Abstract
The thermophysical performance of nanofluids is strongly influenced by nanoparticle morphology, making realistic property modeling essential for accurately predicting coupled flow and heat transfer characteristics in advanced thermal systems. Motivated by this need, the present study investigates magnetohydrodynamic flow of nanofluid between two active parallel plates embedded in a porous medium under quadratic thermal radiation and convective boundary condition. Multiple viscosity and thermal conductivity correlations, including morphology-dependent models for spherical, brick, cylindrical and platelet nanoparticles, are incorporated to quantify their influence on momentum and heat transfer. The governing nonlinear partial differential equations are transformed into coupled ordinary differential equations using similarity transformations and solved through the Tchebichef polynomial collocation method. Numerical accuracy is verified using the Runge–Kutta-Fehlberg fourth-fifth order method, while a Levenberg–Marquardt artificial neural network is employed to predict the nonlinear velocity and temperature fields. Results demonstrate that increasing magnetic field strength and porous resistance suppress velocity, whereas larger radiation and Biot numbers enhance temperature. The proposed framework provides reliable guidance for thermal management technologies.
Keywords: Morphology, Nanofluid, Levenberg–Marquardt algorithm, Tchebichef polynomial collocation, Active parallel plates
Introduction
The transfer of thermal energy from an area of high temperature to an area of low temperature by conduction, convection and radiation. It is one of the basic parameters that affect the thermal performance and energy efficiency of engineering systems. For fluid flow applications, the heat transfer is affected by flow properties, external effects (thermal radiation, diffusion effects), boundary conditions and fluid properties. To optimize performance, reliability and maximize energy utilization, effective control of heat transfer is necessary for thermal management, heat exchangers, rotating machinery, cooling technologies, chemical processing, power generation, microfluidic devices and advanced manufacturing systems. Akbar et al. [1] elucidated the heat transfer and flow of nanofluid (N-F) via a permeable medium. Sarfraz and Khan [2] studied the heat transfer and hybrid N-F flow due to an inclined stretchable surface. Akbar et al. [3] exemplified the effect of nanoparticle aggregation on the flow and heat transfer of N-F. Muhammed et al. [4] deliberated the heat transfer and flow of hybrid N-F flow subjected to coaxial cylinders using the Keller-box scheme. Iqbal et al. [5] scrutinized the heat transfer and hybrid N-F flow through a symmetric channel. The heat transfer and flow of bioconvective Jeffery N-F past a stretchable surface were investigated by Muhammad et al. [6]. Sarfaraz and Khan [7] exemplified the consequence of a heat source/sink on the heat transfer and flow of hybrid N-F via a permeable inclined surface.
N-Fs are novel heat transfer materials developed by combining ultrafine solid particles, usually on the nanometre scale, with conventional fluids such as water or ethylene glycol. Adding nanoparticles to the base fluid changes the effective viscosity, thermal conductivity, density and heat capacity of the fluid, thereby altering momentum and energy transfer properties. Their high surface-to-volume ratio increases interfacial heat exchange and makes them suitable for high-performance thermal systems. The concentration, size distribution and morphology of particles in N-Fs are critical to their overall behavior, determining flow resistance and thermal enhancement. Proper mathematical modeling is therefore necessary to estimate their performance, especially when magnetic fields, porous media or radiative heat transfer effects are involved in the system. Jagadha et al. [8] elucidated the N-F flow in a microtube with slip conditions. Khan et al. [9] probed the stagnation point flow of N-F past a cylindrical disk. Kattimani et al. [10] examined the reactive Casson N-F flow subjected to a permeable stretchable surface. Srilatha et al. [11] investigated the heat transfer and flow of N-F past a spinning disk. Kumar et al. [12] critically reviewed the mathematical models and computational techniques to study the heat transfer in N-Fs. Akbar et al. [13] explored the flow of electrically conducting N-F via a symmetric channel.
A dynamic channel configuration is represented by two parallel active plates, with both bounding surfaces subjected to controlled movement, deformation, suction or injection. In contrast to stationary plate systems, the active plates produce additional kinematic effects that directly influence momentum transfer and the pressure distribution within the confined fluid. These configurations are widely used to model squeezing flows, contraction–expansion processes and mechanically driven transfer phenomena in microfluidics and industrial applications. Wall motion and fluid inertia generate complex velocity gradients that significantly alter boundary layer development. When coupled with thermal effects, porous resistance or magnetic fields, active parallel plate systems provide a realistic framework for analyzing complex heat transfer and flow control systems. Erbay et al. [14] found that transient laminar flow across parallel plates generates the most entropy at high Reynolds numbers during counter motion and the least during parallel motion. Banakar et al. [15] examined the radiative fluid due to two parallel active plates using the collocation technique. Rahbari et al. [16] investigated the heat transfer and Magnetohydrodynamic (MHD) flow of non-Newtonian Maxwell fluid between two parallel plates. Raza and Wang [17] elucidated the Boger N-F flow between two parallel plates. Turkyilmazoglu and Alotaibi [18] scrutinized the fluid flow between two parallel active plates.
Thermal radiation is another form of heat transfer that does not need a medium to transfer heat. Radiative effects may also play a major role in the energy balance and temperature distribution in fluid flow systems experiencing high temperature gradients. Unlike conduction and convection, radiation is strongly influenced by temperature differences and surface emissivity. Linear radiation approximations are often insufficient in engineering designs, particularly where thermal loads are high or the media exhibit semi-transparency. Nonlinear or quadratic thermal radiation (QT-R) formulations provide greater physical realism in thermal modeling under high-temperature conditions. Proper modeling of radiative heat flux is thus necessary for accurately capturing thermal boundary layer behavior and the performance of heat transfer in sophisticated fluid systems. Kumari and Nath [19] analyzed the impact of thermal radiation on the fluid flow due to a horizontal surface. Mahantesh et al. [20] investigated the effect of QT-R on the dusty N-F flow subjected to a vertical plate. Sanju et al. [21] reviewed recent works on the effect of QT-R on heat transfer in micropolar fluids. Rao et al. [22] studied the influence of QT-R on the bioconvective Ree-Eyring fluid across an inclined plate. Mandal and Pal [23] elucidated the tri-hybrid N-F flow due to a permeable disk under the effect of QT-R.
