Abstract
An unsteady two-dimensional magnetized Casson nanofluid flow model is constructed over a wedge under the effect of thermal radiation and chemical reaction. The multiple slip effects are also assumed near the surface of the wedge along with the convective boundary restrictions. This study investigates the application of soft computing techniques to address the challenges posed by the complexity of problem modeling and numerical methods. Traditional approaches incorporating various model factors may struggle to provide accurate solutions. To resolve this issue, Gaussian process regression (GPR) is employed to predict the solution of the proposed flow model. With the help of the numerical shooting method together with Runge–Kutta–Fehlberg fourth-fifth-order (RKF-45) reference data, the GPR model is trained. The numerical simulation illustrated that the Casson fluid parameter and the unsteadiness parameter strengthen the friction factor, and the heat transfer rate is enhanced as the radiation parameter becomes larger. In addition, the Biot numbers lead to strengthen nanoparticle temperature; an opposite behavior is noticed with the skin friction coefficient , heat transfer rate , and nanoparticle transfer rate . The GPR model with the exponential Kernel function provided better performance than other functions on both training and checking datasets to predict , and . Statistical metrics including RMSE, MAE, MAPE, MSE, and R are employed to check the accuracy and convergences of the predicted and numerical solutions obtained through GPR and RKF-45. It is observed that all three GPR models had an value of higher than 0.9. The proposed study demonstrates the advantages of employing soft computing methods (GPR) to effectively analyse the behavior of complex flow models.
1. Introduction
Aerospace engineering and fluid mechanics both benefit significantly from the study of fluid flow over a wedge because they offer important insights into how fluids behave when they interact with solid surfaces. This knowledge aids in the comprehension of boundary layer dynamics, aerodynamic phenomena, and the design of airfoil shapes for effective lift and drag characteristics in a variety of engineering applications. Recently, the study of MHD boundary layer slip flow 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 flow over a static wedge immersed in a viscous fluid and developed the Falkner–Skan equation. Awaludin et al. [2] discovered the repercussions of a magnetic field on the flow of an incompressible and electrically conducting fluid past a stretching/shrinking wedge. The viscous dissipation effects on the MHD boundary layer stream of nanofluid 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 flow 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].
The study of non-Newtonian fluid 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 fluids (blood, salvia, etc.) and foodstuffs (honey, jellies, jams, soups, etc.) are examples of non-Newtonian fluids because of their physical nature. Casson fluid is a type of non-Newtonian fluid that behaves like an elastic solid. Casson model constitutes a fluid model that exhibits shear thinning characteristics, yield stress, and high shear viscosity [7]. These fluids are applied in technical processes, such as biomedical and industrial engineering, energy generation, dynamics, and geophysical fluid 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 field, Soret–Dufour, viscous dissipation, and chemical reactions on the Casson fluid in different flow settings. Mukhopadhyay and Mandal [13] developed a numerical study of the boundary layer forced convection flow of a Casson fluid over a symmetric porous wedge. They found that the Casson fluid parameter tends to control the flow separation. El-dabe et al. [14] used the numerical method (finite difference method) to obtain the solution of the MHD boundary layer flow of Casson fluid on a moving wedge with heat and mass transfer. Mahdy [15] illustrated the impact of slip at the boundary of unsteady two-dimensional MHD flow of a Casson fluid 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 flow of a dissipative Casson fluid over a moving wedge with heat source/sink. Recently, researchers focused on investigating the sundry flow features of Casson nanofluid in different frames [17–20].
One of the massive challenges within the modern science and technology panorama is attaining concrete enhancements regarding the rate of heat transfer of ordinary fluids such as water, lubricants, oils, ethylene glycol, biological fluids, and toluene. These fluids have low thermal conductivity. To enhance the thermal conductivity of regular fluids, Choi[21] were the first who award a novel cohort of heat transfer fluid that is developed by dissolving nonmetallic or metallic tiny particles with a size of under 100 nm in an ordinary fluid. The components of the nanoparticles include chemically stable metals (gold and copper), metal oxides (alumina, zirconia, titania, and silica), metal carbides (SiC), oxide ceramics (Al2O3, CuO, TiO2, and SiO2), metal nitrides (SiN and AIN), carbon in various forms (fullerene, diamond, graphite, carbon nanotubes, and graphene), and other functionalized nanoparticles. The nanofluids can augment the thermal conductivity and upgrade the heat transfer efficiency of ordinary fluids. Nanofluids are used in different fields, 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 diffusion and thermophoresis as slip mechanisms was proposed by Buongiorno [22]. Mustafa [22] demonstrated the insignificant impact of Brownian movement on heat transfer while illuminating the slip influence for rotating flow using the Buongiorno model. A few studies involving the consequence of Brownian and thermophoresis on different types of nanofluid have been specified in Makkar [23], Song et al. [24], and Ragupathi et al. [25].
