Impact of variable thermal conductivity on couple-stress Casson fluid flow through a microchannel with catalytic cubic reactions

Nomenclature

A

constant pressure term

a

microchannel width ( m )

Br

Brinkman number

Bi 1 , Bi 2

Biot numbers for each microchannel plate

Be

Bejan number

C p

specific heat capacity ( J ⋅ kg − 1 ⋅ k − 1 )

C f

skin friction coefficient

D A , D B

diffusion coefficients of species S 1 and S 2

Ec

Eckert number

E 0

characteristic entropy generation

E g

volumetric entropy production rate ( kg ⋅ m 2 ⋅ s − 2 ⋅ k − 1 )

Gr

Grashof number

g

acceleration due to gravity ( m ⋅ s − 2 )

h 1 , h 2

coefficients of convective heat transfer for each microchannel plate

H ss

heat source/sink parameter

K ⁎

porous medium permeability

k ⁎

variable heat conductivity ( W ⋅ m − 1 ⋅ k − 1 )

k ′

mean absorption coefficient

K 1

porous media shape factor

K s

heterogeneous parameter

K c

homogeneous parameter

L

ratio of characteristic temperature

N h

irreversibility due to heat transfer

N v

irreversibility due to viscous dissipation

N s

dimensionless entropy generation

Pr

Prandtl number

P

pressure ( Pa )

Q

coefficient of heat source/sink

q r

radiative heat flux

q w

wall heat flux

N h

entropy production due to heat transfer

N v

entropy production due to viscous dissipation

Nu 

Nusselt number

Rd

radiation parameter

Re

Reynolds number

s 1 , s 2

concentration of chemical species S 1 and S 2

s 0

constant concentration

Sc

Schmidt number

S

couple-stress parameter

T

temperature ( K )

T a

ambient temperature ( K )

T h

hot fluid temperature ( K )

V

axial velocity ( m ⋅ s − 1 )

v

dimensionless axial velocity

x , y

co-ordinate axes

Greek symbols

ϵ

variable thermal conductivity term

η 1

viscosity coefficient that characterizes the couple stress

β

Casson parameter

β ′

thermal expansion coefficient ( k − 1 )

ρ

fluid density ( kg ⋅ m − 3 )

μ

fluid dynamic viscosity ( kg ⋅ m − 1 ⋅ s − 1 )

δ

diffusion coefficient ratio

η

non-dimension transversal coordinate

χ 1 , χ 2

dimensionless concentrations

v 0

wall suction/injection

ν

kinematic viscosity ( m 2 ⋅ s − 1 )

θ

dimensionless temperature

σ

Stefan Boltzmann constant

τ w

wall shear stress

Abbreviation

H–H

homogeneous–heterogeneous reactions

RKF

Runge–Kutta–Fehlberg

1 Introduction

Recent advancements in the study of flow through micro channels have greatly contributed to the fields of thermal systems and fluid dynamics, providing new insights into micro-scale transport phenomena. Micro channels are highly effective at managing small fluid volumes with remarkable efficiency and precision, making them a key technology for innovation and progress. Understanding the underlying principles, design considerations, and applications of microfluidics is crucial for fully exploiting its potential and overcoming challenges related to scaling and fabrication. In particular, vertical micro channels play a vital role in modern technology, offering substantial benefits in heat transfer, fluid control, and reaction management. Due to their compact size and high efficiency, vertical micro channels are essential in various fields, such as heat exchange, microfluidics, chemical reactions, biomedical diagnostics, energy systems, MEMS, and environmental monitoring, where they help improve heat dissipation and overall performance. These micro channels enhance heat exchange by facilitating better fluid distribution, reducing thermal resistance, and allowing for precise temperature control. Their use in modern heat exchangers has become essential for improving energy efficiency and system reliability in applications such as HVAC systems, renewable energy technologies, and advanced manufacturing processes. Entropy analysis is necessary in microchannel flows because it helps to identify and quantify the irreversibilities present in the system, which are often a result of factors like friction, heat transfer, and flow resistance. There is a growing interest in optimizing heating systems through the analysis of entropy production, which is closely tied to the thermodynamic irreversibilities inherent in real processes, as described by the thermodynamics second law. Entropy generation is an unavoidable consequence of these processes. For example, minimizing entropy generation in fluids flowing through a porous channel can significantly reduce the input power required, leading to more efficient heat and mass transfer rates. Therefore, it is crucial to identify and optimize the factors that contribute to entropy generation. This approach helps mitigate its adverse effects and allows researchers to refine microchannel designs, thereby improving heat transfer efficiency, minimizing energy losses, and enhancing overall system performance. Abbas et al. [1] analysed entropy production in the flow of an electrically conducting fluid within a vertical porous channel under the influence of thermal radiation. Their findings show that changes in the magnetic field, suction/injection parameter, and specific values of the slip and temperature jump parameters affect the fluid velocity at the solid–fluid interface. Siddabasappa et al. [2] conducted an analysis of irreversibility and thermal transfer in couple-stress fluid flow within a thermally non-equilibrium vertical porous channel. They found that both heat transport and frictional drag are highest in the case of local thermal non-equilibrium, while they reach maximum values in local thermal equilibrium. Manthesha et al. [3] investigated the production of irreversibility in mixed convection Casson fluid flow within a vertical channel, taking into account the effects of nonlinear thermal radiation and variable viscosity. They found that reducing the values of Casson and mixed convection parameters led to a decrease in entropy generation. Padma and Ontela [4] studied entropy generation in the flow of thermally radiative viscoelastic fluid within a vertical microchannel. They reported that the regulation of entropy and the Bejan number is significantly affected by the Brinkman number. Mishra et al. [5] conducted a computational analysis of irreversibility production in an electro-osmotic fluid flow in a vertical channel with cilia containing gyrotactic microorganisms. Their study shows that the mass transfer rate increases with the thermophoresis parameter, while the opposite effect is observed for the Brownian motion parameter. Various authors have investigated different aspects of the subject in their studies [6,7,8,9,10,11,12].