Artificial neural networks (ANNs) are mathematical models based on biological neural systems, designed to estimate complex, nonlinear relationships between input and output variables. In thermal-fluid investigations, ANNs are used to estimate heat transfer rates, Nusselt numbers and temperature distributions from governing parameters, without repeatedly solving the differential equations. An ANN can also learn to describe hidden interplay among physical variables using numerical or experimental data and make quick, dependable forecasts. This ability is particularly useful in multiparameter systems where analytical solutions cannot be easily obtained. In engineering applications, predictive accuracy must be ensured and overfitting prevented through proper training, validation and testing procedures. Sharma et al. [24] elucidated the N-F flow past a stretchable cylinder utilizing an ANN. Kattimani et al. [25] utilized the ANN technique to examine the reactive Casson fluid flow due to a revolving disk. Kumar and Sharma [26] assessed the heat transfer and Bodewadt flow of fluid using an ANN. Naik et al. [27] used an ANN approach to study the Boger fluid flow across a revolving disk. Kumar et al. [28] explored the consequences of porous structures on Casson N-F flow past a stretching sheet using a neural network. Kaswan et al. [29] elucidated the feature of bioconvective flow over an elongating surface using ANN.
The collocation method is a numerical technique for obtaining approximate solutions to various equations by enforcing the governing equations at selected discrete points in the computational domain. Rather than integrating stepwise, as in shooting or finite difference methods, the unknown function is represented as a finite series expansion in terms of appropriate basis functions and the residual is made to vanish at specified collocation points. This approach transforms nonlinear boundary value problems into efficiently solvable algebraic equations. Collocation methods used to solve fluid flow and heat transfer equations combine high accuracy with comparatively low computational cost when used together with orthogonal polynomials, making them applicable to solving complex fluid flow and heat transfer equations. Kattimani et al. [30] examined the fluid flow between two parallel disks using the collocation method. Sadiq et al. [31] investigated the heat transfer and couple stress fluid flow by employing the collocation method. Khader [32] exemplified the unsteady Casson fluid flow utilizing the collocation method. Gao et al. [33] studied the N-F flow between two parallel surfaces with the aid of the collocation approach. Kumbinarasaiah and Preetham [34] used the collocation technique and examined the boundary layer flow of viscous fluid.
Morphology in N-F systems refers to the geometric shape and structural features of the dispersed nanoparticles, which significantly influence the effective thermophysical properties of the suspension. Particle geometry affects surface area, aspect ratio and interfacial contact with the base fluid, such as spherical, cylindrical, brick and platelet. These factors directly alter the enhancement of thermal conductivity, viscous resistance and momentum diffusion within the flow. Anisotropic heat transfer pathways and distortions of micro-convection processes are common characteristics of non-spherical particles, contrary to the traditional assumption of sphericity. Thus, morphology-dependent models should be integrated to obtain realistic predictions of heat transfer and flow behavior, especially in systems subjected to magnetic fields, porous resistance and radiative influences. Zhang et al. [35] elucidated the morphological effects on the coupled fluid flow. Xu and Yuan [36] utilized the finite difference method to examine the morphology effect on the coupling of heat and fluid flow. The heat transfer and flow of fluid in the presence of the morphology effect were examined by Zhuang et al. [37]. Nichik et al. [38] investigated the effect of morphology on the cross-fluid flow subjected to a circular cylinder. Yasin et al. [39] studied the consequence of morphology on the ternary N-F flow due to a porous disk.
In recent years, N-Fs have been the subject of intense research, with a recent focus on morphology-engineered nanoparticles, high-fidelity thermophysical property correlations and intelligent data-driven prediction techniques to design advanced thermal management systems. However, despite these advances, most current investigations rely on a single viscosity or thermal conductivity correlation and assume spherical nanoparticles, thereby ignoring the effect of particle geometry on coupled momentum and heat transfer. In addition, research on active parallel-plate systems subjected to magnetic fields, porous media, QT-R and convective boundary conditions remains scarce. As a result, the effects of morphology-dependent thermophysical properties, radiative heat transfer, porous resistance and wall deformation have not been comprehensively studied within a single mathematical framework.
To address this gap, the present study provides a new morphology-sensitive MHD N-F model with multiple viscosity correlations, three thermal conductivity models and four nanoparticle geometries: spherical, brick, cylindrical and platelet particles. The nonlinear governing equations are solved numerically using the Tchebichef polynomial collocation method (TPCM), and the results are verified using the Runge–Kutta-Fehlberg fourth-fifth-order (RKF-45) method. Moreover, a Levenberg–Marquardt artificial neural network (LM-ANN) is trained using high-order numerical calculations to accurately predict the nonlinear velocity and temperature distributions, thereby combining machine learning with high-order numerical computation in complex transfer phenomena.
The main goal of this work is to develop a comprehensive computational model capable of accurately predicting how nanoparticle morphology affects coupled flow and heat transfer between two active parallel plates. In particular, the following points are addressed: comparison of various viscosity and thermal conductivity correlations, effects of governing dimensionless parameters on momentum and thermal transfer, robustness and efficiency of the proposed numerical procedure, and the predictive capability of the LM-ANN. Based on that, the following research questions are developed:
How do momentum transfer and thermal diffusion change when nanoparticle morphology is modified in the presence of simultaneous magnetic, porous, radiative, and convective effects?
