Investigation of Magnetized Casson Nanofluid Flow along Wedge: Gaussian Process Regression

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Introduction
Aerospace engineering and fuid mechanics both beneft signifcantly from the study of fuid fow over a wedge because they ofer important insights into how fuids behave when they interact with solid surfaces.Tis knowledge aids in the comprehension of boundary layer dynamics, aerodynamic phenomena, and the design of airfoil shapes for efective lift and drag characteristics in a variety of engineering applications.Recently, the study of MHD boundary layer slip fow of heat and mass transfer performance over a wedge-shaped geometry has been extensively explored due to its wide applications in science and engineering.It is used in industrial processes, including geothermal systems, nuclear reactors, nuclear waste storage, thermal insulation in aircraft cabins, and heat exchangers.Earlier in 1931, Falkner and Skan [1] investigated the fow over a static wedge immersed in a viscous fuid and developed the Falkner-Skan equation.Awaludin et al. [2] discovered the repercussions of a magnetic feld on the fow of an incompressible and electrically conducting fuid past a stretching/shrinking wedge.Te viscous dissipation efects on the MHD boundary layer stream of nanofuid across a wedge embedded in porous medium were examined numerically via the spectral quasilinearization method (SQLM) by Ibrahim and Tulu [3].A few inquiries involving the Falkner-Skan fow with various types of physical characteristics past a wedge can be found in the studies of Kudenatti and Amrutha [4], Haq et al. [5], and Butt et al. [6].
Te study of non-Newtonian fuid model has gained an incredible position among researchers because of their applications in industries and chemical engineering process, such as petroleum and polymer industries, food technology, heat exchangers, paper production, and electronic cooling system.Biological fuids (blood, salvia, etc.) and foodstufs (honey, jellies, jams, soups, etc.) are examples of non-Newtonian fuids because of their physical nature.Casson fuid is a type of non-Newtonian fuid that behaves like an elastic solid.Casson model constitutes a fuid model that exhibits shear thinning characteristics, yield stress, and high shear viscosity [7].Tese fuids are applied in technical processes, such as biomedical and industrial engineering, energy generation, dynamics, and geophysical fuid mechanics.Hussanan et al. [8], Khan et al. [9], Ullah et al. [10], Ullah et al. [11], and Guadagni et al. [12] have scrutinized the consequence of magnetic feld, Soret-Dufour, viscous dissipation, and chemical reactions on the Casson fuid in diferent fow settings.Mukhopadhyay and Mandal [13] developed a numerical study of the boundary layer forced convection fow of a Casson fuid over a symmetric porous wedge.Tey found that the Casson fuid parameter tends to control the fow separation.El-dabe et al. [14] used the numerical method (fnite diference method) to obtain the solution of the MHD boundary layer fow of Casson fuid on a moving wedge with heat and mass transfer.Mahdy [15] illustrated the impact of slip at the boundary of unsteady two-dimensional MHD fow of a Casson fuid over a stretching surface using the very robust computer algebra software MATLAB.From their results, it was observed that the velocity increases and the thermal boundary layer becomes thinner with the increasing slip parameter.Raju and Sandeep [16] used the Runge-Kutta and Newton's methods to obtain the solution of MHD slip fow of a dissipative Casson fuid over a moving wedge with heat source/sink.Recently, researchers focused on investigating the sundry fow features of Casson nanofuid in diferent frames [17][18][19][20].