In today’s world, artificial intelligence (AI) techniques, such as artificial neural network (ANN), adaptive neuro-fuzzy inference system (ANFIS), multiple adaptive neuro-fuzzy 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. These 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 artificial neural network (ANN) model to investigate Darcy–Forchheimer hybrid nanofluid flow heat transfer through a rotating Riga disk. The effect 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 nanofluid flow over biaxial porous stretching/shrinking sheet with heat transfer, radiative heat, and mass flux solved through Bayesian regularization technique based on backpropagation neural networks. Computational fluid dynamic (CFD) AI techniques were employed for Casson nanofluid [29], MHD Carreau nanofluid flow containing gyrotactic microorganisms [30], biomagnetic ternary hybrid nanofluid [31], MHD Sutterby hybrid nanofluid flow with activation energy [32], and nonlinear radiative magnetized Carreau nanofluid [33].
From the above literature survey, no attempt has been discussed before on the presented physical model for multiple slip flow of magnetized Casson nanofluid over a wedge. The following significant characteristics can be used to highlight the goals, novelty, contributions, and insights of the research analysis that have been introduced:(i)The present study introduces an innovative approach to investigate heat transport in fluid models by combining soft computing techniques with numerical computing through the GPR model(ii)An unsteady, incompressible, laminar, viscous, and magnetized Casson nanofluid flow model over a wedge under the effect of thermal radiation, chemical reaction, and multiple slip features with first-order relations is considered(iii)A dataset is constructed through mathematical simulation (Runge–Kutta–Fehlberg fourth-fifth-order method along with shooting technique) for analyzing dimensionless quantities of engineering interest(iv)Convergence of the developed GPR results is examined through statistical metrics, including root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), mean square error (MSE), coefficient of determination , and correlation coefficient (R)(v)Considering statistical metrics such as RMSE, MAE, MAPE, and MSE , the developed GPR models are more accurate in predicting , and values
1.1. Applications
The study of magnetized Casson nanofluid flow along a wedge using Gaussian process regression (GPR) combines the concept of fluid dynamics, nanotechnology, and machine learning. This study is potential contributions to optimizing processes and systems in various industries, ranging from material processing and energy systems to biomedical applications and environmental engineering. Also, this study provides valuable insights that can be leveraged to improve the efficiency and effectiveness of diverse applications where complex fluid dynamics play a crucial role.
2. Modeling
MHD Casson nanofluid flow model over a wedge-shaped geometry with thermal radiation, chemical reaction, and slip effects is considered. The flow over the wedge with velocity and the free stream velocity is the free stream, where are positive constants and is the time. Here, is the wedge angle parameter that corresponds to for the total wedge angle . Temperature and nanoparticle fraction at the wall are and , respectively, and these are greater than that of free stream and , respectively. The variable magnetic field was applied to the flow direction. Figure 1 depicts the mechanism of flow structure.
Cauchy stress tensor for the Casson fluid model is defined by Raju and Sandeep [16]:wherein which plastic dynamic viscosity, product of the rate of strain tensor with itself, , critical value based on Casson non-Newtonian model, and velocity components. Following Song et al. [25], Animasaun et al. [34],and Cao et al. [35], the modification of Buongiorno’s nanofluid 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 fluid transport equations become (Hussanan et al. [8], Khan et al. [9], and Ullah et al. [11])where and indicates the Newtonian and non-Newtonian fluid models, respectively.
The corresponding boundary restrictions with slip conditions are as follows:where with being constants. Suitable similarity variables are introduced as follows:
The stream function satisfies equation (1). Under the transformations, equations (5)–(10) yieldand the associated boundary restrictions becomewhere the governing parameters are as follows:where are, respectively, Lewis number, moving wedge parameter, slip parameter, and Biot numbers. Skin friction coefficient , heat transfer rate , and nanoparticle transfer rate at the wall are defined as follows:where is the Reynolds number.