Which thermophysical property correlations yield realistic predictions of N-F behavior?
Are the proposed TPCM solutions of the governing nonlinear equations accurate, and can the LM-ANN reliably reproduce the numerical solutions?
The results yield new physical understanding of morphology-controlled transfer mechanisms and provide design guidelines for new applications such as microchannel cooling systems, porous heat exchangers, compact electronic cooling systems, thermal energy storage systems, renewable energy technologies, biomedical heat transfer systems, and other next-generation heat transfer systems using engineered N-Fs.
Mathematical analysis
In the present study, heat transfer and flow of an electrically conducting, incompressible N-F confined between two permeable parallel active plates is considered. The flow is exposed to a uniform transverse magnetic field, QT-R and a convective boundary condition, along with wall suction/injection. Multiple correlations for the viscosity and thermophysical behavior of the suspension, including spherical, brick, cylindrical and platelet nanoparticle morphology-based models, are introduced to accurately capture the influence of nanoparticle geometry on the thermophysical behavior of the suspension. The following assumptions are made to examine the flow and heat transfer characteristics,
An incompressible, electrically conducting N-F is considered to flow between two deformable active parallel plates, as illustrated in Fig. 1.
The upper plate is located at
, while the lower plate is positioned at 
, where α represents the relative spacing between the plates.The flow is driven by the relative motion of the plates and is further influenced by wall suction or injection, whose intensity is characterized by the wall transpiration parameter β.
The fluid temperature is denoted by T, with
and 
representing the prescribed reference temperature at the upper and lower plate, respectively.
Fig. 1.
Flow geometry
The corresponding boundary conditions are (see [18]),
 |
5 |
where, 
.
The similarity variables are given by (see [18]),
 |
6 |
The QT-R is given by (see [20]),
 |
7 |
The reduced ordinary differential equations (ODEs) are,
 |
8 |
 |
9 |
Reduced boundary conditions are,
 |
10 |
where,
 |
11 |
The thermophysical characteristics of density, dynamic viscosity, thermal conductivity and specific heat capacity are as follows (see [40]),
 |
12 |
 |
13 |
 |
14 |
 |
15 |
 |
16 |
 |
17 |
 |
18 |
 |
19 |
 |
20 |
Here, 
,
and 
.
Heat transfer is analyzed through thermal radiation and the Biot number, along with three thermal conductivity models: spherical, non-spherical and nanolayer. Additionally, the thermal conductivity model accounts for four different shape factors: spherical 
, brick 
, cylindrical 
and platelet 
. Table 1 represents the thermophysical values of the base fluid and nanoparticles.
Table 1.
Thermophysical values of nanoparticle and base fluid (see [40])
 |
 |
|
|---|---|---|
 |
385 | 997 |
| ρ | 8933 | 4180 |
| k | 401 | 0.613 |
| σ | 5.96 
|
5.50 |
Methodology
Fundamentals note on Tchebichef’s polynomials
Orthogonal polynomials exhibit several useful properties that make them powerful tools in mathematical and physical analysis. They provide an efficient way to represent and solve many differential equations. For instance, the orthogonality property of orthogonal polynomials simplifies mathematical computations, especially in solving equations and approximation problems. They are also widely used in signal and image processing for efficient signal representation, filter design, reconstruction and feature extraction. Tchebichef polynomials are orthogonal polynomials on the interval [ - 1,1] and are highly important in many areas of mathematics and applied sciences, such as numerical analysis, signal and image processing, and computer vision. The 
degree discrete Tchebichef polynomials are defined by (see [41]),
 |
21 |
Here 
denotes the Pochhammer symbol expressed as 
and 
denotes the hypergeometric function given by,
.
Tchebichef polynomials satisfy the recurrence relation specified by,
 |
22 |
for 
with the initial set of polynomials 
and 
.
Now, let 
and suppose it can be approximated in terms of the above polynomials as,
 |
23 |
where, 
and 
.
Construction of orthogonal matrices using Tchebichef’s polynomials
Integration of the polynomial vector 
over the interval [ - 1,τ ] is given by.
 |
24 |
Where, 
is an operational matrix of integration of order and 
is a 
column vector. For the case N = 5, these matrices are obtained as,
,
 |
25 |
Similarly, the representation of integrals of double, triple and fourth integrations of the above polynomials in the vector 
for N = 5 are denoted as.
,
,
 |
26 |
The corresponding matrices of integration are formulated as.
 |
27 |
Where, 
and 
.
Tchebichef’s Polynomials collocation method (TPCM) for handling nonlinear ODEs
This section demonstrates the applicability of the operational matrices by constructing a numerical scheme capable of solving the generalized ODEs. Consider a general 
order ODE of the form,
 |
28 |
Subjected to the initial or boundary conditions
 |
29 |
Now the highest derivative of the unknown function 
in Eq. (28) is approximated as explained in Eq. (23) using Tchebichef’s polynomials as,
 |
30 |
Integrate Eq. (30) n - times and use the initial conditions defined as in Eq. (29) with the help of Sect. 3.2, then Eq. (28) takes the form,
 |
31 |
The collocation method is now utilized to compute the numerical approximation 
to Eq. (31) under the boundary conditions in Eq. (29). Employing the collocation points 
for i = 0,1,2, … ,N to Eq. (31), it generates N + 1 algebraic equations. By solving these N + 1 equations, the unknown values 
in the vector 
are obtained. Substitution of procured unknowns into the expression 
to get the required approximation for the problem Eq. (28).
