Introduction

Micropolar fluids, which incorporate microstructure implications such as particle rotation and deformation, are used in a wide range of thermal transmission applications. They serve as effective models for nanoliquids, which promote heat transfer in electronics cooling and thermal management. Micropolar fluid models in heat exchangers provide insights into heat transmission augmentation strategies that take advantage of novel channels created by particle rotation. Furthermore, in the petroleum industry, micropolar liquid models are used to develop enhanced oil recovery technologies. Eringen [1] developed the idea of micropolar fluids, which define the stream and changes of individual liquid constituents within a specific region. Microstructured liquids have recently gained popularity in a variety of sectors, with mechanical engineering, nanotechnology, and bioscience. Mohapatra et al. [2] scrutinised the manner in which mixed convection influences magnetohydrodynamic (MHD) flowing towards an upwardly permeable interface radiating effects. Das [3] inspected the transmission of thermal and mass attributes of a polar liquid flowing along a widening surface at rotational circumstances, with a continuous heat source, while considering the implementation of chemical reaction along with linear radiation and hydromagnetic free convection. The basic equations are addressed by the perturbation scheme, and the associations of numerous factors are reviewed utilising tabular and visualisations. El-Aziz [4] explored the unstable mixed flow phenomenon of a viscous micropolar liquid across a deformable sheet within incompressible convection circumstances, considering dissipation and buoyancy effects. Furthermore, the study thoroughly investigates the imposition of important factors on flow characteristics and heat transmission. Panda et al. [5] studied the imposition of an magnetic field, linear radiation, activation energy and porosity on a circular segment. The findings are presented and analysed using tables and graphs, providing insights into many aspects of the inquiry.

Balla and Naikoti [6] quantitatively studied the unstable hydromagnetic convective movement caused by mass and thermal transference including the interaction of porosity, electrical conductivity, chemical reactions, and linear radiation. Salawu and Dada [7] inspected the transference of heat in a micropolar liquid moving on a deformable surface, which includes alterations in viscosity and conductivity caused by temperature fluctuations, as well as the joint interaction of radiation, porosity, electric, and magnetic flux. Ojjela et al. [8] deliberated how Hall and slip impacts as well as mass and thermal transference, affect the free convective movement of a micropolar liquid, taking into account Soret and Dufour effects, porosity, and chemical reaction. They used the quasilinearization strategy to numerically simulate the basic nonlinear equations and graphically demonstrate the impact of geometric factors. Reddy et al. [9] deliberated the mass and thermal transference of a magnetohydrodynamic fluid in a porous matrix within an inclined, asymmetric channel, taking into account the wave motion of the fluid stress. They used graphics to clarify their findings. Pal and Biswas [10] inspected the thermal along with mass transference factors of a micropolar liquid flowing over a mobile porous surface, taking into account factors, such as chemical reaction, radiative heat, dissipation, and the effects of magnetic and electric fields. They conducted a thorough analysis of how chemical reaction and micro-rotational velocity influence fluid dynamics. Waqas et al. [11] deliberated the magnetised joint flow of a micropolar liquid over a deformable sheet under convective conditions, accounting for viscous dissipation and Joule heating. Their research entailed using the “Homotopy Analysis Method” to resolve the basic dimensionless equations, and they visually portrayed the effects of various control factors on fluid dynamics using graphical representations.

Hayat et al. [12] conducted a computational investigation on the thermal and mass transmission parameters of convective boundary-layer movement across a nonlinear curved stretchy material, considering chemical reaction effects. They used a shooting strategy to approximate the modified nonlinear equations and ran analysis on numerous variables.

Baag et al. [13] conducted a theoretical study on thermal transference features in magnetised movement of convective 3D liquid along a flexible surface, accounting for dissipation. Sheri and Shamshuddin [14] studied the behaviour of micropolar fluid in a hydromagnetic boundary layer across a semi-infinite upward plate, considering the effects of Hall current, chemical reaction, and viscous dissipation. They used the Boussinesq approximation to renovate the basic PDEs, followed by the variational finite element approach for numerical solution. The study included tabulated changes in flow characteristics for various thermophysical factors, along with thorough explanations. Patel and Singh [15] considered the joint interaction of viscous dissipation, Joule heating, thermophoresis, Brownian motion, thermal radiation, and chemical reactions on thermal transference and mass transference of mixed convection micropolar fluid flow across a stretching surface under a uniform magnetic field. They used similarity conversion to turn the basic equations into dimensionless forms and then solved them using the “Homotopy Analysis Method”, which made it easier to interpret the results.

Goud [16] inspected the incorporation of thermal transference on the steady magnetohydrodynamic (MHD) flow of micropolar fluid along a vertically inclined semi-infinite surface exposed to suction/injection. The renovated basic equations were estimated via the RK fourth-order method joint with shooting methodology. Graphs and tables were implemented to highlight the features of nondimensional factors. Pattnaik et al. [17] deliberates the interaction of buoyancy variables including radiation on magnetism induced micropolar nanofluid movement across a shrinking surface. The impacts of various parameters were tallied and visualised via graphs. Khan et al. [18] anticipated the insertion of bioconvective microstructured nanoliquid movement on a transient needle carrying gyrotactic microorganisms. They utilised the “Homotopy Analysis method” (HAM) to analyse the altered dimensionless basic equations and revealed that the outcomes agreed with existing solutions. Sharma et al. [19] studied the insertion of magnetisation together with thermal radiation across an expanding surface. They deliberate the “Runge–Kutta method” and the shooting scheme to scrutinise the outcomes, which were then examined using graphs and tables. Mishra et al. et al. [20] studied the thermal transmission along a tapered channel, considering the mutual effect of radiation and magnetic field effects. Kumar et al. [21] deliberate the convective movement of a rotating magnetohydrodynamic (MHD) microstructured liquid through an infinity stretchable plate to determine the joint interaction of radiation, Soret, and Dufour. They used the Galerkin method for numerical computation to examine the influence of various regulating elements on fluid flow. The outcomes were visually presented and compared to earlier investigations. Baag et al. [22] presented the transient flow of a viscous liquid, using variable plate condition on an expanding/ contracting sheet. Saraswathy et al. [23] examined micropolar fluid flow at certain boundary conditions known as Newtonian Heating and Prescribed Surface Temperature. The study took into account unsteady viscosity, heat conductivity, and activation energy, with numerical solutions derived using the Runge–Kutta-Fehlberg technique. Graphical representations of the importance of several nondimensional characteristics were offered. Shah et al. [24] explored the incompressible viscous flow and thermal transference of magnetohydrodynamic (MHD) micropolar fluids in a two-dimensional convective flow across a porous sheet. Computational study of the altered nonlinear equations (ODEs) was carried out using MATLAB, with discussions concentrating on the significance of the radiation parameter, among other parameters, as well as system stability. Sharma and Mishra [25] inspected the imposition of thermal radiation and mass transmission on the continuous convective movement of a microstructure fluid along a vertically expanded sheet, incorporating the Soret effect and magnetic field effects. They utilised the “Runge–Kutta-Fehlberg” scheme to resolve the basic standard equations. Mabood et al. [26] developed a micropolar fluid model to mimic reactive movement along a randomly moving plate, which encompassed the Soret effect, Brownian motion, and chemical reactions. The data were analysed via the “Homotopy Analysis Method”, with several nondimensional features depicted as tables and graphs to clarify their responsibilities. Pal et al. [27] developed a micropolar liquid convective model to study heat transport, which incorporated Ohmic dissipation and transverse magnetic flux through a composite material. The basic nonlinear equations were estimated via the “Runge–Kutta-Fehlberg method”, highlighting the importance of basic parameters. Saidulu and Reddy [28] inspected thermal and mass transmission in magnetic-induced micropolar fluid flow over a deformable surface, accounting for viscous dissipation as well as chemical reactions. The adjusted equations were computed via MATLAB using the bvp4c approach, demonstrating how basic factors affect diverse fluid flow patterns.

Research questions

  • What effects do bilateral interactions have on the properties of mass and heat transport in a micropolar fluid flow over a stretching sheet?

  • In the presence of Joule heating, how does thermal radiation affect the micropolar fluid flow's overall thermal behaviour?

  • Is it possible to forecast heat and mass transfer in real-world applications employing micropolar fluids more accurately by incorporating thermal radiation and viscous dissipation into the mathematical model?

  • When dissipation, radiation, and Joule heating are taken into account, what part does the stretching rate of the sheet play in the convective heat and mass transfer processes in a micropolar fluid flow?

Novelty of the proposed study

  • Implementing the results in real-world situations to increase the study's applicability to issues with radiative heat transfer in micropolar fluids.

  • Investigation of free convective hydromagnetic micropolar fluid flow across a stretching sheet with the inclusion of radiative heat transmission.

  • Analysis of the bilateral interactions in the micropolar fluid under particular circumstances.

  • The incorporation of heat radiation, Joule heating, and viscous dissipation into the fluid flow system mathematical model.

Problem description

Consider a continuous laminar micropolar fluid flow induced by a moving sheet as displays in Fig. 1. Let u and v denote the velocity components along the x- and y-axes, respectively. We suppose that the speed of a point on the sheet is inversely related to its distance from the slit. We also discard any influence from Hall impact, and assume no induced magnetic field due to the weak magnetic Reynolds number.

Fig. 1
figure 1

Flow geometry

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The basic equations for the micropolar fluid are described as

$$\partial_{{\text{x}}} u + \partial_{{\text{y}}} v = 0$$
(1)
$$u\partial_{{\text{x}}} u + v\partial_{{\text{y}}} u = \left( {\upsilon + \frac{k}{\rho }} \right)\partial_{{{\text{yy}}}} u + \frac{k}{\rho }\partial_{{\text{y}}} N - \frac{{\sigma B_{0}^{2} u}}{\rho } + g\beta_{{\text{T}}} \left( {T - T_{\infty } } \right) + g\beta_{{\text{c}}} \left( {C - C_{\infty } } \right)$$
(2)
$$u\partial_{{\text{x}}} N + v\partial_{{\text{y}}} N = \frac{\gamma }{j\rho }\partial_{{{\text{yy}}}} N - \frac{k}{j\rho }\left( {2N + \partial_{{\text{y}}} u} \right)$$
(3)
$$u\partial_{{\text{x}}} T + v\partial_{{\text{y}}} T = \frac{{k_{{\text{f}}} }}{{\rho C_{{\text{p}}} }}\partial_{{{\text{yy}}}} T + \frac{{\left( {\mu + k} \right)}}{{\rho C_{{\text{p}}} }}\left( {\partial_{{\text{y}}} u} \right)^{2} + \frac{{\sigma B_{0}^{2} u^{2} }}{{\rho C_{{\text{p}}} }} - \frac{1}{{\rho C_{{\text{p}}} }}\frac{{\partial q_{{\text{r}}} }}{\partial y} + \frac{{Q_{0} (T - T_{\infty } )}}{{\rho C_{{\text{p}}} }}$$
(4)
$$u\partial_{{\text{x}}} C + v\partial_{{\text{y}}} C = D\partial_{{{\text{yy}}}} C - Kr^{2} \left( {C - C_{\infty } } \right)\left( {\frac{T}{{T_{\infty } }}} \right)^{{\text{m}}} \exp \left( {\frac{{ - E_{{\text{a}}} }}{{k_{{\text{b}}} T}}} \right)$$
(5)

As well as boundary restrictions

$$\left. \begin{gathered} As\,y = 0:\left[ {u,v,N,T,C} \right] = \left[ {u_{{\text{w}}} = bx,0, - s\partial_{{\text{y}}} u,T_{{\text{w}}} ,C_{{\text{w}}} } \right] \hfill \\ As\,y \to \infty :\left[ {u,N,T,C} \right] = \left[ {0,0,T_{\infty } ,C_{\infty } \,} \right] \hfill \\ \end{gathered} \right\}$$
(6)

where \(\psi\) function is described by

$$u = \partial_{{\text{y}}} \psi ,\,\,v = - \partial_{{\text{x}}} \psi$$
(7)

The similarity conversion variables are as follows;

$$\left[ {\eta ,u,\phi \left( \eta \right),v,N,\theta (\eta )} \right] = \left[ {\left( {\frac{b}{\upsilon }} \right)^{0.5} y,bxf^{\prime}\left( \eta \right),\frac{{C - C_{\infty } }}{{C_{{\text{w}}} - C_{\infty } }}, - (b\upsilon )^{0.5} f\left( \eta \right),\sqrt {\frac{{b^{3} }}{\upsilon }} xG\left( \eta \right),\frac{{T - T_{\infty } }}{{T_{{\text{w}}} - T_{\infty } }}} \right]$$
(8)