3. 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 flow of magnetized Casson nanofluid over a wedge. The numerical approach of RFK-45 and the background of the GPR model were explained briefly in this section.
3.1. Mathematical Simulation
3.1.1. Explanation of the RKF-45 Scheme
The system of nonlinear differential 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-fifth-order integration scheme. The 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
The numerical simulation is performed until the result is corrected up to the desired accuracy of 10−6 level.
3.1.2. Value of with Variation of
Owing to this validity of solution, a comparative investigation of for various values of m and 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.
3.2. Soft Technique Simulation
3.2.1. Explanation of the GPR Model
Gaussian process regression (GPR) is one of the nonparametric learning algorithms which can model highly complex systems. Every finite subset of data produced by the Gaussian process in a certain domain can adhere to a multidimensional Gaussian distribution. For a given set of observations training samples, , where is the input vector and is the corresponding output. Thus, a Gaussian process (GP) is a collection of random variables and is defined as follows:where represents the mean function of the prior knowledge about the latent function for variable and denotes the covariance or kernel function of the confidence level for . Usually, the value of the mean function of the equation is considered to be 0 in most applications. The relation between the input vector of each data point and its output value in the GP is defined as follows:where denotes the Gaussian distribution noise value that has 0 mean and variance
Moreover, also displays Gaussian behavior, defined as Here, the covariance matrix has components.
conditioned distribution on is represented by Here, is the unit matrix of dimensions.
To estimate the eventual quantity and its covariance for a new input , the joint distribution of and is shown as follows:where and denote the training and checking data phases of a covariance matrix of test samples , respectively.
The conventional method for conditioning Gaussian is used to generate the predictive distribution and is defined as follows:where
3.2.2. Kernel Functions
A kernel (or covariance function) describes the covariance of the GPR variables. Kernel function calculates the closeness and similarity degree among the actual datasets. Therefore, it determines the analyses of GPR in handling systematic prediction error. There are several types of Kernel functions that can be used in GPR. For example, exponential (E), squared exponential (SE), rational quadratic (RQ), Matérn class (MT), ardexponential (ardE), ardsquared exponential (ardSE), and ardrational quadratic (ardRQ), which are defined as follows:where , and indicate the standard deviation, parameter of length scale, the signal, intercept constant, smooth factor, Gamma, and Bessel function, respectively.
4. Results and Discussion
In this section, the important features of the flow, heat transfer, and mass transfer are achieved using Casson fluid flow over a moving wedge with slip effects and also the GPR technique was developed to predict the skin friction coefficient , heat transfer rate , and nanoparticle transfer rate .
4.1. Analysis of Physical Quantities
This section visualizes the physical description of engaged parameters developing in equations (12)–(16). The sixteen distinct nondimensional parameters, such as , and , and the corresponding ranges of constraints of the research are exhibited in Table 3. The numerical illustration for , and is shown in Tables 4–6. A prominent variation in has been noticed for However, is enhanced for more tremendous values of , whereas decreases with and increases with
4.2. Discussion of Results
4.2.1. Velocity Distribution
Figure 3 presents the significant impact of and on . Decreasing completion is perceived in for greater values of . Because they inversely correlate to the yield stress and fluid viscosity rate, the velocity field declines as β upturns. Viscous force, a resistive force, is created and is what causes this distortion. This force’s energy grows as the Casson nanofluid parameter’s strength is enhanced with a decrease in the surface’s thickness in response to fluid movement within the boundary layer. The velocity field tends to improve when Hartree pressure gradient credits are enhanced because they exert an intensity force on the flow and also inverse variation is performed between and the velocity boundary layer thickness. Figure 4 reflects the effect of and on . A raised velocity distribution is examined with unsteadiness parameter. It provides that the velocity boundary layer thickness imperceptibly increases with an increment in . 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 fluid. Figure 5 shows the effects of and on . In both cases, a widening of the momentum boundary layer is inspected. As is evident, the greatest levels of cause greater force on the flow of the velocity field . The influence of on is depicted in Figure 6. With an increase of , the velocity distribution grows up. Therefore, the slip at the wedge surface energetically leads to the closeness of the boundary layer.
