Minimizing frictional irreversibility in a rough-walled tapered bearing with a nanoparticle-enhanced Sutterby lubricant

Abstract

The thermodynamic performance of a tapered roller bearing is governed by the irreversible dissipation within its lubricating film. Minimizing the total entropy generation rate, which arises from fluid friction and heat transfer, is paramount for enhancing mechanical efficiency and operational longevity. This study conducts a numerical investigation into the entropy optimization of a nanoparticle-enhanced Sutterby lubricant within a rough-walled, tapered bearing geometry. The flow is modeled using the continuity, momentum, and energy equations, coupled with an entropy transport equation. The non-Newtonian lubricant behavior is characterized by the Sutterby fluid model, with its shear-thinning intensity governed by the Weissenberg number ($\:Wi$). The analysis focuses on the influence of key dimensionless parameters: the Reynolds number, the Hartmann number for magneto-hydrodynamic effects, and a defined surface roughness parameter. The analysis demonstrates that the total entropy generation, comprising frictional and thermal contributions is highly sensitive to the converging-diverging geometry. It is found that increasing the Weissenberg number significantly reduces the Bejan number, indicating a shift from thermal to viscous flow irreversibility dominance. Furthermore, the nanoparticle volume fraction directly enhances thermal transport, reducing the temperature gradient contribution to entropy generation. Optimal conditions for minimizing entropy are identified for specific combinations of Reynolds number, Weissenberg number and surface roughness. The results provide a rigorous thermodynamic framework for optimizing tapered bearing performance. By quantifying the interplay between rheology, inertia, and surface texture on the entropy generation map, this work establishes design criteria for minimizing irreversibility. This enables the development of high-efficiency lubrication systems utilizing advanced non-Newtonian nano lubricants.

Introduction

Entropy generation is a fundamental concept for evaluating thermal system capabilities, as it quantifies the irreversibility induced by several mechanisms, including heat transfer, fluid friction, and other dissipative mechanisms. Every heat transfer process is intrinsically accompanied by energy dissipation, which contributes to irreversibility and reduces overall system efficiency. Researchers have increasingly adopted entropy generation analysis because of its essential role in quantifying energy losses associated with fluid flow and heat transfer phenomena. Moreover, effective control of heat transfer is of paramount importance in power generation systems and electronic cooling applications. One of the earliest investigations concerning entropy formation and its importance in thermodynamic systems was carried out by Bejan^1,2. He put out the idea of entropy generation minimisation (EGM) as a method for minimising irreversibility in thermal and fluid systems. According to his studies, entropy development results from processes including mixing, heat transmission, and fluid friction-all of which lead to a system’s energy degrading. Following Bejan’s groundbreaking work, numerous studies have been conducted to predict the rate of entropy generation in thermal systems. Das et al.³ examined entropy generation in unsteady MHD nano fluid flow over an accelerating stretching sheet with convective boundary condition and revealed that metallic nanoparticles increased entropy in comparison to ordinary fluids. Rashidi et al.⁴ discovered that total entropy generation is enhanced by an increase in the mean Brinkman number during the flow of third-grade magnetohydrodynamic (MHD) fluid past a linearly stretching surface. Entropy production in a Cross fluid with motile gyrotactic bacteria under the influence of a magnetic field was investigated by Naz et al.⁵. Their findings demonstrated that entropy formation is strongly impacted by temperature change. The entropy production curve for Casson nanofluid flow caused by stretching discs was examined by Khan et al.⁶. Later, Qureshi et al.⁷ investigated entropy generation in the flow of a hybrid Ag/MgO-water nanofluid between coaxial permeable disks, accounting for the effects of suction/injection, viscous dissipation, and Joule heating. Murshid et al.⁸ analysed the squeezed flow of a hybrid nanofluid composed of water and propylene glycol between parallel plates, incorporating the influence of activation energy and entropy generation. They employed the shooting method to solve the governing equations of the system. Rehman et al.⁹ focuses on the entropy minimization during Power-law fluid flow within a wedge-shaped geometry in the presence of thermal transport, highlighting the effects of non-Newtonian behaviour on flow irreversibility. A series of studies over the recent years on entropy generation during various flow problems has been documented in Refs^10,11,12,13.

During the past few decades, a significant amount of research work has been performed in improving heat transport performance in industrial processes including power generation, electronics cooling, solar thermal setups and biomedical engineering. Improving ordinary fluid’s thermal conductivity is pivotal for heat exchange system optimization. Researchers have invented a new class of engineered formulations, “nanofluids” consisting of nanoparticles (NPs) of sizes ranging between 1 and 100 nm suspended in ordinary fluids, like, water, ethylene glycol, glycine, kerosene oil. Choi in 1995¹⁴ initially proposed the concept of nanofluids for better heat transmission. Nanofluids differ significantly from conventional fluids due to their superior thermophysical characteristics. These newly engineered fluids are being utilized in variety of industrial applications such as the petrochemical industry, hydraulic systems, chemical production, drug delivery, power generation, liquid cooled computers and oil refining, etc. These advanced nanofluids have emerged as a promising solution for enhancing thermal conductivity while alleviating the detrimental effects of viscous dissipation on fluid flow. Later, Buongiorno¹⁵ made a significant addition to the theoretical understanding of nanofluid behaviour by providing that the mechanism like Brownian motion and thermophoresis are responsible for improved thermal transport in nanofluids. Buongiorno’s model bridged the gap between theoretical predictions and experimental results by offering a thorough framework for predicting and analysing the intricate fluid dynamics of nanofluids. Sarkar et al.¹⁶ reviewed recent advancements in hybrid nanofluids, focusing on their preparation techniques, thermo-physical properties, and current practical applications. Moradi et al.¹⁷ explored how heat transfer and viscous dissipation influence the Jeffery–Hamel flow characteristics of nanofluids. Rehman et al.¹⁸ discussed the heat and mass transport features of Carreau nanofluids via converging/diverging channels using numerical technique. Alraddadi et al.¹⁹ explored the properties of a ternary hybrid cross bio-nanofluid in an expanding/contracting cylinder under an inclined magnetic field. Ayub et al.²⁰ discussed the quadratic convection-driven thermal transport at infinite shear rate in a magnetized ternary radiative cross nanofluid, incorporating homogeneous–heterogeneous chemical reactions.

