concentration, and heat mass transfer rate) and the fictitious fluid motion across a wedge-shaped surface.The effects of viscous dissipation and convergent divergent opening in the inclined magnetic field existence and velocity slip are considered when conducting the mass and heat transfer assessments of the postulated flow.Additionally, the degradation of heat, mass transmission mechanism take place due to heat, mass transfer irreversibility, liquid friction irreversibility and magnetic force irreversibility are examined.The analysis are captures for convergent and divergent opening of the channel with a convergence and divergent angle (α < 0 and α > 0), respectively.

Flow dynamics
The effects of the Re number on the fluid velocity f (ζ ) of the convergent/divergent flow are shown in Figure 2.
Increasing Re number can lead to a flatter profile in the (1) values of with previous results without nanofluid, for α = −5 0 , M = 0, We = 0, n = 1.channel's middle and steep gradients along the walls.
The thickness of the boundary layer consequently thins out.It is obvious that the backflow is completely prohibited in convergent flow circumstances.On the other hand, the effect of Reynolds number on divergent flow is conflicting.The divergent flow concentrates the volume flux at the centre of channels with lower gradients towards the walls.The results show that the flow reversal is strongly preferred for channels that are only divergent.We observed that as the Reynolds number rises, the fluid phase velocity falls in the left half of the channel and rises in the right half.It is noteworthy that the pressure gradient in the left half of the channel is significantly dominated by inertial force.Furthermore, while the inflow portion experiences an increase in velocity, the outflow part also experiences an increase in velocity.This can be understood as follows: when viscosity increases, the pressure gradient needed to maintain the same flow rate grows.As a result, the magnitude of the outflow velocity grows along with a comparable peak amplitude of the inflow velocity toward the centre.Figure 3 demonstrates the influence of the velocity slip parameter (opposing force of the fluid) γ 1 on fluid velocity.It should be observed that a rise in γ 1 causes the fluid's velocity in the convergent channel to increase, and its velocity in the divergent channel to decrease, even if there is a large increase throughout the entire channel due to the fluid's opposing force.In fact, a lower adhesive force between the channel wall and the nanofluid particles resists the superior γ 1 , which causes the velocity f (ζ ) to decrease.The velocity at the flow centre is reduced when the slip factor is augmented, but it is increased at the sides and on the wall.The larger solid boundaries distribution along the flow direction may be a feasible  explanation for this.Because of the slip effect, all obstacles now have lower higher velocity sections, increasing the velocity gradients at the boundary regions.As seen, for sustaining constant flow rate in the channel, an improvement in the slip velocity (increase in γ 1 ) results in a decrease in velocity gradient at both the wall and the core velocity, in contrast to the no-slip scenario.Substantial values of converging to 'Plug flow,' in which the liquid behaves like a solid and slides in the channel.The velocity profiles inside converging and diverging channels are strongly influenced by magnetic number M, as seen in Figure 4.The result is extremely dramatic since fluid elements close to the wall can be seen accelerating to speeds that are faster than the centerline velocity in convergent channels.More significantly, flow separation in divergent channels may be totally suppressed by the acceleration brought on by the magnetic forces.It may be inferred that the flow becomes ever more stable the higher the magnetic number since a rise in the magnetic field intensity eventually causes the inflection point (which naturally exists in the velocity profiles in a diverging channel) to vanish.Physically, the cross flow that results from the improvement of either the magnetic permeability or the magnetic intensity and predominates the fluid's viscosity within the channel is caused due to the magnetic field strength.According to Figure 5, the Weissenberg number We has an impact on velocity profiles by intensifying the acceleration of fluid elements close to the wall.This behaviour is explained by the high elastic stresses that are produced in the flow direction as an outcome of the fluid elements uniaxial and/or biaxial extensions, which can be elevated by, for instance, enhancing the concentration of the polymeric additive.In reality, as is evident, these elastic strains in diverging channels may be sufficient to prevent flow separation.Physically, by increasing the Weissenberg number, the wall shear stress increases.
The reason for this is that when Weissenberg number rises, viscosity decreases because 1 < n < 2 allows for easy flow movement and an increase in wall shear stress.
When We(= 1, 2, 3) the entire region is subject to negative values of the wall shear stress.When the catheter speed is increased, the wall shear stress likewise rises.It is evident that the size of the vorticity region in the middle of the wedge steadily grows as the Weissenberg number rises.This is because when the Weissenberg number rises, the shear-thickening tendency of the polymer (Carreau model) solution gradually increases.This reduces the polymer solution apparent viscosity, which in turn makes it easier for a vortex to form inside the wedgeshaped channel.Moreover, as the Weissenberg number gradually rises, the velocity magnitude also increases.