One of the massive challenges within the modern science and technology panorama is attaining concrete enhancements regarding the rate of heat transfer of ordinary fuids such as water, lubricants, oils, ethylene glycol, biological fuids, and toluene.Tese fuids have low thermal conductivity.To enhance the thermal conductivity of regular fuids, Choi [21] were the frst who award a novel cohort of heat transfer fuid that is developed by dissolving nonmetallic or metallic tiny particles with a size of under 100 nm in an ordinary fuid.Te components of the nanoparticles include chemically stable metals (gold and copper), metal oxides (alumina, zirconia, titania, and silica), metal carbides (SiC), oxide ceramics (Al 2 O 3 , CuO, TiO 2 , and SiO 2 ), metal nitrides (SiN and AIN), carbon in various forms (fullerene, diamond, graphite, carbon nanotubes, and graphene), and other functionalized nanoparticles.Te nanofuids can augment the thermal conductivity and upgrade the heat transfer efciency of ordinary fuids.Nanofuids are used in diferent felds, including generator cooling, engine and transformer cooling, solar heating, nuclear system cooling, electronic cooling, vehicle thermal management, lubrication, refrigeration, thermal storage, defense, space, biomedical, heat pipe, ships, and drug reduction.A two-phase model with the roles of Brownian difusion and thermophoresis as slip mechanisms was proposed by Buongiorno [22].Mustafa [22] demonstrated the insignifcant impact of Brownian movement on heat transfer while illuminating the slip infuence for rotating fow using the Buongiorno model.A few studies involving the consequence of Brownian and thermophoresis on diferent types of nanofuid have been specifed in Makkar [23], Song et al. [24], and Ragupathi et al. [25].
In today's world, artifcial intelligence (AI) techniques, such as artifcial neural network (ANN), adaptive neurofuzzy inference system (ANFIS), multiple adaptive neurofuzzy inference system (MANFIS), group method of data handling (GMDH), category and regression tree (CART), support vector machine (SVM), genetic algorithm (GA), and particle swarm optimization (PSO), play a vital role for solving system of nonlinear complex models in every domain of science and engineering.Recently, numerous researchers have explored these new computational methods (AI technology) to predict the output responses of nonlinear complex systems.Among those, Gaussian process regression (GPR) is one of the AI techniques to forecast the result responses of nonlinear complex systems.Tese models have widespread application due to their outstanding performance in practice and attractive analytical features, such as machining optimization, machining optimization, analytical sensor calibration, and rehabilitation engineering.Sharma et al. [26] developed an artifcial neural network (ANN) model to investigate Darcy-Forchheimer hybrid nanofuid fow heat transfer through a rotating Riga disk.Te efect of chemical reaction is also included, and a high-performance accurate ANN model was trained to predict thermal energy transfer performance.Raja et al. [28] investigated the 3D hybrid nanofuid fow over biaxial porous stretching/ shrinking sheet with heat transfer, radiative heat, and mass fux solved through Bayesian regularization technique based on backpropagation neural networks.Computational fuid dynamic (CFD) AI techniques were employed for Casson nanofuid [29], MHD Carreau nanofuid fow containing gyrotactic microorganisms [30], biomagnetic ternary hybrid nanofuid [31], MHD Sutterby hybrid nanofuid fow with activation energy [32], and nonlinear radiative magnetized Carreau nanofuid [33].
From the above literature survey, no attempt has been discussed before on the presented physical model for multiple slip fow of magnetized x 0.5(m− 1) /(1 −  εt) 0.5 was applied to the fow direction.Figure 1 depicts the mechanism of fow structure.
Cauchy stress tensor  τ * 1/q for the Casson fuid model is defned by Raju and Sandeep [16]: where  [34],and Cao et al. [35], the modifcation of Buongiorno's nanofuid model was considered in the energy equation and concentration equation since thermomigration and haphazard motion of nanoparticles occur due to variation in the concentration.Based on the aforesaid deliberation, the fuid transport equations become (Hussanan et al. [8], Khan et al. [9], and Ullah et al. [11]) International Journal of Mathematics and Mathematical Sciences where Te corresponding boundary restrictions with slip conditions are as follows: where with N 0 , h 0 , and h 1 being constants.Suitable similarity variables are introduced as follows:  International Journal of Mathematics and Mathematical Sciences Te stream function ψ satisfes equation (1).Under the transformations, equations ( 5)-( 10) yield and the associated boundary restrictions become where the governing parameters are as follows: International Journal of Mathematics and Mathematical Sciences where Le, c, δ, Bi where Re x �  u e x /ϑ is the Reynolds number.

