Thermal analysis of bioconvection flow of CNTs/water based hybrid nanofluid with gyrotactic microbes using artificial neural network

Abstract

This study uses the Levenberg–Marquardt strategy with feed forward neural networks (LMS-FNN) to inspect the Soret-Dufour effect on radiative hybrid nanofluid flow across a Riga plate with gyrotactic microorganisms. The suggested model, which investigates the thermal behavior of bioconvection flow in CNTs/water based hybrid nanofluid, gyrotactic microbes, when considered alongside Soret-Dufour impact, contribute notably to important uses in biotechnology, energy systems, and industrial heat management. It is important to optimize bioreactors that require high microbial activity and heat transfer, to improve the design of innovative cooling systems that use hybrid nanofluid, and to aid in the creation of efficient microfluidic devices. Artificial neural networks provide accurate prediction of complex fluid-microbe interactions, which aids in the design and control of next-generation thermal and bioengineering processes. From reference results, execute LMS-FNN validation, training, and testing to get approximated solutions for variations connected with the physical system and to demonstrate the correctness of the suggested LMS-FNN. The Mean squared error, histograms, and regression analysis are used to examine the performance of LMS-FNN, and the problem is satisfactorily solved. The microorganism profile profile declines as the Peclet number increases.

Introduction

It is evident that calculating the flow field and heat transmission in relation to several input parameters necessitates a complex process. In general, computational intelligence methods that can learn many models—like artificial neural networks (ANN)—may be employed in place of intricate analytical connections. An effective method for forecasting modelling output outputs under various circumstances is ANNs. Numerous attempts have been made in the literature to use this method for various fluid flow problems. In order to investigate tribology of trihybrid nanofluid and thermo-bioconvection flow containing oxytactic microbes, Abbas et al. [1] investigated the Levenberg–Marquardt–driven back-propagation neural intelligence approach. Employing the regression approach, Abdal et al. [2] examined the machine learning research for CCHFM (Cattaneo-Christov heat flux model) in 3D Casson fluid flow: Use in engineering technology and science. Khedher et al. [3] studied the role of hybrid nanoparticles in improving heat transport by using artificial neural networks to scrutinize how the CCHFM and temperature radiation affect Sutterby hybrid nanoliquids. By a numerical and artificial intelligence strategy, Khan et al. [4] scrutinized the generation of entropy and heat transport in l-shaped frames with MHD free convection. Liaqat et al. [5] examine the utilization of artificial neural networks in bio-nanomedicine to optimize the hybrid nanofluid’s chemical reactive flow that generates heat and thermal radiation. Alarabi et al. [6] inspect the artificial intelligence-powered magnetorheological dynamics of a dusty nanofluid: employing artificial neural network. The nanofluid’s thermosolutal convective flow with Marangoni convection was investigated by Aoudia et al. [7] employing an artificial neural networks research that took thermal radiation impacts into account. To predict the thermal behaviour of blood-based THNF (trihybrid nanofluid) flow in biomedical drug delivery systems, Anjum et al. [8] used an intelligent soft computing model. A unified neural network informed by physics is developed by Chandra et al. [9] to study bioconvection NF flow across a surface using the interior point approach. Anjum et al. [10] examined the solitary-peakon analysis of the lax model using the Kolmogorov-Arnold network. Chandra and Das [11] developed a hybrid machine learning method to study magnetized NF flow with gyrotactic microbes on a vertically inclined surface. The dynamics of divorce in social networks were examined by Saqib et al. [12] using Gudermannian neural networks and sophisticated heuristic computing. Awan et al. [13] used clever Bayesian regularization to estimate the mobility of HNPs (hybrid nanoparticles) of AA7075- AA7072 oxide with a permeable surface. For a bio-convective nanofluid flow model that incorporates heat immersion, stratification, viscous dissipation, and gyro-tactic creatures, Awan et al. [15] investigate intelligent Bayesian regularization networks. Using ANN and the Levenberg–Marquardt approach, Farooq et al. [16] study the numerical modeling of radiative NF flow across a nonlinear sheet. Bayesian regularization-based intelligent networks were analyzed by Awan et al. [17] for the thermophysical study of a 3D magnetohydrodynamics nanofluidic flow model across surface. For an effective heat transmission study in a radiative moving permeable triangular fin with heat generation, Saqib et al. [18] examined special recurrent neural networks. Awan et al. [19] investigate the new design of intelligent Bayesian networks and the impact of Joule heating and magnetic fields on the flow of hybrid nanomaterials, which has medicinal uses in blood circulation. The mass and heat transfer in constant 2D flow around a cylinder, Abdal et al. [20] examine optimizing hidden layers. The AI-driven RNN method is analysed by Saqib et al. [21] to investigate thermal properties in magneto-radiative NF (nanofluids) with random microbial flow. The ANN computing architecture is studied by Farman et al. [22] in order to describe and predict the dynamics of a fractional order hepatitis C virus. The assessment of the radial velocity of an electrohydrodynamic ion drag flow model using intelligent LMBPN (Levenberg–Marquardt back-propagated networks) was studied by Awan et al. [23]. Taha et al. [24] evaluate the effectiveness of scaled conjugate, Bayesian, and Levenberg–Marquardt methods, in a comparative study of AI algorithms in boundary layer flow. Abdal et al. [25] use a neural network-based methodology in their investigation of heat transport and bioconvection in a NNF (non-Newtonian fluid) with chemical reaction across a sheet. An inventive computational intelligence method to entropy production and heat transmission in NNF third grade MHD peristaltic transport via asymmetric channels is examined by Rehman et al. [26].