4.2.2. Temperature Distribution
To examine the variation in against various flow parameters, Figures 7–13 are developed. From Figures 7 and 8, it is noticed that the increasing values of result in an augmentation of both the rate of heat transfer and the temperature profile. It can be attributed to alterations in the fluid’s rheological properties, flow dynamics, and the influence of the magnetic field. These changes collectively impact the thermal behavior of the system, leading to enhanced heat transfer and temperature profiles. A reverse phenomenon is perceived for growing values of on . Figures 9 and 10 point out that upon increasing , the decline is made in the heat transfer rate and . Enhancing strengthened the internal energy of nanoliquid which in turn augmented the heat transfer rate. Physically, the 𝐸𝑐 is employed to simulate a relationship between the boundary layer enthalpy difference and kinetic energy. A liquid is only warmed internally by friction between its particles when a wedge expands, converting mechanical energy to thermal energy. The enhancement in at the wedge surface raises the thermal energy associated with fluid motion by raising the temperature of the fluid and producing a thicker boundary layer. The response of to the variation of is illustrated in Figure 11. The detected results show that the amount of is impeded for greater values of . When increases, the momentum diffusivity outweighs the thermal diffusivity, resulting in a decrease in the flow region’s temperature field . Augmentation performance is perceived in for higher values of , since increasing spawns more heat which in turn boosts the fluid temperature. Figure 12 witnesses that increasing increases the kinetic energy of the particles due to the collision; hence, temperature is made immense. An identical configuration is perceived for cultivating values From Figure 13, the observations reveal that is larger with the increasing values of and . Thermal Biot numbers play an important portrayal in the enhancement of nanoparticles temperature, as it is directly associated with the coefficient of heat transfer.
4.2.3. Concentration Distribution
The influence of peculiar flow parameters such as , and on the concentration of nanoparticles field is highlighted in Figures 14–21. Figures 14–18 elucidates the increasing nature in due to increasing values of . Inverse variations are seen for the growing values of Figure 19 shows the behavior of on The increasing value of reduces , and this is due to the fact that Brownian motion makes the fluid mild within the frontier and the absence of particle removal from the fluid regime to the surface results in a reduction in while increasing augmented . Boosting enhances the motion of nanoparticles from higher to lower temperature gradient which in turn exploits the concentration of nanoparticles. Figure 20 illustrates the behavior on for signified and It is discerned that is improved for greater evaluation of and . Because the Biot numbers of nanoparticle concentration are directly correlated with the coefficient of mass transfer, they play a significant role in the enhancement of nanoparticle concentration. Figure 21 demonstrates the influence of and on . It is depicted that the improving credits of cause a decline in because has an inverse relationship with the Brownian dispersion factor. As grows, a Brownian factor of dispersion falls, resulting in a reduction in nanoparticle concentration and boundary layer thickness. Also, boosting influences φ(ζ), which in turn affects mass transport rates, chemical rates, and nanoparticle concentrations, and subsequently, temperature and humidity fields. The consequences include detrimental effects on yields, such as freezing damage, and a shift in energy distribution towards a rainy cooling tower.
4.3. Mathematical Model Using GPR
In this section, we proposed a novel data-driven model based on Gaussian process regression (GPR) technique to predict , and based on numerical output. This model is more flexible 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. This can improve its performance, especially when we have relevant prior information. In the present study, the developed GPR model uses , and R as the input parameters. The 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 workflow of the proposed GPR model for estimating the skin friction coefficient , heat transfer rate , and nanoparticle transfer .
GPR model depends on the choice of kernel function and hyperparameters, which should be carefully selected through cross-validation and grid search. This practice helps avoid overfitting, 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 different kernel functions. The lower error levels and the highest indicate a superior model. From Table 7, we noticed that the exponential Kernel function has better prediction of , and results for both training and checking phases of magnetized Casson nanofluid. Also, the determined values for the exponential kernel function have better performance than other functions in both training and checking phases.
4.3.1. Performance Assessment
Evaluating the model’s performance using various statistical metrics is a standard practice in machine learning and regression analysis. Hence, the performance assessment of error between the GPR model and the numerical data of magnetized Casson nanofluid was employed and compared using statistical metrics including root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), mean square error (MSE), coefficient of determination , and correlation coefficient (R). The mentioned metrics are defined as follows:where , , indicate the number of datasets, the target value, the predicted value, the measured average and mean of targeted and predicted values, respectively.
4.3.2. GPR Model Validation with Numerical Simulation
For better judgement about the developed GPR model, the simultaneous demonstration of numerical and predicted results of , and is depicted in Figures 23–28. The symmetrical straight lines are targeted values from these figures, and the predicted values are represented near and far away from the straight lines. In all the figures, the numerical and measured values of , , and for training and checking phases showed superior predictive performance. The values taped for training and checking phases of are 0.999999 and 0.999999, of are 0.99997 and 0.999999, and of are 0.999823 and 0.999999, respectively. These figures state the high accuracy prediction of engineering physical interest quantities of magnetized Casson nanofluid using GPR models.
5. Conclusions
The flow behavior of magnetized Casson nanofluid over a wedge subject to multiple slip effects, thermal radiation, and chemical reaction was addressed and discussed in detail. The RKF-45 together with the shooting technique was utilized to simulate the numerical steady similarity solutions. The computational outcomes are obtained through the GPR (Gaussian process regression) intelligent soft computing technique for estimating the dynamic behavior of Casson nanofluid models. The computations are shown as follows:(i)From the mathematical simulation, the addition of and devaluates the momentum boundary layer thickness.(ii)The distribution of velocity attains maximum for higher values of , , and (iii)The nanoparticle temperature enhances with the increase of , and (iv)As increases, both the nanoparticle temperature and concentration decrease.(v)When , the nanoparticle concentration rises. Conversely, when , the nanoparticle concentration decreases.(vi)All three employed GPR models have an value higher than 0.9. An value of 0.9 indicates a very strong correlation between the predicted and actual values.(vii)Considering statistical metrics such as RMSE, MAE, MAPE, and MSE, the developed GPR models are more accurate in predicting , and values.(viii)This study suggests that the GPR models are effective in simulating and predicting heat and mass transfer coefficients of complex physical flow problems.
Nomenclature
| : | Stretching and free stream velocity (m/s) |
| : | Positive constants |
| : | Hartree pressure gradient |
| : | Magnetic induction parameter (T) |
| : | Magnetic field |
| : | Time |
| : | Temperature near and far away from the wedge wall (K) |
| : | Concentration near and far away from the wedge surface |
| : | Brownian and thermophoresis diffusion coefficient (m2/s) |
| : | Product of the rate of strain tensor |
| : | Yield stress of the fluid |
| : | Deformation rate |
| : | Critical value based on Casson non-Newtonian model |
| : | Temperature (K) |
| : | Nanoparticle concentration (moles/kg) |
| : | Constants |
| : | Rate of chemical reaction (1/s) |
| : | Velocity components of directions (m/s) |
| : | Distance along the surface (m) |
| : | Distance normal to the surface (m) |
| : | Dimensionless velocity |
| : | Magnetic parameter |
| : | Prandtl number |
| : | Radiation parameter |
| : | Porosity parameter |
| : | Unsteadiness parameter |
| : | Chemical reaction parameter |
| : | Brownian motion parameter |
| : | Thermophoresis parameter |
| : | Eckert number |
| : | Lewis number |
| : | Biot numbers |
| : | Skin friction coefficient (Pascal) |
| : | Heat transfer rate |
| : | Nanoparticle transfer rate |
| : | Reynolds number |
| : | Observations |
| : | Covariance or kernel function |
| : | Parameter of length scale |
| : | Signal, intercept constant |
| : | Similarity variable |
| : | Cauchy stress tensor |
| : | Plastic dynamic viscosity |
| : | Ratio of heat capacity of the nanoparticle |
| : | Stream function |
| : | Electrical conductivity (S/m) |
| : | Stefan–Boltzmann constant (W/m2 K4) |
| : | Mean absorption coefficient (1/m) |
| : | Dimensionless temperature |
| : | Dimensionless concentration |
| : | Moving wedge parameter |
| : | Slip parameter |
| : | Total wedge angle |
| : | Wedge angle parameter |
| : | Casson nanofluid parameter |
| : | Surface shear stress |
| : | Radiative heat flux |
| : | Radiative mass flux |
| : | Kinematic viscosity (m2/s) |
| : | Density (kg/m3) |
| : | Specific heat |
| : | Thermal conductivity (W/m K) |
| : | Heat capacity (kg/m3K) |
| : | Dynamic viscosity (kg/m s) |
| : | Gamma function |
| : | Bessel function |
| : | Smooth factor |
| : | Gaussian distribution noise value |
| : | Quantities at wall |
| : | Quantities at free stream. |
Data Availability
All the data and material used in this research are included in the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.