Neural network approach for thermal analysis of magnetized radiative micropolar hybrid nanofluid flow on a variable porous stretching sheet with Cattaneo-Christov flux model

Abstract

This work examines the micropolar hybrid nanofluid flow on a variable porous elongating sheet using inclined magnetic effects. The fluid is set into motion by stretching nature of the sheet. The effects of heat source/sink and thermal radiation have been utilized in this work together with thermophoresis and Brownian motion effects. Additionally, the Cattaneo-Christov flux model is used to accurately describe mass and heat diffusion by accounting for relaxation time effects, providing a more realistic representation of thermal transport phenomena. The leading equations have been solved using artificial neural network (ANN) method. It has noticed in this work that, when the diamond nanoparticles augments from 0.00 to 0.05 the Nusselt number surges from 0.00% to 8.27% while on the same range of nanoparticles volume fraction it augments from 0.00% to 8.46% for diamond + copper nanoparticles that ensures the dominance of hybrid nanoparticles. Thermal distributions have been increased due to the influence of thermophoresis, Brownian motion, magnetic effects, radiation, and heat source factor while they have decreased with a rise in the thermal relaxation parameter. The linear velocity of the fluid has decreased due to the increase in magnetic factor and variable porous factor. The validation of current work has ensured through comparative analysis of current results with established data set, with a fine agreement among all the results.

Introduction

Nanofluid flow consists of nanoparticles suspended in some base fluids. These nanoparticles, which are typically smaller than 100 nm, considerably augment the thermal properties of the fluid as noticed initially by Choi and Eastman. ¹ In heat transfer applications, nanofluids exhibit enhanced thermal conductance, allowing for proficient heat dissipation compared to conventional fluids. When nanofluids flow through channels or pipes, the suspended nanoparticles increase the fluid’s thermal flow features, causing a decline in temperature gradients, and improved heat transfer efficiency as detected by Khan et al. ² This enhanced heat transfer results in higher surface temperatures being cooled more effectively, making nanofluids useful in applications like electronics coolant, automotive, and energy systems. Mahabaleshwar et al. ³ examined nanofluid flow on an accelerated sheet with effect of inclined magnetic field. However, nanofluid flow also impacts velocity profiles due to changes in viscosity; higher concentrations of nanoparticles increase viscosity, which leads to higher frictional resistance and lower flow velocity as revealed by Boujelbene et al. ⁴ Additionally, the interactions between nanoparticles and the fluid boundary layer can modify flow behavior, such as promoting turbulence or improving mixing. Recently, it has been revealed that when two or more distinct nanoparticles are suspended in a pure fluid, its thermal properties can be significantly enhanced as observed by Lone et al. ⁵ This new class of fluids is termed as hybrid nanofluid and with this an innovative approach in nanofluids research has opened the new possibilities to improve heat transfer. Nanoparticles, like metals, oxides, and carbon-based materials, can be combined in a fluid to create what is known as hybrid nanofluids. Algehyne et al. ⁶ highlighted that the synergy between different types of nanoparticles often results in improved thermal conductance compared to fluids with single nanoparticles. This enhancement is attributed to the diverse shapes, sizes, and thermal conductivities of the particles, which promote better heat dispersion within the fluid. Hybrid nanofluids are being explored in industries like automotive, electronics cooling, and renewable energy systems, where efficient thermal management is crucial. Habib et al. ⁷ discussed nanofluid flow on a paraboloid revolution with effects of variable viscosity by implementing Keller-Box approach. Lone et al. ⁸ analyzed the irreversibility effects for stagnant point hybrid nanofluid flow on an expanding sheet with thermally radiative effects. Bhatti et al. ⁹ examined EMHD hybrid nanofluid flow in pipe with applications in geothermal energy. Alqahtani et al. ¹⁰ discussed computationally the electrically conducted spinning flow hybrid nanofluid through two parallel sheets. Additionally, these fluids exhibit improved stability and reduced viscosity, allowing for better flow characteristics. As a result, the combination of multiple nanoparticles optimizes not only heat transfer but also energy efficiency. ¹¹ This advancement marks a significant step toward the development of high-performance cooling systems for modern technologies.

Fluid flow with inclined magnetic field is a significant topic in magnetohydrodynamics (MHD), which studies electrically conductive fluids by using magnetic effects. When fluid flow in the presence of magnetic field the flow experience Lorentz force that acts in opposite direction to fluid’s motion. Mirzaei et al. ¹² inspected convective thermal transport for MHD fluid flow in a conduit using various flow conditions. Dharmendar Reddy et al. ¹³ discussed thermal generation and absorption effects on MHD fluid flow on a cylinder with permeable medium. Sarma and Paul ¹⁴ highlighted that the inclination of the magnetic field modifies the distribution and intensity of magnetic forces, affecting the flow velocity and pressure distribution within the fluid. This can lead to complex flow patterns, including velocity gradients, secondary flows, and boundary layer effects. The angle of inclination significantly influences these phenomena; when the field is applied perpendicularly, the Lorentz force is maximized, while a parallel field minimizes the effect. ¹⁵ In industrial applications like magnetic control of molten metals, cooling systems for nuclear reactors, and MHD power generation MHD plays a vital role. Mao et al. ¹⁶ discussed electrohydrodynamic driven fluid flow through an immiscible interface with valveless water pump. Ali et al. ¹⁷ examined magneto-hydrodynamics nanoparticles fluid flow through co-axial disks with dynamics of Bogar fluid. The flow behavior under an inclined magnetic field is crucial for optimizing performance of flow system. The inclination angle also affects heat transfer rates and stability in fluid flows, making it important for thermal management applications. ¹⁸ In some cases, the magnetic field can suppress turbulence, stabilizing the flow. Furthermore, in geophysics and astrophysics, inclined magnetic fields influence plasma flows in stars and planetary magnetospheres. Thus, analyzing fluid flow under an inclined magnetic field provides insights into both theoretical physics and practical engineering systems.

Fluid flow on a variable porous surface involves the interaction among the fluid and a surface that has a changing permeability or porosity. The surface’s permeability varies in space, affecting the flow dynamics and the distribution of pressure, velocity, and temperature. In such a scenario, the porous medium can either resist or facilitate the flow depending on its porosity distribution. As the fluid moves across the surface, the rate of mass transfer through the porous material varies, influencing the velocity field of the fluid as perceived by Waqas et al. ¹⁹ In the case of an increasing porosity, the fluid experiences less resistance, and the velocity increases near the surface. Alnahdi et al. ²⁰ discussed stagnation point flow on a variable permeable space using thermal convective effects. The temperature distribution is similarly affected by the variable porosity. Heat transfer in such flows depends on the fluid’s velocity and the porosity of the surface. In regions where porosity is higher, there is generally more heat transfer due to higher velocity gradients and less resistance to fluid movement. Khanam et al. ²¹ inspected MHD convective flow on a porous plate placed vertically with varying suction and slip effects. Salama et al. ²² discussed computationally the fluid flow on a porous sheet with spherical pores. On the other hand, lower porosity regions tend to slow down the fluid, leading to a less efficient heat transfer and potentially higher local temperatures near the surface due to reduced convective cooling. Khan et al. ²³ revealed that the combined effects of the variable porous surface influence both the velocity and temperature fields, leading to more complex, localized thermal and flow behaviors. Higher porosity increases heat transfer efficiency due to higher fluid velocities and enhanced convection, leading to a more uniform temperature profile. In contrast, lower porosity regions slow the fluid, reducing convective heat transfer and causing localized hot spots or higher temperature gradients near the surface. These variations in velocity and temperature are critical in designing systems involving porous media, like heat exchangers, filtration, and geothermal energy applications. ²⁴ Mahabaleshwar et al. ²⁵ examined radiative impacts on a dusty laminar trihybrid nanofluid flow on a permeable shrinking/ stretching sheet. Waseem et al. ²⁶ studied thermally the radiative effects on non-Newtonian Darcy fluid flow on an inclined sheet with effects of Cattaneo-Christov flux model.

Micropolar fluid flow is characterized by micro-rotational effects, where fluid particles exhibit both translational and rotational motion, differing from classical Newtonian fluids. This theory, introduced by A.C. Eringen,^27,28 is useful for describing complex fluids like polymers, suspensions, and biological fluids. When a micropolar fluid flows on an extending sheet, typically modeled with a linear stretching velocity both fluid velocity and micro-rotational velocity are governed by coupled differential equations. Bhatia and Chandrawat. ²⁹ highlighted that near the sheet, boundary conditions often dictate that micro-rotational velocity is proportional to the gradient of the fluid velocity, while far away from the sheet, both velocities diminish. Sulaiman et al. ³⁰ discussed the efficiency of thermal transportations for micro-polar fluid flow with boundary constraints with supervised network and have detected that with surge in material factor there has a reduction in linear velocity while the micro-rotational velocity has augmented with it. Higher couple stress viscosity enhances the micro-rotation, leading to more significant deviations from classical fluid behavior. Additionally, increased micro-inertia or surface spin parameters can further amplify micro-rotational velocity. These rotational effects also influence heat and mass transfer, which is critical in industrial applications like polymer extrusion, metal forming, and lubrication processes. ³¹ Algehyne et al. ³² examined the augmentation of thermal transportation for micropolar nanofluid flow on a flat vertical sheet. Yuan et al. ³³ explored how microfluidic devices had been employed to sort Caenorhabditis elegans, allowing for automated handling, improved sorting accuracy, and enhanced efficiency in biological experiments. The micro-rotation reduces drag, improve energy dissipation, and enhance material processing efficiency. Thus, micropolar fluid dynamics on a stretching sheet is essential for optimizing processes involving non-Newtonian fluids where particle rotation significantly alters the overall fluid behavior.