Particles are transported by a variety of parameters that are calculated using a thermal gradient in a process called the thermos-diffusion mechanism. The Dufour and Soret effects emerge when a noteworthy density differential remains in the flow phase. When mass and heat transmission occur instantly in liquid motion, the linkage among driving forces and resulting fluxes becomes more complicated. This phenomenon arises when the concentration of one chemical species consistently differs from that of the others in the process. The mutual heat and mass transmission in dual systems can be made easier for intermediate atomic weight gases by the Soret-Dufour influence. Theoretical and practical research for significant uses in storage of energy units, geothermal power plants, nuclear waste storage, catalytic reactors, and drying technologies have motivated extensive investigations into solutal and heat transport within permeable media. Sivanandam and Alqurashi [27] examine how radiation, viscosity dissipation, Soret and Dufour impact, and slip affect 3D spinning nanoliquid flow. When HNF flows across a needle in a porous sheet with heat radiation, Srinivasa et al. [28] inspected the Joule heating, Dufour, and Soret effects. Shah [29] examined the effect of MHD influence on the Soret–Dufour impacts in a corrugated triangle with a slope cavity filled with non-Newtonian fluid. Salahuddin et al. [30] emphasis is placed on the thermal and solutal behavior of MHD tangent-hyperbolic fluid with two layers in stagnation-point flow, incorporating Soret–Dufour influences.