In recent years, controlling the flow of electrically conducting fluids has become a key focus for engineers and scientists due to its significance in various industrial and technological processes involving mass and heat transfer. Several techniques have been developed to regulate the movement of such fluids, particularly within magnetohydrodynamic (MHD) frameworks. MHD boundary layers, which describe the behaviour of these fluids near surfaces under the influence of magnetic fields, have wide-ranging technical and geophysical implications. The application of magnetic fields plays a crucial role in modifying the thermophysical properties of fluids, making it a vital aspect of modern fluid mechanics with benefits in both heat and mass transfer enhancement. Dawood and Teamah²¹ investigated magnetic mixed convection in a square chamber with a sliding lid, emphasising the importance of buoyancy and Richardson ratios. Shankar and Naduvinamani²² investigated the combined effects of non-Fourier and non-Fick heat and mass fluxes for squeezing flow of Casson fluid under magnetic influence and generalized double-diffusive transport. The primary focus of their work was to solve the governing problem using adaptive numerical schemes and compared directly against traditional Fourier–Fick approaches. Again, Shankar and Naduvinamani²³ explores the unsteady radiative squeezing flow of a magneto-hydrodynamic Casson fluid between two parallel plates, accounting for thermal radiation and Joule heating. The behaviour of unsteady Williamson fluid flow and heat transfer across a horizontal sensor surface when subjected to external transverse magnetic field is examined by Shankar et al.²⁴. Later, Basha²⁵ used Cattaneo-Christov heat and mass flux models to investigate Casson fluid flow over a stretching cylinder.

The literature at present showcases different works on several physical phenomena, all focused on improving thermal transfer and entropy generation during converging/diverging flow of nanofluids. Various fluid models, like, Newtonian and non-Newtonian have been incorporated to investigate the fluid flows along with heat transport characteristics in nanofluids. Motivated by these considerations, the key novelties of the present study are summarized as follows:

Novelty of the article

• The novelty of the study lies in the formulation of a mathematical model that integrates a nanoparticle-enhanced Sutterby non-Newtonian lubricant flow with frictional and thermal irreversibility in a rough-walled tapered bearing(converging/diverging geometry).

• The incorporation of surface roughness, shear-thinning rheology, and nanoparticle-induced thermal transport provides new insights into entropy generation and the distribution of irreversibility within the bearing.

• The transformed system of nonlinear ordinary differential equations, obtained from the governing partial differential equations via similarity transformations, is solved numerically using the classical fourth-order Runge-Kutta (RK4) method in conjunction with a suitable shooting technique to satisfy the boundary conditions.

• An in-depth investigation of the interplay between Reynolds number, Weissenberg number, and surface roughness through numerical modelling and sensitivity analysis addresses critical factors for optimizing thermodynamic performance and reducing frictional losses.

Modelling and problem description

Flow configuration

As depicted in Fig. 1, a non-uniform channel having converging/diverging walls is taken into consideration with non-Newtonian and incompressible fluid flowing inside the channel. The flow is generated with a source or sink at the intersection point of the walls. The polar coordinates $\:(r,\theta\:)$ having centred on two intersecting walls is chosen to model the two-dimensional, laminar, incompressible and purely radial flow with a symmetric nature. In this coordinate system, $\:r-$ represents the radial distance from the corner apex formed by the intersection of the channel walls, while $\:\theta\:-$ denotes the polar angle, with the planar walls positioned at $\:\theta\:=-\alpha\:$ and $\:\alpha\:$. Moreover, the walls of the channel are taken to be rough and are at an angle of $\:2\alpha\:$. Moreover, the flow is considered purely radial, with the velocity field primarily governed by the $\:r$ and $\:\theta\:$ coordinates. Also, it is clear from Figs. 1,2 that the magnetic field operates in the opposite direction of the flow. This type of configuration commonly encountered in practical lubrication systems such as automotive journal bearings, turbine and compressor shaft supports, and high-speed rotating machinery. The tapered geometry enhances hydrodynamic pressure and load-carrying capacity, while surface roughness represents realistic manufacturing and wear effects. This configuration is particularly relevant for analysing advanced non-Newtonian nanolubricants aimed at reducing frictional losses, improving thermal regulation, and enhancing energy efficiency in modern tribological applications.

Fig. 1

An illustration of schematic diagram.

Sutterby model

The non-Newtonian nature of the working fluid is modelled using the Sutterby fluid formulation. This model effectively captures both shear-thinning and shear-thickening behaviours by incorporating a nonlinear relationship between the shear stress and the shear rate. The relationship between shear stress and rate of deformation is expressed as below²⁶:

$$\:S={\mu\:}_{0}{\left[\frac{{{sin}h}^{-1}(B\dot{\varvec{\gamma\:}})}{B\dot{\varvec{\gamma\:}}}\right]}^{n}{A}_{1}$$

(1)

Here, $\:{(\mu\:}_{0})-$signifies the viscosity at zero shear rate, $\:\left(B\right)-$ means the Sutterby parameter, $\:\left(n\right)-$the power-law index, $\:\dot{(\varvec{\gamma\:}})-$the second-invariant strain tensor, and $\:{(A}_{1})-$the first Rivlin-Erickson tensor.

The first-Rivlin Erickson tensor in terms of fluid velocity is written as:

$$\:{A}_{1}=\nabla\:V+{\left(\nabla\:V\right)}^{\top},\text{}\dot{\varvec{\gamma\:}}=\sqrt{\frac{1}{2}trace\left({A}_{1}^{2}\right)}$$

(2)

Using the binomial expansion for $\:{sinh}^{-1}$, the stress tensor (3) takes the following form:

$$\:S={\mu\:}_{0}{\left[1-\frac{{B}^{2}(\dot{\varvec{\gamma\:}}{)}^{2}}{6}\right]}^{n}{A}_{1}$$

(3)

Governing equations

For mathematical modelling of the described flow phenomenon, the following vector form of the basic flow equations will be considered^27,28.

Continuity equation:

$$\:\nabla\:\cdot\:\overrightarrow{V}=0$$

(4)

Momentum equation:

$$\:\rho\:\frac{d\overrightarrow{V}}{dt}=-\nabla\:p+{div}\overrightarrow{S}+\overrightarrow{J}\times\:\overrightarrow{B}$$

(5)

In above expressions, $\:(\rho\:$, $\:p)$ signifies the fluid density and pressure, $\:\overrightarrow{V}=\left[u,v,w\right]$ means the fluid velocity, $\:\overrightarrow{B}=\left(0,{B}_{0},0\right)$ denotes the applied magnetic field. The term on the right-hand side of Eq. (5) signifies inertial term i.e., the rate of change of momentum of a fluid particle as it moves. The first term on the right-hand express the pressure gradient, while the second term accounts for viscous (internal friction) forces within the fluid. It resists motion and dissipates energy due to the fluid’s internal resistance to deformation. The last term on right-hand side represents the Lorentz force, which arises due to the magnetohydrodynamics. It represents the force experienced by the fluid due to the interaction of electric currents $\:\overrightarrow{J}$ within the fluid and the magnetic field $\:\overrightarrow{B}$.

As, we have assumed a two-dimensional purely radial flow, the below velocity field is taken²⁹:

$$\:V=\left[u(r,\theta\:),\text{0,0}\right]$$

(6)

The component form of the continuity and momentum Eqs. (4 and 5) after utilizing the velocity field (6) becomes³⁰:

$$\:\frac{\partial\:}{\partial\:r}\left(ru\right)=0$$

(7)

$$\:\rho\:u\frac{\partial\:u}{\partial\:r}=\left[\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r{S}_{rr}\right)+\frac{1}{r}\frac{\partial\:}{\partial\:\theta\:}\left({S}_{\theta\:r}\right)-\frac{{S}_{\theta\:\theta\:}}{r}\right]-\frac{\partial\:p}{\partial\:r}-\sigma\:u{{B}_{0}}^{2},$$

(8)

$$\:0=\left[\frac{1}{{r}^{2}}\frac{\partial\:}{\partial\:r}\left({r}^{2}{S}_{r\theta\:}\right)+\frac{1}{r}\frac{\partial\:}{\partial\:\theta\:}\left({S}_{\theta\:\theta\:}\right)\right]-\frac{1}{r}\frac{\partial\:p}{\partial\:r}.$$

(9)

It follows from the continuity Eq. (7) that³¹

$$\:ru(r,\theta\:)=f\left(\theta\:\right).$$

(10)

After obtaining all the stress components of the Sutterby model (3) in view of (10), the governing Eqs. (8) and (9) can take the form:

$$\:\frac{1}{\rho\:}\frac{\partial\:p}{\partial\:r}=\frac{{f}^{2}}{{r}^{3}}+\frac{\nu\:{f}^{{\prime\:}{\prime\:}}}{{r}^{3}}{\left[1-\frac{{B}^{2}}{6{r}^{4}}\left[4{f}^{2}+{f}^{{\prime\:}2}\right]\right]}^{n}-\frac{\nu\:n{B}^{2}}{3{r}^{7}}\left[16{f}^{3}+8f{f}^{{\prime\:}2}+{f}^{{\prime\:}2}{f}^{{\prime\:}{\prime\:}}\right]{\left[1-\frac{{B}^{2}}{6{r}^{4}}\left[4{f}^{2}+{f}^{{\prime\:}2}\right]\right]}^{n-1}-\frac{\sigma\:{B}_{0}^{2}}{\rho\:{r}^{3}}f,$$

(11)

$$\:\frac{1}{\rho\:}\frac{\partial\:p}{\partial\:\theta\:}=\frac{2\nu\:{f}^{{\prime\:}}}{{r}^{2}}{\left(1-\frac{{B}^{2}}{6{r}^{4}}\left[4{f}^{2}+{f}^{{\prime\:}2}\right]\right)}^{n}+\frac{2}{3}\frac{\nu\:n{B}^{2}}{{r}^{6}}\left[{f}^{{\prime\:}3}-f{f}^{{\prime\:}}{f}^{{\prime\:}{\prime\:}}\right]{\left[1-\frac{{B}^{2}}{6{r}^{4}}\left[4{f}^{2}+{f}^{{\prime\:}2}\right]\right]}^{n-1}.$$

(12)

Eliminating the pressure term from above equations by cross differentiation, we get:

$$\begin{gathered} 2ff^{{\prime \:}} - \frac{{\nu \:nB^{2} }}{{3r^{4} }}\left( {32ff^{{\prime \:}} f^{{\prime \:\prime \:}} + 3f^{{\prime \:}} f^{{\prime \:\prime \:2}} + 64f^{2} f^{{\prime \:}} + f^{{\prime \:2}} f^{{\prime \:\prime \:\prime \:}} } \right)\left[ {1 - \frac{{B^{2} }}{{6r^{4} }}\left( {4f^{2} + f^{{\prime \:2}} } \right)} \right]^{{n - 1}} \hfill \\ \quad + \frac{{\nu \:n\left( {n - 1} \right)B^{4} }}{{18r^{8} }}\left( {128f^{4} f^{{\prime \:}} + 64f^{3} f^{{\prime \:}} f^{{\prime \:\prime \:}} + 32f^{2} f^{{\prime \:3}} + 32ff^{{\prime \:3}} f^{{\prime \:\prime \:}} + 2f^{{\prime \:3}} f^{{\prime \:\prime \:2}} - 8f^{{\prime \:5}} } \right)\left[ {1 - \frac{{B^{2} }}{{6r^{4} }}\left( {4f^{2} + f^{{\prime \:2}} } \right)} \right]^{{n - 2}} \hfill \\ \quad - \frac{{\sigma \:B_{0}^{2} }}{{\rho \:}}f^{{\prime \:}} + \nu \:\left( {f^{{\prime \:\prime \:\prime \:}} + 4f^{{\prime \:}} } \right)\left[ {1 - \frac{{B^{2} }}{{6r^{4} }}\left( {4f^{2} + f^{{\prime \:2}} } \right)} \right]^{n} = 0. \hfill \\ \end{gathered}$$

(13)

The non-dimensional form of the mathematical model (13) is derived by introducing the dimensionless variables as follows^31,32:

$$\:f\left(\theta\:\right)=rUF\left(\eta\:\right),\text{}\eta\:=\frac{\theta\:}{\alpha\:}.$$

(14)

The subsequent dimensionless equation after utilizing (14) becomes

$$\begin{gathered} \left( {F^{{\prime \:\prime \:\prime \:}} + 4\alpha \:^{2} F^{{\prime \:}} } \right)\left[ {1 - \frac{{We^{2} }}{6}\left( {4\alpha \:^{2} F^{2} + F^{{\prime \:2}} } \right)} \right]^{n} + 2\alpha \:ReFF^{{\prime \:}} \hfill \\ \quad - \frac{n}{3}We^{2} \left( {32\alpha \:^{2} FF^{{\prime \:}} F^{{\prime \:\prime \:}} + 3F^{{\prime \:}} F^{{\prime \:\prime \:2}} + 64\alpha \:^{4} F^{2} F^{{\prime \:}} + F^{{\prime \:2}} F^{{\prime \:\prime \:\prime \:}} } \right)\left[ {1 - \frac{{We^{2} }}{6}\left( {4\alpha \:^{2} F^{2} + F^{{\prime \:2}} } \right)} \right]^{{n - 1}} \hfill \\ \quad + \frac{{n\left( {n - 1} \right)}}{{18}}We^{4} \left[ {128\alpha \:^{6} F^{4} F^{{\prime \:}} + 64\alpha \:^{4} F^{3} F^{{\prime \:}} F^{{\prime \:\prime \:}} + 32\alpha \:^{4} F^{2} F^{{\prime \:3}} + 32\alpha \:^{2} FF^{{\prime \:3}} F^{{\prime \:\prime \:}} + 2F^{{\prime \:3}} F^{{\prime \:\prime \:2}} - 8\alpha \:^{2} F^{{\prime \:5}}} \right]\left[ {1 - \frac{{We^{2} }}{6}\left( {4\alpha \:^{2} F^{2} + F^{{\prime \:2}}} \right)} \right]^{{n - 2}} \hfill \\ \quad - \alpha \:^{2} M^{2} F^{{\prime \:}} = 0. \hfill \\ \end{gathered}$$

(15)

In order of significance, the following specifications are listed:

The Reynolds number: $\:\left({Re}=\frac{\alpha\:rU}{\nu\:}\right)$, and the local Weissenberg number: $\:\left(We=\frac{BU}{r\alpha\:}\right)$, the magnetic parameter: $\:\left(M=\sqrt{\frac{\sigma\:{r}^{2}{B}_{0}^{2}}{\nu\:\rho\:}}\right)$ .

Boundary conditions

The following lists of boundary conditions are employed in this analysis^29,30:

At centre line:

$$\:u=U,\text{}\frac{\partial\:u}{\partial\:\theta\:}=0,\quad\text{at}\quad \theta\:=0.$$

On the walls:

$$\:\frac{\partial\:u}{\partial\:\theta\:}=-\gamma\:U,\quad \text{at} \quad \theta\:={\upalpha}.$$

(16)

Where $\:U$ is the centre line velocity, $\:\gamma\:\ge\:0$ denotes the friction coefficient factor. The boundary is smooth for $\:\gamma\:=0$, and becomes perfectly rough as $\:\gamma\:\to\:\infty\:$ .

Physically, the condition $\:u=U,$ for $\:\theta\:=0$ imposes a symmetric maximum velocity along the centreline, while $\:\frac{\partial\:u}{\partial\:\theta\:}=0$ for $\:\theta\:=0$ ensures symmetry of the velocity profile about the centreline. At the wall $\:\theta\:={\upalpha\:}$, the velocity gradient $\:\frac{\partial\:u}{\partial\:\theta\:}=-\gamma\:U$ reflects the influence of wall roughness, with $\:\gamma\:\ge\:0$ indicating that rougher walls increase the tangential shear near the boundary.

The new form of boundary conditions in view of dimensionless transformations (14) is:

$$\:F\left(0\right)=1,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{F}^{{\prime\:}}\left(0\right)=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{F}^{{\prime\:}}\left(1\right)=-\gamma\:F\left(1\right).$$

(17)

Heat and mass transport analysis

The heat and mass transfer of nanofluids has become a prominent topic in fluid dynamics due to their enhanced heat transfer capabilities over conventional fluids. To predict and optimize the performance of such fluids in practical systems, mathematical modeling offers a systematic approach for analyzing the coupled behavior of momentum, energy and concentration transport. Unlike conventional fluids, nanofluids exhibit complex thermal behavior due to the presence of nanoparticles, which introduce additional terms such as Brownian motion and thermophoresis. The Buongiorno model subject to viscous dissipation and Joule heating is given by⁹:

Energy equation:

$$\begin{gathered} u\frac{{\partial \:T}}{{\partial \:r}} = \frac{{k_{f} }}{{(\rho \:Cp)_{f} }}\left( {\frac{1}{r}\frac{{\partial \:T}}{{\partial \:r}} + \frac{{\partial \:^{2} T}}{{\partial \:r^{2} }} + \frac{1}{{r^{2} }}\frac{{\partial \:^{2} T}}{{\partial \:\theta \:^{2} }}} \right) + \tau \:D_{B} \left( {\frac{{\partial \:C}}{{\partial \:r}}\frac{{\partial \:T}}{{\partial \:r}} + \frac{1}{{r^{2} }}\frac{{\partial \:C}}{{\partial \:\theta \:}}\frac{{\partial \:T}}{{\partial \:\theta \:}}} \right) \hfill \\ \quad \quad \quad + \tau \:\frac{{D_{T} }}{{T_{0} }}\left[ {\left( {\frac{{\partial \:T}}{{\partial \:r}}} \right)^{2} + \frac{1}{{r^{2} }}\left( {\frac{{\partial \:T}}{{\partial \:\theta \:}}} \right)^{2} } \right] + \frac{{\mu \:_{0} }}{{\left( {\rho \:Cp} \right)_{f} }}\left[ {\frac{{4u^{2} }}{{r^{2} }} + \frac{1}{{r^{2} }}\left( {\frac{{\partial \:u}}{{\partial \:\theta \:}}} \right)^{2} } \right]\left[ {1 - \frac{{B^{2} }}{{6r^{2} }}\left( {4u^{2} + \left( {\frac{{\partial \:u}}{{\partial \:\theta \:}}} \right)^{2} } \right)} \right]^{n} + \frac{{\sigma \:B_{0}^{2} }}{{r^{2} }}u^{2} , \hfill \\ \end{gathered}$$

(18)

Concentration equation:

$$\:u\frac{\partial\:C}{\partial\:r}={D}_{B}\left(\frac{1}{r}\frac{\partial\:C}{\partial\:r}+\frac{{\partial\:}^{2}C}{\partial\:{r}^{2}}+\frac{1}{{r}^{2}}\frac{{\partial\:}^{2}C}{\partial\:{\theta\:}^{2}}\right)+\frac{{D}_{T}}{{T}_{0}}\left(\frac{1}{r}\frac{\partial\:T}{\partial\:r}+\frac{{\partial\:}^{2}T}{\partial\:{r}^{2}}+\frac{1}{{r}^{2}}\frac{{\partial\:}^{2}T}{\partial\:{\theta\:}^{2}}\right).$$

(19)

In above Where the specific heat is $\:Cp$, the thermal conductivity is $\:k$, the fluid temperature is $\:T$, the reference temperature is $\:{T}_{0}$, the nanofluid concentration is $\:C$, the Brownian diffusion coefficient is $\:{D}_{B}$, and the thermophoresis diffusion coefficient is $\:{D}_{T}$ and $\:\tau\:={\left(\rho\:Cp\right)}_{np}/{\left(\rho\:Cp\right)}_{f}$.

We introduce the following similarity variables:

$$\:\frac{T}{{T}_{w}}=\varTheta\:\left(\eta\:\right),\text{}\frac{C}{{C}_{w}}=\varPhi\:\left(\eta\:\right).$$

(20)

We simplify the energy and concentration rules (18) and (19) to:

$$\:{\varTheta\:}^{{\prime\:\prime\:}}+Pr{N}_{B}{\varTheta\:}^{{\prime\:}}{\varPhi\:}^{{\prime\:}}+Pr{N}_{T}{{\varTheta\:}^{{\prime\:}}}^{2}+PrEc\left(4{F}^{2}{\alpha\:}^{2}+{F}^{{\prime\:}2}\right){\left[1-\frac{W{e}^{2}}{6}\left(4{\alpha\:}^{2}{F}^{2}+{F}^{{\prime\:}2}\right)\right]}^{n}$$

$$\:+{\alpha\:}^{2}EcPr{M}^{2}{F}^{2}=0,$$

(21)

$$\:{\varPhi\:}^{{\prime\:\prime\:}}+\frac{{N}_{T}}{{N}_{B}}{\varTheta\:}^{{\prime\:\prime\:}}=0,$$

(22)

with dimensionless boundary conditions,

$$\:\varTheta\:\left(1\right)=1,\text{}{{\Theta\:}}^{{\prime\:}}\left(0\right)=0, \text{}{\Phi\:}\left(1\right)=1,\text{}{{\Phi\:}}^{{\prime\:}}\left(0\right)=0,$$

(23)

where, the Prandtl number is $\:{Pr}=\frac{\mu\:Cp}{k}$, the Eckert number is $\:Ec=\frac{{U}^{2}}{Cp{T}_{w}}$, the Brownian diffusion is $\:{N}_{B}=\frac{{\left(\rho\:Cp\right)}_{p}{D}_{B}Cw}{(\rho\:Cp{)}_{f}{\nu\:}_{f}}$ and the thermophoresis diffusion is $\:\hspace{0.33em}\hspace{0.33em}{N}_{T}=\frac{(\rho\:Cp{)}_{p}{D}_{T}{T}_{w}}{(\rho\:Cp{)}_{f}{\nu\:}_{f}{T}_{0}}$ .

Nondimensional shear stress, mass, and heat transfer rates

The variables of practical and engineering concerns are the shear stress, heat and mass transfer rates, which are mathematically expressed by the relations:

$$\:\left.\begin{array}{c}{\:C}_{f}={\left.{\frac{{\mu\:}_{0}}{{\rho\:}_{f}{U}^{2}}\left[1-\frac{{B}^{2}}{6}\left(2{\left(\frac{\partial\:u}{\partial\:r}\right)}^{2}+\frac{1}{{r}^{2}}{\left(\frac{\partial\:u}{\partial\:\theta\:}\right)}^{2}+\frac{2{u}^{2}}{{r}^{2}}\right)\right]}^{n}\left(\frac{\partial\:u}{\partial\:\theta\:}\right)\right|}_{\theta\:=\alpha\:}\\\:Nu=-\frac{r{k}_{f}}{{T}_{w}}{\left.\left(\frac{\partial\:T}{\partial\:\theta\:}\right)\right|}_{\theta\:=\alpha\:}\\\:Sh=-\frac{r{D}_{B}}{{C}_{w}}{\left.\left(\frac{\partial\:C}{\partial\:r}\right)\right|}_{\theta\:=\alpha\:}\end{array}\right\}$$

(24)

.

Invoking the similarity variables (10), (14) and (20), we get the following non-dimensional forms:

$$\:\left.\begin{array}{c}{C}_{f}=\frac{1}{Re}{{\left[1-\frac{W{e}^{2}}{6}(4{\alpha\:}^{2}{F}^{2}\left(1\right)+{F}^{{\prime\:}2}\left(1\right))\right]}^{n}F}^{{\prime\:}}\left(1\right)\\\:Nu=-\frac{1}{\alpha\:}{{\Theta\:}}^{{\prime\:}}\left(1\right)\\\:Sh=-\frac{1}{\alpha\:}{\varPhi\:}^{{\prime\:}}\left(1\right)\end{array}\right\}$$

(25)

.

Formulations of entropy production

In recent years, minimizing energy waste has emerged as a critical objective for scientists and engineers striving for more efficient thermal systems. Among the primary challenges in thermodynamic systems is the irreversible loss of energy, which directly impacts system performance and sustainability. Entropy generation provides a theoretical and quantitative framework to evaluate and reduce such irreversibility.

Entropy generation analysis is employed to gain a more comprehensive insight into the system’s thermal characteristics. In this analysis, the primary sources of entropy are thermal and mass gradients, viscous dissipation and magnetization. The corresponding equations are expressed as follows:

$$\:{S}_{T}^{*}=\frac{{k}_{f}}{{T}_{w}^{2}}{\left(\nabla\:T\right)}^{2}+\frac{S\:\::\:L}{{T}_{w}}+\frac{{D}_{B}}{{C}_{w}}(\nabla\:C{)}^{2}+\frac{{D}_{B}}{{T}_{w}}(\nabla\:C\cdot\:\nabla\:T)+\frac{\sigma\:{B}_{0}^{2}}{{T}_{w}}{V}^{2}.$$

(26)

The first term is irreversibility due to temperature gradient, the second term is irreversibility due to fluid friction, the third term denotes Entropy generation due to mass diffusion and last term signifies Entropy generation due to magnetic force.

The normalized form of entropy generation for the modelled problem is given by:

$$\begin{gathered} S_{T}^{*} = \frac{{k_{f} }}{{T_{w}^{2} }}\left[ {\left( {\frac{{\partial \:T}}{{\partial \:r}}} \right)^{2} + \frac{1}{{r^{2} }}\left( {\frac{{\partial \:T}}{{\partial \:\theta \:}}} \right)^{2} } \right] + \frac{{\mu \:_{0} }}{{T_{w} }}\left[ {\frac{{4u^{2} }}{{r^{2} }} + \frac{1}{{r^{2} }}\left( {\frac{{\partial \:u}}{{\partial \:\theta \:}}} \right)^{2} } \right]\left[ {1 - \frac{{B^{2} \mathop {\gamma \:}\limits^{.2} }}{6}} \right]^{n} \hfill \\ \quad \quad + \frac{{D_{B} }}{{C_{w} }}\left[ {\left( {\frac{{\partial \:C}}{{\partial \:r}}} \right)^{2} + \frac{1}{{r^{2} }}\left( {\frac{{\partial \:C}}{{\partial \:\theta \:}}} \right)^{2} } \right] + \frac{{D_{B} }}{{T_{w} }}\left[ {\frac{{\partial \:C}}{{\partial \:r}}\frac{{\partial \:T}}{{\partial \:r}} + \frac{1}{{r^{2} }}\frac{{\partial \:C}}{{\partial \:\theta \:}}\frac{{\partial \:T}}{{\partial \:\theta \:}}} \right] + \frac{{\sigma \:B_{0}^{2} }}{{T_{w} }}u^{2} . \hfill \\ \end{gathered}$$

(27)

Invoking the dimensionless variables (14) and (20), the above expression reduces to:

$$\:{N}_{s}=\frac{{r}^{2}{\alpha\:}^{2}}{{k}_{f}}{S}_{T}^{*}={{\varTheta\:}^{{\prime\:}}}^{2}+PrEc\left(4{\alpha\:}^{2}{F}^{2}+{F}^{{\prime\:}2}\right){\left(1-\frac{W{e}^{2}}{6}(4{\alpha\:}^{2}{F}^{2}+{F}^{{\prime\:}2})\right)}^{n}+Md\left({\varPhi\:}^{{\prime\:}2}+{\varPhi\:}^{{\prime\:}}{\varTheta\:}^{{\prime\:}}\right)+{M}^{2}PrEc{\alpha\:}^{2}{F}^{2}.$$

(28)

The dimensionless Bejan number is written as follows:

$$\:Be=\frac{{{\varTheta\:}^{{\prime\:}}}^{2}}{{N}_{s}}=\frac{{{\varTheta\:}^{{\prime\:}}}^{2}}{{{\varTheta\:}^{{\prime\:}}}^{2}+PrEc\left(4{\alpha\:}^{2}{F}^{2}+{F}^{{\prime\:}2}\right){\left(1-\frac{W{e}^{2}}{6}(4{\alpha\:}^{2}{F}^{2}+{F}^{{\prime\:}2})\right)}^{n}+Md\left({\varPhi\:}^{{\prime\:}2}+{\varPhi\:}^{{\prime\:}}{\varTheta\:}^{{\prime\:}}\right)+{M}^{2}PrEc{\alpha\:}^{2}{F}^{2}}.$$

(29)

Solution methodology

The resulting non-linear conservation Eqs. (15), (21) and (23) and equation (24) for boundary conditions are numerically solved and presented in this section. Runge-Kutta Fehlberg and the Nachtsherim-Swigert technique the shooting approach is used for numerical computations because to its versatility and efficiency. This approach solves the initial value problem by converting the governing dimensionless boundary value problem Eqs. (15), (21) (23). For this, let’s introduce the variables listed below:

$$\:F={y}_{1},\text{}{F}^{{\prime\:}}={y}_{2},\text{}{F}^{{\prime\:}{\prime\:}{\prime\:}}={y}_{3},\text{}{\Theta\:}={y}_{4},\:{\varTheta\:}^{{\prime\:}}={y}_{5},\:\varPhi\:={y}_{6},\:{\varPhi\:}^{{\prime\:}}={y}_{7}.$$

The numerical integration is performed using the adaptive Runge-Kutta-Fehlberg method of order five with an embedded fourth-order error estimator. The shooting parameters are iteratively corrected using the Nachtsheim-Swigert technique to satisfy boundary conditions. The overall scheme achieves a global order of accuracy close to four. The complete flow chart of this method is given by Fig. 2.

Fig. 2

Flow chart of RK4 scheme.

Results and discussion

This section presents a detailed analysis of the numerical results, highlighting the effects of key governing parameters such as the Hartmann number, Reynolds number, nanoparticle volume fraction, and porosity on the flow characteristics. Comparative assessments and graphical trends are used to interpret the behaviour of the MHD nanofluid within the porous Jeffery-Hamel configuration.

Validation

To validate the proposed approach, a comparison table is provided, presenting selected results alongside those reported in previous studies under similar conditions. The close agreement between the present findings and the existing literature confirms the accuracy and reliability of the method as shown in Table 1.

Table 1 Validation of present fluid velocity $\:F\left(\eta\:\right)$ against $\:\eta\:$ with $\:We=0,M=0,n=1,\gamma\:=0,\:Re=50,\:\alpha\:={5}^{0}.$.

Velocity distribution

The effects of varying Reynolds number $\:\left(Re\right)$, magnetic parameter $\:\left(M\right)$, and friction coefficient $\:\left(\gamma\:\right)$ on the velocity profiles $\:F\left(\eta\:\right)$ are illustrated in Figs. 3 and 4, and 5, respectively. These figures provide a clear representation of how each parameter influences the flow behaviour within the Jeffery-Hamel configuration. A detailed discussion of each case is presented below to highlight the physical implications of the observed trends. Figure 3 illustrates the dimensionless velocity profiles $\:F\left(\eta\:\right)$ as a function of the Reynolds number $\:\left(Re\right)$ is distinctly for both divergent $\:\left(\alpha\:>0\right)$ and convergent $\:\left(\alpha\:<0\right)$ channels. As depicted in the relevant figures, an increase in Re enhances inertial forces relative to viscous forces, significantly affecting the flow behaviour. In the convergent channel, higher Reynolds numbers lead to a more accelerated flow toward the centreline, resulting in steeper velocity profiles and an increase in peak velocity. This behaviour is attributed to the narrowing geometry, which focuses the flow and amplifies momentum transport. Conversely, in the divergent channel, although the inertial effects are also intensified with increasing $\:Re$, the expanding geometry causes a deceleration of the fluid. This leads to flatter velocity profiles and broader distributions, particularly near the walls. In some cases, flow separation may occur due to adverse pressure gradients. These contrasting effects highlight the critical role of channel geometry in modulating the impact of Reynolds number on nanofluid flow under the Jeffery-Hamel configuration. Figure 4 illustrates the dimensionless velocity profiles $\:F\left(\eta\:\right)$ across the channel width $\:\left(\eta\:\right)$ for varying magnetic field parameters $\:M$(e.g., $\:M=\text{0,5},15$) in case of convergent $\:\left(\alpha\:=-1{0}^{0}\right)$ and divergent $\:\left(\alpha\:=1{0}^{0}\right)$ channels, respectively. The profile decelerates smoothly with $\:\eta\:$, adhering to the wall without separation (for low $\:Re$). Moreover, with rising magnetic parameter $\:M$, the velocity curves are observed to be higher in both divergent and convergent channels. The velocity profiles $\:F\left(\eta\:\right)$ in response to varying friction coefficients $\:(\gamma\:$) for convergent and divergent channels are illustrated in figure 5. The study reveals that increasing the friction coefficient $\:(\gamma\:$) consistently reduces flow velocities in both convergent and divergent channels, though through distinct mechanisms. In convergent channels, higher $\:\gamma\:$ values progressively dampen the characteristic flow acceleration, causing the velocity profiles to flatten and peak magnitudes to decrease significantly. Similarly, divergent channels exhibit reduced velocities with increasing $\:\gamma\:$, but here the effect is compounded by enhanced flow deceleration and earlier separation tendencies. While both geometries show velocity reduction, the convergent channel maintains more attached flow due to favourable pressure gradients, whereas the divergent channel suffers more pronounced velocity losses from combined friction and adverse pressure gradient effects. These findings demonstrate that wall friction universally diminishes flow momentum regardless of channel geometry, though the specific manifestations differ based on the prevailing pressure gradient conditions. The results provide critical insights for systems where friction control is essential for maintaining desired flow characteristics.

Temperature distribution

The temperature distribution profiles $\:\varTheta\:\left(\eta\:\right)$ in Figs. 6, 7 and 8 reveal how the magnetic parameter ($\:M)$, Eckert number $\:\left(Ec\right)$, and friction coefficient ($\:\gamma\:$) influence thermal behaviour in both convergent and divergent channels. It is observed that the temperature decreases with increasing $\:M$ in both geometries. This behaviour is attributed to the damping action of the Lorentz force, which suppresses fluid motion and reduces convective heating, thereby lowering the overall temperature field. In contrast, the temperature increases with higher $\:Ec$ due to viscous dissipation, where a portion of the fluid’s kinetic energy is converted into internal energy, leading to additional heating. As the Eckert number increases, the fluid flow rate along the core portion increases. The fluid temperature may have increased dramatically in both the narrow and opening channels because of higher values of the viscous heating parameter $\:Ec$. Similarly, an increase in the friction coefficient $\:\gamma\:$ enhances the micro-rotational effects of the micropolar fluid, introducing extra resistance and internal friction that contribute to heat generation, thereby elevating the temperature distribution. These trends are consistent in both convergent and divergent channels, though the magnitude of change is influenced by the channel geometry.

Concentration distribution

The physical characteristics of concentration distributions $\:\varPhi\:\left(\eta\:\right)$ are produced by the interaction of the Eckert number $\:Ec$, magnetic field parameter $\:M$, and friction coefficient $\:\gamma\:$, as illustrated in Figs. 9-11These results are computed considering both divergent and convergent channels. Both converging and diverging channels’ concentration patterns seem to be similarly impacted by the Eckert number $\:Ec$, as seen in Fig. 9 The graphical results show that as the value of $\:Ec$ rises, the concentration profiles decrease. Fig. 10 illustrates how magnetic field parameter $\:M$ effects on concentration profiles appear to be consistent in both converging and diverging channels. The graphical results show that when $\:M$ values rise, the concentration profiles decrease. The concentration distributions for converging and diverging channels are plotted against many estimates of the friction coefficient $\:\gamma\:$ at two fixed channel angle values, namely $\:\alpha\:=-1{0}^{0}$ and $\:\alpha\:=1{0}^{0}$, in Fig.11 It is evident that the concentration distribution shrinks as the friction coefficient rises.

Figure 12 shows a rising trend in the entropy profile and Bejan number in contrast to increasing Eckert number $\:Ec$ values. Higher values of the viscous heating parameter Ec resulted in a noticeable increase in entropy in both the narrow and opening channels. The influence of the diffusion parameter $\:Md$ on dimensionless entropy profiles for divergent and convergent channels is seen in Fig. 13. Higher $\:Md$ values are seen in both channels, which considerably raise the entropy profiles. The Bejan number, however, behaves in the opposite way. The Sutterby nano-fluid entropy trends for range magnetic field parameter $\:M$ values are shown in Fig. 14. These figures demonstrate that in both divergent and convergent channels, the dimensionless entropy and Bejan number rise as the magnetic field parameter $\:M$ increases. Fig 15. shows the Bejan number and entropy for convergent and divergent channels with varying friction coefficient $\:\gamma\:$. The graphic shows that the entropy and Bejan profiles for both channels are considerably increased by the friction coefficient $\:\gamma\:$ .

Engineering quantities

The results presented in Figs. 16, 17, 18, 19, 20 and 21 illustrate the variations in the skin-friction coefficient $\:\left({ReC}_{f}\right)$, Nusselt number $\:\left(Nu\right)$, and Sherwood number $\:\left(Sh\right)$ under different governing parameters, like, Reynolds number $\:\left(Re\right)$, power-law index $\:\left(n\right)$, friction coefficient $\:(\gamma\:$), magnetic parameter $\:\left(M\right)$, Prandtl number $\:\left(Pr\right)$ and Eckert number $\:\left(Ec\right)$. These dimensionless quantities provide important physical insight into the momentum, heat, and mass transfer characteristics of the flow. The skin-friction coefficient, which characterizes the wall shear stress, exhibits a dependence on the Reynolds number and boundary layer dynamics. The Nusselt number represents the convective heat transfer rate relative to conduction, indicating the efficiency of thermal transport within the channel. Similarly, the Sherwood number characterizes mass transfer rates and provides a measure of solute distribution in the nanofluid. A detailed discussion of these variations is presented in the following subsections, emphasizing the role of key fluid and physical parameters in shaping the momentum, thermal, and mass transport behaviour of the system.

Figures 16 and 17 depict the influence of the power-law index $\:\left(n\right)$ on the skin-friction coefficient for varying values of the Reynolds number $\:\left(Re\right)$ and friction coefficient $\:(\gamma\:$). The skin-friction coefficient is a critical parameter in fluid dynamics, representing the shear stress exerted by a fluid on a solid surface. The opposing trends in skin-friction for convergent $\:\left(\alpha\:=-1{0}^{0}\right)$ and divergent $\:\left(\alpha\:=1{0}^{0}\right)\:$channels highlight the critical role of flow geometry and inertial effects in viscous fluid dynamics. In the convergent channel, as Reynolds number increases, the flow accelerates toward the centreline, reducing the velocity gradient at the walls. This leads to a decrease in skin-friction, since the shear stress along the wall becomes weaker due to the more streamlined velocity distribution. On the other hand, for the divergent channel, an increase in Reynolds number enhances inertial effects against the adverse pressure gradient caused by the expanding geometry. This results in stronger wall shear and a rise in skin-friction. Therefore, while convergent geometry reduces wall resistance with higher $\:Re$, divergent geometry amplifies it, reflecting the crucial role of channel configuration in controlling momentum transfer.

The Nusselt number $\:\left(Nu\right)$ quantifies convective heat transfer relative to conductive heat transfer at a surface. In Figs. 18 and 19, the Nusselt number is analysed against the Eckert number $\:\left(Ec\right)$ and magnetic parameter $\:\left(M\right)$ for different Prandtl numbers $\:\left(Pr\right)$ and friction coefficients $\:(\gamma\:$) in convergent $\:\left(\alpha\:=-1{0}^{0}\right)$ and divergent $\:\left(\alpha\:=1{0}^{0}\right)$ channels. The results reveal opposing trends in heat transfer behaviour between the two channel geometries. It is observed that the Nusselt number enhances in case of divergent channel with higher values of $\:M$, $\:Pr$, $\:Ec$ and $\:\gamma\:$. This trend indicates that magnetic damping, viscous dissipation, higher thermal diffusivity, and friction coefficient effects collectively enhance heat transfer in the expanding geometry, where fluid spreading promotes stronger thermal gradients. Conversely, in the convergent channel, the Nusselt number decreases with increasing $\:M$, $\:Pr$, $\:Ec$ and $\:\gamma\:$. In this case, the narrowing geometry accelerates the flow, reducing the residence time of fluid particles near the heated surface, thereby weakening convective heat transfer despite the enhanced dissipation and magnetic effects.

The Sherwood number $\:\left(Sh\right)$ characterizes mass transfer rates at a surface, analogous to how the Nusselt number $\:\left(Nu\right)$describes heat transfer. In Figs. 20 and 21, $\:Sh$ is plotted against Eckert number $\:\left(Ec\right)$ and magnetic parameter $\:\left(M\right)$ for different Prandtl numbers $\:\left(Pr\right)$ and friction coefficients $\:(\gamma\:$) in convergent and divergent channels. The results reveal opposite trends in mass transfer behaviour between the two geometries. In the convergent channel, the Sherwood number increases with higher $\:M$, $\:Pr$, $\:Ec$ and $\:\gamma\:$. This indicates that magnetic damping, viscous dissipation, thermal diffusivity, and micropolar effects strengthen concentration gradients at the walls, thereby enhancing mass transfer. On the other hand, in the divergent channel, the Sherwood number decreases as these parameters increase. The expanding geometry leads to fluid deceleration and flow spreading, which weakens concentration gradients near the boundaries and reduces the overall mass transfer rate. These contrasting responses emphasize the significant role of channel geometry in determining solutal transport: convergent channels facilitate mass transfer enhancement, whereas divergent channels suppress it under the influence of magnetic and micropolar effects.

Fig. 3

Variations in velocity $\:F\left(\eta\:\right)$ for various values of $\:Re$ .

Fig. 4

Variations in velocity $\:F\left(\eta\:\right)$ for various values of $\:M$.

Fig. 5

Variations in velocity $\:F\left(\eta\:\right)$ for various values of $\:\gamma\:.$.

Fig. 6

Variations in temperature $\:\varTheta\:\left(\eta\:\right)$ for various values of $\:Ec.$.

Fig. 7

Variations in temperature $\:\varTheta\:\left(\eta\:\right)$ for various values of $\:M$.

Fig. 8

Variations in temperature $\:\varTheta\:\left(\eta\:\right)$ for various values of $\:\gamma\:.$.

Fig. 9

Variations in concentration $\:\varPhi\:\left(\eta\:\right)$ for various values of $\:Ec.$.

Fig. 10

Variations in concentration $\:\varPhi\:\left(\eta\:\right)$ for various values of $\:M.$.

Fig. 11

Variations in concentration $\:\varPhi\:\left(\eta\:\right)$ for various values of $\:\gamma\:.$.

Fig. 12

Variations in entropy $\:Ns$ and Bejan number $\:Be$ for various values $\:Ec.$.

Fig. 13

Variations in entropy $\:Ns$ and Bejan number $\:Be$ for various values $\:Md.$.

Fig. 14

Variations in entropy $\:Ns$ and Bejan number $\:Be$ for various values $\:M$.

Fig. 15

Variations in entropy $\:Ns$ and Bejan number $\:Be$ for various values $\:\gamma\:.$.

Fig. 16

Variations in skin friction coefficient $\:{ReC}_{f}$ for various values of $\:Re.$.

Fig. 17

Variations in skin friction coefficient $\:{ReC}_{f}$ for various values of $\:\gamma\:.$.

Fig. 18

Variations in Nusselt number $\:Nu$ for various values of $\:Pr.$.

Fig. 19

Variations in Nusselt number $\:Nu$ for various values of $\:\gamma\:.$.

Fig. 20

Variations in Sherwood number $\:Sh$ for various values of $\:Pr.$.

Fig. 21

Variations in Sherwood number $\:Sh$ for various values of $\:\gamma\:.$.

Concluding remarks

This comprehensive study investigated the thermo-fluid dynamics of Sutterby nanofluid flow through convergent-divergent channels, analysing key aspects including momentum transfer (skin friction), heat transfer (Nusselt number), mass transfer (Sherwood number), and entropy generation. For the distributions of flow velocity, temperature, concentration and entropy versus various flow parameters, numerical simulation has been done. We came to the following important conclusion after conducting this investigation.

• In divergent channels, high Reynolds numbers resulted in a significant reduction in flow velocity, whereas in convergent channels, the opposite was observed.

• The skin-friction coefficient increases in the divergent channel and decreases in the convergent channel with higher Reynolds number, reflecting the opposing effects of geometry on wall shear stress.

• In both channels, skin-friction showed an opposite tendency when measured against higher Reynolds number values.

• Overall, the results confirm that channel geometry plays a decisive role in controlling momentum, heat, and mass transfer. Convergent channels favour momentum and solutal transport enhancement but suppress thermal transfer, whereas divergent channels enhance thermal transfer while reducing solutal transport.

• Entropy in both channels exhibited an inverse trend when measure to greater power law index, Brownian diffusion parameter, and Weissenberg number.