Thermal pattern
The behaviour of material parameter on temperature profile is visualized in this section.As seen in Figure 6, as the Weissenberg number We is increased in the shear-thickening fluid, n > 1, the dynamics of flow and heat transfer for viscoelastic fluids are identical to those of Newtonian fluids at low values of the Weissenberg number.Such as the Newtonian fluid, for illustration, at We = 0.1 (Gupta et al., 2022).Physically, We is a linear function of the relaxation time and we have assumed ∈ √ < 1.As a result, as the shear rate decreases, high viscosity is produced in the fluid structure, for shear thinning and low viscosity is formed in the fluid that is shear-thickening.The nature of the temperature profile is depicted as diminishing as the We increases.Physically, a higher material component ∈ implied by a stronger dimensionless Weissenberg number.The molecular connection between the fluid and the nanoparticles is destroyed when Weissenberg number is increased, allowing for unfettered interaction.Variation for various Brinkman numbers Br at constant values for other parameters is seen in Figure 7. Increase in Br = PrEc ( = 0.2, 0.5, 0.8) effectively improves (ζ ).
A higher Brinkman number improves viscous heating, which causes a large increase in temperature.The Brinkman number is directly correlated with heat generation by viscous dissipation and inversely correlated with heat conduction by molecules.According to Figure 7, even a minor amount of the viscous dissipation coefficient can cause a noticeably larger increase in the difference between the channel wall and centre temperatures when subjected to an unfavourable pressure gradient.
Even though the dimensionless temperature seems to increase, this does not necessarily mean that more heat is being carried out across the channel wall and towards the channel centre.This is because the reduced local temperature gradient caused by the greater viscous dissipation, which maintains Pr = 7, causes less heat to be carried across the channel wall.Even in this situation, however, certain inferences regarding the calibre of convective heat movement can be made.The local heat transfer rate for shear thickening fluids can be assumed to rise as the strength of the unfavourable pressure gradient increases (for mild to moderate values of Br).Although the degree to which a negative pressure gradient increases local convective heat transfer is noticeably less, shear  thinning fluids behave similarly.The effects of a heat source (δ 0 > 0) and a heat sink (δ 0 < 0) on the temperature profile (ζ ) are elucidated in analogous fashion in Figures 8 and 9. Evidently, (ζ ) increases with the strength of the heat source (δ 0 > 0) and falls with the strength of the heat sink (δ 0 < 0), as expected.Greater thermal profile is anticipated because of the heat generation along the isothermal wall.This is because when heat is absorbed, the fluid densifies and the convection current weakens, causing the fluid temperature to drop.On the other side, increased heat generation strengthens the convection current, causing a decrease in fluid density and an increase in temperature.Figure 10, shows how the thermophoresis parameter Nt impacts the temperature profile.The thermophoresis coefficient Nt( = 0.2, 0.4, 0.6) in nanofluids is calculated as the ratio of momentum diffusion to thermophoretic diffusion.As the thermophoresis parameter increases, the liquid particles start to migrate quickly away from the boundary, indicating that the boundary layers thickness and kinetic energy have increased as well.Therefore, increasing the thermophoretic parameter Nt causes the temperature distribution to grow.Physically, an increase in thermophoresis can be attributed to temperature differences between the surface and ambient.An increase in the thermal diffusion D T , or a larger heat capacity to kinematic viscosity ratio.Figure 11 shows the impact of Nb( = 0.2, 0.4, 0.6) on (ζ ).As Nb values rise, the (ζ ) curves rise as well.Since the quantity of Nb also increases the random mobility of nanoparticles, more nanoparticles collide because of the particles increased velocity, converting their kinetic energy into heat.The parameter for Brownian motion Nt is a measure of the suspended nanoparticle chaotic motion within the nanofluid.This motion is carried out by the microscopic particles randomly colliding, which happens more frequently at higher temperatures.Heat flow to channel walls is restricted when mobile particles, such as nanoparticles, are added to the system.

