In view of the fact that Navier-Stokes equations cannot be explained as non-Newtonian and Newtonian fluids theory, Eringen (1964); Eringen (1966) was the first to introduce the theory of micropolar fluids and developed it into thermomicropolar fluids. Micropolar fluid equations differ from Navier-Stokes equations due to the non-symmetry of the stress tensor and couple stress. Since then, a variety of investigations have been conducted based on micropolar fluids of animal blood, polymeric fluids, oils, paints, and ferrofluids (Ishak, 2010; Kumar and Gupta, 2012; Mahmoud and Waheed, 2012; Postelnicu, 2012; Cortell, 2013; Sajid et al., 2018; Abbas et al., 2020a; Rana et al., 2020; Shezad et al., 2020). Hashem Zadeh et al. (2020) have linked the change in Peclet number and bio-convection Lewis number directly to the coupled stress. Thermal radiation is a branch of heat transfer that generates electromagnetic radiation because of the thermal motion of particles. In industries, thermal radiation is essential in designing aircraft, astrophysical issues, satellites, solar power equipment, and gas turbine applications. A novel finding by Tiwari et al. (2020) is that Grashof number and radiation parameter strongly affect hematocrit and Fahraeus effects. It has been demonstrated that the local skin friction coefficient and the local Nusselt number are directly associated with the permeability parameter stated by Rosali et al. (2012). The decreased hydrodynamic permeability of the membrane can be attributed to the increased micro-rotation viscosity, according to Yu Khanukaeva et al. (2019). The thin film thickness parameter and the Soret number were explored by Ali et al. (2019), directly related to the concentration field. According to Hussanan et al. (2018), enhancing the conjugate parameter decreased the micro-rotation parameter. As Bhattacharjee et al. (2019) showed, the micropolar fluid has a greater stiffness coefficient and mass flow rate than a Newtonian fluid, resulting from the additional microstructural. The heat source/sink is significant in conduction or convection heat transfer. Manufacturing plastic film, wire coating, cooling of a machine tool, and condensers are the broad applications of the heat source/sink. It was found by Mishra et al. (2018) that the heat source has a considerable effect on the hydrodynamic boundary layer. In their study, Ramadevi et al. (2020) demonstrated that Dufour and Soret numbers increase and decrease temperature functions. Moreover, porous stretching sheets are also widely used in the industry. In addition to extrusion from dies, aerodynamic extrusion of plastic and drawing of plastic films and wires are some of the applications supported by technology. The effects of magneto-convection on a stretching sheet were probed by Eswaramoorti et al. (2020). The researchers determined that momentum and thermal boundary layer thicknesses are directly related to buoyancy ratio parameters. A radiative hybrid nanofluid that undergoes a chemical reaction on a stretching surface was researched by Santhi et al. (2020). Magnetic field parameters were inversely linked to skin friction coefficients, Nusselt, and Sherwood numbers. Researchers recently studied stretching sheets in 3-D (Anuar et al., 2020; Dinarvand and Rostami, 2020; Shankar et al., 2020; Shoaib et al., 2020; Thumma and Mishra, 2020). In a study by Hosseinzade et al. (2020), micropolar hybrid ferrofluid was perused in a vertical plane. Abbas et al. (2020b) showed that the reduction in heat transfer is caused by an increase in the Prandtl number and non-linearity parameter, as well as a decrease in heat flux constant. Patel et al. (2019) showed that the micro-rotation profiles increase with increasing volume fractions for stretching and shrinking sheets using a semi-analytical method. Khash’ie et al. (2019) found that temperature enhancement is provided by the thermal stratification parameter, while a reduction in concentration profiles is achieved by enhancing the solute stratification parameter. It was reported in a study by Lu et al. (2018) that the concentration profile is inversely related to the strength of homogeneous and heterogeneous reactions. As Lund et al. (2019) recognized, temperature and concentration profiles directly affect thermophoresis and concentration parameters. Shah et al. (2020) found that the velocity distribution increases as the electric field strength for the stretching sheet increases. On the other hand, when the sheet is shrunk, the micro-rotation parameter distribution decreases with an increase in the micro-rotation parameter. As the value of the solute stratification parameter (s) enhances, micro-rotation distributions decline and rise for opposing flow, as Ramzan et al. (2017) reported. A study by Jalili et al. (2021) used the finite element method and two semi-analytical methods (HPM and AGM) to solve the ODEs and showed that ferrofluid possesses more velocity without the magnetic parameter. It has been shown by Kumar et al. (2019) that the highest velocity is achieved by second-order velocity slip, whereas the highest temperature is achieved by first-order slip. A study conducted by Jalili et al. (2019) demonstrated that the highest temperature occurs when micro-rotation parameters are absent. Dawar et al. (2020) realized that concentration fields tended to increase with increasing Biot number, decreasing with increasing chemical reaction and Schmidt number. Rehman et al. (2021) demonstrated that buoyancy parameters affect velocity and angular velocity differently. Abdal et al. (2019) found that skin-friction coefficients are reduced when slip, magnetic, and unsteadiness parameters are increased. Mandal and Mukhopadhyay (2020) reported that the mixed convection parameter increased velocity, but the angular velocity and temperature declined with the mixed convection parameter. Khan et al. (2020) pointed out that the Nusselt number reduces due to the generation/absorption variable. In addition, the temperature increases with the Biot number. Chamkha (1999) stated that the Hartmann number lowered the wall heat transfer. The Nusselt number and skin friction coefficient increase with the magnetic parameter value and when the domain of dual solutions is widened, as stated by Mustafa et al. (2020). In addition, they concluded that the Schmidt number and slip parameter both boosted the Nusselt number. Other researchers have also studied the impact of many physical parameters on the analysis of entropy production (Jain S and Gupta, 2019; Rashid et al., 2019; Zaib et al., 2019; Sen et al., 2020; Hussain and Jamshed, 2021; Khan et al., 2022a; Khan et al., 2022b; Waini et al., 2022), Micropolar (Damseh et al., 2009; Khedr et al., 2009; Modather et al., 2009; Magyari and Chamkha, 2010), heat generation/absorption (Chamkha, 2000; Reddy and Chamkha, 2016; Krishna and Chamkha, 2019; Krishna et al., 2021), convection (Chamkha, 1997a; Krishna and Chamkha, 2020), MHD (Takhar et al., 1999; Takhar et al., 2002; Kumar et al., 2020; Wakif et al., 2020), radiation (Chamkha et al., 2011; Chamkha et al., 2019; Sreedevi et al., 2020), and inclined surface (Chamkha, 1997b; Krishna et al., 2020).

An extensive literature review indicated that numerous papers investigated MHD micropolar fluid flow using semi-analytical and numerical techniques. Nonetheless, no perusal has been presented on applying Lorentz and buoyancy forces, Rosseland thermal radiation, and heat source/sink using the varying wall temperature over an inclined porous stretching sheet to calculate the entropy production as well as experimental design. The current problem represents a film polymer on a heated/cooled inclined porous stretching sheet in the MHD micropolar fluid flow under the thermal radiation effect. Moreover, a new parameter is defined as a combination of the magnetic, buoyancy, and inclination of the porous stretching sheet. We present analytical solutions for non-Newtonian micropolar fluid flow with the abovementioned properties to fill the literature gap. After solving coupled PDEs by suitable similarity solutions, select exponential type solutions to solve highly nonlinear coupled ODEs.

Assume a 2-D flow of a viscous, incompressible, laminar, and steady micropolar fluid over a stretching sheet continuously stretched in the x-direction. The x-component of the velocity varies linearly u _w (x) = ax, where a is a positive constant and a variable surface temperature T _w (x) = T _∞ +bx, where b is also a positive constant. The sheet is presumed to be subjected to a vertical magnetic field. An illustration of the coordinate system and flow model can be seen in Figure 1.

In two dimensions, the simplified governing equations are (Turkyilmazoglu, 2017): ∂ u ∂ x + ∂ v ∂ y = 0 (1) ρ ( u ∂ u ∂ x + v ∂ u ∂ y ) = ( μ + κ ) ∂ 2 u ∂ y 2 + κ ∂ N ∂ y − σ B 0 2 u + ρ β T g ( T w − T ∞ ) cos ( α ) (2) ρ j ( u ∂ N ∂ x + v ∂ N ∂ y ) = γ ∂ 2 N ∂ y 2 − κ ( 2 N + ∂ u ∂ y ) (3) ρ C p ( u ∂ T ∂ x + v ∂ T ∂ y ) = k ∂ 2 T ∂ y 2 + Q 0 ( T − T ∞ ) − ∂ q r ∂ y , (4) N is the micro-rotation vector, K = κ / μ is the dimensionless viscosity ratio, and is called the material parameter, representing Newtonian fluid and micropolar fluid when it is equal to zero and positive, respectively. Ahmadi (1976) demonstrated γ as γ = ( μ + κ / 2 ) j = μ ( 1 + K / 2 ) j and j = ν / a as a reference length. v w is a constant mass flux velocity, positive and negative related to suction and injection, respectively. The corresponding boundary conditions are: u = u w ( x ) = a x ,       v = v w ,       N = − m ∂ u ∂ y ,       T = T w ( x ) = T ∞ + b x       at       y = 0 (5) u → 0 ,       N → 0 ,       T → T ∞       as       y → ∞ , (6)

Based on Rosseland approximation, radiation heat flux is given by: q r = 4 σ ∗ 3 k ∗ ∂ T 4 ∂ y = 16 σ ∗ T 3 3 k ∗ ∂ T ∂ y (7) where k ∗ and σ ∗ are mean absorption coefficient the Stefan-Boltzmann constant, respectively. The micro-gyration constraint m lies in the range of 0–1. Due to transforming the system of governing boundary layer Eqs. 1–4 into the coupled ODEs, we use the following similarity transformation: η = y a ν f ,       ψ = a ν f x f ( η ) ,       u = a x f ′ ( η ) ,       v = − a ν f f ( η ) ,       N = a x a ν f g ( η ) , θ ( η ) = T − T ∞ T w − T ∞ (8)

Substituting Eqs. 8 into Eqs. 2–4, we acquire the following ordinary differential equations: ( 1 + K ) f ‴ + f f ″ - f ′ 2 + K g ′ - H a f ′ + Λ θ ⁡ cos ( α ) = 0 (9) ( 1 + K 2 ) g ″ + f g ′ - f ′ g - K ( 2 g + f ″ ) = 0 (10) ( 1 + 4 3 R d ) θ ″ + Pr ( f θ ′ - f ′ θ + Q θ ) = 0 (11)

Following the similarity transformation in Eqs. 5, 6 turn into f ( 0 ) = s ,       f ′ ( 0 ) = 1 ,       g ( 0 ) = − m f ″ ( 0 ) ,       θ ( 0 ) = 1 (12) f ′ ( ∞ ) → 0 ,       g ( ∞ ) → 0 ,       θ ( ∞ ) → 0 (13)

Here prime denotes differentiation with respect to the similarity variable η . The dimensionless parameters used in the ODEs are the magnetic parameter H a = σ B 0 2 / ρ a , the buoyancy parameter Λ = Gr / Re 2 = g β T b / a 2 , the Grashof number G r = g β T ( T w − T ∞ ) / ν f 2 , the Reynolds number Re = ( x u w ( x ) / ν f ) 2 , the suction/injection parameter s = ∓ v w / a ν , the thermal radiation parameter R d = 4 σ ∗ T ∞ 3 / k ∗ ν f ( ρ C p ) f , the Prandtl number Pr = ρ C p ν / k , and the heat source/sink Q = Q 0 / a ( ρ C p ) f .

A Chakrabarti and Gupta (1979) figured out the exact solutions for stretching surfaces, and many researchers have worked in this field. It is important to remember that heat sources/sinks, non-Newtonian, mixed convection flow, sheet porosity, magnetic field, micropolar fluid, radiation, and inclined sheet characteristics all play a role in the process. We pick the exponential form of physical solutions to satisfy the boundary conditions as below: f ( η ) = s + 1 − e − λ η λ g ( η ) = − m f ″ ( η ) = m λ e − λ η (14) θ ( η ) = f ′ ( η ) = e − λ η

The momentum, angular velocity, and energy equations produce relations: ( − K m + K + 1 ) λ 2 − s λ + Γ − 1 = 0 (15) ( m + K m 2 ) λ 2 − m s λ − 2 K m + K − m = 0 (16) ( 1 + 4 3 R d ) λ 2 − Pr s λ + Pr Q − Pr = 0 (17)

We combine the mixed convection, magnetic field, and inclination parameters to define a new parameter called the magneto-buoyancy-inclination parameter Γ = Λ ⋅ cos ( α ) - Ha . Eqs. 15–17 could be reorganized as: [ c 1 c 2 2 Pr 2 ] λ 4 + [ ( − 6 K 2 + ( 3 Γ − 9 Q − 12 ) K + 6 Γ − 12 Q ) Pr − ( 8 R d + 6 ) ( Γ − 2 K − 2 Q ) 6 Pr ] λ 2 − ( Γ − Q ) ( 2 K + Q ) = 0 (18)

The parameter λ must be determined, and there are four negative and positive solutions. The negative solutions are mathematically and physically possible and impossible, respectively. On the other hand, the positive solutions are mathematically and physically conceivable. Therefore, the required solutions are positive ones. Solving Eq. 18 results in λ 1 = Pr ( c 3 − 3 c 4 ) 6 c 1 c 2 (19) λ 2 = − Pr ( c 3 + 3 c 4 ) 6 c 1 c 2 m = c 2 K Pr + Γ − Q K λ 2 s = ( 4 3 R d + 1 ) λ Pr + Q − 1 λ where c i , i = 1     to     4 are defined as: c 1 = ( K + 2 ) ⋅ Pr − 8 3 ⋅ R d − 2 (20) c 2 = ( K + 1 ) ⋅ Pr − 4 3 ⋅ R d − 1 c 3 = ( ( 3 ( K + 2 ) Γ ) 2 + 108 Γ K ( K + 2 ) ( K + Q 6 + 2 3 ) + ( 6 K ( K − Q 2 + 2 ) ) 2 ) Pr 2 − 48 ( R d + 3 4 ) ( ( K + 2 ) Γ 2 + 8 Γ K ( K + Q 8 + 1 ) + ( 2 K ) 2 ( K − Q 2 + 2 ) ) Pr + ( 6 ( R d + 3 4 ) ( Γ + 2 K ) ) 2 c 4 = ( − 2 K 2 + ( Γ − 3 Q − 4 ) K + 2 Γ − 4 Q ) Pr − 8 ( Γ − 2 K − 2 Q ) ( R d + 3 4 ) 3

The local volumetric rate of entropy production for the micropolar fluid flow is formulated as (Sayed and Abdel-wahed, 2020): E G = k T ∞ 2 ( ( ∂ T ∂ x ) 2 + ( 1 + 16 σ T ∞ 3 3 k ) ( ∂ T ∂ y ) 2 ) ︸ H e a t   t r a n s f e r I r r e v e r s i b i l i t y + ( μ + κ ) T ∞ ( ∂ u ∂ y ) 2 ︸ F l u i d   f r i c t i o n I r r e v e r s i b i l i t y + σ B 0 2 T ∞ u 2 ︸ J o u l e   d i s s i p a t i o n I r r e v e r s i b i l i t y (23)

By using Eqs. 8, 23 yields: N g = S g e n ′ ′ ′ S 0 ′ ′ ′ = Re L Br Ω ( 1 + K ) ( f ′ ′ ) 2 + ( ( θ ) 2 X 2 + Re L ( 1 + 4 3 R d ) ( θ ′ ) 2 ) + Re L Br Ω H a ( f ′ ) 2 (24)

Here Re L = a L 2 / ν f , Br = μ f u w 2 / ν f k f Δ T , Ω = Δ T / T ∞ and X = x / L stand for Reynolds number, Brinkman number, dimensionless temperature ratio, and axial distance, respectively.

In this section, we conducted a quadratic regression to estimate the value of the entropy production number in a MHD micropolar fluid flow. To better understand the influences of K, Λ, and Rd, 3-D figures are presented. In accordance with the quadratic estimation equation, assuming Ha = 10 , α = 75 o , Pr = 0.7 , Q = 10 , X = 0.5 , Re = 2 , B r = 3 , Ω = 0.6 , would yield the following entropy generation number: N g = 172.87784887509 + 15.916483724999 ⋅ K + 5.5316012437495 ⋅ Λ + 78.1614713 ⋅ R d − 0.021906 ⋅ K ⋅ Λ − 0.65886122 ⋅ K ⋅ R d + 0.331027715 ⋅ Λ ⋅ R d + 0.0431583 ⋅ K 2 − 0.0115246125 ⋅ Λ 2 + 2.60163884 ⋅ R d 2 (25)

Seventeen different sets of values of the material parameter between [90, 100], buoyancy parameter between [60, 100], and thermal radiation parameter between (Mahmoud and Waheed, 2012; Abbas et al., 2020a) were computed using regression analysis. In Eq. 25, it can be seen that K, Λ, and Rd directly correlate with Ng.

Here, the results are summarized in a tabular and graphical format based on the MHD mixed convection micropolar fluid flow with dual solutions. Additionally, the local couple stress was calculated. The dual exact explicit solutions are derived analytically. A comparison table supporting the validity of our current analytical solutions is also provided in Table 1, and this table shows perfect agreement with previous results.

The following Figures 2A–C illustrates the existence domain of (λ, m, s) curves versus Γ for the various material parameters. The current analysis makes it possible to determine the overall trend based on the ranges displayed. Figure 2A indicates the changes in the heat transfer coefficient to the Γ with the increase of the material parameter. By fixing the Prandtl number, thermal radiation, and heat generation/absorption parameters, the range of the Γ parameter value expands with the increase of the material parameter. For example, for K = 70, the critical value of the Γ parameter is Γ _c = 16.4. The implication is that if Γ _c <16.4, there are dual solutions, and if Γ _c >16.4, there are no solutions. So, this is one of the necessary constraints for dual solutions. In this way, the justification of the choosing values for the physical parameters is based on this kind of calculation.

As shown in Figure 3, increased K is physically related to increasing vortex viscosity (micro-rotation motion) of the flow, which, in turn, boosts N_g(η). Also, by increasing the magnetic parameter 50 times, the generation of entropy is enhanced. Rising the magnetic field parameter boosted the resistant force against the fluid movement, so the heat transfer rate in the boundary layer was raised. When the buoyancy parameter is high, the heat is absorbed, and the stretching porous sheet inclination is 45 ° , and the thermal radiation is high; by increasing the magnetic field parameter, the entropy generation near the sheet decreases, but it improves after a short distance. Low buoyancy parameters exhibited the opposite behavior.

A simultaneous presentation of the effects of the magneto-buoyancy-inclination and thermal radiation parameters on entropy generation is shown in Figure 4. Entropy was raised by increasing the thermal radiation parameter and lowering the magneto-buoyancy-inclination parameter, which includes boosting Ha and Λ from 1 to 30 and the sheet orientation from horizontal to 60 ° . Considering a high internal heating source, increasing thermal radiation results in higher temperature, so the disorder in MHD micropolar fluid flow boosts. Therefore, setting the magneto-buoyancy-inclination parameter to zero is beneficial to minimizing entropy production.

Figure 5 shows the effects of thermal radiation and buoyancy parameters on entropy generation under conditions of minor heat sources and 30 ° inclination. This shows that the buoyancy and thermal radiation parameters directly relate to the entropy generation number near the sheet. As the heat transfer shifts from conduction to convection, more entropy is produced in the MHD micropolar fluid flow. So, by reducing Γ, we get closer to the main goal of the second law of thermodynamics, which is to minimize entropy production.

As shown in Figure 6, the heat source generates more entropy than the heat sinks due to the direct relationship between temperature and N_g(η). Under intense thermal radiation, by boosting Q from -10 to 10, the entropy generation over the horizontal and vertical sheets increased by 14.82% and 12.68%, respectively. Thus, the porous stretching sheet, that is, oriented horizontally causes more entropy.

In Figures 7A–C, you can observe how K, Λ, and Rd affect Ng in 3-D plots. The 3-D plots clarify the role of parameter coefficients in Eq. 25. For instance, since the Rd coefficient in Eq. 25 is greater than the Λ, in Figure 7C, the effect of Rd on the Ng is more than the Λ.

Figure 8 compares the analytical solutions (Actual) from Eq. 19 with the estimated entropy generation number (Predicted) from Eq. 25. The values of regression coefficients for Eq. 25 are quite suitable, and R² = 0.9999 confirms this.

The numerical data have been set in Table 2 to illustrate the effect of K versus the heat source and sink for the M x Re x 1 / 2 . According to Table 2, the magnitude of the M x Re x 1 / 2 for heat sink is slightly greater than that of the heat source. Since the K directly influences the coefficient.

According to the model summary statistics (Table 3), a quadratic model can be constructed based on the significant value of the adjusted R2 value. Calculating the quadratic model coefficients was done using the regression analysis method in the RSM.

The following Table 4 is a statistical evaluation of the fit-response surface model based on the coefficient of determination (R²) and the results of an analysis of variance. The regression model is deemed highly significant when the p-value is low (less than 0.05). The F-value of 1,105.71 and the p-value of less than 0.0001 in Table 4 prove that the model achieved is reliable. There is only a 0.01% chance that this large F-value could happen due to noise.

As shown in Table 5, the R² value is 0.9999, which has in good agreement with the Predicted R² of 0.9989. The adequate precision is 371.3933, which is quite good (since if it is greater than 4, the model has a powerful signal to be employed for optimization). By subtracting the Predicted R² from the Adjusted R², the difference is 0.000935, which is pretty decent (if the difference is less than 0.2, then the model matches the data and can confidently be applied to interpolate). Table 5 contains indicators proving that the model is effective in the experimental range.

All solutions to the dimensionless velocity, angular velocity, temperature, local skin friction coefficient, local couple stress coefficient, and local Nusselt number are explicitly derived. In addition to presenting analytical solutions, these solutions facilitate a deeper understanding of flow behavior, heat transfer, and entropy production. Additionally, the optimization process for entropy production was performed by experimental design (BBD) and yielded excellent results based on the predicted (analytical) values. The entropy production correlation on the sheet is estimated through a quadratic regression that involves three independent parameters. Among the key findings of this work are:

• Magnetic field, mixed convection, and inclination phenomena are governed by the magneto-buoyancy-inclination parameter.