Methodology
In this study, two methodologies, namely, shooting technique together with Runge-Kutta-Fehlberg 4-5th order (RKF-45) and Gaussian process regression (GPR), have been used to perform the mathematical and soft technique simulation for the fow of magnetized Casson nanofuid over a wedge.Te numerical approach of RFK-45 and the background of the GPR model were explained briefy in this section.

Mathematical Simulation
3.1.1.Explanation of the RKF-45 Scheme.Te system of nonlinear diferential equations ( 12)-( 14) with the boundary restrictions equations ( 15) and ( 16) are solved mathematically with the assistance of shooting technique together with Runge-Kutta-Fehlberg fourth-ffth-order integration scheme.Te mathematical simulation of the RKF-45 scheme is presented in Figure 2. Initially, we reduce the order of the equation by using the following procedure: with the boundary restrictions 6 International Journal of Mathematics and Mathematical Sciences Te numerical simulation is performed until the result is corrected up to the desired accuracy of 10 −6 level.

Value of 􏽥 S fx Re 0.5
x and  H tx Re 0.5 x with Variation of m and Pr.Owing to this validity of solution, a comparative investigation of  S fx Re 0.5 x for various values of m and  H tx Re 0.5 x for various values of Pr with earlier published results (Ishak et al. [36] and Ullah et al. [37]; Kuo [38] and Raju and Sandeep [16]) is reported in Tables 1 and 2 which validate the current code.

Explanation of the GPR Model.
Gaussian process regression (GPR) is one of the nonparametric learning algorithms which can model highly complex systems.Every fnite subset of data produced by the Gaussian process in a certain domain can adhere to a multidimensional Gaussian distribution.For a given set of n observations training samples, S � (x i , y i )|i � 1, 2, . . .n  , where x i ∈ R n is the input vector and y i ∈ R is the corresponding output.Tus, a Gaussian process (GP) is a collection of random variables G(x) and is defned as follows: where m(x) � Ε[G(x)] represents the mean function of the prior knowledge about the latent function for variable x and denotes the covariance or kernel function of the confdence level for m(x).Usually, the value of the mean function of the equation is considered to be 0 in most applications.Te relation between the input vector (x i ) of each data point and its output (y i ) value in the GP is defned as follows: where ϵ denotes the Gaussian distribution noise value that has 0 mean and T also displays Gaussian behavior, defned as p(G|x i ) � N(0, K).
Here, the covariance matrix Here, I is the unit matrix of n dimensions.
To estimate the eventual quantity y * and its covariance cov(G * ) for a new input X * , the joint distribution of y and G * is shown as follows: where K(X, X) and K(X * , X * ) denote the training and checking data phases of a covariance (n × 1) matrix of test samples X * , respectively.
Te conventional method for conditioning Gaussian is used to generate the predictive distribution and is defned as follows: where

Kernel
where σ, L, c, ξ, Γ, and K ς indicate the standard deviation, parameter of length scale, the signal, intercept constant, smooth factor, Gamma, and Bessel function, respectively.

Results and Discussion
In this section, the important features of the fow, heat transfer, and mass transfer are achieved using when Hartree pressure gradient m credits are enhanced because they exert an intensity force on the fow and also inverse variation is performed between m and the velocity boundary layer thickness.Figure 4 refects the efect of S and M on f ′ (ζ).A raised velocity distribution is examined with unsteadiness parameter.It provides that the velocity boundary layer thickness imperceptibly increases with an increment in S. Also, the broadening magnetic parameter is taking over the force to dwindle the velocity component.
Physically, this occurs due to the fact that by boosting the values of M, the Lorentz force diminishes, which leads to the retarding force on the movement of the fuid.Figure 5 shows the efects of K and c on f ′ (ζ).In both cases, a widening of the momentum boundary layer is inspected.As is evident, the greatest levels of c cause greater force on the fow of the velocity feld f ′ (ζ).Te infuence of δ on f ′ (ζ) is depicted in Figure 6.With an increase of δ, the velocity distribution grows up.Terefore, the slip at the wedge surface energetically leads to the closeness of the boundary layer.