The creation of hybrid nanofluids occurs when two or more kinds of NPs disperse in base fluids. When compared to unitary nanofluids, hybridized nanofluids exhibit greater heat dispersion, molecular concentration density, and thickness and diameter of nanomaterial. In the fields of biomedicine, heat exchangers, nuclear systems, solar systems, and generator cooling, this has many applications. It is impossible to estimate the temperature conductivity of hybrid nanofluids using a specific model. Many scientists are, nonetheless, motivated to investigate various nanofluids due to the growing industrial demand for nanomaterials for better technological improvement. Alharbi et al. [31] examined the MHD influence and permeable surface interactions in the nano-confined multi-scale examination of thermal transmission in HNF flow. The Cattaneo-Christov model to illustrate the significance of elastic deformation on tangent hyperbolic HNF radiative flow was analyze by Abbas et al. [32]. Malik [33] used a paired ANN and mathematical analysis of thermal radiation, bioconvection, and MHD to study the hybrid nanofluid flow in a round cylinder. Moyes and Anala [34] evaluated entropy production in MHD-driven third-grade HNF through a sheet with non-uniform heating effects and heat radiation. Wahid et al. [35] used a computational and statistical method to investigate the radiative flow of an HNF with heat production across a porous convective moving surface. Muhammad et al. [36] inspected the thermal efficiency of NF in an artificially roughened solar heater. In a computational study of a peristaltic flow with internal heat production, Qureshi et al. [37] examined the impact of radial magnetic field characteristics. Suhaib et al. [38] analyzed the peristaltic rheology of nanomaterials under temperature radiation and a fluctuating heat source. Ontela and Enamul [39] examine the DFF (Darcy-Forchheimer flow) of a non-Newtonian second-grade HNF that is limited by double-revolving discs with different thermal conductivities. Thumma et al. [40] examine the impacts of DFF and temperature-dependent viscosity in the thermal enhancement of C2H6O2-MoS2 nanofluid flow over a slippery surface. Electroosmotic slip flow in peristaltic transport of third-grade non-Newtonian MHD fluids is investigated by Ellahi et al. [41] using sensitivity analysis based on RSM.

Microorganisms, which include bacteria, viruses, fungus, archaea, and protists, are microscopic life forms that are found in a wide variety of habitats and are essential to industrial operations, human health, and ecological equilibrium. Microorganisms have an impact on fluid flow, resulting in a complex and dynamic interplay between fluid particles and germs. Through their movement and interactions, microorganisms like bacteria and algae have an impact on fluid dynamics in aquatic environments. Microorganisms have an impact on blood flow in the human body. White blood cells and platelets interact with pathogens to affect fluid rheology and support vascular health. In environmental science, engineering, and medicinal research, the intricate link between fluid flow and microbes is essential because it provides information about natural processes as well as how to design systems that are vulnerable to microbial effects. Microorganisms have a complex effect on thermal transfer in fluid flow, affecting heat exchange processes in a range of settings. Microorganisms can cause biofouling in industrial settings, including heat exchangers, by growing biofilms on surfaces that reduce the effectiveness of heat transfer. This biofouling layer may serve as an insulating barrier, decreasing heat exchanger efficiency and raising energy usage. By changing the water’s thermal characteristics through their metabolic processes, microbes can affect thermal dynamics in natural aquatic systems. The mass and heat transportation in MHD boundary-layer flow with mobile microbes through permeable surfaces in varied thermal conditions was evaluated by Ashfaq et al. [42]. An ANN model to investigate the hydromagnetic bioconvection of gyrotactic microorganisms in micropolar nanofluid through an inclined permeable sheet was inspected by Khan et al. [43]. Gupta et al. [44] explored EMHD flow behavior of HNF incorporating gyrotactic microbes using the CCHFM. Afzal et al. [45] used streamline-based approach and multilinear regression to evaluate microbe dynamics in temperature magnetohydrodynamics Casson fluid flow on a sheet under source of generation. The mass and heat transfer of magnetized NF under the effect of temperature radiation and motile microbes was studied by Chaudhry et al. [46]. Abo-Zaid et al. [47] inspected the Entropy generation in MHD natural bioconvection in a permeable cavity with a NF enriched with gyrotactic bacteria. The Hall-induced bioconvective Jeffrey HNF flow in a squeezing channel is dynamically modified by Samal et al. [48] using activation energy: sensitivity analysis. Parveen et al. [49] investigate the thermophysical properties of chemotactic microorganisms, including viscous dissipation, Joule heating, and slippage, in the bio-convective peristaltic rheology of nano-liquid. Samal et al. [50] examine the entropy and heat transfer sensitivity of a HNF squeezing flow which includes bioconvection and is electroosmotic-Casson radiative. Chandra et al. [51] use finite element analysis to examine the effects of gyrotactic bacteria in NF via porous medium along an inclined stretching plate. A time-dependent MHD squeezing flow of radiative Eyring-Powell HNF containing gyrotactic microbes is optimized and its sensitivity is examined by Samal et al. [52]. Squeezing a bioconvective-Casson HNF with different conductivity: a model-based investigation of heat transfer and irreversibility analysis by Samal and Ontela [53].