Solution for formulated ODEs using TPCM
The above proposed TPCM is employed in Eqs. (8–9) for N = 5. Let the approximations to unknown functions in Eqs. (8–9) be considered as,
 |
32 |
 |
33 |
Where, 
and 
are the unknown vectors to be found out.
Integrating Eq. (32) four times with respect to η over the interval [ - 1,η ] leads to,
,
 |
34 |
In a comparable way, integrating Eq. (33) two times with respect to η over the interval [ - 1,η ] leads to,
 |
 |
35 |
Substitute the boundary conditions Eq. (10) into Eqs. (34–35) and replace the obtained derivative values in the resulting equations along with the Eqs. (32–33) into Eqs. (8–9) yields,
 |
 |
Plug in the collocation points 
in the above two equations to generate two systems of equations for 
and 
respectively. The unknown values in the vectors 
are accomplished by solving these systems separately for f and θ. Then the corresponding approximations are given by,
 |
 |
36 |
Validation of the present TPCM
In this section, the accuracy of the present numerical method is ensured by the absolute error (A.E.) table. Apply the present TPCM to Eq. (8–9) and compare the results with the RKF-45 method throughout the interval 
in Table 2. The proposed schemes demonstrate remarkable accuracy relative to the RKF-45 method.
Table 2.
Comparison of TPCM results with RKF-45
| η | Absolute error for 
|
Absolute error for 
|
||||
|---|---|---|---|---|---|---|
| TPCM | RKF-45 | A.E | TPCM | RKF-45 | A.E | |
| − 1 | 0.100000125 | 0.100000000 | 1.24E-07 | 0.157050187 | 0.157162782 | 1.13E-04 |
| − 0.6 | 0.213669488 | 0.213672701 | 3.21E-06 | 0.124957279 | 0.125056971 | 9.97E-05 |
| − 0.2 | 0.428726996 | 0.428731858 | 4.86E-06 | 0.092624793 | 0.092699478 | 7.47E-05 |
| 0.2 | 0.676281009 | 0.676284555 | 3.55E-06 | 0.060633901 | 0.060682788 | 4.89E-05 |
| 0.6 | 0.888547056 | 0.888548283 | 1.23E-06 | 0.029586460 | 0.029610312 | 2.39E-05 |
| 1 | 1.000000038 | 1.000000000 | 3.80E-08 | − 0.000000013 | 0.000000000 | 1.33E-08 |
To further verify the accuracy and robustness of the proposed collocation method, the numerical solutions are compared with those obtained using other orthogonal polynomial-based collocation methods, such as Legendre and Chebyshev. Comparison of 
values from the present method is done with the same values obtained from the Legendre polynomial collocation method (LPCM) as well as the Chebyshev polynomial collocation method (CPCM) in Table 3. The comparison reveals excellent agreement among the computed solutions over the entire computational domain, indicating that the proposed approach provides results consistent with established orthogonal polynomial collocation techniques. This close correspondence further confirms the reliability, stability, and convergence of the present method.
Table 3.
Comparison of present TPCM results of 
with their values from different polynomial collocation methods
| η | TPCM | LPCM | A.E | CPCM | A.E |
|---|---|---|---|---|---|
| − 1 | 0.100000125 | 0.100000000 | 1.2450E-07 | 0.100000000 | 1.2450E-07 |
| − 0.6 | 0.213669488 | 0.213669975 | 4.8700E-07 | 0.213669976 | 4.8710E-07 |
| − 0.2 | 0.428726996 | 0.428727827 | 8.3020E-07 | 0.428727827 | 8.3030E-07 |
| 0.2 | 0.676281009 | 0.676281645 | 6.3600E-07 | 0.676281645 | 6.3610E-07 |
| 0.6 | 0.888547056 | 0.888547282 | 2.2620E-07 | 0.888547282 | 2.2630E-07 |
| 1 | 1.000000038 | 1.000000000 | 3.8000E-08 | 1.000000000 | 3.8000E-08 |
As an additional verification, the numerical solutions obtained using the proposed method are compared with the corresponding results available in the literature for the particular values of the parameters given in Table 4. The computed values exhibit a good agreement with the published results, with only negligible differences attributable to numerical approximation and computational precision, thereby validating the accuracy and consistency of the proposed collocation scheme.
Table 4.
Verification of TPCM results by comparison with the work of [18]
| A = B = 0 | Reference [18] | Present | A.E | Reference [18] | Present | A.E | |||
|---|---|---|---|---|---|---|---|---|---|
 |
S | β | κ |  |
 |
||||
| 0 | 0 | 0 | 0 | 1.5 | 1.5 | 0 | -1.5 | -1.5 | 0 |
| 0 | 0.5 | 2 | 0 | − 1.303 | − 1.4545 | 0.1515 | 1.303 | 1.4525 | 0.1495 |
| 0.5 | 0.5 | 2 | 0 | − 1.773 | − 1.8736 | 0.1006 | 0.973 | 1.1552 | 0.1822 |
| 1 | 1 | 0.5 | 0 | 0.765 | 0.8617 | 0.0967 | − 0.416 | − 0.5771 | 0.1611 |
| 0 | − 0.5 | 2 | 0 | − 1.703 | − 1.5551 | 0.1479 | 1.703 | 1.5525 | 0.1505 |
| 0.5 | − 0.5 | 2 | 0 | − 2.168 | − 2.0501 | 0.1179 | 1.379 | 1.1965 | 0.1825 |
| − 1 | − 1 | − 0.5 | − 0.5 | 3.211 | 3.1316 | 0.0794 | − 2.781 | − 2.1582 | 0.6228 |
Compared to the traditional numerical and semi-analytical methods, the combination of TPCM and LM-ANN possesses several advantages in solving the present nonlinear transfer problem. The TPCM is different from finite difference and finite element methods that typically need fine spatial discretization and repeated iterative computation to get high accuracy; it transforms the governing nonlinear differential equations directly into a system of algebraic equations by using orthogonal polynomial approximations and collocation points. This method can obtain fast convergence, high numerical accuracy and low computational complexity and solve the nonlinear coupled governing equations efficiently. Moreover, unlike the perturbation-based analytical methods, which are usually restricted to small or large parameter values, the TPCM is valid in the entire range of the parameters investigated without any approximation. In addition, the LM-ANN further complements the numerical methodology by learning the nonlinear relationship between the governing physical parameters and the corresponding numerical solutions generated by the TPCM. After training, the ANN can be used to predict the velocity and temperature distributions quickly, which can save a lot of computational time in extensive parametric studies without repeatedly solving the governing equations. Therefore, the TPCM is integrated into the LM-ANN, which is a robust numerical method that can be used for the morphology-dependent analysis of N-F transfer with the high computational efficiency of data-driven prediction.