International Journal of Mathematics and Mathematical Sciences
Brownian dispersion factor.As Le grows, a Brownian factor of dispersion falls, resulting in a reduction in nanoparticle concentration φ(ζ) and boundary layer thickness.Also, boosting R infuences φ(ζ), which in turn afects mass transport rates, chemical rates, and nanoparticle concentrations, and subsequently, temperature and humidity felds.Te consequences include detrimental efects on yields, such as freezing damage, and a shift in energy distribution towards a rainy cooling tower.

Mathematical Model Using GPR.
In this section, we proposed a novel data-driven model based on Gaussian process regression (GPR) technique to predict  S fx Re 0.5 x ,  H tx Re 0.5 x , and  C tx Re 0.5 x based on numerical output.Tis model is more fexible and can handle uncertainty in data.GPR is rooted in a Bayesian framework, which allows for the incorporation of prior knowledge or domain expertise into the model.Tis can improve its performance, especially when we have relevant prior information.In the present study, the developed GPR model uses m, M, c, β, S, K, δ, R d , Pr , Ec, Nb, Nt, Bi 1 , Bi 2 , Le, and R as the input parameters.Te data have been collected from the numerical results using RKF-45.Here, 70% of the dataset is used in the training phase and 30% is used in the checking phase.Figure 22 gives the workfow of the proposed GPR model for estimating the skin friction coefcient (  S fx ), heat transfer rate (  H tx ), and nanoparticle transfer (  C tx ).GPR model depends on the choice of kernel function and hyperparameters, which should be carefully selected through cross-validation and grid search.Tis practice helps avoid overftting, where the model performs well on the training data but fails to generalize to new, unseen data.Table 7 represents the prediction error of the developed GPR model for diferent kernel functions.Te lower error levels and the highest R 2 indicate a superior model.From Table 7, we noticed that the exponential Kernel function has better prediction of  S fx Re 0.5 x ,  H tx Re 0.5 x , and  C tx Re 0.5 x results for both training and checking phases of magnetized Casson nanofuid.Also, the determined R 2 values for the exponential kernel function have better performance than other functions in both training and checking phases.
where n ds , y numerical i , y predicted i , μ, y numerical i , and y predicted i indicate the number of datasets, the target value, the predicted value, the measured average and mean of targeted and predicted values, respectively.

GPR Model Validation with Numerical Simulation.
For better judgement about the developed GPR model, the simultaneous demonstration of numerical and predicted results of  S fx Re 0.5 x ,  H tx Re 0.5 x , and  C tx Re 0.5 x is depicted in Figures 23-28.Te symmetrical straight lines are targeted values from these fgures, and the predicted values are represented near and far away from the straight lines.In all the fgures, the numerical and measured values of f ″ (0), θ ′ (0), and ϕ ′ (0) for training and checking phases showed superior predictive performance.Te R 2 values taped for training and checking phases of f ″ (0) are 0.999999 and 0.999999, of θ ′ (0) are 0.99997 and 0.999999, and of ϕ ′ (0) are 0.999823 and 0.999999, respectively.Tese fgures state the high accuracy prediction of engineering physical interest quantities of magnetized Casson nanofuid using GPR models.

. Conclusions
Te fow behavior of magnetized Casson nanofuid over a wedge subject to multiple slip efects, thermal radiation, and chemical reaction was addressed and discussed in detail.Te RKF-45 together with the shooting technique was utilized to simulate the numerical steady similarity solutions.Te computational outcomes are obtained through the GPR (Gaussian process regression) intelligent soft computing technique for estimating the dynamic behavior of Casson nanofuid models.Te computations are shown as follows: (i) From the mathematical simulation, the addition of β and M devaluates the momentum boundary layer thickness.(ii) Te distribution of velocity attains maximum for higher values of K, c, and δ. (iii) Te nanoparticle temperature enhances with the increase of R d , Ec, Nb, Nt, Bi 1 , and Bi 2 .
(iv) As Le increases, both the nanoparticle temperature and concentration decrease.
(vi) All three employed GPR models have an R 2 value higher than 0.9.An R 2 value of 0.9 indicates a very strong correlation between the predicted and actual values.

