Evaluating energy transmission characteristics of Non-Newtonian fluid flow in stratified and non-stratified regimes: A comparative study

Results in Engineering 22 (2024) 102157 Available online 21 April 2024 2590-1230/© 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Evaluating energy transmission characteristics of Non-Newtonian fluid flow in stratified and non-stratified regimes: A comparative study S. Bilal d , Asadullah a , * , Muhammad Bilal Riaz b , c a Department of Mathematics, Air University, Sector E-9, P.A.F Complex, P.O. 44000, Islamabad, Pakistan b IT4Innovations, VSB – Technical University of Ostrava, Ostrava, Czech Republic c Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon d Dept. of Mechanical Engineering, College of Engineering, Prince Mohammad Bin Fahd University, Al Khobar 31952, Kingdom of Saudi Arabia ARTICLE INFO Keywords: MHD Thermosolutal stratification Williamson fluid Levenberg-marquardt and bayesian regularization schemes Convective inclined surface Chemical reaction ABSTRACT Considering the natural and industrial importance of flow characterization in stratified media, the current study is articulated. This work highlights the influence of linear stratification as well as convective surfaces in both thermal and solutal fields on the rheological attributes of Williamson fluid flow through an inclined surface. Novel physical aspects of a uniformly provided magnetic field of strength B and chemically reactive species are also included. The concerned transport equations are derived from the associated conservation laws in dimensional forms. Modification in the developed couple system is achieved by using a set of similar variables. Levenberg-Marquardt Scheme (LMS) and Bayesian Regularization Scheme (BRS) are utilized in comparative manner to analyze initial data accessed for quantities of interest. The data used in the generation of MLP was 80 percent for model training and 20 percent for testing and validation. Error histograms, performance plots, fitness curves, and regression plots for training, testing, and validation are presented. Data in the form of tables and graphs are presented, which express an excellent match between the ANN-predicted and targeted values. It is revealed that an artificial neural network approach can provide highly efficient forecasting for such problems by providing accurate data for quantities of interest. It is noticed that Nusselt number and Sherwood number enhances up to 33 % and 29 % versus respective stratification parameters. Velocity profile declines against magnetic field parameter (M) whereas, skin friction coefficient increments up to 25 %. Appliance of convective boundary constraints at the surface of inclined sheet tends to enhance the temperature and concentration fields. 1. Introduction Over the years, Navier Stokes equation has been utilized to predict the behavior of fluids against appliance of shear stresses. Experts have experimentally verified that owing to the distinctive response of liquids to external forces, they are categorized into viscoelastic and viscoinelastic. Through rheological characterization of the mentioned subclasses, extensive dissimilarities have been found in their physiological attributes. Viscoinelastic fluids have been considered more realistic and applicable to real-world applications owing to the prediction of flow behavior at zero shear rates. Some mesmerizing applications of viscoelastic fluids include food manufacturing, oil and gas refinement processes, power generation systems, protective systems, medicinal devices, electronic devices, and crystal growth. On the basis of such exclusive significance, various flow models have been proposed to utilize it in different situations; few models among them are power law, Casson, and Williamson fluids. Among these frameworks, the fittest is the Williamson fluid because it falls into the fluid category that possesses a pseudo-plastic nature. This type of fluid is utilized for many industrial and engineering purposes, such as blood cells, photographic films, and food processing. A few recent studies depicting the flow behavior of viscoinelastic materials under different physical constraints