Concentration field
The concentration profiles (ζ ) of the polymeric liquid are displayed in Figures 12-14 along with various values for the Schmidt number Sc, Soret number Sc and chemical reaction parameter χ .According to Figure 12, as the Schmidt number rises, the fluid concentration levels drop.This is in line with the idea that a rise in Sc implies a fall in molecular diffusivity, which causes the thickness of the concentration boundary layer to decrease.The ratio of viscous width to mass diffusivity distinguishes the Schmidt number.The Schmidt number quantifies the virtual effectiveness of momentum and mass transport via dispersion in the velocity and singular boundary layers.As Sc values increased, species were absorbed, and the thickness of the boundary layer significantly decreased.The definition of Sr indicates that a rising Soret effect corresponds to a rising molar mass diffusivity.The concentration increases as molecular mass diffusivity increases.This suggests that the fluid species concentration tends to increase as Soret number increases (Figure 13).Since complex combinations sometimes contain a significant proportion of materials that are different from one another and polar in nature, such as petroleum organisms, the appearance of the mass flux that results from thermal gradients is still not completely under control.The affinities of each component for warming or freezing determine whether thermal diffusion will attenuate or amplify those effects, despite the fact that gravity is inclined to separate the foundation of the reservoirs and the luminosity to those tops.As a result, the Soret effect develops because of the interaction between particles and fluids.The mobility of fluid molecules in the hottest zone and the maximum energy level in this area shift the particles toward the coldest area, which is how temperature gradients produce this phenomenon.Moving the hottest particles from the hottest to the coldest region will therefore speed up the heat transfer process.These forces are important only at the very least fluid velocity, notably in free or natural convection, due to the sizes of the fluid molecules and the particles.The An increase in χ , a chemical reaction measure, indicates similarity to Schmidt number (Figure 14).concentration distribution quickly decreased as the chemical reaction parameter χ improved.As the chemical reaction parameter rises, the number of solute species that were affected by the chemical reaction grows, which causes the concentration distribution to narrow.Consequently, a detrimental chemical reaction significantly reduces the width of the solutal border layer.

Energy degradation (the effect of channel design)
Figures 15-19 offer the values of global entropy generation rate E G at the wall isoflux case, while axial heat transmission is absent.According to the findings (Figure 15), a spike in Brinkman number causes an increase in E G .A fluidic system with less Br would therefore function more effectively.According to the description of the Brinkman number, the magnitude of velocity and the heat flow as well as the channel width are inversely related to each other and directly proportional to velocity.In other words, higher heat flux and larger channel width should be considered when minimizing the global entropy generation rate (i.e.irreversibility).Additionally, values for shear-thickening fluids are higher than for shear-thinning fluids due to greater velocity and temperature gradients near walls, are more important factors for entropy production distributions.Although temperature and velocity gradients are negligible at the channel centre, they are at their highest along the channel wall.E G the rate of entropy generation, consequently, achieves its highest and minimum values near the wall and the centre, respectively.Improvements in diffusion characteristics D 1 lead to more scattered nanoparticles, and results in an increase in entropy generation (Figure 16).Originally, entropy formation takes place at the microscopic level when heat transmission occurs.When heat is transferred, additional movements take place, such as spin moment, internal molecule migration, molecular vibration, kinetic energy, molecular friction, etc., which cause heat to be lost and turned into less work.This increased movement both within the system and in the fluid leads to chaos.Entropy is referred to be a 'metric of chaos'.The magnetic effects on the fluid flow and heat  transmission in channels were inspected for a number of years (Mohyud-Din et al., 2015;Sadeghy et al., 2007).As has been previously shown, magnetic resultant forces effect the thermophysical properties of the converging diverging channel system.The main force occurring in channel fluid flow comes from the interplay of the electric current density and the magnetic field, which produces the Lorentz body force.However, investigations on the  entropy generation rate within channels in which magnetic forces play a role are relatively limited.To date, diverse channel geometries with electromagnetic forces are reflected for analyses of entropy generation rate.The results show that the volumetric entropy rate is gradually augmented with an upturn in the magnetic field (Figure 17).According to the findings (Figure 18), the involvement of slip condition or an upsurge in slip coefficient both result in a reduction in the global entropy generation rate because slip condition results in relatively small velocity and temperature gradients, especially at the solid surface, which enhances the system performance.Consequently, entropy formation adjacent to the walls is significantly influenced by Brinkman number and slip coefficient.Figure 19 establishes the volumetric entropy contours for diverse enclosure angles.Two significant parameters converging and diverging opening angle play a fundamental role in Entropy generation (ETG) include abrupt changes in Carreau fluid (CF) temperature and velocity.Sudden changes in velocity in a small location as well as high temperature gradients in diverse parts cause irreversibility and ETG.Therefore, in the parts where the velocity gradients or the temperature gradients are high, the amount of ETG is extraordinary.Temperature differences significantly affect total ETG since total ETG is strongly dependent on thermal entropy and has a very low value due to the increased generation of thermal entropy in this condition.Thus, when there is a large temperature change, there is unlimited quantity of entropy formation.The higher values of the enclosure with α > 0 angle have solid ETG rates.In the other enclosure angles α < 0, the ETG is lower.In these regions, the hot fluid either collides with the isothermal or the cold wall.Due to the outsized temperature differential produced near to the wall by this impact, a lot of entropy is produced.However, in the core region, the entropy of system slightly decreases.It has been determined that a rise in flow resistance which consists of frictional force along the flow path in the channel and local resistance brought on by a rapid spike in flow area causes a drop in EG.The local resistance brought on by the abrupt constriction of the flow region as the fluid exits the outlet depends on the inlet and outlet channel angles.Assuming a consistent volume flow rate, the narrowing of the outlet leads to a wider channel at the inlet and outflow, which reduces local resistance.Temperature gradients significantly affect total ETG because of the extremely low value of ETG and the absence of thermal entropy.

Heat, mass transportation rate
The physical justification for heat transfer and mass transfer rates against pertinent parameters in a converging-diverging channel flow of Carreau nanofluid is shown in Tables 4 and 5. Growing Prandtl and Eckert numbers enhance the heat transfer rate in the converging channel, whereas rising Reynolds and Weissenberg numbers depicts detrimental effect.The Nusselt number tends to increase due to the thermophoresis parameter because of the heat flux at the wall and the drift velocity engendered by the constituent particles towards the walls, but the random motion of the nanoparticles tends to decrease the heat transfer rate as they absorb heat from the surface.The nanoparticles random motion in the convergent channel unfavourably impacts the heat transfer rate, but it is positively impacted in the divergent channel.Heat transfer is inhibited by both slip flow and viscous dissipation.The major aspect is that while viscous dissipation diminishes with rarefaction, slip flow reducing effect on heat transfer grows.The Nusselt number is elevated by the thermophoretic behaviour in the convergent channel but diminished in the divergent channel.For both divergent and convergent channels, the heat generation parameter speeds up the rate of heat and mass transfer.The chemical reaction parameter augmented the mass transfer rate in divergent channel while a slight reduction is perceived in convergent channel.

Conclusion and future outlooks
In this study, we have applied the entropy minimization technique to the magnetohydrodynamic flow that takes place in a wedge-shaped channel with slip and realistic conditions of the geometry.Therefore, nonequilibrium thermodynamics was utilized to assess the entropy generation because the system under consideration was not to state the global equilibrium.An equation for the local entropy generation of heat, mass-diffusion Joule heating systems was effectively established using nonequilibrium thermodynamics.The novelty of this research examined how design dimensions affects thermal optimization and entropy degradation.The Carreau model is the most efficient and is capable of accurately forecasting the thermal and flow performances within the thermal system discussed here.Two distinct heat sources are considered in the thermal transport phenomenon.While creating a flow model for the flow of nanofluids and their thermal characteristics in heat sinks, the dispersion and random motion of the particles, clustering, and other aspects are considered.The governing equations are transformed into a system of ordinary differential equations by making specific modifications, and under genuine geometry conditions, the Bvp4c algorithm is used to solve equations numerically.The flow pattern and heat transfer presences are revealed accordingly via multiple portrays, whose upshots are deliberated comprehensively for various emerging parameters.The main significant outcomes of the studies follow as: • An increase in inertial forces significantly promotes the liquid motion in opposite manner.• The Weissenberg number predicts that elastic instabilities eventually move into the domain of elastic turbulence.The flow field inside the channels becomes chaotic and unstable due to the presence of this elastic turbulence, elevating the fluid velocity.• Slip factor shows greater change in the streamlines pattern of Convergent/ divergent in the geometry in quite opposite manner.• The magnetic field is strong enough, it is also possible to delay flow separation for non-Newtonian fluid in a diverging channel.• The decline in concentration is distinguished for higher impact of Schmidt number because Schmidt number has reverse relation with mass diffusion.
While the Soret number results in upshot.• Increasing the chaotic motion and thermophoresis parameter, external heat source, and Brinkman number elevate the thermal field.• The results emphasize that irreversibility due to fluid friction is high near the walls of the channel for enlarged values of magnetic parameter and Brinkman number.• The minimum entropy generations are determined by the optimum inclined angle.• The thermal hydraulic performance and irreversibility degradation of this configuration is related to the optimal geometrical parameters such as slip flow and width of the channel.
The current work on entropy generation analysis may be oriented toward effective use of energy by minimizing the loss of available energy in such a small confinement channel to the practical selection of interfacial electrokinetic and exterior thermo-fluidic body forces.
Future studies will optimize the converging diverging channel design to optimize its thermal and hydraulic performance.It is feasible to optimize the divergence angle of the divergent design as well as the angle, width, lubricated walls in the current design, for instance.Also, the current approach can be expanded to additional models of nanofluid flow in various base fluids in a channel with porous, moving, and converging diverging heated walls.

Figure 1 .
Figure 1.Physical configuration of radial flow of non-Newtonian fluid.

Figure 15 .
Figure 15.Volumetric rate of entropy production E G with Brinkman number Br.

Figure 16 .
Figure 16.Volumetric rate of entropy production E G with diffusion parameter D 1 .

Figure 17 .
Figure 17.Volumetric rate of entropy production E G with magnetic parameter M.

Figure 18 .
Figure 18.Volumetric rate of entropy production E G with slip parameter γ 1 .

Figure 19 .
Figure 19.Volumetric rate of entropy production E G with channel semi-angle α.

Table 4 .
Numerical values of heat and mass transfer rate against pertinent parameters in a convergent channel.

Table 5 .
Numerical values of heat and mass transfer rate against pertinent parameters in a divergent channel.