In thermal systems, radiation and heat sources/sinks play a crucial role in optimizing heat transfer. Radiation transfers heat via electromagnetic waves, impacting thermal behaviour by adding extra heat fluxes and altering temperature distributions. On the other hand, heat sources and sinks directly influence heat levels, affecting local temperature gradients and the overall energy balance. The combined effects of these factors are crucial for optimizing system performance, allowing precise control over thermal conditions, improving energy efficiency, and enhancing the effectiveness of various thermal applications, such as microchannel heat exchangers, chemical reactors, biomedical devices, energy systems, environmental monitoring, advanced materials processing, microelectronic devices, heat spreaders, spacecraft thermal control, thermal sensors, and advanced heat exchangers. Jha et al. [13] investigated the effects of uniform heat source constraints on free convection flow in a vertical microchannel. They found that both the flow and thermal fields are improved with the point line source parameter. Madhukesh et al. [14] studied the thermal and mass transport in nano-fluids within a microchannel under the impact of non-uniform heat source/sinks. Their work reported that the magnetism, heat sources/sinks, and the porous medium directly influences the thermal field. Furthermore, they observed that the mass transfer rate improves with an increase in the local external source constraint. Sharma and Jain [15] examined electrically conducting fluids in a vertical porous channel influenced by thermal radiation. They observed that the solid–fluid interface is affected by the magnetic field. Hamza et al. [16] analysed the radiation impact on magnetized flow within a vertical super hydrophobic microchannel. They found that the thermal radiation drives fluid motion and heat gradients in both super hydrophobic surface and no slip surface (NSS) cases, leading to a decrease in fluid temperature at the NSS.

Investigating heat transfer performance in porous media with variable thermal conductivity is crucial for numerous scientific and engineering applications. Managing thermal performance in porous media becomes particularly challenging when thermal conductivity changes with factors like temperature or pressure. These variations can significantly influence heat transfer processes, affecting the efficiency of thermal insulation, energy storage, and chemical reactions within the medium. Accurate modeling requires a deep understanding of both variable thermal conductivity and the complex nature of porous media. Advances in this field could lead to significant improvements in energy systems, heat transfer, chemical reactors, heat-driven desalination, biomedical applications, high-performance cooling, environmental management, and material science, highlighting the need for continued research and innovative solutions to address the challenges of heat transfer in these systems. Enyadene et al. [17] conducted a numerical study on thermal and mass transport in Casson fluid flow within a microchannel that includes porous media. They stated that a decrease in the thermal field is noticed with increasing porous media, while the heat transfer rate improves with variable viscosity and Eckert number. Kaita et al. [18] investigated thermal and mass transport in a channel with porous media, emphasizing the effects of varying thermal conductivity. They found that the concentration decreases as the chemical reaction rate and Schmidt number increase. Khan et al. [19] investigated the intricate dynamics of incompressible fluid flow in a vertical microchannel containing porous media. Their work reports that both the flow and thermal fields increase with the Grashof number and Reynolds number. Recently, many authors have examined the impact of porous media by addressing different aspects of the subject [20,21,22,23,24].