have been conducted. For instance, the flow behavior of viscoinelastic fluids over external surfaces employing the similarity approach was investigated by Darji and Timol [1]. In an investigation, peristaltic flow of Williamson fluid in the small intestine with the insertion of an endoscope was manifested by Nadeem et al. [2]. The flow was confined between two concentric tubes. They observed the pattern of formed streamlines and attained that trapping occurs, and the size of the trapped bolus varies by varying physical parameters. Heat transfer analysis of a steady 2D Williamson fluid flow through a permeable exponentially enlargeable * Corresponding author E-mail addresses: [email protected] (Asadullah), [email protected] (M.B. Riaz). Contents lists available at ScienceDirect Results in Engineering journal homepage: www.sciencedirect.com/journal/results-in-engineering https://doi.org/10.1016/j.rineng.2024.102157 Received 11 November 2023; Received in revised form 6 April 2024; Accepted 18 April 2024 Results in Engineering 22 (2024) 102157 2 sheet was demonstrated by Nadeem and Hussain [3]. In this study, they considered two regimes for heat transfer, that is, the prescribed exponential order heat flux (PEHF) case and the prescribed exponential order surface temperature (PEST) case. They concluded that, for both cases, the thermal boundary layer depreciates by flourishing the Prandtl number. A numerical investigation of Williamson fluid flow over an extendable cylinder with variable thermal conductivity and heat generation/absorption was manifested by Malik et al. [4]. They confirmed that the temperature profile was enhanced by amplifying the magnitude of the thermal conductivity parameter, while the opposite trend was attained against the Prandtl number. Kumaran et al. [5] illustrated the flow of Williamson fluid incorporated with the impacts of magnetohydrodynamic and viscous dissipation through an upper paraboloid of revolution. They observed that the velocity distribution declined with increasing Hartmann number. Shah et al. [6] discussed Williamson fluid flow over time dependent extendable surface incorporated in permeable media with heat transmission and thermal radiation. The heat and mass transfer analysis of magnetized Williamson fluid through a curved surface for unsteady and steady flow regimes was addressed by Kumar et al. [7]. In addition, the flow was also characterized by convective boundary constraints, chemical reaction and thermal radiation. Raju et al. [8] explored Williamson and Casson fluid flow over stretchable surface along with transfer of mass and heat. In addition, the flow is also characterized by homogeneous–heterogeneous reactions. Magnetized Williamson fluid flow over an extendable surface by taking the impacts of thermal radiation, chemical reaction and viscous dissipation was demonstrated by Kumar et al. [9]. The flow characteristics of Williamson fluid over a nonlinear extendable surface embedded in a porous medium were disclosed by Abbas et al. [10]. Some recent developments concerning the described non-Newtonian model in various physical constraints are encapsulated in Refs. [11–14]. Heterogeneity among fluid layers is generated owing to density and viscosity differences caused by temperature and concentration variations. The interplay between fluid heterogeneity and gravity results in a striking phenomenon known as stratification. Comprehension of the movement of objects in a stratified environment involves pervasive environmental, geophysical, ecological, and industrial procedures. For instance, conversion of air pollutants (dust, aerosols, weather balloons, matter) resides in the lower atmospheric region where they encounter low temperature gradients, and by providing stratification, they will scatter in the upper zone, which forms clouds and improves the quality of air. In addition, the removal of accumulated toxicants in oceanic layers is efficiently eradicated by producing stratification. Other than ecological utilizations, stratified environments are also immensely applicable in energy conversion and storing systems. In view of such splendid implications, the phenomenon of stratification through both