Results and discussion
The current section presents a parametric study of an N-F system modeled between two active, parallel, deformable plates under the influence of MHD and porous medium. Similarity transformations reduce the governing partial differential equations for continuity, momentum and energy conservation, with QT-R and convective boundary condition to coupled nonlinear ODEs. Multiple correlations for viscosity and thermal conductivity model the N-F’s thermophysical properties, including morphology-dependent correlations for spherical, brick, cylindrical and platelet nanoparticle geometries, as well as nanolayer effects. This multiphase modelling framework enables systematic comparison of property variations within the same mathematical context. The TPCM is used to solve the reduced boundary value problem numerically, and the solutions are compared with those from RKF-45 to assess computational consistency. Additionally, the LM-ANN model is used to test its potential in estimating the numerical solutions of the governing equations. The discussion is structured to evaluate the impact of key dimensionless parameters, i.e., the magnetic field parameter, porosity parameter, Reynolds number, squeeze/expansion parameter, radiation parameter, Biot number and temperature ratio, within the morphological and radiative modelling assumptions adopted.
Figure 2 demonstrates how the magnetic parameter M affects the distribution of the axial velocity 
of various models of viscosity. The velocity consistently decreases as M increases. The magnetic parameter is always raised to suppress the velocity consistently, since the induced Lorentz force is opposite to the velocity of the fluid, which leads to the decrease of the kinetic energy of the fluid and the increase of the electromagnetic resistance of the fluid, which leads to the decrease of the momentum transfer. The decrease is more significant in the middle of the channel where the flow momentum is highest. This behaviour is also modified by the influence of the nanoparticle morphology, which also alters the hydrodynamic interaction. The enhanced particle–fluid momentum exchange and the disruption of the local shear field caused by particles with larger surface area and higher aspect ratio intensify effective momentum diffusion and enhance the magnetic damping effect. Thus, morphology and magnetic braking are coupled to control the momentum transfer in the electrically conducting N-Fs. The impact of the porosity parameter 
on the velocity distribution is given in Fig. 3. As 
is raised, 
is evidently decreased across the channel. As the porosity parameter is increased, the velocity is decreased as more Darcy resistance is created, which decreases the momentum transfer and fluid flow through the channel. At the same time, the shape of the nanoparticles affects the value of this resistance, which depends on the interaction between the particles and the fluid. Particles with a non-spherical shape create more hydrodynamic disturbances due to their increased interfacial area and aspect ratio, which means they have a larger momentum diffusion than spherical particles. Thus, the effect of porous drag and momentum transfer by morphology is more significant in attenuating the velocity field. Figure 4 indicates how 
varies with Reynolds number 
. The velocity initially enhances as 
increases, then it decreases throughout the channel. A high 
implies that the forces of inertia dominate over the forces of viscosity, causing increased axial acceleration. As a result, the fluid is moving faster, and the momentum boundary layer is reduced. The enhancement, however, is related to the morphology of nanoparticles. The momentum diffusion in the suspension is modified by the high-aspect-ratio particles due to increased particle–fluid interactions, which changes the local shear distribution. Therefore, the effective viscosity is not the only effect of morphology; it is also the redistribution of momentum in the flow domain.
Fig. 2.
Effect of M on 
Fig. 3.
Effect of 
on 
Fig. 4.
Effect of 
on 
The effect of the radiation parameter Rd on the temperature distribution 
is shown in Fig. 5. An increase in Rd leads to an increase in temperature within the channel. Improved thermal radiation provides more thermal energy to the fluid, creating a thicker thermal boundary layer and increasing thermal diffusion. This enhancement varies depending on the morphology of the nanoparticles, as a larger surface area gives higher interface contact and thus enables the creation of continuous conductive pathways within the suspension. Thus, the effective thermal conductivity and also the internal energy transfer efficiency increase, and a more uniform temperature distribution is obtained, which is beneficial to morphology. Figure 6 shows how the temperature changes with the squeeze/expansion parameter S. An increase in S increases the temperature in the channel. When you squeeze harder, the gap between the plates becomes smaller and the heat from dissipation and compression rises. The small flow field increases thermal concentration and the average temperature of fluids. The morphology of the nanoparticles also affects the response due to local hydrodynamic interactions and shear transfer, which are modified by the anisotropy of the particles. The differences in velocity development between the viscosity models are more visible for the particles with high aspect ratios, as this aspect ratio has a significant influence on the momentum transfer between the fluid and the dispersed phase. Figure 7 gives the effect of the ratio of the wall temperature 
on the temperature field 
. When the value of 
goes higher, the whole temperature profile moves on the x-axis upwards. The difference in temperature between the walls and the ambient temperature increases the strength of the thermal driving force and, consequently, additional heat enters the fluid. The bulk temperature and the thermal boundary layer increase in the same proportion. This is further enhanced by morphology-dependent nanoparticles, which increase the contact area of particles and fluid and minimize the thermal resistance at the interface. This results in enhanced transfer from the heated walls to the fluid and an increase in the thickness of the thermal boundary layer and bulk temperature.
Fig. 5.
Effect of Rd on 
Fig. 6.
Effect of S on 
Fig. 7.
Effect of 
on 
The relation of temperature with the Biot number Bi is illustrated in Fig. 8. As Bi raises the temperature distribution within the channel rises. Increased wall convection provides extra thermal energy to the fluid, which diffuses into the fluid, thereby increasing the size of the thermal boundary layer. The efficacy of this heat transfer also depends on the form of the nanoparticles. The thermal energy transfer is more rapid through the suspension of non-spherical particles than through the spherical particles due to more efficient conductive networks formed by non-spherical particles as a result of their higher aspect ratio and larger interfacial area. Figure 9 illustrates the contour distribution of the velocity field with diameter viscosity model 1 when β = - 1, β = 0, and β = 1. This flow pattern varies greatly with β, negative β gives a symmetric saddle-like pattern, whereas positive β makes the flow more directionally oriented. The neutral case 
is intermediate, and it suggests that the parameter of viscosity can have a strong impact on the topology of the flow and the magnitude of the gradients.
Fig. 8.
Effect of Bi on 
Fig. 9.
Contour plots
ANNs are mathematical models that are based on biological neural systems and are created to estimate complicated nonlinear correlations among numerous input variables and output variables. They are especially useful in the modeling of fluid flow and heat transfer problems with highly coupled and nonlinear governing equations. The ANN works by using the interconnected neurons formed in the form of input, hidden and output layers, with each connection having an adjustable weight. These weights are changed by backpropagation (iteratively) during training with the goal of reducing a specified error measure, commonly the mean square error (MSE). The Levenberg–Marquardt algorithm will be used in the current research because it has a fast convergence rate and has a hybrid optimization nature; it combines the merits of the gradient descent and the Gauss–Newton methods. The technique is particularly appropriate in the nonlinear regression system with large data sets. To guarantee generalization and avoid overfitting, the numerical data obtained by the use of the RKF-45 scheme is split into three parts: 80% training, 10% validation and 10% testing. Predictive performance of the model is measured by MSE, gradient magnitude, damping parameter 
, validation check, regression coefficient (R) and absolute error distribution. Figure 10 demonstrates the design of the created LM-ANN model, which is comprised of a single input neuron, a hidden layer, including 10 neurons and a single output neuron. In this setup, the nonlinear mapping between the input parameters and the predicted flow or thermal profile converges predictably.
Fig. 10.
Architecture of the developed LM-ANN model
The proposed LM-ANN is a single hidden-layer feedforward network consisting of ten neurons to set up the nonlinear relation between the governing physical parameters and the numerical solutions. This architecture was selected because of its simplicity of calculation, convergence property and ability to predict, even for moderately sized engineering datasets, nonlinear regression and function approximation problems. The dimensionless velocity and temperature profiles from the RKF-45 are the target outputs, while the governing dimensionless parameters controlling the transfer phenomena are the ANN inputs. Numerically generated datasets were used for training the network with the various ranges of Reynolds number, magnetic parameter, radiation parameter and volume fraction of nanoparticles investigated to ensure the reliability of the network to interpolate the results within the computational domain investigated in this study. The present work aims to show the usefulness of the LM-ANN as an efficient surrogate model for the computed numerical solutions and not to optimize the architecture of the neural network; therefore, a systematic study of other possible hidden layer structures was not performed. Similarly, robustness against noisy or incomplete datasets and extrapolation beyond the range of investigated parameters were not included since in the numerical training data, the uncertainties of measurements are zero. These are interesting areas for further study using experimental data and/or wider operating conditions.
Figure 11 shows the validation performance of the LM-ANN model of the velocity gradient 
, and temperature profile 
, namely the trend of the MSE versus training epochs. The validation performance of the velocity gradient 
is shown in Fig. 11a. The lowest validation performance occurs at the epoch of 205, with the lowest MSE of 1.4369e-10. The training, validation and testing curves decline rapidly in the initial epochs and slow down to a constant value, which implies effective weight modification and stable convergence. The validation performance with respect to the temperature profile 
is illustrated in Fig. 11b. The best validation, in this case, happens at epoch 526, and even a smaller MSE of 3.6015e-13. The continuous and linear decreasing trend of the error curve also proves that the network is able to represent the nonlinear thermal history without oscillatory chaos. Figure 12 represents the error histogram of the LM-ANN predictions of 
and of 
, showing how errors of prediction between the network outputs and targets are distributed. Both profiles’ error histograms show that there is no error at -1.3e-06 and -9.5e-08 of 
and of 
profiles, respectively. The case of 
, Fig. 12a with most errors concentrated around zero, and the distribution is almost symmetric over the distribution. Most of the training, validation, and testing errors lie within a very thin band, which implies that there is very little variation between the predicted and target values. In the case of 
(Fig. 12b), the same effect of symmetrical distribution of errors about the zero point can be seen. The fact that there are no big outliers proves that there is no variation in the predictions of the ANN throughout the data set. The histogram analysis shows that the residual errors are quite small and evenly distributed, which indicates valid network performance. Figure 13 represents the training condition of the LM-ANN model with 
and 
as well as the magnitude of the gradient, damping parameter and validation measures at convergence. In Fig. 13a, the training state of 
is shown. The gradient steadily declines and reaches a value of about 9.9543e-08 at epoch 205, which means that the optimization reaches a stationary point. The damping parameter μ approaches 1e-10 and indicates steady Levenberg–Marquardt convergence. Checks of validation are also controlled to ensure appropriate termination of criteria. Figure 13b displays the state of training 
. The gradient value is reduced to about 9.9746e-08 around epoch 526, and 1e-11 becomes stable. The absence of overfitting is confirmed by the smooth convergence of the two parameters.
Fig. 11.
Validation performance of the LM-ANN model for 
and 
Fig. 12.
Error histogram for the LM-ANN predictions of 
and 
Fig. 13.
Training state of the LM-ANN model for 
and 
Figure 14 demonstrates the performance of the LM-ANN model of 
and 
where the expected outputs are compared to the target outputs. Figure 14a provides the ANN results and target values of 
. The forecasted curve is almost coincident with the numerical targets, and the bottom subplot ensures that the error in the residue is very small all over the domain. The same analysis is presented in Fig. 14b of 
. Once again, we can see that the target and predicted curves are almost identical, which proves that the network recreates the nonlinear temperature variation with high fidelity. The regression analysis of the LM-ANN prediction of the 
and 
are as illustrated in Fig. 15, which shows the correlation between the network outputs and the numerical targets. Figure 15a shows the regression plots of 
. R is 1 in both training, validation, testing, and the general data. The points are considered to be practically on the diagonal reference line, which would indicate the complete linear agreement between predicted and target values. Regression analysis of 
is available in Fig. 15b and the correlation coefficient is also equal to one. This means that ANN predictions are almost proportional to the numerical predictions. Regression outcomes show that predictive reliability and systematic error are strong and negligible, respectively.
Fig. 14.
Function fitting plots for 
and 
Fig. 15.
Regression analysis for 
and 
Figure 16 illustrates the absolute error distribution of the LM-ANN model for 
and 
across the similarity domain. Figure 16a provides the distribution of error in 
at absolute error. Its maximum deviation is also of the order 
, which means that the numerical deviation in the similarity domain is very small. Figure 16b gives the absolute error of 
in which the error is even less, of the order 
. The variations are not very substantial and there is no instability. The fact that the magnitude of absolute errors is small proves the great approximation ability. Figure 17 compares the LM-ANN predicted profiles with the RKF-45 numerical solutions for 
and 
, confirming the accuracy of the trained network. In Fig. 17a, the LM-ANN predictions are compared to the RKF-45 solutions of 
. Numerical consistency is verified by the fact that the curves almost coincide within the domain. The comparison of 
is shown in Fig. 17b. The correlation between ANN and RKF-45 is also very high, and the apparent deviation is quite low. This last comparison confirms that the LM-ANN is able to reproduce the deterministic numerical solutions. Overall, the LM-ANN model has a high predictive power of the velocity gradient 
and 
. Highly small values of MSE 
, low values of the gradient (around 
), and stable values of damping parameters ensure fast and sound convergence. The regression coefficient value of one and the symmetric error distribution demonstrate that there is a high level of generalization and no overfitting. Moreover, the fact that the LM-ANN solution and the RKF-45 solution match each other nearly perfectly confirms that the constructed neural network has the ability to predict the nonlinear flow and heat transfer of the entire similarity range. Table 5 shows the Nusselt number values for various dimensionless parameters.
Fig. 16.
Absolute error distribution of the LM-ANN model for 
and 
over the similarity variable
Fig. 17.
Comparison of LM-ANN predictions with RKF-45 numerical solutions for 
and 
Table 5.
The values of 
and 
for various parameters
| M |  |
Rd |  |
 |
 |
|---|---|---|---|---|---|
| 0.1 | 0.1 | 0.1 | 0.2 | − 0.110205050627050 | − 0.00175526598907803 |
| 0.2 | − 0.110203525199448 | − 0.00175557813868243 | |||
| 0.3 | − 0.110202006974957 | − 0.00175588300722709 | |||
| 0.2 | 0.2 | 0.1 | 0.1 | − 0.111563558426038 | − 0.00511917442285543 |
| 0.4 | − 0.111561583523378 | − 0.00511965265579060 | |||
| 0.6 | − 0.111559632256885 | − 0.00512010659605354 | |||
| 0.1 | 0.3 | 0.1 | 0.1 | − 0.111563558426038 | − 0.00511917442285543 |
| 0.15 | − 0.109601335356573 | − 0.00109137224013378 | |||
| 0.2 | − 0.106956270485597 | − 0.0000105538136878085 | |||
| 0.2 | 0.2 | 0.1 | 0.1 | − 0.111563558426038 | − 0.00511917442285543 |
| 0.18 | − 0.110434349478935 | − 0.00217502542564679 |
Conclusion
In the current study, a detailed mathematical model is developed to analyze the MHD N-F flow and heat transfer between active parallel plates in a porous medium with QT-R and convective boundary condition. The research systematically quantified the effect of particle shape on coupled momentum and thermal transfer and included several viscosity and thermal conductivity correlations, as well as a variety of nanoparticle shapes. The TPCM was employed to solve the nonlinear governing equations with high accuracy, and, for comparison, an LM-ANN was able to predict the nonlinear flow and temperature fields with high accuracy. The developed framework provides an enhanced physical description to replace traditional single-property models and serves as a reliable computational tool for studying morphology-dependent transfer phenomena in advanced thermal engineering systems. The main new findings from the present study are briefly summarized below:
The simultaneous study of the effects of nanoparticle morphology, magnetic field, porous medium, QT-R, and convective boundary condition was conducted within a single mathematical model, yielding a more realistic description of N-F flow than classical models.
Multiple viscosity correlations and three thermal conductivity models were systematically compared, and it was found that the choice of thermophysical property can greatly affect momentum transfer and heat transfer predictions.
The magnetic field and porous resistance were effective flow-control mechanisms, and it was found that they weakened the velocity field, whereas the increasing Rd, Bi, wall temperature ratio and squeezing effect strengthened thermal energy transfer and thus enhanced the temperature distribution.
The TPCM proposed in this paper was shown to be robust and efficient in solving the nonlinear boundary value problems encountered, achieving very high accuracy and excellent agreement with the RKF-45.
The LM-ANN predicts generated numerical velocity and temperature distributions with good accuracy, demonstrating its potential as a reliable surrogate model for fast prediction of complex morphology-dependent thermal transfer problems.
The results are useful for the design and optimization of morphology-engineered N-Fs for microchannel cooling systems, porous heat exchangers, compact electronic cooling systems, thermal energy storage systems and other advanced thermal management technologies.
The present study offers an extensive investigation of morphology-dependent N-F flow and heat transfer with active parallel plates; however, the mathematical model developed in the present study is based on a set of assumptions that limit the scope of the study. The analysis is limited to steady, laminar, single-phase, incompressible flow with uniformly dispersed nanoparticles. Particles are assumed to be in a state of agglomeration, and sedimentation, Brownian diffusion, and thermophoretic transfer are ignored. The medium is assumed to be porous, homogeneous and isotropic, and the induced magnetic field, Joule heating, Hall current, ion-slip effects and viscous dissipation are neglected to keep the problem mathematically tractable. In addition, only a few correlations for viscosity and thermal conductivity corresponding to certain nanoparticle morphologies are explored. Although the numerical solutions are validated against the RKF-45, experimental validation is beyond the scope of the present work. Thus, the results reported should be viewed under the adopted assumptions and parameter ranges.
Based on the proposed mathematical model, several future research directions are suggested to enhance the physical realism and engineering applicability of morphology-dependent N-F models. Future studies could include the effects of unsteady, three-dimensional, or turbulent flow, as well as other transfer phenomena (Brownian motion, thermophoresis, viscous dissipation, Joule heating, Hall current, and ion-slip effects). The effects of hybrid and multi-hybrid N-Fs, variable thermophysical properties, and more complex porous media models can also be analyzed to gain a broader perspective on the coupled transfer phenomena. The proposed morphology-dependent correlations could be experimentally validated to further establish their practical reliability. Furthermore, combining high-order numerical methods with advanced machine learning algorithms may enable fast prediction, optimization and real-time control of complex thermal systems, such as electronic cooling systems, renewable energy systems, microchannel heat exchangers and thermal energy storage systems.
Abbreviations
- ANN
Artificial neural network
- LM-ANN
Levenberg-Marquardt artificial neural network
- RKF-45
Runge-Kutta-Fehlberg’s Fourth-Fifth order
- TPCM
Tchebichef polynomial collocation method
- MHD
Magnetohydrodynamics
- QT-R
Quadratic thermal radiation
- N-F
Nanofluid
List of symbols
Wall deformation rates
Distance
Magnetic field strength
- Bi
Biot number
- β
Specific heat
Velocity profile
Heat transfer coefficient
- A
Thermal conductivity
Permeability
Mean absorption coefficient
- M
Magnetic field parameter
- P
Pressure
Porosity parameter
- \Pr
Prandtl number
Radiative heat flux
- Rd
Radiation parameter
Reynolds number
- S
Squeeze/expansion parameter
- t
Time
- T
Temperature field
Temperature of the upper plate
Temperature of the lower plate
- u,v
Velocity components
Deformation rate of walls
- x,y
Directions
Greek symbols
- α
Distance between two plates
- β
Wall suction/blowing ratio
- η
Similarity variable
Stefan-Boltzmann constant
- μ
Relative distance between plates
Temperature profile
Temperature ratio
- μ
Dynamic viscosity
- ν
Kinematic viscosity
- ν
Density
- ϕ
Solid volume fraction
- σ
Electrical conductivity
Subscripts
- bf
Base fluid
- nf
Nanofluid
- s
Solid nanoparticle
- np
Nanoparticle
- f
Fluid
Nanolayer of nanofluid
Author contribution
“Conceptualization: R. J. Punith Gowda, Vishwanatha Rajeev Banakar, K M Anil kumar, Data curation: R. J. Punith Gowda, Vishwanatha Rajeev Banakar, Formal analysis: Vishwanatha Rajeev Banakar, Shahbaz Juneja, Sachin Singh, Investigation: Vishwanatha Rajeev Banakar, Shahbaz Juneja, Sachin Singh, Methodology: R. J. Punith Gowda, K M Anil kumar, Software: R. J. Punith Gowda, Vishwanatha Rajeev Banakar, Supervision: R. J. Punith Gowda, K M Anil kumar, Validation: R. J. Punith Gowda, K M Anil kumar, Writing—original draft: Vishwanatha Rajeev Banakar, Shahbaz Juneja, Sachin Singh, Writing—review & editing: R. J. Punith Gowda, K M Anil kumar.”
Funding
Open access funding provided by JSS Academy of Higher Education and Research, Mysore. Not applicable.
Data availability
The findings of this work are based on theoretical analysis; no external data were required. Hence, all the data generated or analysed during this study are included in this article.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Clinical trial
Not applicable.
Competing interests
The authors declared no conflict of interest.
Footnotes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
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Data Availability Statement
The findings of this work are based on theoretical analysis; no external data were required. Hence, all the data generated or analysed during this study are included in this article.