Research on non-Newtonian fluids has garnered a considerable attention due to their wide range of applications across various fields. In biological contexts, non-Newtonian fluids encompass substances such as apple sauce, emulsions, muds, blood, soaps, sugar solutions, and shampoos. These real-world examples highlight the diverse materials that fall under the category of non-Newtonian fluids, including micropolar fluids, Maxwell fluids, second-grade fluids, third-grade fluids, Casson fluids, Power law fluids, and Carreau fluids. Among these, Casson fluids are particularly important in industrial applications because of their complex flow behaviours, which differ from the simple linear relationship between shear stress and shear rate seen in Newtonian fluids. Casson fluids are distinguished by their yield stress and shear-thickening properties. Studying Casson fluids provides key insights into their complex flow behaviours, which are distinct from Newtonian fluids. These fluids are important in industries like food processing, cosmetics, and pharmaceuticals due to their yield stress and shear-thickening properties. Understanding their behaviour helps optimize fluid flow models, improving processes in pumps, valves, and pipelines. In chemical, industrial, and biomedical applications, this knowledge enhances efficiency in mixing, heat transfer, and blood flow simulation. Couple-stress Casson fluid is a non-Newtonian fluid that combines the yield stress behaviour of Casson fluid with internal torque effects, aiding in the understanding of material flow in industrial and biomedical applications where shear and couple stresses impact efficiency and performance. Overall, research on Couple-stress Casson fluids enables the design of more efficient systems across various fields. Gireesha and Sindhu [24] investigated the entropy analysis of Casson fluid flow in a vertical channel, taking into account the impacts of Joule heating, magnetic fields, viscous heating, and radiative heat flux. Their study found that the rate of heat entropy generation decreases as the ratio of the wall-to-ambient temperature difference increases. Murtaza et al. [25] presented a finite difference simulation to explore a fractal fractional model of electro-osmotic Casson fluid flow in a vertical microchannel. Rasheed et al. [26] examined the unsteady Casson fluid over a vertical surface, considering the effects of chemical reactions and Hall current. They observed that an increase in the mass flow rate with the rise in Schmidt number, while the heat transfer rate exhibited a decline as the Prandtl number and Eckert number increased. Abbas et al. [27] studied thermal and mass transport in the convective flow of non-Newtonian fluids within a microchannel by applying the Caputo Fabrizio derivative method. Numerous researchers have examined the flow behaviour and thermal transfer of Casson fluid, considering various aspects [28,29,30,31,32,33,34,35,36,37,38,39,40,41].

Many industrial processes involve chemical reactions, which can be either heterogeneous or homogeneous. Recent research has focused on developing novel catalytic mechanisms that operate at higher temperatures. In homogeneous catalysis, the catalyst and reactants are in the same phase, typically liquid, allowing for uniform interaction and high reaction rates. In contrast, heterogeneous catalysis involves a catalyst in a different phase, usually solid, which presents challenges such as surface interactions and diffusion limitations but allows for easier catalyst separation and recovery. Studying both types of reactions enhances understanding of reaction mechanisms and efficiency, benefiting applications in industrial synthesis and environmental technology. The initial work on the boundary layer mathematical model of fluid flow with chemical reactions over a flat sheet was discussed by Merkin [42]. They observed that the homogeneous reaction prevails downstream, occurring in a narrow region situated far from the surface. Yang et al. [43] elucidated the effect of oxygen concentration on the characteristics of chemically coupled reactions involving methane in a microchannel. Their work reported that, an increase in O 2 concentration enhances the homogeneous reaction, leading to the consumption of CH 4 and H radicals, while also inhibiting the heterogeneous reaction rate of CH 4 and H on the catalytic surface. Noor [44] investigated the influence of homogeneous-heterogeneous chemical reactions on the colloidal behaviour of Casson fluids. They found that the homogeneous reactions in the fluid domain are governed by first-order kinetics, along with isothermal cubic autocatalytic kinetics. Aslam et al. [45] conducted a second law scrutiny of homogeneous and heterogeneous reactions in an electrohydrodynamic environment. Their study reports that the Grashof number, viscous dissipation, and Joule heating enhance both the flow and entropy generation. Numerous researchers have investigated homogeneous and heterogeneous reactions under various conditions and applications [41,46–55].

Despite extensive research on microchannel flows, entropy generation, and non-Newtonian fluids, several critical aspects remain unexplored. Previous studies have mainly focused on Newtonian and Casson fluids under thermal and magnetic influences [24,25,26]. However, a comprehensive analysis of entropy generation, heterogeneous-homogeneous reactions, and non-Newtonian fluid flow in vertical porous microchannels with heat source/sink effects and convective boundary conditions is still lacking [27,38,39]. While some works address entropy production in microchannels, the combined impact of variable thermal conductivity, internal heat generation, and convective constraints on energy dissipation remains underexplored [16, 17, 18]. Additionally, studies on homogeneous and heterogeneous reactions often overlook their simultaneous influence on thermal transport and entropy generation [36,37]. This study bridges these gaps by analysing couple-stress Casson nanofluid flow in a vertical porous microchannel, incorporating entropy generation, variable thermal conductivity, mixed convection, and convective boundary conditions. The novelty lies in integrating entropy analysis with chemical reactions, systematically evaluating irreversibilities under the combined effects of heterogeneous–homogeneous reactions, heat source/sink variations, and convective conditions. It also extends existing models by assessing the influence of porosity, and inverse couple stress variations on flow irreversibility.

The objective of the current study is to answer the following research questions:

1. How do heterogeneous–homogeneous reactions influence entropy generation and heat transfer in a couple-stress Casson nanofluid flowing through a vertical porous microchannel?

2. What is the effect of variable thermal conductivity and convective boundary conditions on flow irreversibilities and energy dissipation?

3. How do radiation variations, porosity, and internal heat generation impact thermodynamic optimization in microfluidic systems?

Addressing these aspects contributes to optimizing heat transfer performance and minimizing energy losses in microfluidic and thermal management systems.

2 Mathematical formulation of the flow problem

The geometrical interpretation of the formulation is shown in Figure 1. Here we consider steady, laminar, incompressible couple-stress Casson fluid flow in a vertical porous microchannel. The combined effects of radiative heat flux, variable thermal conductivity, and uniform heat source/sink are incorporated in the thermal equation. The two vertical microchannel plates are separated by a distance “ 2 a ” in the presence of constant longitudinal pressure gradient d p d x . The two plates are embedded with porous media. The co-ordinated axis “ x ” is in flow direction and no quantities depend on it. Further, the coordinate axis “ y ” is perpendicular to the microchannel plates and the quantities depend solely on the y -axis. The left microchannel plate is placed at y = − a and is maintained at a hot fluid temperature “ T h .” The right microchannel plate is placed at y = a and is maintained at an ambient temperature “ T a .” Without loss of generality, assume that both microchannel plates are infinitely long, ensuring that the velocity and thermal fields are fully developed. To strengthen this analysis, convective boundary conditions are employed. The isothermal cubic autocatalytic model has been employed for homogeneous and heterogeneous reactions.

Figure 1

Geometric configuration of the flow problem.

Based on the previously mentioned hypotheses, the governing equations for momentum, energy, and concentration can be formulated as follows [24,32,54,56].

3 Numerical procedure

The reduced non-dimensional system of boundary value problems (equations from (8) to (11), (21) to (22) and (24)) was transformed into a system of initial value problems using the shooting technique. These initial value problems were then solved using the Runge-Kutta-Fehlberg method of fourth and fifth order. A step size of h = 0.001 was used for the numerical analysis. The following formulas illustrate the technique employed in the solution process [32]:

The approximate solution was computed using the fourth-order Runge-Kutta method.

The refined solutions were obtained using the fifth-order Runge-Kutta method.

The error term is determined by subtracting the two previously obtained values. If a significant error term is observed, the step size is reduced to ensure the required accuracy and satisfy the convergence criterion of 10 − 6 . Table 1 compares the exact solution with the numerical results for the momentum profile. The close agreement between them confirms the reliability of the current results.

4 Results and discussion

We conduct numerical simulations to examine the effects of isothermal chemical reactions (H–H) and variable thermal conductivity on the flow of couple-stress Casson fluids in a vertical microchannel, under convective boundary conditions and various heat source constraints. The physical parameters used for the numerical solutions are set as follows: Re = A = KS = 1 ,  Gr = 0.3 , K 1 = 0.3 , S = 0.4 , KC = Sc = L = β = Rd = 0.5 , Br = 0.2 , H ss = 0.2 , and ϵ = 0.1. The influence of these physical parameters on flow, temperature distribution, skin friction coefficient, Nusselt number, entropy generation, and the irreversibility ratio is analysed and displayed through graphical representations.

Figure 2(a)–(c) illustrates the effects of the Casson parameter on flow behaviour, thermal fields, and the Bejan number, respectively. As the Casson parameter increases, the fluid’s yield stress becomes more significant, requiring higher stress to initiate flow. Physically, this leads to increased resistance against motion under a given driving force, such as a pressure gradient. Consequently, the overall flow rate decreases, as observed in Figure 2(a). This effect is particularly relevant in yield-stress fluids, where flow initiation depends on overcoming an inherent material resistance. Regarding the thermal field, an enhancement in yield stress alters the temperature distribution. A more dominant Casson parameter causes the thermal boundary layer to thicken, as shown in Figure 2(b). This thickening results from the reduced convective motion in the fluid, which in turn enhances heat transfer from the walls to the fluid. Figure 2(c) illustrates the variation in the Bejan number with a more significant Casson parameter, which can be explained by considering the balance between heat transfer irreversibility and fluid friction irreversibility. As the Casson parameter intensifies, the dominant factor in entropy generation shifts towards heat transfer irreversibility due to the thickened thermal boundary layer. Since heat transfer plays a more substantial role relative to viscous dissipation, the Bejan number exhibits an upward trend; however, due to greater fluid friction irreversibilities, the effect remains slightly uniform in the central region of the channel. These results have practical implications in optimizing flow and heat transfer processes in industries where yield-stress fluids are prevalent, such as chemical reactors, electronic cooling systems, and food or pharmaceutical processing. Additionally, they are relevant in biomedical applications like artificial organs, as well as in lubrication systems and heat exchangers, where controlling flow resistance and thermal performance is critical.

Figure 2

Influence of Casson parameter ( β ) on (a) velocity, (b) thermal field, and (c) Bejan number.

Figure 3(a)–(c) shows the effect of Biot number on the thermal field, entropy production, and Bejan number. The Biot number represents the balance between internal thermal resistance within the fluid and thermal resistance at the boundary. When the Biot number is high, the thermal resistance at the boundary becomes dominant, restricting heat flow from the boundary into the fluid. This results in a steeper temperature gradient near the boundary and lower temperatures at the boundary regions, as shown in Figure 3(a). The reduction in temperature at the boundary occurs because heat is not efficiently conducted from the surface into the fluid due to the higher external thermal resistance. Figure 3(b) demonstrates the variation in entropy production with a larger Biot number. Entropy generation remains uniform near the left plate of the microchannel and increases towards the right plate. This is due to the fact that the thermal resistance at the left boundary is significant, meaning that heat transfer from the plate to the fluid is restricted. As a result, the temperature gradient remains nearly constant, limiting heat transfer and thermal irreversibilities, keeping entropy generation unchanged near the left plate and up to half of the microchannel ( η = 0.3 ). Figure 3(c) depicts the variation in the Bejan number for larger estimations of the Biot number, exhibiting a dual behaviour. On the left plate, a higher Biot number increases thermal resistance at the boundary, restricting heat transfer into the fluid and reducing heat transfer irreversibilities, leading to a lower Bejan number. Conversely, on the right plate, enhanced heat conduction improves heat transfer efficiency relative to entropy generation, causing the Bejan number to rise, reflecting the shifting balance between heat transfer and fluid friction irreversibilities. By adjusting the Biot number, engineers can optimize heat transfer performance and minimize irreversibility in critical regions of thermal systems.

Figure 3

Influence of Bi on (a) thermal field, (b) entropy production, and (c) Bejan number.

Figure 4(a)–(d) illustrates the physical mechanisms governing the impact of the Brinkman number on fluid velocity, temperature distribution, entropy generation, and the Bejan number. As shown in Figure 4(a), an increase in the Brinkman number corresponds to greater viscous dissipation, which injects additional energy into the fluid, enhancing momentum transfer and raising the fluid velocity. This occurs because viscous dissipation generates internal heating, reducing resistance to flow and facilitating greater motion within the fluid. Figure 4(b) reports the variation in the thermal field for larger Brinkman number. As viscous dissipation intensifies, the heat generated within the fluid surpasses conduction effects, leading to a more pronounced thermal field. This results in an elevated temperature profile, as the excess energy from dissipation accumulates within the system rather than being efficiently conducted away. Figure 4(c) explains the influence of the Brinkman number on entropy generation. As viscous dissipation becomes more pronounced with enhanced Brinkman number effects, the additional energy input increases the overall irreversibilities within the system, leading to higher entropy production. The effect is most prominent at η = 0.5 , where the balance between heat conduction and viscous dissipation shifts. At this point, the intensified dissipation significantly impacts the thermal field, creating a more uniform entropy generation profile. Figure 4(d) reveals a dual behaviour in the Bejan number across the microchannel. Near the right plate, where viscous dissipation exerts a stronger influence, the intensified internal heat generation surpasses the systems’ ability to transfer heat efficiently, causing heat transfer efficiency to weaken relative to entropy generation and resulting in a lower Bejan number. Conversely, near the left plate, where the effects of dissipation are weaker, heat transfer becomes relatively more efficient, resulting in a higher Bejan number. This dual behaviour reflects the localized competition between thermal and frictional irreversibilities. Understanding these effects is crucial in optimizing thermal management systems, such as microchannel heat exchangers, where regulating the Brinkman number can help control heat transfer efficiency and minimize energy losses due to irreversibilities.

Figure 4

Influence of Brinkman’s number ( Br ) on (a) velocity, (b) thermal field, (c) entropy production, and (d) Bejan number.

Figure 5(a)–(d) illustrates the effect of the Grashof number on fluid velocity, temperature distribution, entropy generation, and the Bejan number. The Grashof number characterizes the interplay between buoyancy and viscous forces within the fluid. When buoyancy forces become more dominant, they generate stronger convective currents, which enhance fluid motion within the system, leading to a greater velocity magnitude, as seen in Figure 5(a). Figure 5(b) presents the effect of the Grashof number on the thermal field, demonstrating a more pronounced temperature profile. As buoyancy-driven convection strengthens relative to viscous resistance, the transport of thermal energy within the fluid intensifies, contributing to an augmented temperature gradient in the microchannel. Figure 5(c) shows entropy generation variations for larger Grashof numbers, with higher dissipation near the channel plates and uniform distribution in the central region. Stronger buoyancy forces steepen thermal and velocity gradients near the plates, intensifying heat transfer and frictional irreversibilities. In contrast, the central region maintains a balanced convection effect, leading to minimal variation. This reflects the interaction between buoyancy, thermal gradients, and viscous dissipation in shaping entropy generation. Figure 5(d) depicts the dual behaviour of the Bejan number for increasing Grashof number. Near the plates, the dominance of viscous dissipation and conductive heat transfer over natural convection results in a reduced Bejan number. However, within the central region of the channel, where buoyancy-driven convection plays a more influential role, heat transfer irreversibilities become more significant relative to frictional effects, leading to an enhancement in the Bejan number. Figure 5(d) shows the dual behaviour of the Bejan number for larger Grashof numbers. Near the plates, dominant viscous dissipation and conductive heat transfer limit heat transfer irreversibilities, reducing the Bejan number. In the central region, stronger buoyancy-driven convection enhances heat transfer relative to frictional effects, leading to a more pronounced Bejan number. These observations highlight the crucial role of buoyancy in shaping fluid dynamics and thermal transport within the system. A deeper understanding of these interactions allows for improved thermal management strategies, minimizing energy losses and enhancing system efficiency in applications such as chemical reactors and heat exchangers.

Figure 5

Influence of Grashof number ( Gr ) on (a) velocity, (b) thermal field, (c) entropy production, and (d) Bejan number.

Figure 6(a)–(c) shows that the impact of the porous parameter on velocity, temperature, and the Bejan number. As fluid moves through the porous medium, the increased resistance impedes motion, limiting velocity, as observed in Figure 6(a). This opposition to flow arises due to the interaction between the fluid and the porous matrix, restricting movement and influencing practical applications such as filtration and heat exchangers, where flow control is crucial for performance. The thermal field intensifies with larger estimations of the porous shape factor by altering heat conduction and convection, is as shown in Figure 6(b). The Bejan number reducing in the channel central region while becoming more prominent near both microchannel plates is as shown in Figure 6(c). The impact is reduced near the channel central region due to the decreased heat transfer irreversibilities relative to viscous dissipation. However, the localized variations near the left plate result in a slight enhancement in the Bejan number, highlighting spatially dependent flow and heat transfer dynamics. These findings are essential in optimizing systems such as filtration, geothermal reservoirs, and heat exchangers, where flow resistance and heat transfer efficiency play a critical role. Additionally, applications in biomedical engineering, wastewater treatment, and electronic cooling systems benefit from understanding the interplay between porous media and thermal performance.

Figure 6

Influence of magnetic parameter ( K 1 ) on (a) velocity, (b) thermal field, and (c) Bejan number.

Figure 7(a)–(c) illustrates the impact of H ss on temperature distribution, entropy generation, and Bejan number within the microchannel. As H ss increases, it induces stronger thermal gradients, which intensifies heat conduction and promotes convective transport within the fluid. This leads to an elevated temperature distribution, as seen in Figure 7(a), due to the enhanced energy exchange between the fluid and the channel walls. The increased temperature differentials contribute to higher heat flux, influencing the thermal field within the microchannel. Figure 7(b) depicts the variation in the entropy generation for higher estimations of H ss . Figure 7(b) depicts the variation in entropy generation for higher estimations of H ss . This increase in entropy generation occurs due to the combined effects of intensified thermal gradients and enhanced viscous dissipation. However, the effect appears slightly uniform over the central region of the channel because, in this region, the temperature gradients and shear stress distribution reach a relative balance. As H _ss increases, the temperature differentials within the microchannel become more pronounced, leading to greater heat flux and, consequently, higher thermal irreversibilities. Additionally, the intensified convective motion within the fluid results in increased internal friction, further contributing to entropy production. Figure 7(c) reveals the discrepancies in the Bejan number. Near the left plate, where heat transfer irreversibilities dominate, the Bejan number exhibits a notable increase, indicating that heat conduction plays a primary role in entropy generation. In contrast, near the right plate, the influence of viscous dissipation becomes more pronounced, reducing the Bejan number in this region. This variation suggests a shift in the governing entropy mechanisms, highlighting the necessity of balancing conductive and viscous effects to optimize system performance.

Figure 7

Influence of heat source/sink parameter H ss on (a) thermal field, (b) entropy production, and (c) Bejan number.