thermal and solutal exchanges has accounted for the pervasive intent of researchers’ fraternity. Rishabh and Ardekani [15] delineated sedimentation of an isolated object influenced by thermal stratification in shallow water and discussed their hydrodynamic mechanism. The application of a stratified flow regime in gas pipelines, by investigating the hydrothermal attributes of viscoelastic liquid flow over an extending surface subjected to thermally variant stratification, was divulged by Lone et al. [16]. The role of variable magnetic flux in generating stratified flow of a non-Newtonian fluid over an exponential surface by employing a numerical scheme was explicated by Singh et al. [17]. Couette flow of stably stratified dusty Walter liquid through a permeable medium and under physical insight of a uniformly applied transverse magnetic field through an analytical approach was contemplated by Dey [18]. Hydrothermal attributes of stratified Eyring-Powel fluid with physical aspects of variable thermo physical characteristics (conductivity and viscosity) over a non-linearly extendable surface were determined by Wahab et al. [19]. Density variations produced in the stratified flow of micropolar fluid due to the non-persistent impact of buoyancy forces over a plate were scrutinized by Waqas et al. [20]. The stratified flow of Powell-Eyring fluid over an extendable permeable surface with dissipation and radiative heat flux was investigated by Abbas and Megahed [21]. Some relevant studies disclosing the efforts made in the direction of stratified flows are presented in Refs. [22–26]. In the current century, ANN has played an important role in solving complex problems in different domains, such as science and engineering. The ANN model comprises interconnected layers or neurons that work based on the working mechanism of the human brain. It is due to the fact that neural network obtained information from input data; they recognize input data and organize the data to predict the outcomes. Currently, ANN are used extensively in the field of non-Newtonian fluids. ANN based time dependent study of Williamson fluid flow through a permeable extendable surface with Levenberg-Marquardt back propagation scheme was carried out by Shafiq et al. [27]. Shayya et al. [28] developed ANN model for the estimation of friction factor for Herschel-Bulkley liquids under turbulent and laminar flow situation through closed pipes. Machine learning analysis of bioconvective flow of Williamson fluid over an enlargeable sheet with Brownian motion and thermopherotic impacts was manifested by Priyadharshini et al. [29]. Shoaib et al. [30] analyzed the flow features of magnetized third-grade fluid through an enlargeable sheet with activation energy and convective boundary constraints by utilizing a machine learning algorithm based on the Levenberg–Marquardt scheme. Hussain et al. [31] developed an ANN model based on the Levenberg–Marquardt algorithm to scrutinize the features of Casson fluid flow over a nonlinear slanted surface. For the scrutinization of flow properties of mixed convective non-Newtonian fluid flow over stretching media, an intelligence algorithm based on the Bayesian regularization approach was formulated by Shah et al. [32]. The phenomenon of stratification has numerous applications in various fields such as geology and earth science, ecology and environmental science, market segmentation, atmospheric science, and many more. In view of such splendid implications, the phenomenon of stratification through both thermal and solutal exchanges has been accounted. In addition, vast range of applications concerned with stratified flows are found in multiple scientific and engineering processes, such as Nomenclature U1,V1 Velocity components υ Kinematic viscosity Γ Time constant α Thermal diffusivity cp Specific heat constant βT1 Thermal expansion coefficient β1 Inclination angle K1 Chemical reaction coefficient B Uniform magnetic field βC1 Solutal expansion coefficient A1 Williamson fluid parameter D Diffusion species coefficient T1∞ Ambient temperature k Thermal conductivity ρ Density of fluid C1f Fluid Concentration g Gravitational acceleration T1f Fluid Temperature hf Heat transfer coefficient C1∞ Ambient concentration M Magnetic field parameter hc Mass transfer coefficient λ Thermal buoyancy parameter λ1 Solutal buoyancy parameter S1 Thermal stratification parameter S2 Solutal stratification parameter Sc Schmidt number kr Chemical reaction parameter Pr Prandtl number γ1 Thermal Biot number Cfx Skin friction coefficient γ2 Solutal Biot number τ w Wall share stress Nux Nusselt number qw Heat flux Shx Sherwood number qm Mass flux S. Bilal et al. Results in Engineering 22 (2024) 102157 3 heat transfer in buildings, aerospace engineering, environmental engineering, electronic cooling, and oceanography. Subsequently, consideration of convective boundary constraints makes the present communication more impactful. From the above-mentioned literature survey, it is worth mentioning that no contribution has yet been made to explore the features of stratified Williamson fluid flow with magnetohydrodynamic (MHD) and chemical reaction effects. The principal aim of the present study is to scrutinize the aspects of stratified Williamson fluid flow over an enlargeable surface with convective boundary conditions in the existence of magnetohydrodynamic and chemical reaction effects. Moreover, a machine learning algorithm based on the Levenberg Marquardt and Bayesian Regularization Schemes is also employed in a comparative sense to check the fitness of the attained data. The innovative contributions of the study are as below. 1. The governing equations are constituted for stratified Williamson fluid flow over a convective inclined enlargeable sheet. 2. Data sets are obtained numerically by utilizing shooting method in combination with the RK-4 scheme, and a comparative analysis of the data set is performed to train through LMS and BRS. 3. Validation of the attained outcomes is shown by error histogram, mean square error, regression, and fitness plots. 4. The impacts of significant flow parameters on the temperature, concentration and velocity distributions, drag coefficient, and heat and mass fluxes are observed numerically and graphically. 2. Problem formulation In this section, a brief description of the implemented methodologies for the proposed governing problem, that is, stratified Williamson fluid flow over a convective surface with magnetic field and chemical reaction aspects, is presented. Quantities of engineering interest such as the skin friction coefficient (SFC), heat flux coefficient (HFC), and mass flux coefficient (MFC) are calculated. For this purpose, the initial dataset mentioned quantities are produced through the shooting method (SM) and then divided into training, testing, and validation. Subsequently, artificial back-propagated neural networking (ABPNN) based on renowned schemes (LMS) and (BRS) is employed. A graphical visualization of the study is depicted in Fig. 1. By utilizing aforementioned suppositions, the governing continuity, momentum, energy and concentration equations take following form. Continuity equation [33]. ∂ U1 ∂ X1+ ∂ V1 ∂ Y1=0,(1) Momentum equation for non-Newtonian Williamson fluid model is represented as under [33]. U1 ∂ U1 ∂ X1+V1 ∂ U1 ∂ Y1=ϑ ∂ 2U1 ∂ Y1 2+ 2 √ϑΓ ∂ U1 ∂ Y1 ∂ 2U1 ∂ Y1 2− σ B2 ρ U1 +(βT1(T1−T1∞)+βC1(C1−C1∞))gsinβ1,(2) In the momentum equation, the last two terms were formulated by employing the Boussinesq approximation, which expresses the features of convection arising from temperature and concentration differences caused by linear stratification [34]. Energy equation [34]. U1 ∂ T1 ∂ X1+V1 ∂ T1 ∂ Y1=k ρ cp ∂ 2T1 ∂ Y1 2,(3) Concentration equation [34]. U1 ∂ C1 ∂ X1+V1 ∂ C1 ∂ Y1=D ∂ 2C1 ∂ Y1 2−K1(C1−C1∞).(4) The relevant boundary conditions are expressed as under [34]. U1=U1w(X1)=cX1,V1=0,−k ∂ T1 ∂ Y1=hf(T1f−T1),−D ∂ C1 ∂ Y1 =hc(C1f−C1),at Y1=0,U1→0,T1→T1∞(x),C1→C1∞(x)as Y1→∞, (5) Where, T1f(X1)=T10+d1X1,C1f(X1)=C10+e1X1,T1∞(X1)=T10+d2X1,C1∞(X1)=C10 +e2X1. (6) In above equations (2)–(6), U1 and V1 represents velocity component along X1and Y1-axis, respectively. U1w stretching velocity, T1f heated fluid temperature, T1∞ variable ambient fluid temperature, C1f heated fluid concentration, C1∞ variable ambient concentration, hf and hc represents heat and mass transfer coefficients, D is coefficient of diffusion species, k shows thermal conductivity, d1,d2,e1 and e2 highlights dimensional constants. Dimensionless form of governing equations is attained by employing following transformations [34]. Fig. 1. Physical domain of the problem. S. Bilal et al. Results in Engineering 22 (2024) 102157 4 U1=cX1F ′ (ξ),V1= −  cϑ √F(ξ),ξ= c ϑ √Y1,θ(ξ) = T1−T1∞ T1f−T10 ,φ(ξ) =C1−C1∞ C1f−C10 .(7) After employing similar variables continuity equation is identically satisfied and the remaining equations subject to boundary constraints are as follows F F ″ +F ‴ +A1F ″ F ‴ −F ′ 2−MF ′ +λθ sin β1+λ1φ sin β1=0,(8) θ ″ +Pr Fθ ′ −Pr S1F ′ −Pr F ′ θ=0,(9) φ ″ +ScFφ ′ −ScF ′ φ−ScS2F ′ −Sckrφ =0.(10) Dimensionless form of boundary constraints is as under F ′ (0)=1,F(0)=0,θ ′ (0)= − γ1(1−S1−θ(0)),φ ′ (0)= − γ2(1−S2 −φ(0)),F ′ (∞)=0,θ(∞)=0,φ(∞)=0, (11) here, A1 Williamson fluid parameter, λ and λ1 are thermal and solutal buoyancy parameters, respectively, M is magnetic field parameter. Pr shows Prandtl number, S2 and S1 depicts solutal and thermal stratified parameters, Sc highlights Schmidt number, kr highlights chemical reaction parameter, γ2 and γ1 depicts solutal and thermal Biot numbers. These parameters are expressed as under A1=ΓX1 2c3 ϑ √,M= σ B2 c ρ ,λ=gβT1(T1f−T10) X1c2,λ1=gβC1(C1f−C10) X1c2,Pr =ϑ α S1=d2 d1 ,S2=e2 e1 ,Sc =ϑ D,kr =K1 c,γ1=hf k c ϑ √,γ2=hc D c ϑ √. (12) The parameters of physical interests i.e. drag coefficient, Nusselt number and Sherwood number are illustrated by the following expressions Cfx = τ w 1 2 ρ U1 2 w ,Nux=X1qw k(T1f−T1∞),Shx=X1qm D(C1f−C1∞),(13) where, qm wall mass flux, τ w wall shear stress, and qw wall heat flux are defined as follows τ w= μ 0[ ∂ U1 ∂ Y1+Γ  2 √( ∂ U1 ∂ Y1)2],qw= − k( ∂ T1 ∂ Y1),qm= − D( ∂ C1 ∂ Y1).(14) After the implementation of dimensionless variables in eq. (7), drag coefficient, heat flux and mass flux in dimensionless form are as follows 1 2CfxRe1 2 x=[F ″ (0)+A1 2(F ″ (0))2],Nux=−θ ′ (0) Re−1 2 x(1−S1) ,Shx=−φ ′ (0) Re−1 2 x(1−S2) . (15) 3. Numerical scheme To solve analytically the translated nonlinear ODE’s is not easy. Approximate solution of these nonlinear ODEs along with boundary conditions is obtained by implementing shooting approach in combination with Runge-Kutta method of order four, which is widely used numerical technique to originate approximate solution of ODEs with adaptive controlled step size. To use this technique, higher-order ODEs are transformed into first order by introducing new variable F=t1, F ′ =t2,F ″ =t3,F ‴ =t ′ 3,θ=t4,θ ′ =t5,θ ″ =t ′ 5,φ=t6,φ ′ =t7,φ ″ =t ′ 7. First-order ODEs are t ′ 1=t2, t ′ 2=t3, t ′ 3=t2 2+Mt2−t1t3−λt4sin β1−λ1t6sin β1 1+A1t3 , t ′ 4=t5 t ′ 5=Pr t2t4+Pr t2S1−Pr t1t5, t ′ 6=t7, t ′ 7=Sckrt6+ScS2t2+Sct2t6−Sct1t7, where relevant boundary conditions are ξ=0:t1(ξ)=0,t2(ξ)=1,t5(ξ)= − γ1(1−S1−t4(ξ)),t7(ξ)= − γ2(1−S2 −t6(ξ)), ξ→ ∞ :t2(ξ)→ 0,y4(ξ)→ 0,y6(ξ)→0. To attain the numerical solution of the initial value problem firstly initial guesses are selected, then solution process is carried out to attain the outcomes. The key factor of the shooting method is to select appropriate value of ξ∞.For this, initial guesses for the for particular physical parameters are selected to obtain f ″ (0),θ ′ (0)and φ ′ (0).Modified the initial guesses until the difference between two adjacent outcomes of f ″ (0),θ ′ (0)and φ ′ (0)is close to required digit. Last computation for ξ∞ is considered as adequate for the given data set. Afterwards, solution is processes further. After attaining the results for f ″ (0),θ ′ (0)and φ ′ (0)RK-4 approach is utilized to attain the results. This process repeated until the required degree of accuracy is attained. The steps involved in the utilized numerical approach is manifested in the following flowchart portrayed in Fig. 2(a). Implementation of utilized numerical scheme on currently modelled problem is step wisely shown in Fig. 2(b), whereas working mechanism of shooting method is depicted in Fig. 2(c). In this mechanism, we used the hit and trial technique to convert the boundary value problem into an initial value by choosing the initial guesses through the hit and trial approach. 4. Artificial neural networking (ANN) Artificial neural networking (ANN) is a machine learning algorithm whose working mechanism is highly based on the working structure of the human brain. After receiving input information, ANN is an effective tool for nonlinear statistical data modeling. The structure of an ANN consists of three layers that are correlated with each other. All input layers are hidden, and after receiving information, these layers alter them from layer to layer through a series of transformations. To understand more complex objects, each layer is treated as both input and output for the ANN. Overall, these inner layers are renowned as neuron layers. To predict the hydrodynamic flow attributes of Williamson Fluid in a stratified environment, an ANN model developed over a convective surface is designed. The ANN model comprises back propagation (BP), feed forwarding (FF), and multilayer perception (MLP) networking. The MLP network consists of three layers: input, hidden, and output layers (as shown in Fig. 3). In the input layer data for SFC, HFC and MFC against physical parameters like Hartmann number M,Williamson fluid parameter A1,Prandtl number Pr,thermal Biot number γ1,Schmidt number Sc and chemical reaction parameter kr is entered into system for training purpose. The hidden layer comprises the number of neurons to be specified for training the dataset that predicts data with high accuracy. Finally, the output layer in which data are to be tested is attained S. Bilal et al. Results in Engineering 22 (2024) 102157 5 for (SFC), (HFC) and (MFC) for various scenarios. Data validation is attained for the construction of neural networking, and testing data is applied to check unbiased input performance. A working diagram of multilayer perception (MLP) is illustrated in Fig. 3. In the present study, three quantities of interest that are useful in many engineering problems are computed in the output and input layers. After accessing the input data via numerical approaches, they are categorized into three parts, that is (training, testing, and validation) to predict the solution through (LMS) and (BRS). Data is distributed in a percentage wise manner such that 80 % is assigned to train the network and 20 % for validation and testing modules. The performance of (LMS) and (BRS) on the developed ANN model for Williamson fluid flow over a convective surface by estimating the mean square error (MSE) for the training, testing, and validation modules related to the scenarios of (SFC), (HFC) and (MFC) is Fig. 2. a. Steps involved in numerical procedure. b Description of implemented scheme on present problem. c Working mechanism of shooting procedure. Fig. 3. Multilayer perception of ANN model. S. Bilal et al. Results in Engineering 22 (2024) 102157 6 Fig. 4. Flow chart for developed ANN model [37]. Table 1 Numerical results through LMS for the ANN model. Parameters Physical Quantity MSE Epochs Grad Performance Mu Time Training Testing Validation M,A1 SFC 3.53E-8 4.66E-7 7.61E-7 218 9.02E-08 3.53E-08 1.00E-08 0:00:03 Pr,γ1 HFC 4.68E-8 2.46E-7 8.39E-8 33 9.69E-08 4.68E-08 1.00E-08 0:00:00 Sc,kr MFC 8.61E-9 1.60E-8 5.70E-9 19 9.39E-08 8.91E-09 1.00E-09 0:00:00 S. Bilal et al. Results in Engineering 22 (2024) 102157 7 illustrated in Table 1 for (LMS) and Table 2 for (BRS). Furthermore, (MSE) is also evaluated for (SFC), (HFC) and (MFC) versus variation in (M),(A1),(Pr),(γ1),(Sc)and (kr)ranges from 0.1 to 2 comparatively through (LMS) and (BRS). A reduction in the magnitude of (MSE) is revealed in (BRS) as compared to (LMS) which signifies that (BRS) predicts more accurate values. The results for the flow field, thermal gradient, concentration distribution, skin friction, and heat and mass fluxes were also manifested in a numerical and graphical manner (see Fig. 4). Fig. 5(a–h) demonstrates the error histogram, training, performance analysis, and regression plots for (SFC) against Hartmann number (M) and Williamson fluid parameter A1 for (LMS) and (BRS). The data are categorized into three modules: training (80 %), testing (20 %), and validation. Multiple samples are obtained, the model is applied to each sample to forecast the output, and the target or actual values are then