A numerical analysis of heat and mass transfer in water-based hybrid nanofluid flow containing copper and alumina nanoparticles over an extending sheet

Nomenclature

u , v

velocity components

x , y

coordinates

u w ( x ) = a x

stretching velocity

a

constant

χ

micro-rotation velocity

T

temperature

C

Concentration

κ

vortex viscosity

μ

dynamic viscosity

ρ

density

j

micro-inertial factor

k

thermal conductivity

C p

specific heat

D B , D T

Brownian and thermophoresis diffusion coefficients

Q t

coefficient of heat source

ν f

kinematic viscosity

k r

coefficient of chemical reaction

E a

coefficient of activation energy

h f

heat transfer rate

n

micro-gyration constraint

ϕ 1 , ϕ 2

nanoparticles volume fractions

S 1 , S 2

solid nanoparticles

1 Introduction

Viscous dissipation describes the conversion of kinetic to thermal energy within a fluid with the action of viscous forces. Lund et al. [1] explored that this process occurs when fluid layers with different velocities slide past one another, generating frictional forces that result in the production of heat. Idris et al. [2] proved that the dissipation of energy due to viscosity becomes particularly significant in flows where high-velocity gradients are present, such as in boundary layers or turbulent flows. In fluid dynamics, when viscous dissipation is considered in the energy equation, it directly affects the temperature distribution within the fluid, as noticed by Hafed et al. [3]. Specifically, in cases where fluid flow involves viscous dissipation, the heat generated by viscous forces causes an upsurge in the fluid’s internal energy, thereby raising its temperature. The influence of viscous dissipation on thermal and velocity profiles is quite significant. Mahesh et al. [4] proved that in the case of velocity, viscous forces tend to retard fluid’s velocity, causing a more gradual velocity profile, especially near solid boundaries where no-slip conditions apply. This deceleration near the walls contrasts with a relatively faster-moving core, creating a steeper velocity gradient. Lone et al. [5] explored that in the case of temperature transportation, the heat produced by dissipation advances the fluid’s temperature, especially in regions with high-velocity gradients, such as near the walls. This results in a nonlinear temperature distribution, where the temperature rises near the wall due to the heat generated by friction, leading to a higher temperature gradient in these regions. Consequently, with the use of viscous dissipation, the overall thermal performance of the system can be significantly altered, with potential implications for heat transfer rates, thermal boundary layer development, and fluid stability, especially in systems like microchannels, where viscous effects are more pronounced due to the small scales involved [6,7]. Hussain et al. [8] analyzed three-dimensional (3D) radiative nanofluid flow with impressions of viscous dissipation and microorganisms. Razzaq et al. [9] examined in a non-similar manner the magneto-hydrodynamics (MHD) nanofluid flow on a porous surface subject to radiative and Joule heating effects.

Fluid flow with Brownian motion comprises the random association of microscopic particles mixed in a fluid, resulting from collisions with molecules of fluid. This process is mainly important in nanofluids, where nanoparticles are mixed in the base fluid to enhance their thermal features. Thabet et al. [10] showed that the continuous and erratic motion of these nanoparticles due to Brownian motion significantly influences the fluid’s thermal behavior. Lone et al. [11] highlighted that the Brownian motion causes a redistribution of heat, as the random motion of nanoparticles leads to enhanced thermal conductivity. This effect is especially prominent in cases where the particle size is very small, and the fluid’s viscosity is relatively low, allowing for greater particle mobility. The impact of Brownian motion on thermal distribution is multifaceted. It stimulates more unvarying thermal panels in a fluid by enhancing the dispersion of heat, especially in regions where temperature gradients exist, as noticed by Duguma et al. [12]. This leads to a reduction in localized hot spots, which is beneficial in applications like cooling systems, where uniform heat removal is desired. Yang et al. [13] established that the Brownian motion augments the thermal conductance of a fluid, allowing for more well-organized heat transfer. However, the extent of this impact is dependent on factors like particle concentration, size, and the base fluid’s properties. At high particle concentrations or in highly viscous fluids, the enhanced thermal conductance due to Brownian motion causes a more noticeable thermal layer at the borderline, affecting the overall heat transfer rate, as observed by Jalili et al. [14]. Additionally, the random nature of Brownian motion causes fluctuations in the local temperature field, potentially influencing the stability and predictability of thermal distribution in sensitive applications. Madhura and Babitha [15] debated computationally on MHD fluid flow with impacts of Brownian motion and proved that thermal profiles of fluid have amplified with growth in Brownian factor. Waqas et al. [16] have also determined the computational effects of Brownian motion on nanomaterial fluid flow. Kumar et al. [17] examined the impressions of Brownian numbers on unsteady fluid flow with flux constraints and varying fluid characteristics.

Fluid flow with thermophoresis describes the movement of particles within a fluid with the effects of the gradient in temperature. Thermophoresis causes particles to travel from regions of greater to zones of lower temperatures. This phenomenon is particularly significant in systems involving aerosols, colloidal suspensions, or nanofluids, where particles are subjected to a temperature gradient. Abbas et al. [18] explored that the driving force behind thermophoresis is the differential impact of thermal energy on particles, causing them to drift along the temperature gradient, which results in a redistribution of particles within the fluid. Dong et al. [19] proved that thermophoresis is highly dependent on particle size, fluid viscosity, and the magnitude of the temperature gradient, making it a critical factor in applications involving thermal management, filtration, and environmental control. Yasir et al. [20] studied that for temperature profiles, the thermophoresis itself does not directly alter the fluid’s temperature distribution, but it indirectly influences it by affecting particles’ concentration, which, in turn, modifies the fluid’s thermal properties. As particles accumulate in cooler regions, the local thermal conductance of fluid’s particles changes, potentially leading to a more complex and non-uniform temperature distribution, as noticed by Sharma et al. [21]. Regarding concentration profiles, thermophoresis creates a gradient in particle concentration, with a higher concentration of particles in cooler regions and a lower concentration in hotter areas, as noticed by Srilatha et al. [22]. This leads to a non-uniform distribution of particles within the fluid, which impacts various transport properties of the fluid. Karthik et al. [23] debated impressions of thermophoresis and thermal radiations on trihybrid nanofluid flow by inspecting mass and thermal transportations. Almeida et al. [24] studied MHD nanomaterial fluid flow using the impacts of thermophoresis and perceived that thermal profiles have augmented with growth in the thermophoresis factor. Kopp et al. [25] examined thermal transportations for trihybrid nanofluid flow on a contracting and elongating sheet. Alqahtani et al. [26] numerically evaluated hybrid nanofluid flow and its stability on a porous wedge using thermal flow simulations, providing insights into fluid behavior and heat transfer characteristics. Their study highlights the potential of hybrid nanofluids in enhancing thermal performance in porous systems under specific flow conditions. Khan et al. [27] studied computationally the Williamson magnetized nanoparticles flow on a shrinking/elongating subject to the influence of thermal and velocity slip constraints. Zeeshan et al. [28] computationally analyzed mass and thermal transport phenomena on a permeable elongating medium, considering the impacts of varying magnetic fields and heat source influence. Their study provides valuable insights into how magnetic effects and heat sources affect fluid flow and heat transfer, enhancing understanding of transport mechanisms in permeable media under dynamic conditions. Nisar et al. [29] conducted a mathematical analysis of magnetohydrodynamic radiative flow in Bingham nanofluids. Their study focused on fluid dynamics influenced by radiation and MHD forces, considering the non-Newtonian behavior of Bingham fluids combined with nanoparticles. Their research provides insights into heat and mass transfer mechanisms in industrial applications, enhancing understanding of such complex fluid systems. Abrar et al. [30] studied slip factor’s effects on micro-spinning bio-convective nanofluid flow on an elongating sheet by implementing the machine learning approach.

Fluid flow with Joule heating involves the production of heat inside the fluid due to the passage of an electric current through a conductive medium. This process is significant in applications where fluids are electrically conductive, like in electrolytes, plasma, or certain nanofluids. Rilwan et al. [31] noticed that the heat generated by Joule heating results in the fluid’s temperature upsurge, affecting both the thermal distribution and flow characteristics. Rafique et al. [32] examined that the intensity of Joule heating depends on factors like the electrical conductance of the fluid, the magnitude of the current, and the geometry of the system through which the current flows. The impact of Joule heating on thermal distribution is substantial, often leading to a non-uniform temperature field within the fluid. As the current passes through the fluid, regions with higher resistance or lower electrical conductance tend to generate more heat, resulting in localized temperature rises, as observed by Irfan et al. [33]. Waqas et al. [34] studied nanofluid flow on a stretchable sheet using the impressions of Joule heating and microorganisms. Khan et al. [35] discussed the impacts of Joule heating on nanomaterial fluid flow on a gyrating cylinder using dissipative effects. Rasool et al. [36] explored that fluid flows with Joule heating impacts generate thermal gradients, particularly near regions of high current density or in parts of the fluid where the electrical conductivity is variable. These temperature gradients, in turn, influence the fluid flow, as variations in temperature affect the fluid’s viscosity and density, potentially leading to convective currents that further modify the flow pattern. Khedher et al. [37] highlighted that the increased temperature due to Joule heating augments the rate of thermal transportation between the fluid and its surroundings, impacting the overall thermal management of the system. In applications like microfluidic devices, where precise control of temperature is crucial, Joule heating can be both an advantage and a challenge. It can be used deliberately to heat fluids in controlled environments, but it can also lead to undesirable temperature increases that affect the performance and stability of the system. Pasha et al. [38] inspected nonlinear mixed convective fluid flow on a surface using Joule heating effects. Nisar et al. [39] investigated the thermal performance of radiative and mixed convective peristaltic flow in Bingham nanofluids. Their work explored the collective effects of radiation and convection on non-Newtonian fluid flow, incorporating nanoparticles. Zeeshan et al. [40] discussed two-dimensional (2D) nanofluid flow on a permeable elongating sheet subject to nonlinear thermally radiative and slip conditions at the boundary. Nasir et al. [41] studied Casson MHD nanoparticles flow with the influence of thermal radiations and thermal sink/source. Zeeshan et al. [42] explored heat transfer enhancement in Maxwell fluid by integrating molybdenum disulfide and graphene nanoparticles into an engine oil base fluid. Their study analyzed the impact of isothermal wall temperature conditions, emphasizing the role of nanomaterials in improving thermal performance. Their research highlights the effectiveness of these nanoparticles in optimizing heat transfer properties, offering valuable insights for advanced thermal management in complex fluid systems. Abrar et al. [43] analyzed thermal flow in micro-polar hyperbolic tangent fluids using a hybrid nanofluid approach. Their study examined heat transfer characteristics, fluid dynamics, and the impact of nanoparticles on thermal conductivity. The findings highlighted improved thermal efficiency and flow stability, contributing to advancements in nanofluid applications and engineering. Abrar and Kosar [44] examined entropy generation for MHD fluid flow on a wedge using the effects of thermal radiations and viscous dissipation.

Thermal radiation is the release of electromagnetic waves from a body due to its temperature. The intensity and wavelength distribution of the radiation depend on the temperature of the emitting surface, with hotter objects emitting more intense and shorter-wavelength radiation. When thermal radiation interacts with fluid flow, the thermal exchange amid the fluid and its surroundings significantly alters the fluid’s thermal distributions, as highlighted by Rehman et al. [45]. Obalalu et al. [46] explored that fluid flow using thermal radiation, especially with high-temperature radiation, contributes to the energy balance of the system. This is particularly relevant in applications like combustion, atmospheric science, and high-temperature industrial processes, where radiation can dominate the heat transfer mechanism. Akbar et al. [47] highlighted that the impacts of thermal radiation on thermal profiles in fluid flow are complex and depend on factors like the fluid’s optical properties, the temperature of surroundings, and the nature of the flow. In thick fluids, where radiation is absorbed and re-emitted multiple times, the thermal radiation leads to additional uniform temperature panels of fluid, smoothing out temperature gradients, as noticed by Wahid et al. [48]. However, in thin fluids, where radiation passes through with little absorption, the impact on the temperature profile might be more localized, affecting only the regions near the radiating surfaces. Abrar [49] investigated the thermal behavior of Casson hybrid nanoparticle flow over a permeable surface using the Cattaneo-Christov heat flux model. Their study explored the heat transfer characteristics, fluid stability, and the impact of hybrid nanoparticles, highlighting improvements in thermal efficiency and advanced heat conduction mechanisms for engineering applications.

This study introduces a novel investigation of 2D micropolar flow over an extending sheet using water-based hybrid nanofluid flow with magnetic effects. Unlike previous studies, our research comprehensively considers the collective effects of heat source, thermal radiation, viscous dissipation, thermophoresis, Joule heating, Brownian motion, and activation energy on fluid dynamics. The use of hybrid nanofluids over traditional nanofluids offers enhanced friction forces and heat transfer rates, providing paper insights into the optimization of fluid flow systems in engineering applications. The findings not only validate existing models but also expand the idea of the influence of micropolar parameters and micro-gyration on the micro-rotational velocity and skin friction force, making this analysis a significant advancement in the field of fluid mechanics.

2 Problem formulation

Let us assume the 2D micropolar flow of a water-based hybrid nanofluid comprising alumina and copper nanoparticles over an elongating surface. The velocity component u is chosen along the x -direction, and the velocity component v is chosen along the y -direction. The sheet is stretched with velocity u w ( x ) = a x , where a is a positive constant. The surface of the sheet is retained hot by assuming a hot functioning fluid that has heat transfer h f and temperature T f . The molar concentration of nanoparticles is denoted by C , and the zero-mass flux condition eliminates the surface concentration (i.e. C w = 0 ) and so does the mass transfer rate at the sheet surface. The temperature and molar concentration at the free stream are symbolized by T ∞ and C ∞ . A magnetic field having strength B 0 is assumed along the y -direction. In view of the above assumptions, the main equations can be written as [50,51] (Figure 1):

Figure 1

Geometrical view of flow.

with conditions at the boundary

The thermophysical relations are depicted as follows, and their experimental values are described in Table 1:

The variables used for transformation are described as follows:

Making use of equation (8) we have from the above

with boundary conditions:

Here

The quantities of interest are defined as follows:

where C f = Re x C f x and Nu = Nu x Re x .

Next, the solution method is explained in the following.

3 Numerical solution

We resolve the modeled ordinary differential equations (ODEs) using the numerical approach referred to as the shooting technique. The MATLAB 2020a software executes the built-in bvp4c function. The method adjusts the mesh dynamically to ensure accuracy, refining the points as needed. Users define the differential equations, boundary conditions, and an initial guess for the solution. The convergence of bvp4c is guaranteed under smooth problem conditions due to its underlying fourth-order method, which ensures that the error decreases as the step size is refined. The solver uses an adaptive mesh refinement process to control error estimates, ensuring that the solution converges to the desired accuracy specified by the user. This reliability makes bvp4c a robust choice for solving boundary value problems. Since first-order differential equations can be solved using the shooting methodology, the leading equations are altered to first-order ODEs as follows:

Using equation (16), the dimensionless ODEs are transformed as

with boundary conditions

The applied technique can solve both linear and nonlinear ODEs easily. The users can define the error tolerance by their own choice. The users can find the solution of ODEs much faster than other numerical and analytical techniques.

4 Validation

Our results are validated with the published data as depicted in Table 2. The results of − θ ′ ( 0 ) for varying values of Pr and other factors are ignored (i.e., Nb = 0.0 , Ec = 0.0 , M = 0.0 , Nt = 0.0 , Q = 0.0 , Rd = 0.0 , ϕ 1 = 0.0 , and ϕ 2 = 0.0 ). Furthermore, the convective condition is replaced by constant surface temperature to provide a proper comparison. This adjustment is made in order to compare the published results, which are based on calculations made for a constant wall temperature, with the present results. We may thus conclude from this table that there is a fair degree of consistency between the present and published results, validating the current model and code execution.

5 Discussion of results

This section contracts with the physical interpretation of embedded factors on the hybrid nanofluid flow profiles, which include friction force at the surface, heat transfer rate, velocity distribution, micro-rotation profile, thermal distribution, and molar concentration profile. The data of these results are given in Tables 3 and 4 and Figures 2–16. Table 3 shows the variations in C f via M , K , and n . Here, we perceive that greater M enhances C f while higher K and n decline C f . As M upsurges, the opposing Lorentz force also augments, leading to a higher resistance to the fluid motion. This resistance acts as a friction force which leads to the enhancement of the overall surface drag force. Table 4 depicts the variations in Nu via Nt , Q , Bi T , Ec , and M . From this table, we can see that greater Nt reduces Nu , whereas higher Q , Bi T , Ec , and M enhances Nu . The thermophoresis factor reduces the Nusselt number by altering the particle distribution within the fluid flow. For higher Nt , the temperature gradient at the wall reduces which results in a reduction in the rate of heat transfer. Furthermore, the rate of reduction is greater in hybrid nanofluid flow in comparison to nanofluid flow. As we increase the heat source factor, it enhances the local temperature of the fluid flow. It produces a steeper temperature gradient between the surrounding cooler regions and heat fluid regions and as a result, the rate of heat transfer is enhanced. As the thermal Biot number enhances, the internal resistance in a fluid flow system becomes significant, resulting in a higher rate of heat transfer. Finally, the greater magnetic factor enhances the rate of heat transfer as greater magnetic parameter affects the ionized particles which enhance the friction among nanoparticles. This enhanced friction among nanoparticles enhances the rate of heat transfer. Figures 2 and 3 show the variations in velocity ( f ′ ( η ) ) and temperature ( θ ( η ) ) profiles via magnetic factor ( M ) . Here, we observe that higher M lessens f ′ ( η ) while enhances θ ( η ) . The reduction in f ′ ( η ) due to the presence of magnetic factor is attributed to the MHD effect. An electromagnetic force called Lorentz force is generated in fluid flow in the existence of magnetic factor. This force acts in a perpendicular direction to the flow of fluid. The Lorentz force creates a drag force that opposes flow and consequently declines the velocity of fluid. The greater M , the greater Lorentz force, leading to the more pronounced deceleration of the hybrid nanofluid velocity. This effect is important in applications such as electromagnetic flow control, magnetic drug targeting, etc. On the contrary, the reduction in θ ( η ) is closely related to the reduction in velocity distribution caused by the magnetic factor. The hybrid fluid’s particles have more time to remain nearby the heated surface as a result of the opposing Lorentz force decreasing the fluid’s velocity and enhancing the convective transmission of heat. The fluid’s temperature rises as a result of the longer residence time, which permits more heat transfer between the fluid and the surface. Additionally, the dissipative impacts of electromagnetic force can lead to viscous heating, where the mechanical energy of the hybrid nanofluid flow is converted into thermal energy which results in a higher thermal distribution. This phenomenon is significant in cooling systems in electronic devices or nuclear reactions where increased thermal performance is especially required to ensure efficient temperature regulation and heat dissipation. Figures 4 and 5 show the variations in velocity ( f ′ ( η ) ) and micro-rotation ( G ( η ) ) profiles via micropolar factor ( K ) . From these figures, we observe that the greater K enhances f ′ ( η ) while reduces G ( η ) . This observation highlights the role of vortex viscosity in enhancing the velocity profile of micropolar hybrid nanofluid. Vortex viscosity results from the fluid’s microstructural elements, like nanoparticles, interacting with the properties of the base fluid and influencing its resistance to flow. In micropolar fluids, these interactions give rise to the creation of micro-rotational inertial moments, in which the microelements rotate. These micro-rotational effects are crucial because they transform a portion of the fluid’s energy, minimizing the energy dissipated as viscous drag. Consequently, the fluid faces less resistance, increasing its total velocity. This phenomenon exemplifies the intricate dynamics of micropolar nanofluids, where there is a fine balance between translational and rotational motion, and the induced micro-rotational effects help to improve the fluid’s velocity profile. Therefore, the greater K enhances f ′ ( η ) while reduces G ( η ) . Figures 6 and 7 portray variations in ( f ′ ( η ) ) and micro-rotation ( G ( η ) ) profiles via micro-gyration constraint ( n ) . Here, we observe that higher n diminishes f ′ ( η ) and enhances G ( η ) . Here, it is important to mention that n = 1 represents turbulent flow, while n = 0.5 specifies weak concentration, and n = 0 corresponds to strong concentration. The parameter n affects the dynamics of micropolar hybrid nanofluid flow by imposing restrictions on the micro-rotation of the hybrid nanofluid particles. This constraint declines the velocity profile of hybrid nanofluid flow due to the upsurge in coupling between the micro-rotational motion and translational flow. More energy is diverted into micro-rotational modes when the micro-gyration constraint is applied, leading to the increased rotational activity of the hybrid nanofluid particles. Consequently, less energy is available for the translational motion of the fluid which results in decreasing the overall velocity distribution f ′ ( η ) of the hybrid nanofluid flow. Simultaneously, the micro-rotational velocity is enhanced due to the fact that the micro-gyration constraint promotes greater rotational motion within the fluid’s microstructure. This increased rotational activity leads to more pronounced micro-rotation of the hybrid nanofluid elements, thereby increasing the micro-rotational velocity next to the stretching surface. The relationship between these microstructural rotations and the stretching surface provides a complicated flow dynamic where the energy allocation between translational and rotational modes is changed, lowering the linear flow velocity while magnifying the rotational motion. Thus, the greater n reduces f ′ ( η ) while enhances G ( η ) . Figure 8 shows the variation in thermal distribution ( θ ( η ) ) via heat source factor ( Q ) . From this figure, we observe that the greater Q enhances θ ( η ) . With the enhancing Q , the system gets additional thermal energy which results in higher thermal distribution. This additional heat energy raises the temperature of the hybrid nanofluid near the stretching surface and causes a more significant temperature gradient across the fluid flow domain. The presence of solid nanoparticles in hybrid nanofluid, which have higher thermal conductivity, further enhances the additional energy throughout the nanofluid flow domain, resulting in higher thermal distribution. This phenomenon is particularly important in cooling systems, chemical processing, thermal management in electronic devices, and enhanced thermal transfer. Figure 9 shows the variation in thermal distribution ( θ ( η ) ) via Eckert number (Ec) . From this figure, we observe that the greater E c enhances θ ( η ) . The greater Ec means that more heat is produced from the friction between the hybrid nanofluid layers as they move each other. This generated heat enhances the hybrid nanofluid temperature and as a result the thermal distribution increases. This effect is particularly pronounced in nanofluid flows with greater velocities or strong shear forces where viscous dissipation can significantly contribute to the overall thermal energy of the nanofluid flow. Consequently, the hybrid nanofluid temperature rises, allowing for a more even distribution of temperature over the fluid domain particularly in the vicinity of the stretching surface. This is essential for applications like heat exchangers, industrial cooling procedures, thermal management systems, and others where regulating the temperature distribution is critical. Figure 10 shows the variation in thermal distribution ( θ ( η ) ) via Biot number ( Bi T ) . From this figure, we observe that the greater Bi T enhances θ ( η ) . The thermal energy from the surface is more effectively transferred to the hybrid nanofluid leading to a higher fluid temperature near the surface and a more noticeable thermal gradient. This results in an increased temperature distribution as the hybrid nanofluid closer to the heated surface becomes significantly warmer and the heat diffuses more effectively throughout the hybrid nanofluid. In real-world applications, managing the Biot number helps in maximizing thermal management. Examples include raising heat exchanger performance, cooling system efficiency, and establishing uniform distribution of temperature in heat transfer operations. Therefore, a higher Biot number enhances the hybrid nanofluid ability to absorb and transport heat energy from the surface, improving the total thermal distribution. Figures 11 and 12 show the variations in thermal ( θ ( η ) ) and molar concentration ( φ ( η ) ) distributions via thermophoresis factor (Nt) . From these figures, we observe that the greater Nt enhance both θ ( η ) and φ ( η ) . Both profiles enhance due to the effects of thermophoretic force on the particle’s movement. A phenomenon in which the particles in hybrid nanofluid flow move from a higher temperature region to a lower temperature region is called thermophoresis. When Nt is enhanced, the force driving the particles away from hot regions toward colder regions becomes stronger. As a result, the hybrid nanofluid flow particles migrate away from the hot surface which causes a buildup of particles in cooler regions and creates a steeper concentration gradient. This leads to an increment in the molar concentration distribution, as there is a stronger accumulation of particles in areas away from the zone having higher temperatures. Also, the particles move from hot regions then they carry heat away which effectively enhances the temperature gradient and distributes the thermal energy more evenly throughout the hybrid nanofluid. Consequently, the temperature profile becomes more prominent, with higher temperature difference between the regions nearer the heated surface and those farther away. To sum up, a higher thermophoresis factor increases the gradients in temperature and molar concentration by encouraging particle migration from heated to colder areas, which results in improved temperature and concentration distributions in the hybrid nanofluid. Figures 13 and 14 show the variations in thermal ( θ ( η ) ) and molar concentration ( φ ( η ) ) distributions via Brownian motion factor ( Nb ) . From these figures, we observe that the greater Nb enhances θ ( η ) while reduces φ ( η ) . When Nb increases, the nanoparticles experience more frequent and energetic collisions which enhances the kinetic energy. The increased movement assists better thermal energy transfer throughout the hybrid nanofluid as the nanoparticles are effective sources of heat carriers. The enhanced motion extends the thermal energy more quickly across different regions of the hybrid nanofluid especially from hotter to colder regions. As a result of this, the thermal distribution becomes more enhanced. On the contrary, the increasing Nb enhances the nanoparticles diffusion which leads to a decrease in the concentration gradient. With higher Nb , the nanoparticles disperse more uniformly throughout the hybrid nanofluid which reduces the concentration differences. This dispersion causes the nanoparticles to spread out more quickly which minimizes areas of higher concentrations and leads to a more homogenized mixture. The concentration gradient diminishes which results in reduction in molar concentration distribution. Therefore, the higher Nb enhances θ ( η ) while reduces φ ( η ) . Figure 15 shows the variation in molar concentration distribution ( φ ( η ) ) via chemical reaction factor ( K r ) . From this figure, we observe that the greater K r reduces φ ( η ) . This indicates that the reactant particles, such as solutes or nanoparticles, are consumed more quickly during the reaction process in the context of a reactive flow. The concentration of reactants lowers more quickly as the reaction rate rises because they are transformed into the products more quickly. As a result, the reactant species’ total concentration declines as the fluid’s molar concentration profile of the reactants diminishes. Over time, the reactants become less in quantity in the areas where the reaction is most active due to this decrease. Therefore, the greater K r reduces φ ( η ) . Figure 16 shows the variation in molar concentration distribution ( φ ( η ) ) via activation factor ( E ) . From this figure, we observe that the greater E enhances φ ( η ) . The greater E means that a greater amount of energy is needed for reactants to reach the transition state and subsequently from products. Fewer molecules, especially at lower temperatures, have the energy required to cross this barrier when the activation energy is large. Because fewer molecules of the reactant may participate in the reaction, the chemical reaction progresses more slowly as a result. The reactants can sustain larger concentrations for a longer amount of time because of this slower reaction rate, which results in a reduced rate of absorption. As fewer molecules are transformed into products, greater activation energy essentially slows down the reaction process and improves the molar concentration profile of the reactants. Therefore, the greater E enhances φ ( η ) .

Figure 2

Impact of M on f ′ ( η ) .

Figure 3

Impact of M on θ ( η ) .

Figure 4

Impact of K on f ′ ( η ) .

Figure 5

Impact of K on G ( η ) .

Figure 6

Impact of n on f ′ ( η ) .

Figure 7

Impact of n on G ( η ) .

Figure 8

Impact of Q on θ ( η ) .

Figure 9

Impact of Ec on θ ( η ) .

Figure 10

Impact of Bi T on θ ( η ) .

Figure 11

Impact of Nt on θ ( η ) .

Figure 12

Impact of Nt on φ ( η ) .

Figure 13

Impact of Nb on θ ( η ) .

Figure 14

Impact of Nb on φ ( η ) .

Figure 15

Impact of K r on φ ( η ) .

Figure 16

Impact of E on φ ( η ) .

6 Conclusions

In this article, we have investigated the 2D micropolar flow of water-based hybrid nanofluid containing alumina and copper nanoparticles over an extending sheet. The magnetic field impact is taken into consideration to investigate its influence on water-based hybrid nanofluid containing alumina and copper nanoparticles. Furthermore, the influences of viscous dissipation, Brownian motion, thermophoresis, Joule heating, chemical reaction, thermal radiation, heat source, and activation energy are taken into consideration. A numerical solution to the considered problem by means of the shooting method is determined. The following key points are found during this analysis:

1. The higher values of magnetic factor enhance skin friction, while the higher values of micropolar parameter and micro-gyration constraint reduce skin friction.

2. The higher values of heat source, thermal Biot number, Eckert number, and magnetic factor enhance the rate of heat transfer.

3. The rates of drag force and heat transfer are greater for hybrid nanofluid flow when compared to nanofluid flow.

4. The greater magnetic factor and micro-gyration constraint reduce the velocity profiles, and the greater micropolar factor enhances the velocity profiles.

5. The greater micropolar factor reduces the micro-rotational velocity profiles, and the greater micro-gyration enhances the micro-rotational velocity profiles.

6. The greater magnetic factor, thermophoresis factor, Brownian motion factor, Eckert number, heat source, and thermal Biot number enhance the thermal distribution.

7. The greater thermophoresis factor and activation energy factor enhance the molar concentration distribution, and the greater Brownian motion factor and chemical reaction factor reduce the molar concentration distribution.

7 Future scope and limitations

The present problem focuses on the 2D micropolar flow of a hybrid nanofluid under various physical effects. However, certain research gaps and limitations are acknowledged in this study:

• The analysis assumes an extending sheet with 2D flow, which simplifies real-world geometries. Future work can explore more complex geometries such as 3D flows, flows over cylindrical surfaces, etc.

• Other nanoparticles in water-based nanofluid can extend the hybrid combination, or base fluids can provide more insights into the behavior of hybrid nanofluids.

• The present analysis focuses on steady and laminar flow. Incorporating unsteady phenomena could further increase the applicability of the model.

• The present analysis is validated with previously published results, and direct experimental validation is not included in this work. Future analyses can combine the computational and experimental approaches for more robust validation.

These limitations indicate opportunities for future research to extend the current results and explore the modeled problem in greater detail.