The current study’s innovative component uses the Yamada-Ota model to examine the Soret and Dufour effect on the bioconvection flow of radiative HNF across a Riga plate when gyrotactic microbes are present. This assessment is brand-new and has never been taken into consideration. By the shooting strategy, mathematical solutions are generated and the effects of the significant components are scrutinized. The goal of this scrutiny is:

• Gyrotactic microbes are used to measure the result of Soret and Dufour on the radiative flow of HNF (Hybrid nanofluid).

• A unique design of intelligent computing simulations integrated by Levenberg–Marquardt scheme with feed forward neural networks (LMS-FNN) is constructed to analyze the research on suggested model through a Riga plate in the existence of gyrotactic bacteria.

• The goal is to ascertain how the heat source influences the heat transport of the HNF.

• RHNF is modeled for several scenarios, and the LMS-FNN is tested, trained, and validated to ensure its correctness.

• The developed LMS-FNN’s presentation for successfully resolving radiative flow of a hybrid nanofluid is supported by regression analysis, a histogram convergence plot, and mean square error.

Developing the model and equations transformation

Consider the outcomes of Soret-Dufour effect on gyrotactic bacteria in radiative flow of hybrid nanofluid through a Riga plate using YOM (Yamada-Ota model) (Fig. 1). The two-dimensional flow.

Fig. 1.

Flow description

Equation [1, 54, 55] describe the flow of the HNF. Below the body forces and rules stated earlier:

1

2

3

4

5

The boundary conditions are [11, 54, 55]:

6

7

Similarity variables [55].

8

9

The following are reduced PDEs into ODEs:

10

11

12

13

The BCs are

14

15

where thermal radiation , bio-convection Schmidt number, Peclet number, Prandtl number, Dufour number, Schmidt number, Soret number. The thermos-physical features of HNF (hybrid nanofluid) are shown in Tables 1 and 2.

Table 1.

Demonstrations the thermos-physical assets of the NPs and the based liquid [56, 57]

Properties constituents
Water

Table 2.

Thermos-physical properties of hybrid nanofluid [56, 57]

Characteristics	HN

The thermal gradient , gyrotactic microbes gradient and concentration gradient are defined as:

16

17

18

19

20

Yamada–Ota model thermal conductivity [58].

21

Levenberg–Marquardt scheme

This study’s ANN is a feed-forward fitting network with 10 neurons. The Levenberg–Marquardt method, which offers quick and steady convergence for nonlinear problems, is used for training. The dataset is divided into 76% for training, 14% for validation, and 10% for testing to guarantee strong generalization and avoid over-fitting. Fitness curves, regression graphs, mean squared error, and error histograms are used to assess the network’s performance. To train the Levenberg–Marquardt scheme the relevant data must be collected and organized into a well-structured dataset that is suited to the suggested solutions in order to tackle a fluid dynamics problems. This dataset is used to train and assess a variety of techniques, including feed forward neural networks. The process entails carefully choosing representative samples, compiling data, and validating to ensure accuracy, dependability, and consistency. The diversity and composition of the dataset are important to the model’s capacity to provide credible and generalizable conclusions.

Analytical and ANNs’ results discussion

Figures 2, 3, and 4 depict how the Levenberg–Marquardt strategy feed forward neural network (LMS-FNN) affects the Dufour and Soret on radiative flow of HNF with thermal radiation in I–IV situations. Figure 2 depicts the function approximation findings, as well as error histograms of the model’s residual distribution and prediction accuracy for examples I–IV. Figure 3 depicts regression analysis and neural statistic transition graphs, which demonstrate the network’s resilience and learning dynamics for examples I-IV. Figure 4 depicts the validation performance curves for each scenario, demonstrating how well the LMS-FNN generalizes to new parameter combinations in instances I–IV. Table 3 provides a detailed description of each scenario’s essential performance parameters, including mean squared error (mse), execution time, training length, and time complexity. These metrics offer a full assessment of the proposed framework’s computational efficacy and training quality. Scenarios I–IV go over these repercussions in detail, as well as the model’s sensitivity to various parameter changes. To investigate different transport behaviors within the fluid system, each scenario is designed by a unique combination of controlling factors and boundary conditions. The reference case is represented by Scenario I, thermal impacts are introduced by Scenario II, mass transfer effects are the subject of Scenario III, and the fully coupled scenario—in which both thermal and solutal transport are active at the same time—is represented by Scenario IV. The error histograms in Fig. 3 show a narrow error distribution with good precision, which provides further information about residual behavior throughout the testing phase. The narrow performance and test value ranges displayed in Fig. 4 emphasize the exceptional generalization characteristics of the Levenberg Marquardt approach, which extend within training values include [1.47 × E⁻⁰⁸, 5.21 × E⁻⁰⁹, 1.26 × E⁻⁰⁷, and 3.19 × E⁻⁰⁷]. Regression graphs (Fig. 4) with correlation coefficients near to unity show that the LMS-FNN has high prediction and modeling accuracy in all studied cases. These high R-values imply a very good match between predicted outputs and HAM benchmark results. To verify the model, the LMS-FNN results are directly compared to those produced using the HAM solver. Table 2 shows that the LMS-FNN design consistently delivers low MSE values on the order of , highlighting its resilience and computational efficiency across diverse operating scenarios. To summarize, the LMS-FNN approach offers a high-precision, dependable, and computationally economical solution framework for the model, as demonstrated by graphical and numerical results. It also has great convergence qualities and dependability across a large parameter range. The proposed model, which studies the thermal behavior of bioconvective flow in a CNT-based hybrid nanofluid including gyrotactic microorganisms under Soret-Dufour impact, has significant implications for biotechnology, energy systems, and industrial heat management. It can be used to optimize bioreactors that demand high microbial activity and heat transmission, to improve the design of novel cooling systems that use hybrid nanofluids, and to help create efficient microfluidic devices. The effect of Hartmann number on field are display in Fig. 5a, b. The velocity field increase as enhances the values of . Figure 6a, b illustrate the on It is clear that the microbe profile increases along with an rise in . This causes a dramatic increase in the advective transfer rate, which raises the profile of mobile microorganisms.

Fig. 2.

Outcome of fitness curves and error statistics for scenarios s1–s4

Fig. 3.

State transition solution and regression analysis of the suggested LMS-FNN

Fig. 4.

Outcome of mse validation has been achieved by using the LMS-FNN of s1–s4

Table 3.

Outcome of LMS-FNN Technique

Scenarios	Performance	Training	Validation	Testing	Gradient	Time	Epoch
I	4.73 × E−08	1.47 × E−08	1.93 × E−07	7.61 × E−08	1.13 × E−05	26	907
II	5.33 × E−09	5.21 × E−09	8.20 × E−09	1.95 × E−09	2.17 × E−05	29	1000
III	3.54 × E−07	1.26 × E−07	1.23 × E−06	7.67 × E−07	2.60 × E−05	28	1000
IV	3.45 × E−07	3.19 × E−07	4.48 × E−07	4.01 × E−07	1.19 × E−05	22	745

Fig. 5.

a Comparision plot of on . b Absolute error plot on parameter

Fig. 6.