compared with the obtained outcomes. Errors are calculated by categorizing into 20 bins and represented through a histogram, as shown in Fig. 5(a–b); all 20 bins are displayed versus the number of samples containing errors, where error bins are depicted on the X-axis and the Yaxis represents the instance at which error occurs or samples that contain error in that bin. In the bar diagram, it is noticed that from the training data, seven and five samples have almost zero error, and from the test data, three and one samples contain zero error for (LMS) and (BRS) respectively. In addition, one sample of validation possesses zero error, which shows that most of the data have zero errors, which certified well training of data through the Levenberg Marquardt Scheme (LMS). Subsequently, it is also revealed that range of zero error lies between 10 −5 – 10 −3 and 10 −6 – 10 −4 which also assures excellent training of data through (LMS) and (BRS) respectively. Fig. 5(c–d) shows a visualized graph of the training state for the ANN model through (LMS) and (BRS) and provides crucial information related to the obtained model by estimating the gradient, mu, and optimal epochs. (MSE) is utilized by taking the squared difference between the target and predicted estimations to attain convergence of the model. The gradient value, which suggests that the simulation model has reached the local minimum of the objective function at the lowest possible level, is also discussed. Finally, the controlling parameter (Mu) that controls the ANN model training mechanism is also elaborated, which directly affects the error convergence. Fig. 5(c) reveals that the value of Mu is 1.0E-08 with a gradient of 9.02E-08, and Fig. 5(d) depicts that the outcome for Mu is 0.5, with a gradient of 7.92E-08. It is revealed that as the epochs increase, the estimations for Mu and the gradient shrink, and as a result, the rate of convergence will occur rapidly. Fig. 5(e–f) shows validation performance plots to test the model on the dataset after each epoch to adjust the network weights simultaneously through (LMS) and (BRS). At the start of the training process, MSEs are at their peak, with the passage of time errors declining as the model moves towards the optimal state. The best validation performance is represented by dotted lines. From Fig. 5(e), it is noticed that up to 218 epochs, the model’s validation performance is very high, and its best validation performance was achieved with an MSE of 7.61E-07 through (LMS) while from Fig. 5(f), it is observed that the best validation performance of the model is attained at 174 epochs with an MSE of 4.69E-08 through (BRS). It is deduced that from analysis that (BRS) gives more accurate result than (LMS) by reducing the error between target and predicted values. Fig. 5(g–h) divulges the linear regression graphs which are attained when the ANN model is implemented on all datasets to make prediction through (LMS) and (BRS) and then compared with actual values. The fitness of model is checked by developing linear relations formulated by Output = R*target +bias between outputs and targets. From Fig. 5(g–h), R =1 is observed in all regression plots used for training, validating, and testing datasets, indicating that all formulated ANNs are performing flawlessly. The training data were obtained by solving the governing physical problems for (SFC), (HFC) and (MFC) against the involved physical parameters by implementing shooting and Runge-Kutta numerical approaches. From the sketch (see Fig. 5(g–h)), it is noticed that the output is nearly identical to the target, although the bias varies in all four cases. Overall, it is inferred that ANNs provide excellent outcomes for regression analysis. Assessment of errors between predicted and target values via histogram, performance, regression and fit plots are is portrayed in Fig. 6(a–j) for HFC against Prandtl number (Pr) and thermal Biot number (γ1)in comparative manner through (LMS) and (BRS). The orange line represents zero error. Fig. 6(a–b) show the difference between the predicted output attained from (LMS) and (BRS) and the actual values obtained for (HFC) in the form of all three datasets (training, testing, and validation) on which sampling is executed. Fig. 6(a) calculates errors in each sample collected for (HFC) through (LMS) categorized in the form of 20 bins along with samples constituting those errors. From the histogram, it is revealed that for the training data, five samples have an error of almost zero, whereas for the validation data, three samples contain zero error and one sample of test data possesses zero error. This figure shows that most samples have zero error from all datasets. In comparison to the previous diagram, Fig. 6(b) shows that the zero error in the samples collected for the heat flux coefficient through (BRS) increases for training data up to six and testing up to two samples. Information about the training state plot to evaluate the trained model from the ANN by showing the gradient, mu, optimal epoch, and how they reach the optimal state for both (LMS) and (BRS) is collected in Fig. 6(c)–(d). MSE is utilized for the convergence analysis of the developed model by finding average squared differences between targets and predicted estimations by ANN. The gradient value is also discussed, which indicates that the local minimum of the objective function hits at the lowest point. Finally, the controlling parameter (Mu) for ANN model training is discussed, which directly influences error convergence. Fig. 6(c) reveals that the value of Mu is 1.0E-08 with gradients 9.68E-08 through (LMS) while; Fig. 6(d) shows that the value of Mu is 0.5 with gradient 9.63E-08 through (BRS). It is revealed that as the epochs increase, the estimations for (Mu) and gradient deprecate, and as a result, the rate of convergence will occur rapidly. The performance plot to validate model testing after each epoch to adjust the network weights for data received for (HFC) through (LMS) and (BRS) is shown in Fig. 6(e–f). The linear regression plot describing the use of trained ANN data for all training, validation, and testing modules to predict the heat flux coefficient (HFC) through (LMS) and (BRS) along with a comparison with actual values received from numerical approaches, is shown in Fig. 6(g–h). This graph determines the goodness of the model and follows a linear relationship between the target and input. From both figures, it is observed that R =1 in all three modules, which ensures that ANN has worked excellently. In Fig. 6(g), for all four datasets, the model gives R =1. This means that the output is nearly identical to the target, but the bias varies in each instance. Fig. 6(h) also shows that the training data have R =1, which also reveals perfect accuracy, and the model still performs the best. Fig. 6(i–j) illustrates plots of fitness for the developed ANN model for the Table 2 Numerical results through BRS for ANN model. Parameters Physical Quantities MSE Epochs Effective Parameter Grad Performance Mu Sum of Square parameter Time Training Testing M,A1 SFC 4.69E-8 1.15E-7 174 15.0 7.93E-08 4.70E-08 0.500 44.5 0:00:02 Pr,γ1 HFC 2.45E-9 1.17E-8 162 14.8 9.63E-08 2.45E-09 0.500 74.0 0:00:02 Sc,kr MFC 1.52E-9 7.52E-10 142 14.4 9.76E-08 1.53E-09 0.500 60.8 0:00:02 S. Bilal et al. Results in Engineering 22 (2024) 102157 8 Fig. 5. Graphs of the formulated ANN model for skin friction through LMS and BRS. (a) LMS Error histogram. (b) BRS Error histogram. (c) LMS Training state. (d) BRS Training state. (e) LMS Performance. (f) BRS Performance. (g) LMS Regression. (h) BRS Regression. S. Bilal et al. Results in Engineering 22 (2024) 102157 9 Fig. 6. Plots of the ANN model for the Nusselt number through LMS and BRS. (a) LMS Error histogram. (b) BRS Error histogram. (c) LMS Training state. (d) BRS Training state. (e) LMS Performance. (f) BRS Performance. (g) LMS Regression. (h) BRS Regression. (i) LMS Fitness curve. (j) BRS Fitness curve. S. Bilal et al. Results in Engineering 22 (2024) 102157 16 source/sink, in: IOP Conference Series: Materials Science and Engineering, vol. 263, 2017. [6] Z. Shah, E. Bonyah, S. Islam, W. Khan, M. Ishaq, Radiative MHD thin film flow of Williamson fluid over an unsteady permeable stretching sheet, Heliyon 4 (2018) e00825. [7] K.A. Kumar, J.R. Reddy, V. Sugunamma, N. 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