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International Journal of Mathematics and Mathematical Sciences heat transfer rate.Physically, the ?? is employed to simulate a relationship between the boundary layer enthalpy diference and kinetic energy.A liquid is only warmed internally by friction between its particles when a wedge expands, converting mechanical energy to thermal energy.Te enhancement in Ec at the wedge surface raises the thermal energy associated with fuid motion by raising the temperature of the fuid and producing a thicker boundary layer.Te response of θ(ζ) to the variation of Prand R d is illustrated in Figure 11.Te detected results show that the amount of θ(ζ) is impeded for greater values of Pr.When Pr increases, the momentum difusivity outweighs the thermal
Casson nanofuid over a wedge.Te following signifcant characteristics can be used to highlight the goals, novelty, contributions, and insights of the research analysis that have been introduced: International Journal of Mathematics and Mathematical Sciences 1 , and Bi 2 are, respectively, Lewis number, moving wedge parameter, slip parameter, and Biot numbers.Skin friction coefcient  S fx �  τ w /ρ u 2 e , heat transfer rate  H tx �  xq w /k(  T w −  T fs ), and nanoparticle transfer rate  C tx �  xd w /D B (  C w −  C fs ) at the wall ((ie)ζ � 0) are defned as follows:

Table 1 :
Comparison of values for  S fx Re 0.5 x for various values of m.

Table 2 :
Comparison of values for  H tx Re 0.5x for various values of Pr.
and the corresponding ranges of constraints of the research are exhibited in Table3.Te numerical illustration for  S fx Re 0.5x ,  H tx Re 0.5 x , and  C tx Re 0.5 x is shown in Tables4-6.A prominent variation in  S fx Re 0.5x has been noticed for β and S.However,  H tx Re 0.5 x is enhanced for more tremendous values of R d , Ec, Nb, Nt, whereas  C tx Re 0.5 x decreases with Nb, Le, R, and Bi 2 and increases with Nt and Bi 2 .
Casson fuid fow over a moving wedge with slip efects and also the GPR technique was developed to predict the skin friction coefcient (  S fx ), heat transfer rate (  H tx ), and nanoparticle transfer rate (  C tx ).4.1.Analysis of Physical Quantities.Tis section visualizes the physical description of engaged parameters developing in equations (12)-(16).Te sixteen distinct nondimensional parameters, such as m, M, c, β, S, K, δ, R d , Pr , Ec, Nb, Nt, Bi 1 , Bi 2 , Le, and R, impact of β and m on f ′ (ζ).Decreasing completion is perceived in f ′ (ζ) for greater values of β.Because they inversely correlate to the yield stress and fuid viscosity rate, the velocity feld f ′ (ζ) declines as β upturns.Viscous force, a resistive force, is created and is what causes this distortion.Tis force's energy grows as the Casson nanofuid parameter's strength is enhanced with a decrease in the surface's thickness in response to fuid movement within the boundary layer.Te velocity feld f ′ (ζ) tends to improve

Table 3 :
Nondimensional parameters on the physical features.

Table 4 :
Numerical outcome of skin friction coefcient (  S fx Re 0.5

Table 5 :
Numerical outcome of heat transfer rate (  H tx Re 0.5x ).

Table 6 :
Numerical outcome of nanoparticle transfer rate (  C tx Re 0.5x ).

Table 7 :
Prediction errors of the GPR model with diferent kernel functions.
International Journal of Mathematics and Mathematical Sciences (MAPE), mean square error (MSE), coefcient of determination (R 2 ), and correlation coefcient (R).Te mentioned metrics are defned as follows: vii) Considering statistical metrics such as RMSE, MAE, MAPE, and MSE, the developed GPR models are more accurate in predicting  S fx Re 0.5x ,  H tx Re 0.5x , and  C tx Re 0.5 x values.International Journal of Mathematics and Mathematical Sciences (viii) Tis study suggests that the GPR models are effective in simulating and predicting heat and mass transfer coefcients of complex physical fow problems.
p /(ρc) f : Ratio of heat capacity of the nanoparticle ψ: