2.1 Problem statement and underlying assumptions

This section addresses a mathematical treatment of the physical problem. The laminar, incompressible flow of non-Newtonian fluid from a source or sink located at the channel apex as displayed in Fig. 1.

1. The flow is assumed in cylindrical coordinates $(r,\phi ,z)$ , where $r$ is measured from the wedge's axis, $\phi$ from appropriate meridian plane, and $z$ is along the wedge's axis.

2. Assume that only radial motion affects the flow parameter.

3. Assuming that the two rigid, non-parallel plane walls are taken at an angle of $2\alpha$ .

4. The flow of a non-Newtonian fluid inside a converging divergent wedge originates from a source and driven by a constant pressure gradient.

5. It is assumed that $T_w$ and $C_w$ respectively the wall temperature and concentration and $T$ and $C$ are the fluid temperature and concentration.

6. The walls of the channel are isothermal i.e., no heat exchange occur between fluid and the channel walls.

7. The conservation of energy equation is constructed using traditional Fourier-law coupled with Buongiorno model and viscous heat mechanism.

8. The entropy generations equation is derived in term of velocity, heat energy and dissipations.

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Fig. 1

Power-law wedge shape flow model and coordinate system.

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In radial coordinates the 2-D the field of velocity, temperature, concentration, and Cauchy stress tensor between intersecting rough plates for Carreau fluid are settled as.

2.2 Mathematical analysis

The conservation of mass, momentum equation for non-Newtonian fluid is written as:

Where $\frac{D}{\mathit{Dt}}\left(\ast \right)=\frac{\partial }{\partial t}\left(\ast \right)+\left(V\bullet \mathrm{\nabla }\right)\left(\ast \right)$ , is the material time derivative. On the right-hand side of the equation, the first term is the standard Eulerian derivative (i.e., the derivative on a fixed reference frame), and the second term denotes the modifications brought on by the flowing fluid. Advection is the term for this effect. $-\mathrm{\nabla }p$ is the pressure gradient, ${\rho }_f$ is the fluid density. For Carreau fluid, we take $a=2$ and ${\eta }_{\infty }=0$ (practical situation) and the model become:

The shear stress tensor $\tau$ is defined as

where $\eta$ is the generalized viscosity of the Carreau fluid and ${\dot{\mathrm{Y}}}_{\mathit{ij}}$ is the shear strain rate. After invoking Eq. (1), the shear strain rate can be described as follows

The stresses components in view of Eq. (2), (5) and (7), we have

The continuity and Navier-Stokes equations in component form are summarized as follows for the two-dimensional flow (Ahmad et al., 2021), (Garimella et al., 2022), (Ramesh et al., 2022)

Applying Eq. (5), into Eqs. (7), and (8), we have

2.3 Thermal analysis

The heat transfer equation for the current problem while taking the combined contributions of the viscous dissipation, thermal radiation effect, and Buongiorno's model (Owhaib and Al-Kouz, 2022), (Chu et al., 2021)

In above Eq. (13), ${\tau }_{\mathit{ij}}:\mathrm{\nabla }V$ is the viscous dissipation expression, the Rosseland approximation for non-linear radiation, the radiative heat flux $q_r$ is simplified as (Shashikumar et al., 2020), (Bhaskar et al., 2022), (Veera Krishna, 2020)

By invoking Eq. (14) in Eq. (13), the energy conservation equation reduces to (Krishna et al., 2019), (Krishna et al., 2020), (Krishna et al., 2018)

2.4 Mass concentration equation

The nanoparticles concentration via Buongiorno model is written as:

2.5 Boundary conditions

The governing boundary conditions for the flow model are (Khan et al., 2015), (Saif and Jasim, 2019):

At the center of channel:

At the channel walls:

2.6 Normalization mechanism

Introducing similarity variables to normalize the governing equations (Khan et al., 2015), (Al-Saif and Jasim, 2019), (Bhaskar et al., 2022)

The converted system of governing ODEs become:

with reduced conditions:

The reduced parameters are:

2.7 Entropy mechanism and Bejan analysis

Within the context of the to the second law of thermodynamics the current investigation, for entropy generation which measure the heat sink performance is focused in this communication. Entropy generation in this context is divided into three categories: heat, mass, and frictional entropy generation. The transmitted heat often causes thermal entropy generation (TEG). On the other side, the irreversibility caused by viscous effects during the flow is referred to as frictional entropy formation (FEG). According to these definitions, it is anticipated that TEG and FEG rates will differ greatly depending on the type of fluid used as well as how the morphologies of the nanoparticles in the nanofluid are taken into account. Because these parameters have a significant impact on the flow structure and heat transfer characteristics, which also has an impact on the generated entropy. The totoal rate of entropy generation is obtained by adding the thermal and frictional components of entropy generation. However, energy destruction or created entropy are the main causes of low-efficiency systems. The total entropy generation rate $N^{″\prime }\left(W/m^{3}K\right)$ is constituted of the thermal, mass diffusivity, and frictional entropy generation rates. The total entropy generation rate resulting from the time-averaged steady state, incompressible flow of Carreau fluid within converging diverging wedge is written as (see (Bejan, 1996), (Chen et al., 2018), (Makinde and Bég, 2010), (Shukla et al., 2020), (Weigand and Birkefeld, 2009)), (Abbasi et al., 2022), (Abbasi et al., 2022):

Where $T_0=\frac{T_w+T}{2}$ and $C_0=\frac{C_w+C}{2}$ are referred to the average temperature and concentration of the central line and channel wall. Three primary factors contribute to the rate of entropy generation: the temperature difference-driven heat transfer entropy generation rate, the concentration difference-driven mass transfer entropy generation rate, and the frictional heat loss entropy generation rate. Each term can be determined by integrating over the entire wedge computational domain $\left[-1,1\right]$ and are given as:

In coordinates system the above equation can be settled as

The normalized entropy generation equation in the context of similarity variables reduces to

Here $D_f$ denote the diffusion parameter.

Bejan number $\mathit{Be}$ , which is the ratio of thermal entropy generation rate caused by thermal gradients to the overall entropy generation rate, is used to assess the contribution of thermal generation rates to total entropy generation rates. The confined Bejan number $\mathit{Be}$ measures the ratio of total irreversibility to irreversibility in heat transport. To determine the comparative importance of irreversibility due to heat transmission throughout the entire enclosure, the average Bejan number is utilized.

The range of the Bejan number is restricted to $0$ to 1. If $\mathit{Be}<0.5$ indicates that the total entropy generation rate dominates the entropy generation rate and $\mathit{Be}>0.5$ indicates that the thermal entropy generation rate dominates the total entropy generation rate.

2.8 Curiosity in engineering parameters

Skin drag force.

The skin drag-force i.e., the skin friction for Carreau fluid is

Where ${\tau }_{r\theta }$ denote shear stress and is defined as

Thus, the dimensionless expression become

Heat transfer rate

The local Nusselt Number is demarcated as:

Where $q_w$ signify surface heat flux and expressed as:

Employing the dimensionless variables, we get

Mass transfer rate

The local Sherwood Number is interpreted as:

Where $j_w$ characterizes surface mass flux and written as:

Thus

3.1 Results validation

We compare the current numerical values for velocity field $f\left(\eta \right)$ when $\eta$ is taken in the range $\left[-1,1\right]$ and other parameters are omitted to resemble with earlier work by (Sari et al., 2016) in order to assess the accuracy of our computational system. Table 1 shows a respectable degree of agreement between the current numerical results and the results of earlier studies.

4.1 Flow dynamics within wedge configuration

The fluctuation of velocity profile $f\left(\eta \right)$ for different Reynolds intensities $\mathit{Re}$ is seen in Fig. 2. In a convergent channel, the fluid velocity matching the Reynolds number, producing solid gradients at the medium's center walls and a reduction in the boundary layer thickness momentum. The findings reveal that backflow is completely prevented in convergent channel. A low Reynolds number indicates that the viscous force is dominant, and since the boundary layer region does not extend very deep into the flow region, which consequently accelerate the flow as a result the velocity dominates. Indicative of turbulent flow patterns, such as those seen in turbulent flows, are due to high Reynolds numbers. The results were reversed in the case of a convergent channel and increasing the Reynolds number increased the velocity profiles without causing any apparent reverse flow. Furthermore, the velocity profile along the center line stayed virtually constant at high Reynolds numbers while abruptly dropping to zero at the wall. In a divergent channel, the velocity profile decreases as the Reynolds number increases. Additionally, as the Reynolds number rises, the flow changes and a backflow region may be seen close to the wall. The velocity profile differs between the two channels. The velocity boundary layer thickness falls as the Reynolds number rise. On the other hand, in diverging channels, a rise in the Reynolds number can cause the flow to reverse (see Fig. 3). It's interesting how strong inertial forces influence the flow domain within diverse geometries. That is, in convergent channels, fluid elements near the wall accelerate as the Reynolds number rises, presumably due to an expansion of the beneficial pressure gradient. In diverging-channel flows of Carreau fluids, the inflection point shows that the velocity profiles at high $\mathit{Re}$ may be easily prone to hydrodynamic instability. Since the velocity profile contains detailed information on the flow field, it is particularly significant to us. According to Fig. 4, the velocity in a diverging section is a constantly decreasing function of $\mathit{We}$ , while in a converging section the consequences are conflicting. The dimensionless parameter $\mathit{We}$ distinguish the fluid's viscoelastic characteristics by relating the elastic forces to the viscous forces. In fact, the narrower channel and a higher viscoelasticity, the velocity of the Carreau fluid is suppressed. The most plausible explanation for this is that the shear-thinning property of the fluid is enriched at higher values of the Weissenberg number, which causes a decrease in the apparent viscosity of the fluid. Additionally, fluid radial velocity gradually decreases because of the viscosity drop. The influence of Weissenberg numbers on the normalized radial velocity will undoubtedly be due to main hemodynamic variables, such as shear stress at the wall (WSS), resistive impedance, and volumetric flow rate which exhibit oscillatory behavior. Fig. 4 depict the Weissenberg variation as a function of nanofluid velocity $f\left(\eta \right)$ . It can be perceived that $\mathit{We}$ appears as a mixed derivatives in the momentum conservation equations. In the absence of Weissenberg ( $\mathit{We}=0$ ), the model is reflected to be a Newtonian fluid, while the model extends to elastic solids model as ( $\mathit{We}\to \infty$ ). However, this rheological fluid model displays the viscoelastic assets of the polymer to adequate values. The velocity of the nanofluid declines with increasing $\mathit{We}$ , while in a convergent channel the influence is conflicting. In diverging case, the elastic force behaves as a dominant effect compared to the viscous force; therefore, the flow resistance is established in the boundary layer. Velocity $f\left(\eta \right)$ decline is therefore considered because of the growing $\mathit{We}$ . Physically, this reduction originates from the direct relationship between Weissenberg and relaxation rate, which led to decreases in fluid movement at higher relaxation rates. Fig. 5 is plotted to scrutinize the change in velocity $f\left(\eta \right)$ for various values of power index $n$ (Shear thickening case). It is perceived that dimensionless radial velocity dwindle with increase of $n(=1.2,1.5,1.8)$ and after attaining its maximum value radial velocity declines to zero in a converging zone. In addition, it is depicted that dimensionless radial velocity profile increases its occurrence in diverging channel. The momentum boundary layer thickness rises with the power-law index and the channel width. Therefore, the velocity of the fluid is improved as a consequence of power indexed. While slight reduction is attained in convergent channel within the narrower zone of the convergent channel. Maximum velocity is seen within the central region of both channel while slight reduction near to the wall of the channel is seen.

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Fig. 2

Performance of velocity $f\left(\eta \right)$ and flux shape in a convergent channel against augmented values of $\mathit{Re}$ .

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Fig. 3

Performance of velocity $f\left(\eta \right)$ and flux shape in a divergent channel against augmented values of $\mathit{Re}$ .

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Fig. 4

Performance of velocity $f\left(\eta \right)$ for $\alpha =-{10}^o$ and $\alpha ={10}^o$ against augmented values of $\mathit{We}$ .

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Fig. 5

Performance of velocity $f\left(\eta \right)$ for $\alpha =-{10}^o$ and $\alpha ={10}^o$ against augmented values of $n$ .

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4.2 Thermal distribution phenomena

The flow and temperature phenomena are combined with each other in the case of viscous heating. Consequently, the flow pattern inside the wedge, debated in the subsection, will significantly influence the corresponding temperature phenomena. In this subsection, we have deliberated the heat transfer features in detail in terms of the non-dimensional temperature distribution. The effect of the Weissenberg number ( $\mathit{We}=1.2,1.5,1.8$ ) on temperature distribution $\mathrm{\Theta }\left(\eta \right)$ is depicted in Fig. 6. The relationship between the fluid's kinematic viscosity/process time and its relaxation time is acknowledged as the Weissenberg number. Therefore, increasing the Weissenberg number makes it easier for fluid particles to relax over time, which advances friction and fluid thickness and, simultaneously, raises the fluid temperature. The flow and heat transfer features of non-Newtonian fluids are comparable to those of Newtonian fluids at low Weissenberg number values. For instance, at ( $\mathit{We}=0$ ), like the Newtonian fluid, the Carreau fluid similarly generates a large vortex zone that is virtually in the channel's center, along with two smaller vortex regions that are formed in the channel's up right $\eta =1$ and down $\eta =-1$ corners. We arrived at the conclusion that the measurement of the fluid's Weissenberg number ( $\mathit{We}>1$ ) would be severely hampered by polymer breakdown and the relative change in fluid relaxation time. Finally, fluid temperature has a significant impact on all the rheological parameters taken into consideration. The fluid's temperature was sampled at the beginning and end of the channel to determine the magnitude of the temperature shift brought on by the mechanical churning (inlet and outlet of the channel). The modification in temperature distribution as power-index ( $n$ ) becomes higher is demonstrated in Fig. 7. In a convergent channel, a decrease in thermal boundary layer thickness and temperature distribution is perceived versus $n$ . This tendency is demonstrated by the argument that when fluid viscosity increases, the kinetic energy of the fluid molecules reduces, resulting in a narrowing of the temperature distribution. While in diverging geometry scenario, a little degradation is achieved. In fact, large value of $n>1$ , (shear thickening fluid) the thickness of boundary layer augmented and corresponding the fluid layer thickened, consequently reduced the temperature. The dissipation of heat in the domain is significantly influenced by thermal radiation ( $R$ ). The system is heated by extra heat from a different source. The effective thermophysical characteristics of flow fluids make a strong effort to eliminate these temperatures. Fig. 8 shows that the thermal distribution increases with increasing radiation temperature. Similar trend for radiation parameter is seen in both channels. The higher the radiation parameter $R$ , the more heat is provided to the liquid, raising both temperature and thermal boundary-layer density. This is due to the increased interaction in the thermal boundary layer caused by an increase in the thermal radiation parameter. In fact, significant amount of heat caused by higher radiation parameter values, the nanofluid temperature profile and thermal boundary layer width is also increased.

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Fig. 6

Behavior of temperature $\mathrm{\Theta }\left(\eta \right)$ for $\alpha =-{10}^o$ and $\alpha ={10}^o$ against diverse values of $\mathit{We}$ .

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Fig. 7

Behavior of temperature $\mathrm{\Theta }\left(\eta \right)$ for $\alpha =-{10}^o$ and $\alpha ={10}^o$ against diverse values of $n$ .

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Fig. 8

Behavior of temperature $\mathrm{\Theta }\left(\eta \right)$ for $\alpha =-{10}^o$ and $\alpha ={10}^o$ against diverse values of $R$ .

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4.3 Concentration profile

The concentration profile inside the wedge shape channel at various Weissenberg values is exposed in Fig. 9. There is no mixing of the fluids with low Weissenberg numbers. The middle of the diverging channel where the fluid mingles the most, as the Weissenberg number rises to $1<We<2$ . This is because the elastic instability is present at this Weissenberg number, increasing chaotic advection and, thus, enhances the mixing phenomenon. The chaotic advection inside the convergent channel becomes weaker as the Weissenberg number rises due to the relaxation period decreasing further, which lessens the mixing phenomenon between the two plates. As the Weissenberg number ( $\mathit{We}$ ) rises, the polymer chain extension and elastic stress increase, which substantially influence the concentration dynamics. It can be observed that as $\mathit{We}$ increases, the concentration profile tends to rise closer to the wedge surface while, in the boundary layer region, it tends to decline far away from the wall. The nanoparticle concentration against power index $n>1$ is depicted in Fig. 10. Near the solid wedge wall, the concentration profile decreases with higher n values, while the opposite tendency is seen farther out from the wedge wall. The graphic for both converging diverging channel conveys that the nanoparticles concentration diminishes as the power indexed is augmented. Physically, the shear thickening fluid has a low concentration as compared to the shear thinning fluid.

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Fig. 9

Behavior of concentration $\mathrm{\chi }\left(\eta \right)$ for $\alpha =-{10}^o$ and $\alpha ={10}^o$ against diverse values of $\mathit{We}$ .

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Fig. 10

Behavior of concentration $\mathrm{\chi }\left(\eta \right)$ for $\alpha =-{10}^o$ and $\alpha ={10}^o$ against diverse values of $n$ .

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4.4 Irreversibility analysis of Power-Law fluid

In this section, entropy-generation rate $N_{\mathit{gen}}$ and Bejan analysis $\mathit{Be}$ is carried out to study the irreversibility of the channel with wedge shape structure. The elastic force in converging system is increased when Weissenberg number ( $\mathit{We}$ ) grows. A dominant nature is also expressed by the elastic force in comparison to the viscous force. $\mathit{We}$ increase entropy $N_{\mathit{gen}}$ and decrease $\mathit{Be}$ because of this behavior. Physically, viscosity difference grows for greater ( $\mathit{We}$ ), which raises resistance and finally causes the system to become more disordered (entropy generation) (Fig. 11). While the effects are conflicting for entropy in divergent channel as clear from the Fig. 12. In the region $\eta =-1$ to $0$ , the entropy generation curves are constant, while they change in other parts of the channel (see Figs. 11 and 12). The increased friction towards the walls may be the reason of this. In expanding channel, the entropy generation diminish due to large channel opening against rising trend in $\mathit{We}$ values led to a decline both profiles. Figs. 13 and 14 are plotted for entropy production $N_{\mathit{gen}}$ and Bejan profile $\mathit{Be}$ against power index $n$ in convergent divergent channel respectively. On higher estimates of the power law material constant $n$ , the entropy generation $N_{\mathit{gen}}$ exhibits an uprising tendency and reverse influence for the Bejan profile. The effects of mass and heat transport grow with greater $n$ , but viscous effects significantly rise and entropy decreases (see Figs. 13 and 14). Physically, large estimation of power index $n>1$ , the shear thickening effects dominates the system discordances in both geometries, consequently entropy distribution is higher, while reduction in Bejan profile is diminished. Entropy generation $N_{\mathit{gen}}$ and Bejan number $\mathit{Be}$ are examined and found to be improved for augmented radiation parameter $\left(R\right)$ . Higher intensities of radiation parameter enhance the system's internal energy $.$ This is why systemic disorganization increases (see Fig. 15 and Fig. 16). Higher radiation causes the irreversibility of heat transfer to become more pronounced than the irreversibility of viscous dissipation, raising the value of ( $\mathit{Be}$ ). The non-linear thermal radiation causes an increase in both local and overall entropy generation, which reaches its maximum levels in the areas with the strongest thermal gradients and consequently the greatest energy loss. The analysis of the Bejan number demonstrates that in all sections where conduction is the principal heat transport mechanism, are also with the highest values of entropy generation, the influence of heat transfer irreversibility to total entropy generation is close to 1, whereas it is nearly zero outside these regions.

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Fig. 11

Distribution of entropy generation rate $N_{\mathit{gen}}$ and Bejan profile $\mathit{Be}$ in a converging channel $\alpha =-{10}^o$ against diverse values of $\mathit{We}$ .

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Fig. 12

Distribution of entropy generation rate $N_{\mathit{gen}}$ and Bejan profile $\mathit{Be}$ in a diverging channel $\alpha ={10}^o$ against diverse values of $\mathit{We}$ .

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Fig. 13

Distribution of entropy generation rate $N_{\mathit{gen}}$ and Bejan profile $\mathit{Be}$ in a converging channel $\alpha =-{10}^o$ against diverse values of $n$ .

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Fig. 14

Distribution of entropy generation rate $N_{\mathit{gen}}$ and Bejan profile $\mathit{Be}$ in a converging channel $\alpha ={10}^o$ against diverse values of $n$ .

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Fig. 15

Distribution of entropy generation rate $N_{\mathit{gen}}$ and Bejan profile $\mathit{Be}$ in a converging channel $\alpha =-{10}^o$ against diverse values of $R$ .

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Fig. 16

Distribution of entropy generation rate $N_{\mathit{gen}}$ and Bejan profile $\mathit{Be}$ in a converging channel $\alpha ={10}^o$ against diverse values of $R$ .

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4.5 Quantities of physical interest

Tables 2 and 3 illustrates the effects of physical flow parameter on the rate of mass and heat transmission in the converging and diverging channel. Additionally, a comparison of the outcomes for both geometries are included. From these table, one can infer that as the Eckert number $\mathit{Ec}$ increases, the rate of heat transfer $\mathit{Nu}$ decreases, whereas the rate of heat transfer improves as $\mathit{Ec}$ increases. Consequently, for increasing values of ( $R$ ) and ( $n$ ), the heat transfer rate increases at the convergent channel, but the opposite tendency is observed for ( $\mathit{We}$ ). Physically, it is seen that $\mathit{Nu}$ and $\mathit{Sh}$ increase as $\mathit{We}$ increases, however the relaxation time's declining trend reflects an increasing trend with larger values of $\mathit{We}$ . Tensile stresses in the pseudoplastic nanofluid are accelerated by the lengthening relaxation time. Additionally, heat and mass transfer effects against various parameter are dominate in convergent channel. While significant decrease is seen for Nusselt and Sherwood against various parameter.