Figure 8(a)–(c) illustrates the impact of the radiation parameter on the temperature, entropy generation, and Bejan number within the microchannel. As the radiation parameter becomes more dominant, radiative heat transfer plays a significant role in the energy balance of the system. The increased emission and absorption of thermal radiation lead to enhanced heat dissipation, which facilitates greater energy exchange with the surroundings. This mechanism reduces the overall fluid temperature, as shown in Figure 8(a), since radiation allows energy to be transported away more effectively compared to conduction and convection alone. Figure 8(b) shows the variation in the entropy generation for higher radiation parameter. Near the left plate, entropy generation remains uniform due to a relatively small temperature gradient and the dominance of conductive heat transfer. However, as the fluid moves towards the right plate, the temperature gradient increases due to the stronger influence of radiation, resulting in higher entropy generation. Additionally, viscous dissipation effects contribute to irreversibilities, particularly near the right plate, where velocity gradients are steeper. The combined effects of conduction, radiation, and viscous dissipation lead to an asymmetric entropy distribution, with higher entropy production occurring towards the right plate of the microchannel. The dual behaviour of Bejan number for larger estimations of radiation parameter is as shown in Figure 8(c). Near the left plate, the temperature gradient remains relatively small, and the dominant mode of entropy generation is due to thermal conduction. As the radiation parameter increases, radiative heat transfer enhances the thermal transport, reducing the relative contribution of heat transfer irreversibility compared to viscous dissipation, leading to a decrease in the Bejan number. However, near the right plate, the effects of radiation significantly intensify the temperature gradient, causing heat transfer irreversibility to become more dominant. This results in an increase in the Bejan number in this region. This understanding is crucial for applications such as electronics cooling, energy systems, and microfluidic technologies, where maintaining optimal thermal conditions directly impacts performance and reliability.

Figure 8

Influence of radiation parameter ( Rd ) on (a) thermal field, (b) entropy production, and (c) Bejan number.

The effect of the inverse couple-stress parameter on fluid velocity, temperature, entropy generation, and the Bejan number is illustrated in Figure 9(a)–(c). An increase in the inverse couple-stress parameter typically reduces the fluids’ resistance to deformation. This reduction in resistance lowers the fluids effective viscosity, allowing for improved fluid velocity as the flow experiences less obstruction, as shown in Figure 9(a). Further, it has been noted that a rise in the inverse couple-stress parameter eventually reduces the flow by increasing the dynamic viscosity of the non-Newtonian fluid. With higher fluid velocity, heat is transported more effectively through the microchannel, resulting in lower fluid temperatures, as depicted in Figure 9(b). If the velocity increases with the inverse couple-stress parameter, the minimization of the Bejan number in Figure 9(c) can be explained by the shifting balance between thermal and viscous irreversibilities. As the inverse couple-stress parameter increases, the fluid experiences lower resistance to deformation, leading to enhanced velocity. This results in a higher shear rate, which can contribute to increased viscous dissipation. However, the simultaneous increase in velocity enhances convective heat transfer, reducing temperature gradients and lowering thermal entropy generation. Since the Bejan number represents the ratio of thermal irreversibility to total irreversibility, the relative dominance of heat transfer irreversibility decreases as viscous effects become more significant. Consequently, the Bejan number decreases throughout the microchannel, indicating that the system transitions towards a state where viscous dissipation plays a more prominent role in total entropy generation.

Figure 9

Influence of inverse couple-stress parameter on (a) velocity, (b) thermal field, and (c) Bejan number.

Figure 10(a)–(c) illustrates the impact of variable thermal conductivity on the thermal field, entropy generation, and Bejan number. An increase in variable thermal conductivity decreases the thermal field because higher thermal conductivity enhances the ability of the fluid and porous medium to conduct heat more efficiently. As a result, heat spreads more uniformly throughout the microchannel, reducing temperature gradients. When the thermal gradient decreases, the overall temperature distribution appears lower because the system achieves better thermal equilibrium (Figure 10(a)). This effect is particularly significant in regions where conduction is the dominant heat transfer mode, as the improved thermal conductivity reduces localized heating and promotes a more uniform temperature profile. Consequently, the thermal field diminishes as heat is more effectively distributed across the microchannel. For larger thermal conductivity, entropy generation remains uniform near the left plate as conduction dominates heat transfer. Increased thermal conductivity improves heat distribution, reducing temperature gradients in this region. Towards the right plate, convection becomes more dominant, leading to stronger temperature variations. This enhances heat transfer irreversibilities, causing a rise in entropy generation. The presence of a heat source further amplifies this effect, making entropy production higher near the right plate is shown in Figure 10(b). The Bejan number decreases near the left plate as higher thermal conductivity enhances heat diffusion, reducing temperature gradients and thermal irreversibility. Towards the right plate, stronger convection and higher temperature gradients increase heat transfer irreversibility, intensifying the Bejan number (Figure 10(c)).

Figure 10

Influence of variable thermal conductivity ( ϵ ) on (a) thermal field, (b) entropy production, and (c) Bejan number.