Replacing Eq. (7) from Eqs. (2) to (5) and then with the conversion mentioned above

$$\left( {1 + K} \right)f^{\prime\prime\prime} + ff^{\prime\prime} - \left( {f^{\prime}} \right)^{2} + KG^{\prime} - Mf^{\prime} + \lambda_{1} \theta + \lambda_{2} \phi = 0$$
(9)
$$\left( {1 + \frac{K}{2}} \right)G^{\prime\prime} + fG^{\prime} - f^{\prime}G - K\left( {2G + f^{\prime\prime}} \right) = 0$$
(10)
$$\left( {1 + Rd} \right)\theta^{\prime\prime} + \Pr f\theta^{\prime} + \left( {1 + K} \right)\Pr Ec\left( {f^{\prime\prime}} \right)^{2} + \Pr EcM\left( {f^{\prime}} \right)^{2} + \Pr Q\theta = 0$$
(11)
$$\phi^{\prime\prime} + Le\Pr f\phi^{\prime} - KcLe\Pr \phi \left( {1 + \delta_{1} \theta } \right)^\text{m} \exp \left( {\frac{ - E}{{1 + \delta_{1} \theta }}} \right) = 0$$
(12)

Associate boundary constraints

$$\left. \begin{gathered} \eta = 0:f\left( \eta \right) = 0,f^{\prime}\left( \eta \right) = 1,G = - sf^{\prime\prime}\left( \eta \right),\theta \left( \eta \right) = 1,\phi \left( \eta \right) = 1 \hfill \\ \eta \to \infty :f^{\prime}\left( \eta \right) = 0,G\left( \eta \right) = 0,\theta \left( \eta \right) = 0,\phi \left( \eta \right) = 0 \hfill \\ \end{gathered} \right\}$$
(13)

Shear stress and couple stress can be described as:

$$\tau_{{\text{w}}} = \left[ {(\mu + k)\left( {\partial_{\eta } u} \right) + kN} \right]_{{\upeta = 0}} = (\mu + k)bx\left( {\frac{b}{\upsilon }} \right)^{0.5} f\prime \prime (0)$$
$$M_{{\text{w}}} = \left( {\gamma \partial_{\eta } N} \right)_{{{\upeta = 0}}} = \mu u_{{\text{w}}} \left( {1 + \frac{k}{2}} \right)G^{\prime}(0).$$

By implementing the similarity conversion, coefficient of skin friction is obtained;

$$C_{{\text{f}}} = \frac{{\tau_{{\text{w}}} }}{{\rho u_{{\text{w}}}^{2} }} = \frac{{(1 + K)f^{\prime\prime}(0)}}{{\left( {{\text{Re}}_{{\text{w}}} } \right)^{0.5} }}$$

Heat flow of local surface is defined as follows:

$$q_{{\text{w}}} (x) = - k_{{\text{f}}} \left( {\partial_{\upeta } N} \right)_{{\upeta = 0}} = - k_\text{f} (T_\text{w} - T_{\infty } )\left( {\frac{b}{\upsilon }} \right)^{0.5} \theta \prime (0)$$

The local surface heat flux transfer coefficient is \(h(x) = \frac{{q_{{\text{w}}} (x)}}{{(T_{{\text{w}}} - T_{\infty } )}} = - k_{{\text{f}}} \sqrt {\frac{b}{\upsilon }} \theta^{\prime}(0)\).

And the rate of heat transmission (Nusselt number) defined as.

\(Nu_{{\text{x}}} = \frac{xh(x)}{{k_{{\text{f}}} }} = - \left( \frac{b}{v} \right)^{0.5} x\theta^{\prime}(0)\) also, can be written as \(\frac{{Nu_{{\text{x}}} }}{{\sqrt {{\text{Re}}_{{\text{w}}} } }} = - \theta^{\prime}(0)\).

The Sherwood number can be described as.

\(Sh_{{\text{x}}} = \frac{{J_{{\text{w}}} (x)}}{{D(C_{{\text{w}}} - C_{\infty } )}} = - \left( \frac{b}{v} \right)^{0.5} x\phi^{\prime}(0)\) and it gives \(\frac{{Sh_{{\text{x}}} }}{{\sqrt {{\text{Re}}_{{\text{w}}} } }} = - \phi^{\prime}(0)\).

Here \({\text{Re}}_{{\text{w}}} = \frac{{u_{{\text{w}}} x}}{\upsilon }\) indicates for local Reynolds number.

Results and discussion

The behaviour of a micropolar fluid flowing across a stretched sheet in the existence of a magnetism is investigated in this work. Dissipation and bilateral reactions are two essential components to be considered. Standard conversion rules are implanted to renovate the governing equations into a nondimensional form. Furthermore, this system of equations is solved with the help of numerical methods. The numerical results shown in Table 1 offer a comparative analysis with the earlier research effort undertaken by Saidulu and Reddy [28]. However, the behaviour of several characterising factors are obtained and presented briefly through graphs.

Table 1 Comparison of rate of heat transfer of different values of Prandtl number with \(K = M = Ec = s = Rd = E = \delta_{1} = m = 0\)
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Figure 2 depicts the effect of the magnetic parameter (Solid line) and the micropolar parameter (Dash line) on the axial velocity profile. The magnetic parameter is confined to numerical values within the range of \(0 \le M \le 2\). A value of M = 0 indicates no magnetization, whereas nonzero values imply considerable magnetic parameter behaviour. When a magnetic field interacts with a conducting fluid, it produces Lorentz forces that contradict the fluid's flow, causing it to slow down. The micropolar parameter has a numerical range of 0 to 2. In this situation, K = 0 denotes the Newtonian nature of the profile, whereas nonzero values allocated to the micropolar fluid show its non-Newtonian property. In physical words, K = 0 denotes the absence of vortex viscosity, leading to the disappearance rotational impact. The presence of micro-rotation in a micropolar fluid introduces additional momentum transmission mechanisms causing augmentation in the fluid’s momentum. Figure 3 depicts the association of the magnetic parameter (Solid line) and the micropolar parameter (Dash line) on the velocity profile. Both parameters have a numerical range from 0 to 2. The value of \(\lambda_{1} = \lambda_{2} = 0\) implies negligible bouyancy impact, but nonzero values suggest significant thermal and solutal bouyancy effect behaviour. As the temperature difference or gradient increases, the buoyancy force strengthens, resulting in stronger convective currents and higher fluid velocities. This increased fluid motion caused by thermal buoyancy enhances the velocity profile. Similar to thermal buoyancy, an increase in the solute concentration gradient enhances the buoyancy force, amplifying fluid motion. As solutal buoyancy increases, it also enhances the velocity profile. Figure 4 demonstrates the implications of the micropolar parameter (Solid line) and the wall condition parameter (Dash line) on the angular momentum profile. The micropolar parameter ranges from 0 to 2, while the wall condition parameter ranges from 0 to 1. The increased impact of microscale effects on the flow causes the angular velocity profile to rise as the micropolar parameter upsurges. As the micropolar parameter increases, it indicates a stronger presence of microscale rotational motion within the fluid, resulting in higher angular velocities which enhances the angular velocity profile. Specifically, when n = 0, it represents a condition of strong concentration with no rotation, resulting in linear behaviour. Additionally, n = 0.5 deliberates the scenario of low concentration, which has a higher impact on improving the profile of microrotation throughout. The decrease in the angular velocity profile as the wall condition parameter increases can be attributed to changes in the boundary conditions next to the surface. When the wall condition parameter is amplified, the wall has a greater impact on the flow dynamics. This increased influence frequently leads to increased frictional effects and constraints on fluid movement near the surface, resulting in a reduction in angular velocity. Figure 5 exemplifies the association of thermal radiation (solid line) and magnetic parameter (dash line) on the temperature profile. Both parameters have ranges from 0 to 2. Thermal radiation causes an elevation of the temperature profile. This is due to increased levels of thermal radiation, which indicate a higher contribution of radiative heat transfer to the total thermal energy balance. As a outcome, a larger amount of heat is transmitted through radiation, causing an overall rise in the temperature profile. An increase in the magnetic parameter indicates either a greater magnetic field or a higher conductivity of the fluid. These conditions can lead to increased Joule heating within the fluid, as electrical currents release energy as heat. As a result, this additional heat generation adds to an overall rise in the fluid's temperature profile. Figure 6 illustrates the variation of Eckert number (solid line) and micropolar parameter (dash line) on the temperature profile. Both parameters have ranges from 0 to 2. When the Eckert number augments, the temperature profile rises. This is because the Eckert number reflects the kinetic energy and enthalpy of the fluid flow. A higher Eckert number specifies an increased fraction of kinetic energy compared to enthalpy in the flow. As a result, more kinetic energy is converted into thermal energy, which raises the overall temperature profile of the flow. As the micropolar parameter intensifications, the temperature profile rises due to improved microscale rotational motion allowing for more effective heat transmission. As a result, the increased micropolar parameter causes more intense thermal effects and a higher temperature profile inside the flow. Figure 7 represents the role of heat source/sink parameter on thermal profile. The positive value of Q indicates heat source, whereas negative sign indicates heat sink. The heat source parameter indicates the quantity of heat delivered to the system. When this value increases, more thermal energy is contributed, causing the temperature to rise. Figure 8 exhibits the consequence of the Lewis number (solid line) and the chemical reaction parameter (dash line) on the concentration distribution. The Lewis number has a numerical range of 1 to 3, whereas the chemical reaction parameter ranges from -3 to 3. The Lewis number is determined as the ratio of heat diffusivity to mass diffusivity. When the Lewis number rises, thermal diffusion takes precedence over mass diffusion, causing the substance to disperse more quickly in response to temperature differences. As a result, the concentration profile diminishes. An upsurge in the chemical reaction parameter causes a faster reaction rate, resulting in accelerated consumption or synthesis of the species involved. This, in turn, reduces the concentration profile of the fluid. Figure 9 displays the consequence of the activation energy (solid line) and the temperature difference parameter (dash line) on the concentration profile. Both parameters have a numerical range of 0 to 1. The increased activation energy slows the reaction kinetics, allowing a larger proportion of the species to remain in the fluid without reacting. This causes a growth in the concentration profile. An upsurge in the temperature difference parameter frequently accelerates reaction rates, resulting in more efficient species consumption or transformation and, eventually, a reduction in the fluid's concentration profile. Figure 10 deliberates the implication of the magnetic parameter and the micropolar parameter on the skin friction coefficient. In MHD flows, the magnetic field creates a Lorentz force that interacts with the fluid momentum, leading to increased drag and skin friction. Similarly, the micropolar parameter reflects the microstructural properties of micropolar fluids, which include additional micro-rotations and micro-inertia. Hence frictional forces enhance and resulting in higher skin friction. Figure 11 illustrates the insertion of thermal and solutal buoyancy on the rate of shear stress coefficient. Skin friction decreases as thermal and solutal buoyancy increase, owing to changes in boundary layer dynamics. Increased buoyancy helps to mix and disperse momentum and mass inside the boundary layer. This enhanced mixing reduces the velocity gradient near the wall, lowering skin friction. Figure 12 exhibitions the impact of both the micropolar and wall condition parameters on couple stress. As the micropolar parameter grows, the couple stress increases. This is because an increased micropolar characteristic indicates a higher frequency of microscale rotational motion inside the fluid. As a result, neighbouring fluid elements interact more strongly, resulting in increased rotational forces and, as a result, increased couple stress. Couple stress rises in proportion to the wall condition parameter. This is because a higher wall condition parameter suggests a greater impact of the surface on flow dynamics. As a result, surface roughness or irregularities may increase, amplifying fluid–solid surface interactions. This greater interaction enhances the rotating forces inside the fluid, causing an increase in couple stress. Figure 13 demonstrations the impact of both the radiation and magnetic parameters on Nusselt number. Increasing thermal radiation increases the Nusselt number by augmenting radiative heat transfer, which improves overall heat transfer. As the magnetic parameter upsurges, the Nusselt number decreases. This is because a greater magnetic value indicates a stronger effect of magnetic fields on fluid flow dynamics. These magnetic fields often reduce the rate of heat transfer. Figure 14 represents the interaction of both the Ec and K on Nusselt number. As the Eckert number rises, suggesting a greater influence of fluid kinetic energy on the transfer of heat, the Nusselt number falls because convective heat transfer takes precedence, diminishing the relative importance of conductive heat transfer. As the micropolar parameter increases, the disruptive effect of micro-rotations on convective heat transport outweighs any potential improvements, resulting in a fall in the Nusselt number. Figure 15 displays the variation of heat source on Nusselt number. With increased internal temperature, the thermal boundary layer may thicken. This also contributes to a smaller temperature gradient at the surface, which results in a lower Nusselt number. Figure 16 depicts the interaction of both Le and Kc on the Sherwood number. An enhancement in the Lewis number, which indicates a higher incidence of thermal diffusion, typically leads to enhanced mass transfer. A spike in the chemical reaction parameter can result in sharper concentration gradients, which drives more efficient convective mass transport and hence raises the Sherwood number. Figure 17 portrays the interaction of both Activation energy and the temperature difference parameter on the Sherwood number. The Sherwood number tends to drop when the temperature difference parameter grows and the convective mass transfer coefficient decreases. When the activation energy increases, resulting in slower reaction rates, the convective mass transmission coefficient decreases, potentially increasing the Sherwood number.

Fig. 2
figure 2

Presentation of \(M,K\) on \(f^{\prime}(\eta )\)

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Fig. 3
figure 3

Presentation of \(\lambda_{1} ,\lambda_{2}\) on \(f^{\prime}(\eta )\)

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Fig. 4
figure 4

Presentation of K, s on \(G(\eta )\)

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Fig. 5
figure 5

Presentation of Rd, M on \(\theta (\eta )\)

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Fig. 6
figure 6

Presentation of Q on \(\theta (\eta )\)

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Fig. 7
figure 7

Presentation of Ec, K on \(\theta (\eta )\)

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Fig. 8
figure 8

Presentation of Le, Kc on \(\phi (\eta )\)

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Fig. 9
figure 9

Presentation of E, \(\delta_{1}\) on \(\phi (\eta )\)

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Fig. 10
figure 10

Presentation of \(M,K\) on skin friction

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Fig. 11
figure 11

Presentation of \(\lambda_{1} ,\lambda_{2}\) on skin friction

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Fig. 12
figure 12

Presentation of K, s on couple stress

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Fig. 13
figure 13

Presentation of Rd, M on Nusselt number

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Fig. 14
figure 14

Presentation of Ec, K on Nusselt number

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Fig. 15
figure 15

Presentation of Q on Nusselt number

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Fig. 16
figure 16

Presentation of Le, Kc on Sherwood number

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Fig. 17
figure 17

Presentation of E, \(\delta_{1}\) on Sherwood number

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Conclusions

The behaviour of a micropolar fluid flowing along a stretched sheet in the presence of a magnetic field is examined in this work. The study carefully takes into account two important parameters: bilateral reaction and dissipation. The implication of viscous dissipation, thermal radiation, and Joule dissipation on different flow profiles are carefully examined in this work. Furthermore, the preceding sections thoroughly explore and explain the parametric behaviour of the flow phenomena. The following is a summary of noteworthy findings:

  • In this instance, the comparison analysis offers a convergence analysis of the expected approach in addition to showing a robust correlation with the earlier study.

  • Mixed convection, the result of the joint effects of thermal buoyancy and solutal buoyancy, increases fluid velocity.

  • When compared to its Newtonian equivalent, the non-Newtonian micropolar parameter demonstrates more noteworthy properties. As a consequence, the axial fluid velocity is increased, and in certain areas, microrotation exhibits two effects.

  • The heat transmission rate is accelerated by thermal radiation and decelerated by the magnetic parameter.

  • The temperature difference parameter lowers the concentration distribution, whereas the activation energy increases it.

  • The Lewis number and the chemical reaction are essential to treat as a controlling parameter for the concentration distribution.

  • The rate of mass transfer is increased by the chemical reaction parameter and the Lewis number, but it is decreased by activation energy.

Develop and execute sophisticated optimisation algorithms, like machine learning-based methods or genetic algorithms, to find the ideal conditions for maximising the rate of heat transfer. Examine optimisation in transient or time-dependent flows, where the rate of heat transfer is optimised over a period of time while taking external influences or time-varying boundary conditions into account.