a Comparision plot of on . b Absolute error plot on parameter

Figures 7a, b and 8a, b shows the impacts  on temperature profile and . As value ups, the thermal profile and rate of heat transfer increase significantly using Yamada-Ota model. The experimental rise in the hybrid’s thermal profile can be examined by the liquid flow decreasing if the liquid’s nanoparticle solutal exceeds the NF’s density. The effects on and are displayed in Figs. 9a, b and 10a, b. The heat flux over a solutal gradient can be explained by the DE (Dufour effect). The thermal fields behave more strongly and negatively, respectively, when the Dufour influence is present. The flow through the boundary layer and the depth of the temperature boundary layer both significantly increase as the Dufour number rises. The rate of heat transfer rises as Du values increase. Figures 11a, b and 12a, b illustrate the properties of on and . As levels rise, the concentration profiles get better. As rises, both molar mass diffusivity and rise. Figure 13a, b illustrate the on As the values of increase, so does the local density of microbes. The heat transfer rate evaluation among the current investigation and other studies that solely employed mutual parameters and the particular situation is shown in Table 4. There is good concordance between the new data and previous conclusions.

Fig. 7.

a, b Properties of on field 3D and 2D

Fig. 8.

a, b Effect of on field 3D and 2D

Fig. 9.

a, b Properties of on field 3D and 2D

Fig. 10.

a, b Result of on field 3D and 2D

Fig. 11.

a, b Result of on field 3D and 2D

Fig. 12.

a, b Effect of on field 3D and 2D

Fig. 13.

a, b Result of on field 3D and 2D

Table 4.

Comparison with previous works of literature

	Present results	Shafiq et al. [59]

Conclusions

The proposed model for the thermal study of bioconvection flow in CNT-based hybrid nanofluid, which includes gyrotactic microorganisms and Soret-Dufour effects, has vital uses in a variety of advanced fields. It can be utilized to increase the role of bioreactors and microfluidic devices in which temperature management and microbial activity are important. In energy systems, the model can improve the efficiency of cooling solutions, particularly in the electronic and automotive sectors. Furthermore, it helps to increase heat transport processes in industrial applications, while the use of artificial neural networks enables precise predictions of complex fluid dynamics and bio-thermal interactions, accelerating the creation of effective and sustained heat management solutions. The following is a list of the investigation’s conclusions and recommendations.

• The flow profile increases as the Hartmann number rises.

• Increasing Peclet number leads to a reduction in the microbial profile.

• The rate of heat transfer and thermal profile rise with the volume friction of nanoparticles.

• The solutal profile, thermal profile, mass and heat transport rate rise as improve the values of Soret and Dufour numbers.

• The LMS-FNN is expected to have a mean square error of when compared to the reference solution and proposed data, indicating close agreement.

• The density of gyrotactic bacteria increases with higher Peclet number.

Abbreviations

Nusselt number

Mean absorption coefficient

Ambient concentration

Density for gyrotactic microorganism

Volume friction of nanoparticles

Thermal conductivity

Ambient gyrotactic microbes

Bio-convection Schmidt number

Schmidt number

Mass flux )

Dynamic viscosity

Mass diffusivity

Thermal

Sherwood number

Local density of bio convection

Concentration

Ambient temperature

Pecklet number

Hartman number

Flow fields of fluid

Kinematic viscosity

Skin friction

Fluid density

Heat flux

Fluid density

Thermal radiation

Prandtl number

Constants

Ambient solutal

Microorganism solutal difference factor

Gyrotactic microbe diffusivity

Sherwood number

CCHFM

Cattaneo–Christov heat flow model

THNF

Trihybrid nanofluid

NF

Nanofluids

HNF

Hybrid nanofluids

NPs

Nanoparticles

SDE

Soret-Dufour effect

Author contributions

Mohamed Bechir Ben Hamida and Munawar Abbas: Supervision, Methodology, Conceptualization. Ali Akgül: Methodology, Formal analysis. Murad Khan Hassani: Writing—review and editing.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2603).

Data availability

All data used in this manuscript have been presented within the article.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.