Figure 11(a) and (b) illustrates the variation in concentration with respect to the homogeneous and heterogeneous strength parameters. As the heterogeneous strength parameter varies, the concentration gradient is influenced by the efficiency of surface reactions in consuming reactants at the catalytic interface. Stronger surface reactions enhance the rate at which reactants are depleted, leading to a more uniform distribution of concentration across the system (Figure 11(a)). This reduces the disparity between the bulk fluid and the reactive surface, altering the diffusion-driven transport of species. Additionally, intensified surface reactions modify the local flow structure by altering shear forces and diffusion rates, which in turn affect the mass transport process within the system. The effect of the homogeneous strength parameter on concentration is illustrated in Figure 11(b). The concentration decreases with variations in the homogeneous strength parameter because stronger homogeneous reactions enhance the rate of reactant consumption within the bulk fluid. As reaction activity intensifies, more reactant molecules participate in chemical transformations before reaching the system boundaries, leading to a reduction in overall concentration. Additionally, increased reaction rates alter the fluid’s thermodynamic properties, such as density and diffusivity, which can further influence the transport and distribution of species. These combined effects modify mass transfer dynamics, leading to a lower concentration profile, as shown in Figure 11(b). These effects influence overall heat and mass transport, crucial for optimizing system performance. These mechanisms play an important role in various applications particularly, chemical reactors, energy systems, catalytic converters, cooling of high-power systems, and heat exchangers, where both the thermal and concentration profiles are influenced by the reaction kinetics.

Figure 11

Influence of (a) heterogeneous and (b) homogeneous strengths on concentration.

Figure 12(a) and (b) represents the variation in the drag force and heat transfer rate for different parameters, respectively. Figure 12(a) depicts the variation in skin friction with different values of the inverse couple-stress parameter and Brinkman number. The drag force decreases with an increasing inverse couple-stress parameter. This occurs because the inverse couple-stress parameter characterizes the influence of the fluids’ microstructural behaviour, particularly the resistance to shear deformation in non-Newtonian fluids. As the inverse couple-stress parameter increases, the fluid becomes more resistant to the applied shear, which leads to a reduction in the velocity gradient near the wall, thereby decreasing the drag force. In practical applications, such as lubrication or microfluidic flow, this behaviour can be useful for controlling frictional forces and optimizing flow efficiency by adjusting the fluids rheological properties. Figure 12(b) shows that the heat transfer rate is affected by variations in the variable thermal conductivity and heat source/sink. As variable thermal conductivity increases, heat spreads more efficiently, leading to a more uniform temperature distribution and a reduced temperature gradient, which weakens the driving force for heat transfer. Likewise, the presence of a heat source/sink alters the system’s thermal equilibrium, disrupting the natural heat flow and reducing the efficiency of heat dissipation. Consequently, both factors contribute to a decline in overall heat transfer effectiveness.

Figure 12

Variation in skin friction coefficient and Nusselt number for different physical parameters. (a) Brinkman number and (b) heat source/sink strength.

5 Conclusion

This study examines the influence of heat sources/sinks, variable thermal conductivity, radiation, and catalytic chemical reactions (H–H) on the convective flow of couple-stress Casson fluid in a vertical porous microchannel. The key findings of this research are summarised as follows:

Velocity behaviour

• The Casson parameter and porous medium parameter reduce fluid velocity, indicating increased flow resistance within the microchannel.

• Higher values of the inverse couple-stress parameter, Brinkman number, and Grashof number enhance velocity due to stronger viscous dissipation and buoyancy forces. Increased thermal effects and reduced resistance facilitate smoother fluid motion, promoting convective transport

  Thermal field

• The presence of a heat source/sink and a higher Casson parameter promote heat retention, increasing the thermal field.

• Variable thermal conductivity, porous medium parameter, inverse couple-stress parameter, and radiation improve heat dissipation, leading to a decrease in the thermal field by facilitating greater heat conduction and transfer away from the system.

  Entropy generation

• Entropy production is enhanced with increasing Biot number, variable thermal conductivity, radiation, heat source/sink, Grashof number, and Brinkman number, suggesting improved heat transfer efficiency and thermodynamic irreversibilities.

• Porous medium parameter, inverse couple-stress parameter, and Casson parameter have minimal effects on entropy generation.

  Bejan number behaviour

• The Bejan number exhibits dual behaviour, reflecting the interaction between thermal and viscous effects. Its variation depends on system parameters, flow regime, and material properties.

  Concentration Profiles

• Both homogeneous and heterogeneous reactions reduce concentration levels, as stronger reactions accelerate reactant consumption, decreasing overall reactant concentration.

  Heat transfer and drag force

• The heat transfer rate decreases with increasing heat source/sink strength ( H ss ) and variable thermal conductivity ( ϵ ) , as enhanced thermal diffusion reduces temperature gradients, weakening the heat transfer driving force.

• The drag force decreases with a higher Brinkman number ( Br ) and couple-stress parameter ( S ) due to reduced fluid resistance and increased momentum diffusion, allowing smoother fluid motion.