Artificial Neural Networks (ANNs) are numerical models intended to distinguish patterns and make predictions. Khan et al. ³⁴ has proved that in fluid flow problems, ANNs are particularly useful due to their capability to tackle complex, nonlinear problems which are challenging to evaluate using traditional analytical methods. ANNs can be trained on large datasets generated through experiments or simulations, allowing them to predict flow behavior, temperature distribution, and pressure variations under different conditions. ³⁵ In fluid dynamics, ANNs are often employed to model turbulence, optimize designs, and predict the behavior of nanofluids which are fluids containing nanoparticles that enhance thermal conductance. ANNs can analyze Casson hybrid nanofluids, which exhibit complex, non-Newtonian characteristics, to determine flow behavior in porous media, or between gyrating plates as noticed by Ul-Haq et al. ³⁶ By learning from input data like fluid properties, temperature gradients, and boundary conditions, ANNs provide real-time solutions, reducing the need for computationally expensive numerical simulations. ANNs are also used in conjunction with optimization techniques to improve heat transfer efficiency, which is crucial in industries like aerospace, chemical engineering, and energy systems as observed by Yuan et al. ³⁷ Additionally, their adaptability allows researchers to explore dynamic scenarios, such as varying porosity, viscosity, or magnetic field effects on fluid flow. The integration of ANNs in fluid flow research enhances accuracy and efficiency, enabling better-informed decisions in system design and performance evaluation. This makes ANNs a powerful tool for advancing fluid dynamics and solving real-world engineering challenges. Jegan et al. ³⁸ inspected computationally the bio-magnetic nanofluid flow with impacts of Darcy Forchheimer theory by evaluating modeled equations with ANN. Aslam et al. ³⁹ examined incompressible flow of fluid through diverging and converging channel using the ANN approach. Bairagi et al. ⁴⁰ discussed this technique (ANN) to simulate MHD radiative fluid flow on a gyrating cylinder.

Research gap

Despite considerable progress in understanding the properties of hybrid nanofluids for biomedical applications, several research gaps remain in the existence literature:

• Study of blood based hybrid nanofluid flow with micropolar effects on a variable porous stretching sheet is unexplored by implementing neural network approach. The inclusion of porous medium in such flows is a novel work that has not discussed before.

• The motion of the fluid subject to the influence of inclined magnetic field on variable porous space has not revealed.

• This study further enhances its novelty by investigating mass and thermal diffusion, governed by the Cattaneo-Christov flux model, over a variable permeable surface under the influence of an inclined magnetic field.

Significance of the work

This work holds significant value in advancing thermal engineering. This study integrates artificial neural networks to effectively solve complex, nonlinear differential equations, governing hybrid nanofluid flows influenced by magnetic fields, radiation, and micropolar effects, on a stretching porous surface. By incorporating the Cattaneo-Christov flux model, which accounts for non-Fourier heat conduction, the work enhances the accuracy of heat transfer predictions in advanced materials. The findings have practical implications in improving thermal systems in aerospace, biomedical devices, and energy systems, where precise thermal control and efficient cooling are critical.

Biomedical applications Cattaneo-Christov flux model

The Cattaneo-Christov flux model, an extension of Fourier’s law that incorporates thermal relaxation time and eliminates the paradox of infinite speed of heat propagation, has valuable biomedical applications, especially in thermal therapies and blood flow analysis. In treatments like hyperthermia used for cancer therapy, accurate modeling of heat distribution within tissues is crucial to avoid damage to healthy cells. The model is also useful in understanding heat transfer in biofluids such as blood, where pulsatile flow and temperature regulation are important for patient safety during surgeries. Additionally, it can help design medical devices like thermal sensors or artificial organs by predicting how heat diffuses in biological environments with more realism than classical models. Moreover, in biomedical applications, the heat transfer rate is checked in areas where temperature regulation is vital for effective treatment and patient safety. This includes cancer therapies like hyperthermia, where heat must be accurately delivered to tumors without harming nearby tissues, and in cryotherapy, where controlled cooling is used to destroy abnormal cells.

Problem formulation

Consider two dimensional micro-polar hybrid nano-fluid flow with nanoparticles of copper and diamond mixed in blood taken as base fluid. Magnetic effects with strength $B_0$ have applied to flow filed in inclined direction as portrayed in Figure 1(a). Flow is mainly induced by variable porous stretching nature of sheet’s surface. The sheet stretches with velocity $u_1=a{x}_1$ as portryaed in Figure 1(a), where $a>0$ is a fixed number. The temperatures and concentrations of fluid, sheet’s surface and free stream are respectively given by $T$ , $T_w$ , $T_{\infty }$ and $C$ , $C_w$ , $C_{\infty }$ . Thermal and mass diffusions are regulated by using the Cattaneo-Christov flux phnomenon, along with the effects of Brownian motion and thermophoresis. Thermal radiations and heat sink/source effects have also been used in flow filed.

Figure 1.

(a) Geometrical view of flow problem. (b) Dissemination of layers for current problem.

Based on the abovementioned suppositions, we have^41–43:

$$\frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}=0,$$

(1)

$${\overline{\rho }}_{\mathrm{hnf}}(u_1\frac{\partial u_1}{\partial x_1}+u_2\frac{\partial u_2}{\partial x_2})=({\overline{\mu }}_{\mathrm{hnf}}+{\kappa }_a)\frac{{\partial }^2{u}_1}{\partial {x_2}^2}+{\kappa }_a\frac{\partial N}{\partial u_2}-({\overline{\sigma }}_{\mathrm{hnf}}{B}_0^2\sin \left(\gamma \right)-\frac{{\overline{\mu }}_{\mathrm{hnf}}}{K\left(x_2\right)})u_1,$$

(2)

$${\overline{\rho }}_{\mathrm{hnf}}(u_1\frac{\partial N}{\partial x_1}+u_2\frac{\partial N}{\partial x_2})=\frac{{\chi }_{\mathrm{hnf}}}{j}\frac{{\partial }^{2}N}{\partial {x_2}^2}-\frac{{\kappa }_a}{j}(2N+\frac{\partial u_1}{\partial x_2}),$$

(3)

$$\begin{array}{c}(u_1\frac{\partial T}{\partial x_1}+u_2\frac{\partial T}{\partial x_2})+{\tau }_1\left[\begin{array}{c}{u}_2\frac{\partial T}{\partial x_2}\frac{\partial v}{\partial x_2}+{u_1}^2\frac{{\partial }^{2}T}{\partial {x_1}^2}+{u_2}^2\frac{{\partial }^{2}T}{\partial {x_2}^2}+u_1\frac{\partial T}{\partial x_1}\frac{\partial u_1}{\partial x_1}\hfill \\ +2u_1{u}_2\frac{{\partial }^{2}T}{\partial x_1\partial x_2}+u_1\frac{\partial T}{\partial x_2}\frac{\partial u_2}{\partial x_1}+u_2\frac{\partial T}{\partial x_1}\frac{\partial u_1}{\partial x_2}\hfill \end{array}\right]=\frac{{\overline{k}}_{\mathrm{hnf}}}{{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{hnf}}}\frac{{\partial }^{2}T}{\partial {x_2}^2}-\frac{\partial q_r}{\partial x_2}\hfill \\ +\left(\frac{Q_1}{{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{hnf}}}\right)(T-T_{\infty })+\frac{{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{np}}}{{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{hnf}}}[D_B\frac{\partial T}{\partial x_2}\frac{\partial C}{\partial x_2}+\left(\frac{D_T}{T_{\infty }}\right){\left(\frac{\partial T}{\partial x_2}\right)}^2]\hfill \end{array}$$

(4)

$$u_1\frac{\partial C}{\partial x_1}+u_2\frac{\partial C}{\partial x_2}+{\tau }_2\left(\begin{array}{c}{u}_2\frac{\partial C}{\partial x_2}\frac{\partial u_2}{\partial x_2}+{u_1}^2\frac{{\partial }^{2}C}{\partial x_1^2}+u_2^2\frac{{\partial }^{2}C}{\partial x_2^2}+u_1\frac{\partial C}{\partial x_1}\frac{\partial u_1}{\partial x_1}\hfill \\ +2u_1{u}_2\frac{{\partial }^{2}C}{\partial x_1\partial x_2}+u_1\frac{\partial C}{\partial x_2}\frac{\partial u_2}{\partial x_1}+u_2\frac{\partial C}{\partial x_1}\frac{\partial u_1}{\partial x_2}\hfill \end{array}\right)=D_B\frac{{\partial }^{2}C}{\partial x_2^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial x_2^2}.$$

(5)

In above equations, the intensity of magnetic field is $B_0$ , ${\overline{\mu }}_{\mathrm{hnf}}$ is viscosity, ${\overline{\rho }}_{\mathrm{hnf}}$ is density, ${\overline{\sigma }}_{\mathrm{hnf}}$ is electrical conductivity of hybrid nanofluid, $u_1,u_2$ are flow components, ${\kappa }_a$ is vortex viscosity $D_B,D_T$ are Brownian and thermophoresis diffusions, $Q_1$ is heat source coefficient, $N$ is micro-rotation velocity, $j$ is micro-inertial factor.

The conditions at boundaries are ⁴²

$$\{\begin{array}{c}{u}_1(x_2\to 0)=u_w=a{x}_1,N(x_2\to 0)=0,u_2(x_2\to 0)=0,T(x_2\to 0)=T_w,C(x_2\to 0)=C_w,\hfill \\ u_1(x_2\to \infty )=0,N(x_2\to \infty )=0,T(x_2\to \infty )=T_{\infty },C(x_2\to \infty )=C_{\infty }.\hfill \end{array}$$

(6)

Further

$$q_{\mathrm{rd}}=\frac{4{\sigma }_x}{3k_e}\left(\frac{\partial T^4}{\partial x_2}\right)$$

(7)

$$K(x_2)=\frac{m^2\epsilon {(x_2)}^3}{{(1-\epsilon (x_2))}^2},\epsilon (x_2)={\epsilon }_0(1+{\epsilon }_1{e}^{\frac{x_1{\epsilon }_2}{m}}).$$

(8)

The variables used for transformations are ⁴³

$$\{\begin{array}{c}\eta =x_2\sqrt{a/v_{\mathrm{bf}}},\psi =\sqrt{a{v}_{\mathrm{bf}}}{x}_{1}f\left(\eta \right),{\mathrm{u}}_1=\frac{\partial \psi }{\partial x_2}=a{x}_{1}f\prime \left(\eta \right),-\frac{\partial \psi }{\partial x_1}=u_2=-\left(\sqrt{v_{\mathrm{bf}}a}\right)f\left(\eta \right)\hfill \\ N=a\sqrt{a/v_{\mathrm{bf}}}{x}_{1}g\left(\eta \right),\vartheta \prime \left(\eta \right)=\frac{C-C_{\infty }}{C_w-C_{\infty }},\theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }}.\hfill \end{array}$$

(9)

The computational values of copper, blood and diamond are described in Table 1

Table 1.

Computational values of copper, blood, and diamond.^41,44,45

Properties	$\overline{\rho }\left(\mathrm{kg}{m}^{-3}\right)$	$C_p\left(J{\left(\mathrm{kgK}\right)}^{-1}\right)$	$\sigma (\mathrm{\Omega }.m^{-1})$	$k\left(W{\left(\mathrm{mK}\right)}^{-1}\right)$
Diamond	3100	516	34.84 × 106	1000
Copper $\mathrm{Cu}$	8.933 × 103	385	5.96 × 107	400
Blood	1063	3594	6.67 × 10−1	9.492

Below are the thermophysical features of mono and hybrid nanofluids ⁴⁶ :

$$\begin{array}{c}\frac{{\overline{\mu }}_{\mathrm{nf}}}{{\overline{\mu }}_{\mathrm{bf}}}=\frac{1}{{(1-{\mathrm{\Lambda }}_1)}^{2.5}},\frac{{\left(\overline{\rho }{\hat{C}}_p\right)}_{\mathrm{nf}}}{{\left(\overline{\rho }{\hat{C}}_p\right)}_{\mathrm{bf}}}=(1-{\mathrm{\Lambda }}_1)+\frac{{\left(\overline{\rho }{\hat{C}}_p\right)}_{a1}{\mathrm{\Lambda }}_1}{{\left(\overline{\rho }{\hat{C}}_p\right)}_{\mathrm{bf}}},\hfill \\ \frac{{\left(\overline{\rho }\beta \right)}_{\mathrm{nf}}}{{\left(\overline{\rho }\beta \right)}_{\mathrm{bf}}}=\frac{{\mathrm{\Lambda }}_1{\left(\beta \overline{\rho }\right)}_{a1}}{{\left(\beta \overline{\rho }\right)}_{\mathrm{bf}}}+(1-{\mathrm{\Lambda }}_1),\frac{{\overline{\sigma }}_{\mathrm{nf}}}{{\overline{\sigma }}_{\mathrm{bf}}}=1+\frac{3\left[{\mathrm{\Lambda }}_1(\frac{{\overline{\sigma }}_{a1}}{{\overline{\sigma }}_{\mathrm{bf}}}-1)\right]}{(\frac{{\overline{\sigma }}_{a1}}{{\overline{\sigma }}_{\mathrm{bf}}}+2)-(\frac{{\overline{\sigma }}_{a1}}{{\overline{\sigma }}_{\mathrm{bf}}}-1){\mathrm{\Lambda }}_1},\hfill \\ \frac{{\overline{k}}_{\mathrm{nf}}}{{\overline{k}}_{\mathrm{bf}}}=1+\frac{3\left[(\frac{{\overline{k}}_{a1}}{{\overline{k}}_{\mathrm{bf}}}-1){\mathrm{\Lambda }}_1\right]}{(\frac{{\overline{k}}_{a1}}{{\overline{k}}_{\mathrm{bf}}}+2)-(\frac{{\overline{k}}_{a1}}{{\overline{k}}_{\mathrm{bf}}}-1){\mathrm{\Lambda }}_1},\frac{{\overline{\rho }}_{\mathrm{nf}}}{{\overline{\rho }}_{\mathrm{bf}}}=(1-{\mathrm{\Lambda }}_1)+\frac{{\overline{\rho }}_{a1}{\mathrm{\Lambda }}_1}{{\overline{\rho }}_{\mathrm{bf}}},\hfill \end{array}]$$

(10)

$$\begin{array}{c}\frac{{\overline{\mu }}_{\mathrm{hnf}}}{{\overline{\mu }}_{\mathrm{bf}}}=\frac{1}{{(1-{\mathrm{\Lambda }}_2-{\mathrm{\Lambda }}_1)}^{2.5}},\frac{{\overline{\rho }}_{\mathrm{hnf}}}{{\overline{\rho }}_{\mathrm{bf}}}=[(1-{\mathrm{\Lambda }}_2)+\frac{{\overline{\rho }}_{a1}{\mathrm{\Lambda }}_1}{{\overline{\rho }}_{\mathrm{bf}}}](1-{\mathrm{\Lambda }}_2)+\frac{{\overline{\rho }}_{b2}{\mathrm{\Lambda }}_2}{{\overline{\rho }}_{\mathrm{bf}}}\hfill \\ \frac{{\left({\hat{C}}_p\overline{\rho }\right)}_{\mathrm{hnf}}}{{\left({\hat{C}}_p\overline{\rho }\right)}_{\mathrm{bf}}}=[\frac{{\left({\hat{C}}_p\overline{\rho }\right)}_{a1}{\mathrm{\Lambda }}_1}{{\left({\hat{C}}_p\overline{\rho }\right)}_{\mathrm{bf}}}+(1-{\mathrm{\Lambda }}_1)](1-{\mathrm{\Lambda }}_2)+\frac{{\mathrm{\Lambda }}_2{\left({\hat{C}}_p\overline{\rho }\right)}_{b2}}{{\left({\hat{C}}_p\overline{\rho }\right)}_{\mathrm{bf}}},\hfill \\ \frac{{\left(\overline{\rho }\beta \right)}_{\mathrm{hnf}}}{{\left(\overline{\rho }\beta \right)}_{\mathrm{bf}}}=(1-{\mathrm{\Lambda }}_2)[(1-{\mathrm{\Lambda }}_1)+\frac{{\left(\overline{\rho }\beta \right)}_{a1}{\mathrm{\Lambda }}_1}{{\left(\overline{\rho }\beta \right)}_{\mathrm{bf}}}]+\frac{{\left(\overline{\rho }\beta \right)}_{b2}{\mathrm{\Lambda }}_2}{{\left(\overline{\rho }\beta \right)}_{\mathrm{bf}}}\hfill \\ \frac{{\overline{\sigma }}_{\mathrm{hnf}}}{{\overline{\sigma }}_{\mathrm{bf}}}=1+\frac{3\left(\frac{{\overline{\sigma }}_{a1}{\mathrm{\Lambda }}_1+{\overline{\sigma }}_{b2}{\mathrm{\Lambda }}_2}{{\overline{\sigma }}_{\mathrm{bf}}}\right)-3({\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2)}{2+\left\{\frac{{\overline{\sigma }}_{a1}{\mathrm{\Lambda }}_1+{\overline{\sigma }}_{b2}{\mathrm{\Lambda }}_2}{({\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2){\overline{\sigma }}_{\mathrm{bf}}}\right\}-\{\frac{{\overline{\sigma }}_{a1}{\mathrm{\Lambda }}_1+{\overline{\sigma }}_{b2}{\mathrm{\Lambda }}_2}{{\overline{\sigma }}_{\mathrm{bf}}}-({\mathrm{\Lambda }}_2+{\mathrm{\Lambda }}_1)\}},\hfill \\ \frac{{\overline{k}}_{\mathrm{hnf}}}{{\overline{k}}_{\mathrm{bf}}}=\frac{\frac{{\overline{k}}_{a1}{\mathrm{\Lambda }}_1+{\overline{k}}_{b2}{\mathrm{\Lambda }}_2}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2}+2({\overline{k}}_{a1}{\mathrm{\Lambda }}_1+{\overline{k}}_{b2}{\mathrm{\Lambda }}_2)+2{\overline{k}}_{\mathrm{bf}}-2{\hat{k}}_{\mathrm{bf}}({\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2)}{\frac{{\overline{k}}_{a1}{\mathrm{\Lambda }}_1+{\overline{k}}_{b2}{\mathrm{\Lambda }}_2}{{\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2}-2({\overline{k}}_{a1}{\mathrm{\Lambda }}_1+{\overline{k}}_{b2}{\mathrm{\Lambda }}_2)+2{\overline{k}}_{\mathrm{bf}}+{\overline{k}}_{\mathrm{bf}}({\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2)}\hfill \end{array}]$$

(11)

Above we have, $a1=$ diamond nanoparticles with volume fraction ${\mathrm{\Lambda }}_1$ and $b2=$ copper nanoparticles with volume fraction ${\mathrm{\Lambda }}_2$ .

By implanting equations (7)–(9) we have from above:

$$\frac{{\overline{\mu }}_H}{{\overline{\rho }}_H}(1+\frac{K1}{{\overline{\mu }}_H})f^{″\prime }-{f\prime }^2+\frac{1}{{\overline{\rho }}_H}({\overline{K}}_{H}g\prime -{\overline{\sigma }}_{H}M\sin \left(\gamma \right)f\prime -{\overline{\mu }}_H\frac{1}{\mathrm{Da}}\left(\frac{{[1-{\epsilon }_0(1+{\epsilon }_1{e}^{\frac{-{\epsilon }_2\eta }{\sqrt{\mathrm{Da}}}})]}^2}{{\left[{\epsilon }_0^3(1+{\epsilon }_1{e}^{\frac{-{\epsilon }_2\eta }{\sqrt{\mathrm{Da}}}})\right]}^3}\right)f\prime )+ff″=0,$$

(12)

$$\frac{{\overline{\mu }}_H}{{\overline{\rho }}_H}(1+\frac{K1}{2{\overline{\mu }}_H})g″+\mathrm{fg}\prime -\mathrm{gf}\prime -\frac{K1}{{\overline{\rho }}_H}(2g+f″)=0,$$

(13)

$$\begin{array}{c}\frac{1}{{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_H}.\frac{1}{\mathrm{Pr}}({\overline{K}}_H+\frac{4}{3}\mathrm{Rd})\theta ″+\theta \prime f+\frac{1}{{\left(\rho {\overline{C}}_p\right)}_H}(\mathrm{Nb}\vartheta \prime \theta \prime +\mathrm{Nt}{\theta \prime }^2)\hfill \\ +\frac{Q}{{\left(\rho {\overline{C}}_p\right)}_H}\theta -{\mathrm{\Omega }}_1(\mathrm{ff}\prime \theta \prime +f^2\theta ″)=0,\hfill \end{array}$$

(14)

$$\vartheta ″+\mathrm{Scf}\vartheta \prime +\frac{\mathrm{Nt}}{\mathrm{Nb}}\theta ″-\mathrm{Sc}{\mathrm{\Omega }}_2[\mathrm{ff}\prime \vartheta \prime +f^2\vartheta ″]=0.$$

(15)

The subjected BCs are described as:

$$\begin{array}{c}f\left(0\right)=0,f\prime \left(0\right)=1,g\left(0\right)=0,\theta \left(0\right)=1,\vartheta \left(0\right)=1,\mathrm{at}\eta =0\hfill \\ f\prime \left(\infty \right)=0,g\left(\infty \right)=0,\theta \left(\infty \right)=0,\vartheta \left(\infty \right)=0\mathrm{as}\eta \to \infty \hfill \end{array}$$

(16)

Above

$${\overline{\rho }}_H=\frac{{\overline{\rho }}_{\mathrm{hnf}}}{{\overline{\rho }}_{\mathrm{bf}}},{\overline{\mu }}_H=\frac{{\overline{\mu }}_{\mathrm{hnf}}}{{\overline{\mu }}_{\mathrm{bf}}},{\overline{\sigma }}_H=\frac{{\overline{\sigma }}_{\mathrm{hnf}}}{{\overline{\sigma }}_{\mathrm{bf}}},{\overline{K}}_H=\frac{{\overline{k}}_{\mathrm{hnf}}}{{\overline{k}}_{\mathrm{bf}}},{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_H=\frac{{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{hnf}}}{{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{bf}}}.$$

(17)

In above equations, several substantial factors arise, as explained below:

$\mathrm{Da}=v_f/\alpha m^2=$ variable porous factor, $\mathrm{Sc}={\nu }_{\mathrm{bf}}/D_B=$ Schmidt number, $M={\overline{\sigma }}_{\mathrm{bf}}{B_0}^2/{\overline{\rho }}_{\mathrm{bf}}a=$ magnetic parameter, $\mathrm{Pr}={\overline{k}}_{\mathrm{bf}}/{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{bf}}{v}_{\mathrm{bf}}=$ Prandtl number, $j={\nu }_{\mathrm{bf}}/a=$ micro inertial coefficient, $Q=\frac{Q_1}{a{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{bf}}}$  = heat source parameter $\mathrm{Nb}={\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{np}}{D}_B(C_w-C_{\infty })/{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{bf}}{\nu }_{\mathrm{bf}}=$ Brownian motion parameter, $\mathrm{Rd}=4{\sigma }_x{T_{\infty }}^3/k_e{\overline{k}}_{\mathrm{bf}}=$ radiation factor, $K1={\kappa }_a/{\overline{\mu }}_{\mathrm{bf}}=$ material factor, $\mathrm{\Omega }2$  =  $a{\lambda }_2$  = solute relaxation parameter, $\mathrm{\Omega }1=a{\lambda }_1=$ thermal relaxation factor, $\mathrm{Nt}={\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{np}}{D}_T(T_w-T_{\infty })/{\left(\overline{\rho }{\stackrel{⌢}{C}}_p\right)}_{\mathrm{bf}}{\nu }_{\mathrm{bf}}{T}_{\infty }=$ thermophoresis factor, and ${\chi }_{\mathrm{hnf}}$  =  $({\overline{\mu }}_{\mathrm{hnf}}+{\kappa }_a/2)j$  = spin gradient viscosity.

Quantities of engineering interest

The main quantities are skin friction, Nusselt and Sherwood numbers which are described as follows:

$$\begin{array}{c}{C}_{fr}=\frac{{\tau }_w}{\frac{1}{2}{\overline{\rho }}_{\mathrm{bf}}{u_w}^2},N{u}_x=\frac{x{q}_w}{{\overline{k}}_{\mathrm{bf}}(T_w-T_{\infty })},\hfill \\ S{h}_x=\frac{x{q}_m}{D_B(C_w-C_{\infty })},\hfill \end{array}$$

(18)

With ${\tau }_w=\left(({\overline{\mu }}_{\mathrm{hnf}}+{\kappa }_a)\frac{\partial u_1}{\partial x_2}\right)=$ shear stress, $q_w={\overline{k}}_{\mathrm{hnf}}(-\frac{\partial T}{\partial x_2})=$ heat flux and $q_m=-D_B\frac{\partial C}{\partial x_2}=$ mass flux

So equation (18) becomes

$$\begin{array}{c}{C}_{fr}=\frac{1}{{\overline{\rho }}_{\mathrm{bf}}{u_w}^2}{\left(({\overline{\mu }}_{\mathrm{hnf}}+{\kappa }_a)\frac{\partial u_1}{\partial x_2}\right)|}_{y=0},\hfill \\ N{u}_s=\frac{x_1{\overline{k}}_{\mathrm{hnf}}}{{\overline{k}}_{\mathrm{bf}}(T_w-T_{\infty })}{(-\frac{\partial T}{\partial x_2})|}_{y=0},\hfill \\ S{h}_z=\frac{x_1}{(C_w-C_{\infty })}{(-\frac{\partial C}{\partial x_2})|}_{y=0},\hfill \end{array}$$

(19)

Implementing equation (9) in equation (19) we have as follows:

$$\begin{array}{c}{\left({\mathrm{Re}}_{x_1}\right)}^{0.5}{C}_{fr}=2({\overline{\mu }}_H+K1)f″\left(0\right),\hfill \\ \multicolumn{1}{c}{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x=-{\overline{k}}_H(1+\frac{4}{3}\mathrm{Rd})\theta \prime \left(0\right),}\\ \multicolumn{1}{c}{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}S{h}_x=-\vartheta \prime \left(0\right).}\end{array}\}$$

(20)

${\mathrm{Re}}_{x_1}=\frac{u_w{x}_1}{{\nu }_{\mathrm{bf}}}$ is local Reynolds number.

Method for solution

We applied the ANN technique to solve equations (12)–(15) simultaneously in connection of equation (16). The strategy uses the Levenberg-Marquardt Scheme of a Backpropagation Neural Network Algorithm (LMS-BNNA) to handle the problem efficiently. Levenberg-Marquardt is an optimization technique that improves the backpropagation process to converge faster. In this method, the weights of a Multi-Layer Perceptron (MLP) are adjusted when trained to minimize the cost function, which is most commonly Mean Squared Error (MSE). The LMS-BNNA trains over several training epochs, gradually reducing errors and enhancing network accuracy. Incorporating Levenberg-Marquardt optimization, allows the backpropagation algorithm to minimize the errors faster and tackle complex data efficiently. This optimization becomes particularly helpful when the network closes the optimum results, promoting faster convergence. Figure 1(b) clarifies the process of the neural network with input and output values.

The activation used in this work is defined as follows:

$$\tilde{f}\left(x\right)=\frac{1}{1+\exp (-x)}$$

(21)

The solution presented in this article has been obtained using the Levenberg-Marquardt Scheme of a Backpropagation Neural Network Algorithm (LMS-BNNA) to efficiently handle the complex nonlinear governing equations of the problem. In this approach, we first generated a comprehensive numerical dataset by solving the mathematical model. This dataset includes input-output pairs representing the physical behavior of the micropolar hybrid nanofluid flow under various parameter conditions. Once the dataset was prepared, it was systematically divided into three distinct parts: training, validation, and testing datasets. The training set was used to teach the ANN how to map the input parameters to the corresponding output solutions. The validation set was employed to fine-tune the network’s performance and prevent overfitting, while the testing set was used to evaluate the accuracy and generalization capability of the trained model. Through this method, the ANN was successfully trained on the dataset using the Levenberg-Marquardt optimization algorithm, which is known for its fast convergence and high precision in nonlinear regression tasks. The LMS-BNNA enabled us to obtain an optimal and efficient solution to the problem, demonstrating both robustness and reliability in handling highly nonlinear and coupled physical phenomena.

Validation

For the purpose of validation, the current results have compared with published work of Ali et al., ⁴⁷ Ishak et al. ⁴⁸ and Abdal et al. ⁴⁹ In all these three Refs^47,48 the authors have used numerical schemes like finite element method, finite difference method and RK-4 approach. This comparison has conducted in Table 2 for variations in common parameter (i.e. Prandtl number) while setting all the other factors as zero, that is, $\mathrm{Nb}=0,\mathrm{Nt}=0,Q=0,{\mathrm{\Omega }}_1=0$ . A closed agreement has noticed among all the results that ensures the accuracy and reliability of current results.

Table 2.

Comparison of current results with Refs^47,48 for $\{-\theta \prime \left(0\right)\}$ .

Pr	Ali et al. 47	Ishak et al. 48	Abdal et al. 49	Current results
0.72	0.80860	0.80860	0.80860	0.80860
1.00	1.00000	1.00000	1.00000	1.00000
3.00	1.92360	1.92370	1.92370	1.92371
10.00	3.72060	3.72070	3.72070	3.72070
100.0	12.2946	12.2941	12.2941	12.2942

Results and discussion

This work examines the micropolar hybrid nanofluid flow on a variable porous elongating sheet using inclined magnetic effects. The fluid is set into motion by stretching nature of the sheet. The effects of heat source/sink and thermal radiation have utilized in this work. The well-known Cattaneo-Christov flux model has also employed in the study to regulate the mass and thermal diffusions. The leading equations have evaluated by using ANN method.

Analysis of ANN graphs portrays

The process of neural network has portrayed in Figure 1(b), where the LMS-BNNA process heightens the network structure. Figures 2 to 5 demonstrate the linear velocity, micro-rotational velocity, temperature and concentration scenarios against various substantial factors. The convergence of MSE for various cases has examined in Figures 2 to 5(a). The best performance is achieved at epochs 111, 225, 220, and 399 as noticed in these figures. The gradient of the model for LMS-NNA design has illustrated in Figures 2 to 5(b) with clear evaluation through different stages. These graphical views portray the evaluation of model through initialization, training, optimization and evaluation phases. They identify variations in proficiency measures and the modifications that have been made to increase the correctness and effectiveness of the model. Figures 2 to 5(c) present the error histograms for the design of LMS-NNA. These histograms present a graphical view of error distribution, with the frequency of different magnitudes of errors. From these graphical views the performance and precision of the design can be evaluated. Subject to the relation of EA structures, the curve fitness of the design problem has illustrated in Figures 2 to 5(d). The fitness curves describe how the solutions generated of the modeled problem behave over various iterations, showing how well the evolutionary algorithm (EA) solves the problem. These curves give indications about the algorithm’s convergence, solution quality, and performance as a whole throughout the building process of modeled design. Figures 2 to 5(e) exemplifies the model’s capability to perform regression tasks subject to diverse conditions. The plots represent the model’s success in capturing input-output variable relationship in every scenario. Training, testing, and validation values for regression are all closely clustered around one, indicating high model performance.

Figure 2.

Graphs of LMS-NNA design for velocity scenario. (a) MES outcome. (b) Transition state. (c) Error Histogram. (d) Curve fitting. (e) Regression.

Figure 3.

Graphs of LMS-NNA design’s for micro-rotational velocity scenario. (a) MES outcome. (b) Transition state. (c) Error Histogram. (d) Curve fitting. (e) Regression.

Figure 4.

Graphs of LMS-NNA design’s for temperature scenario. (a) MES outcome. (b) Transition state. (c) Error Histogram. (d) Curve fitting. (e) Regression.

Figure 5.

Graphs of LMS-NNA design’s for concentration scenario. (a) MES outcome. (b) Transition state. (c) Error Histogram. (d) Curve fitting. (e) Regression.

Linear and micro-rotational velocity distributions $\left\{f^{\prime }\left(\eta \right)pg\left(\eta \right)\right\}$

The behavior of velocity $f\prime \left(\eta \right)$ for variations in distinct factors has illustrated in Figure 6(a) to (d), while Figure 6(e) to (h) examine the behavior of micro-rotational velocity $g\left(\eta \right)$ .

Figure 6.

(a) Behavior of $f\prime \left(\eta \right)$ versus $\mathrm{Da}$ . (b) AE versus $\mathrm{Da}$ . (c) Behavior of $f\prime \left(\eta \right)$ versus $M$ . (d) AE versus $M$ . (e) Behavior $g\left(\eta \right)$ versus $M$ . (f) AE versus $M$ . (g) Behavior of $g\left(\eta \right)$ versus $K1$ . (h) AE versus $K1$ .

Impacts of $\mathrm{Da}$ on $f\prime \left(\eta \right)$ : The influence of variations in variable porous factor $\left(\mathrm{Da}\right)$ on velocity profiles $\{f\prime \left(\eta \right)\}$ is described in Figure 6(a). A rise in the variable porous factor $\left(\mathrm{Da}\right)$ designates that the medium on which the fluid is flowing has greater resistance because of the existence of larger pores that interrupts the flow on the surface of stretching sheet. Consequently, the fluid faces greater friction and has trouble sustaining its velocity. This added resistance causes the velocity of the fluid to decline as it flows on the surface of stretching sheet. Physically, this can be likened to blood flowing through a coarse mesh versus a fine mesh: the higher the mesh (higher porous factor), the slower the blood flows as there is more obstruction and loss of energy. The AE in $\{f\prime \left(\eta \right)\}$ for hike in $\left(\mathrm{Da}\right)$ is described in Figure 6(b). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-8}$ . Variable porosity enables simulation of patient-specific tissue morphology, enhancing accuracy in heat distribution in hyperthermia. Accounting for differing tissue porosity, practitioners are able to better control temperature to allow optimal tumor heating with the preservation of healthy tissues.

Impacts of $M$ on $f\prime \left(\eta \right)$ : The effects of variations in magnetic factor $\left(M\right)$ on $\{f\prime \left(\eta \right)\}$ are portrayed in Figure 6(c). With the increase of $\left(M\right)$ , in blood-based hybrid nanofluid during motion on the stretching sheet, velocity $\{f\prime \left(\eta \right)\}$ reduces. This happens under the MHD effect in which the collaboration of the magnetic field and conductive nanofluid produces an opposing force termed the Lorentz force. This force works against the movement of fluid in a frictional drag fashion. Thus, the flow speed of the nanofluid is reduced as the magnetic field constrains the kinetic energy of the fluid. This phenomenon finds particular importance in biomedical and engineering applications, such as targeted drug delivery and cooling systems, where fluid dynamics should be controlled accurately under magnetic effects. The AE (absolute error) in velocity $\{f\prime \left(\eta \right)\}$ for higher $\left(M\right)$ is described in Figure 6(d). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-8}$ .

Impacts of $M$ on $g\left(\eta \right)$ : The influence of variations in magnetic factor $\left(M\right)$ on velocity $g\left(\eta \right)$ is portrayed in Figure 6(e). A rise in $\left(M\right)$ affects the dynamics of $g\left(\eta \right)$ . In the close vicinity of the surface of sheet on the interval $0\le \eta <1.0$ , the magnetic field causes more enhanced Lorentz forces, which produce further torque on the nanofluid particles, leading to a rise in micro-rotational velocity $g\left(\eta \right)$ . This is caused by the direct impact of the magnetic force and the shearing action of the stretching sheet. Conversely, away from the sheet’s surface on the interval $1.0<\eta \le 6.0$ , the influence of these forces becomes weaker, and micro-rotational velocity $g\left(\eta \right)$ is decreased. Magnetic influence decreases with augmented distance from the sheet’s surface, resulting in reduced torque on the particles and a slower rate of micro-rotation. This double behavior indicates the spatial nature of magnetic field influence within the fluid flow. The AE in velocity $\{f\prime \left(\eta \right)\}$ for higher $\left(M\right)$ is described in Figure 6(f). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-3}$ – ${10}^{-6}$ .

Impacts of $K1$ on $g\left(\eta \right)$ : The influence of variations in material factor $\left(K1\right)$ on micro-rotational velocity $g\left(\eta \right)$ is portrayed in Figure 6(g). When material factor $\left(K1\right)$ grows, the micro-rotational velocity $g\left(\eta \right)$ also surges. Physically, this means that the nanoparticles, being part of the hybrid nanofluid, will have more rotational motion due to increased material properties, like viscosity and elasticity, or intensified interactions between nanoparticles and blood. This increased micro-rotational speed can be explained as a result of the sheet stretching effect, producing shear forces that affect not only the translational motion but also the rotational behavior of the suspended nanoparticles in blood. This improved micro-rotation contributes to increased mixing efficiency and heat transfer inside the fluid, which is extremely important in biomedical applications and nanofluid-based cooling devices. The AE (absolute error) in velocity $\{f\prime \left(\eta \right)\}$ for higher $\left(K1\right)$ is described in Figure 6(h). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-3}$ – ${10}^{-7}$ .

Temperature distribution $\left\{\theta \left(\eta \right)\right\}$

The influences of distinct emerging factors on temperature distribution $\left\{\theta \left(\eta \right)\right\}$ are portrayed in Figure 7(a) to (l).

Figure 7.

(a) Behavior of $\theta \left(\eta \right)$ versus $\mathrm{Nb}$ . (b) AE $\theta \left(\eta \right)$ versus $\mathrm{Nb}$ . (c) Behavior of $\theta \left(\eta \right)$ versus $\mathrm{Nt}$ . (d) AE $\theta \left(\eta \right)$ versus $\mathrm{Nt}$ . (e) Behavior of $\theta \left(\eta \right)$ versus $M$ . (f) AE $\theta \left(\eta \right)$ versus $M$ . (g) Behavior of $\theta \left(\eta \right)$ versus $\mathrm{Rd}$ . (h) AE $\theta \left(\eta \right)$ versus $\mathrm{Rd}$ . (i) Behavior of $\theta \left(\eta \right)$ versus ${\mathrm{\Omega }}_1$ . (j) AE $\theta \left(\eta \right)$ versus ${\mathrm{\Omega }}_1$ . (k) Behavior of $\theta \left(\eta \right)$ versus $Q$ . (l) AE $\theta \left(\eta \right)$ versus $Q$ .

Impacts of $\mathrm{Nb}$ on $\left\{\theta \left(\eta \right)\right\}$ : The influence of Brownian motion factor $\left(\mathrm{Nb}\right)$ on $\left\{\theta \left(\eta \right)\right\}$ is described in Figure 7(a). A rise in $\left(\mathrm{Nb}\right)$ signifies that diamond and copper nanoparticles suspended get more erratic, random movement due to sheet’s surface collision with nearby blood molecules. The greater Brownian motion increases diamond and copper nanoparticle kinetic energy, resulting in better thermal diffusion of the fluid. Physically, this means more effective heat transfer since the nanoparticles are more actively distribute the thermal energy within the fluid. The AE in temperature $\left\{\theta \left(\eta \right)\right\}$ for higher $\left(\mathrm{Nb}\right)$ is showed in Figure 7(b). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-8}$ . Brownian motion results in the nanoparticles to be transported randomly throughout the fluid, tending to spread thermal energy more consistently. This enhances the efficiency of heating in hyperthermia therapy for cancer treatment. The greater thermal conductivity due to higher Brownian motion makes temperature control more specific in the targeted areas, with less damage being done to healthy tissues surrounding the target during hyperthermia.

Impacts of $\mathrm{Nt}$ on $\left\{\theta \left(\eta \right)\right\}$ : The impacts of thermophoresis factor $\left(\mathrm{Nt}\right)$ on $\left\{\theta \left(\eta \right)\right\}$ are portrayed in Figure 7(c). A rise in $\left(\mathrm{Nt}\right)$ is an indication that nanoparticles in the fluid are more strongly pushed by temperature gradients. Physically, this implies that with a larger temperature difference, nanoparticles have a tendency to migrate from hotter regions toward cooler regions on sheet’s surface. For blood-based hybrid nanofluid, comprised of copper and diamond nanoparticles mixed with the bio-medium blood, improved thermophoresis can enhance thermal conductivity, and efficiency in heat transfer. The stretching sheet puts a force of shear that affects fluid flow, and the increased thermophoresis assists the even distribution of nanoparticles, avoiding clattering, and ensuring improved thermal performance. This equilibrium between forces is important in order to sustain both effective heat distribution and fluid stability. The AE (absolute error) in temperature $\left\{\theta \left(\eta \right)\right\}$ for higher $\left(\mathrm{Nt}\right)$ is revealed in Figure 7(d). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-3}$ – ${10}^{-6}$ .

Impacts of $M$ on $\left\{\theta \left(\eta \right)\right\}$ : The impacts of magnetic factor $\left(M\right)$ on $\left\{\theta \left(\eta \right)\right\}$ are portrayed in Figure 7(e). A rise in magnetic factor $\left(M\right)$ for fluid flow on a stretching sheet affects the fluid dynamics because of the magnetic nature of the nanoparticles dispersed in the blood. Surge in $\left(M\right)$ generates a Lorentz force that acts against the fluid motion, resulting in an increased resistance to flow. This resistance results in a reduction in fluid velocity in the vicinity of the stretching sheet but also increases thermal conductance because of nanoparticle clustering under magnetization, enhancing heat transfer performance. In this phenomenon the thermal profiles $\left\{\theta \left(\eta \right)\right\}$ are ultimately augmented. The combined characteristics of the hybrid nanofluid with diamond and copper nanoparticles further enhance these effects, providing more control over both mechanical and thermal behaviors of fluid. The AE (absolute error) in temperature $\left\{\theta \left(\eta \right)\right\}$ for higher $\left(M\right)$ is represented in Figure 7(d). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-7}$ . Physically, surge in magnetic factor $\left(M\right)$ is very useful in hyperthermia treatment, where fluid flow and heat distribution in the bloodstream need to be controlled.

Impacts of $\mathrm{Rd}$ on $\left\{\theta \left(\eta \right)\right\}$ : The impacts of radiation factor $\left(\mathrm{Rd}\right)$ on $\left\{\theta \left(\eta \right)\right\}$ are portrayed in Figure 7(g). When the radiation factor $\left(Rd\right)$ rises for blood-based hybrid nanofluid flow over stretching sheet, it shows greater intensity of thermal radiations in the system. Physically, this illustrates that, more thermal energy is being discharged and absorbed by nanoparticles (diamond and copper nanoparticles) dispersed in blood. With increasing radiation factor, the boundary width of the thermal layer increases since the raised radiation increases the rate of heat transfer from the fluid to the ambient environment on the surface of the sheet. This leads to a higher temperature of the fluid, which reduces the nanofluid viscosity, encouraging smoother, and quicker fluid flow along the elongating sheet. The AE (absolute error) in temperature $\left\{\theta \left(\eta \right)\right\}$ for higher $\left(M\right)$ is portrayed in Figure 7(h). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-7}$ .

Impacts of ${\mathrm{\Omega }}_1$ on $\left\{\theta \left(\eta \right)\right\}$ : The impacts of thermal relaxation parameter $\left({\mathrm{\Omega }}_1\right)$ on $\left\{\theta \left(\eta \right)\right\}$ are portrayed in Figure 7(i). The thermal relaxation parameter $\left({\mathrm{\Omega }}_1\right)$ is actually a measure of delay in the reaction of the fluid temperature to a change in the thermal conditions. As the values of this parameter increase, the heat transfer mechanism does not regulate immediately for variation in temperature and the thermal inertia of the fluid becomes more intense. Physically speaking, this implies that the hybrid nanofluid (diamond, copper nanoparticles, and blood) is slower to transfer heat due to this delay in relaxation. This implies that the fluid becomes cooler at a slower rate, hence reducing the total temperature profile close to the stretching sheet. This phenomenon can be attributed to the reason that with increased thermal relaxation $\left({\mathrm{\Omega }}_1\right)$ , the system is more resistant to the diffusion of heat, which decreases the rate at which the thermal energy is transferred across the nanofluid close to sheet’s surface. The AE (absolute error) in temperature $\left\{\theta \left(\eta \right)\right\}$ for higher $\left({\mathrm{\Omega }}_1\right)$ is portrayed in Figure 7(j). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-7}$ .

Impacts of $Q$ on $\left\{\theta \left(\eta \right)\right\}$ : The impacts of heat source factor $\left(Q\right)$ on $\left\{\theta \left(\eta \right)\right\}$ are portrayed in Figure 7(k). As heat source factor $\left(Q\right)$ rises the thermal profiles $\left\{\theta \left(\eta \right)\right\}$ of the fluid are influenced enormously. Physically, the source of heat adds extra thermal energy to the fluid flow system, and the nanoparticles of the hybrid nanofluid absorb and distribute this heat even in better manner. Because blood has a greater thermal conductivity than regular fluids, its mixture with hybrid nanoparticles (diamond and copper) increases the efficiency of heat transfer. When the stretching sheet moves, it produces shear forces that enhance the internal mixing of the fluid, permitting the heat energy to diffuse more evenly. This leads to an increase in the temperature distribution $\left\{\theta \left(\eta \right)\right\}$ in the vicinity of the sheet as a whole, and leads to enhancement of the temperature profiles $\left\{\theta \left(\eta \right)\right\}$ . The AE (absolute error) in temperature $\left\{\theta \left(\eta \right)\right\}$ for higher $\left(Q\right)$ is portrayed in Figure 7(l). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-7}$ .

Concentration distribution

The influence of diverse substantial factors on concentration panels $\vartheta \left(\eta \right)$ is explained in Figure 8(a) to (h).

Figure 8.

(a) Behavior of $\vartheta \left(\eta \right)$ versus $\mathrm{Nb}$ . (b) AE versus $\mathrm{Nb}$ . (c) Behavior of $\vartheta \left(\eta \right)$ versus $\mathrm{Nt}$ . (d) AE versus $\mathrm{Nt}$ . (e) Behavior of $\vartheta \left(\eta \right)$ versus $\mathrm{Sc}$ . (f) AE versus $\mathrm{Sc}$ . (g) Behavior of $\vartheta \left(\eta \right)$ versus ${\mathrm{\Omega }}_2$ . (h) AE versus ${\mathrm{\Omega }}_2$ .

Impacts of $\mathrm{Nb}$ on $\left\{\vartheta \left(\eta \right)\right\}$ : The effect of heat source factor $\left(\mathrm{Nb}\right)$ on $\left\{\vartheta \left(\eta \right)\right\}$ is portrayed in Figure 8(a). The decrease in concentration profiles $\left\{\vartheta \left(\eta \right)\right\}$ for growth in Brownian motion factor $\left(\mathrm{Nb}\right)$ for fluid flow on the stretching sheet indicates that with surge in $\left(\mathrm{Nb}\right)$ , the dispersion of diamond and copper nanoparticles in the fluid becomes more intense. Actually, growth in Brownian motion factor $\left(\mathrm{Nb}\right)$ results in a better mixing of the nanoparticles in the fluid leading to a higher distribution of particles. In blood-based hybrid nanofluid, this enhanced diffusion causes a reduction in the nanoparticle concentration close to the stretching sheet surface. Fundamentally, the enhanced Brownian motion increases the rate of nanoparticles dispersal across the fluid, causing reduced localized concentrations at the surface, which affects the fluid’s thermal and flow properties close to the sheet. The AE (absolute error) in concentration $\left\{\vartheta \left(\eta \right)\right\}$ for higher $\left(\mathrm{Nb}\right)$ is revealed in Figure 8(b). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-8}$ .

Impacts of $\mathrm{Nt}$ on $\left\{\vartheta \left(\eta \right)\right\}$ : The effect of $\left(\mathrm{Nt}\right)$ on $\left\{\vartheta \left(\eta \right)\right\}$ is described in Figure 8(c). The increase in $\left(\mathrm{Nt}\right)$ for nanoparticles fluid flow results in concentration intensification. Thermophoresis actually defines the particles’ motion because of the temperature gradient, whereby the fluid’s particles normally transferred from the warmer region to the cooler region. The enhancement of the thermophoresis factor implies that diamond and copper nanoparticles have a higher tendency to shift toward areas of lower temperature. Consequently, at the stretching sheet, where fluid is usually exposed to a temperature gradient as a result of stretching and cooling, the particles accumulate more, thus resulting in an increased nanoparticle concentration. This phenomenon plays a significant role in enhancing the heat transfer capability of the nanofluid, which could be useful for applications such as cooling systems or biomedical engineering where the control of nanoparticle dispersion is vital. The AE (absolute error) in concentration panels $\left\{\vartheta \left(\eta \right)\right\}$ for higher $\left(\mathrm{Nt}\right)$ is represented in Figure 8(d). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-7}$ .

Impacts of $\mathrm{Sc}$ on $\left\{\vartheta \left(\eta \right)\right\}$ : The influence of heat source factor $\left(\mathrm{Sc}\right)$ on $\left\{\vartheta \left(\eta \right)\right\}$ is portrayed in Figure 8(e). With increasing $\left(\mathrm{Sc}\right)$ , the mass diffusivity (pertaining to the transport of concentration) is smaller than the momentum diffusivity (which rules fluid flow). This implies a reduced rate of concentration transport in the fluid. The concentration profiles $\left\{\vartheta \left(\eta \right)\right\}$ of solute or nanoparticles close to the sheet surface thus reduce due to the inefficiency of mass transfer. In other words, the rise in $\left(\mathrm{Sc}\right)$ , leads to the slow spreading of the fluid being transported, causing a lower concentration at the surface of sheet. This phenomenon is especially significant in operations where the distribution of particles or solute has to be managed, such as in biomedical processes involving blood-based nanofluids. The AE (absolute error) in concentration panels $\left\{\vartheta \left(\eta \right)\right\}$ for higher $\left(\mathrm{Sc}\right)$ is described in Figure 8(f). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-4}$ – ${10}^{-7}$ .

Impacts of ${\mathrm{\Omega }}_2$ on $\left\{\vartheta \left(\eta \right)\right\}$ : The impacts of solute relaxation parameter $\left({\mathrm{\Omega }}_2\right)$ on $\left\{\vartheta \left(\eta \right)\right\}$ are represented in Figure 8(e). The increase in $\left({\mathrm{\Omega }}_2\right)$ implies that the solute responds more slowly to a change in flow conditions. When this relaxation parameter $\left({\mathrm{\Omega }}_2\right)$ increases, the solute’s capability to respond rapidly to the flow dynamics is diminished. This causes to decrease the profiles $\left\{\vartheta \left(\eta \right)\right\}$ , which implies that the solute is no longer uniformly distributed throughout the fluid. Physically, this can be explained as the solute particles facing higher resistance to move as a result of interactions between the stretching surface and the nanofluid. As a result, the fluid flow becomes stable with reduced mixing of the solute, resulting in reduced solute concentration over the flow region. The AE (absolute error) in concentration panels $\left\{\vartheta \left(\eta \right)\right\}$ for higher $\left({\mathrm{\Omega }}_2\right)$ is revealed in Figure 8(f). These absolute errors are determined by applying LMS-NNA design which falls within the range ${10}^{-3}$ – ${10}^{-7}$ .

Discussion of tables

Table 3 illustrates a comprehensive overview of MSE values of testing, training, and validation phases for implementation of LMS-NNA model. Apart from the MSE, it also emphasizes the key performance matrics, such as the mu parameter and the gradient, that are part of the learning process of the model. The table identifies that the model achieves best performance at epochs 111, 225, 220, and 399. These are the top performance points from four unique scenarios of flow profiles. These epochs correspond to the instances during training when the performance of the model was tested with respect to the given metrics. The fluctuations that are noticed in the MSE at various epochs are essential to comprehend the model training dynamics. The fluctuations illustrate about the convergence pattern of the model, the efficiency of parameter, and the modifications that are applied on the internal weights of the network over these epochs. Such precise data is necessary to refine the model and choose the best parameters to maximize the particular artificial neural network design so that it can work optimally for every situation under study. The influences of diverse factors on skin friction $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{0.5}{C}_{fr}\right\}$ are portrayed in Table 4. The physical interpretation of results in this table is that magnetic factor and porous variable factor enhance resistance to fluid flow near the sheet’s surface, and hence declined the skin friction $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{0.5}{C}_{fr}\right\}$ . This influence is more substantial in hybrid nanofluid because it has better thermal and momentum transport capabilities than conventional nanofluids. The existence of two types of nanoparticles in hybrid nanofluid causes stronger interaction and greater effective viscosity, thus greater shear stress at the surface of sheet. However, a surge in the material parameter, which is associated with fluid elasticity or relaxation characteristics, reduces the velocity gradient close to the wall, which causes skin friction $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{0.5}{C}_{fr}\right\}$ to decrease for both fluids. The influences of diverse factors on Nusselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ are portrayed in Table 5. The physical meaning of the results in this table is that the increase in the magnetic factor, Brownian motion factor, radiation factor, and heat source factor increases the thermal energy transport in the fluid, causing an increase in the Nusselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ , which signifies better heat transfer at the surface of stretching sheet. This improvement is greater in hybrid nanofluid because of its better thermal conductivity and synergistic actions of several nanoparticles, permitting more effective heat dissipation. An increase in the thermophoresis parameter, which expels nanoparticles from hot areas, and in the thermal relaxation time, which retards heat conduction response, both these factors decrease the surface temperature gradient of fluid. Consequently, the Nusselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ reduces for nanofluid and hybrid nanofluid, indicating reduced heat transfer effectiveness. Table 6 portrays the influences of diverse factors on Sherwood number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}S{h}_x\right\}$ that can be physically interpreted as, an increase in thermophoresis, Brownian motion factors, Schmidt number, and mass relaxation time factor improves mass transfer rate on the surface and causes an increase in the Sherwood number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}S{h}_x\right\}$ . Actually, thermophoresis and Brownian motion factors result in higher nanoparticle motion, facilitating stronger mixing and concentration gradients close to the surface. A greater Schmidt number, which represents lower mass diffusivity in comparison to momentum diffusivity, will thin the concentration boundary layer, leading to greater mass transfer. Furthermore, a greater mass relaxation time means a sluggish diffusion response, which will enhance concentration gradients and cause larger Sherwood number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}S{h}_x\right\}$ for more effective mass transport.

Table 3.

LMS-NNA design results against various cases.

Scenarios	MSE			Performance	Gradient	Mu	Epoch	Time (s)
	Training	Validation	Testing
1	$1.5346\times {10}^{-9}$	$1.2987\times {10}^{-9}$	$2.4321\times {10}^{-9}$	$1.23\times {10}^{-9}$	$2.54\times {10}^{-8}$	$1.00\times {10}^{-8}$	111	02
2	$5.4563\times {10}^{-9}$	$6.5342\times {10}^{-9}$	$5.5342\times {10}^{-9}$	$5.87\times {10}^{-9}$	$8.56\times {10}^{-8}$	$1.00\times {10}^{-8}$	225	03
3	$6.6345\times {10}^{-10}$	$6.3452\times {10}^{-10}$	$7.8743\times {10}^{-10}$	$6.87\times {10}^{-9}$	$8.98\times {10}^{-8}$	$1.00\times {10}^{-8}$	220	02
4	$2.5342\times {10}^{-9}$	$2.1763\times {10}^{-9}$	$3.1764\times {10}^{-9}$	$2.65\times {10}^{-9}$	$8.83\times {10}^{-8}$	$1.00\times {10}^{-7}$	399	01

Table 4.

Impacts of various factors on skin friction $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{0.5}{C}_{fr}\right\}$ .

$M$	$K_1$	$\mathrm{Da}$	${\left({\mathrm{Re}}_{x_1}\right)}^{0.5}{C}_{fr}$
			Nanofluid	Hybrid nanofluid
0.1			0.75332522	0.85797434
0.2			0.76687422	0.86875464
0.3			0.77475312	0.870743897
0.4			0.78643735	0.88436489
	0.1		0.97548535	0.99785422
	0.2		0.95774747	0.96531534
	0.3		0.93576448	0.94221543
	0.4		0.91975419	0.92467376
		0.1	0.62965716	0.82345644
		0.2	0.65215727	0.86367734
		0.3	0.69689927	0.89087554
		0.4	0.71357428	0.93786322

Table 5.

Impacts of various factors on Nusselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ .

$\mathrm{Nt}$	$Q$	$\mathrm{\Omega }1$	$\mathrm{Rd}$	$M$	$\mathrm{Nb}$	${\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x$
						Nanofluid	Hybrid nanofluid
0.1						1.49754732	1.69646733
0.2						1.48086976	1.68659423
0.3						1.47086522	1.67437454
0.4						1.46086398	1.66797833
	0.1					1.23864809	1.50753734
	0.2					1.28649590	1.55865856
	0.3					1.33754322	1.60875677
	0.4					1.38976533	1.65764387
		0.1				1.28976965	1.67549878
		0.2				1.26597555	1.65965388
		0.3				1.24098645	1.63875877
		0.4				1.22965945	1.61467378
			0.1			1.19754845	1.29754878
			0.2			1.21879565	1.32674777
			0.3			1.23087666	1.34986587
			0.4			1.25880976	1.37008678
				0.1		1.68985467	1.86547877
				0.2		1.70537688	1.88535765
				0.3		1.72557576	1.90643755
				0.4		1.74325753	1.92643758
					0.1	1.46086334	1.66797898
					0.2	1.47086523	1.67437488
					0.3	1.48086944	1.68659477
					0.4	1.49754754	1.69646768

Table 6.

Impacts of various factors on Sherwood number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}S{h}_x\right\}$ .

$\mathrm{Nt}$	$\mathrm{\Omega }2$	$\mathrm{Sc}$	$\mathrm{Nb}$	${\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}S{h}_x$
0.1				1.65473212
0.2				1.78697611
0.3				1.83652223
0.4				1.88863822
	0.1			1.29594512
	0.2			1.40864511
	0.3			1.65755522
	0.4			1.89696512
		0.1		1.18956512
		0.2		1.30766624
		0.3		1.58097643
		0.4		1.97484522
			0.1	1.69546765
			0.2	1.75768856
			0.3	1.75757644
			0.4	1.78575334

Percentage growth in heat transfer rate

Figure 9 demonstrates the percentage growth in Nesselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ for ${\mathrm{\Lambda }}_1$ (diamond nanoparticles) and ${\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2$ (diamond + copper nanoparticles) on the range from 0.00 to 0.05. It has noticed from this figure that when the diamond nanoparticles augments from 0.00 to 0.05 the Nesselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ surges from 0.00% to 8.27% while on the same range of nanoparticles volume fraction it augments from 0.00% to 8.46% for diamond + copper nanoparticles that ensures the dominance of hybrid nanoparticles. Actually the thermal conductivity of the fluid increases as a result of increased concentration of the nanoparticles, showing more escalation in heat transfer rate. Here, thermal efficiency is presented in percentage form, indicating the relative efficiency with which the fluid is able to transfer heat with respect to the base fluid. The findings of this figure show that hybrid nanofluid, made up of a mixture of diamond and copper, has better heat transfer properties than nanofluid made from a single material. The higher thermal performance of hybrid nanofluid results from the synergistic action of the materials making up the hybrids, which enhances heat conduction and dispersion.

Figure 9.

Percentage growth in Nesselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ for ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_1+{\mathrm{\Lambda }}_2$ .

Conclusions

This research is particularly significant for biomedical applications like hyperthermia in cancer treatment. The study examines micropolar hybrid nanofluid flow on a variable porous elongating sheet with impacts of inclined magnetic field. The fluid is put into motion by stretching nature of the sheet using certain flow conditions. The famous Cattaneo-Christov flux model has also used in the work to control the thermal and mass diffusions. The main equations have evaluated using ANN technique. The below mentioned points have deduced in this study:

• The linear velocity of the fluid has decreased due to the increase in magnetic factor and variable porous factor, with AE varying through the range ${10}^{-4}$ – ${10}^{-8}$ .

• For the four scenarios the gradient values fall at $2.54\times {10}^{-8}$ , $8.56\times {10}^{-8}$ , $8.98\times {10}^{-8}$ and $8.83\times {10}^{-8}$ with corresponding epochs 111, 225, 220, and 399.

• The distribution of micro-rotational velocity increased as the material factor grew. This change indicates that this factor had a significant impact on the velocity distribution, with a noticeable rise observed as the factor intensifies.

• For surge in magnetic parameter, micro-rotational velocity profiles have intensified in the region on the interval $0.0\le \eta <1.50$ near the sheet’s surface, due to the support of Lorentz forces, while they are weakened away from the surface on the domain $1.5\le \eta <6.0$ due to the opposing effect on micro-rotations.

• Thermal distributions have increased due to the influence of thermophoresis, Brownian motion, magnetic effects, radiation, and heat source factor while they have decreased with a rise in the thermal relaxation parameter.

• The concentration panels have amplified with a rise in the thermophoresis parameter and decreased with an increase in solute relaxation factor, Schmidt number and Brownian motion factor.

• It has noticed in this work that when the diamond nanoparticles augments from 0.00 to 0.05 the Nesselt number $\left\{{\left({\mathrm{Re}}_{x_1}\right)}^{-0.5}N{u}_x\right\}$ surges from 0.00% to 8.27% while on the same range of nanoparticles volume fraction it augments from 0.00% to 8.46% for diamond + copper nanoparticles that ensures the dominance of hybrid nanoparticles.

• The current work has validated through matching its outcomes with established results by observing a fine agreement among all the results.