Integrated intelligent computing application for effectiveness of Au nanoparticles coated over MWCNTs with velocity slip in curved channel peristaltic flow

Estimation of the effectiveness of Au nanoparticles concentration in peristaltic flow through a curved channel by using a data driven stochastic numerical paradigm based on artificial neural network is presented in this study. In the modelling, nano composite is considered involving multi-walled carbon nanotubes coated with gold nanoparticles with different slip conditions. Modeled differential system of the physical problem is numerically analyzed for different scenarios to predict numerical data for velocity and temperature by Adams Bashforth method and these solutions are used as a reference dataset of the networks. Data is processed by segmentation into three categories i.e., training, validation and testing while Levenberg–Marquart training algorithm is adopted for optimization of networks results in terms of performance on mean square errors, train state plots, error histograms, regression analysis, time series responses, and auto-correlation, which establish the accurate and efficient recognition of trends of the system.

• A novel computing infrastructure via feed-forward artificial neural networks aided with backpropagation of Levenberg-Marquardt method (LMM) for training/learning, i.e., ANN-LMM, is adopted to find the solution of mathematical model for Au nanoparticles concentration in peristaltic flow through a curved channel. • The fitness function based on mean square error is developed for the execution of ANN-LMM for prediction of the solution for the proposed fluidic system through targeted data set by means biased input data points of training and unbiased inputs for testing and validation. • Merit functions representing different scenarios to predict numerical data for velocity and temperature profiles to measure the effect of sundry physical parameters of MWCNT based peristaltic flow model involving gold nanoparticles with different slip conditions are effectively implemented with ANNs-LMM with reasonable precision. • Performance via convergence, precision and stability of the designed ANNs-LMM for the proposed problem of Au nanoparticle concentration in peristaltic flow through a curved channel is authenticated through histogram studies and regression measures.

Model formulation
Consider a curved channel of width 2b 1 filled with Newtonian hybrid nanofluid having temperature T. The center and radius of curvature of the circle, in which the channel is coiled, be labeled by O and R * , accordingly as illustrated in the schematic flow geometry presented in Fig. 1. A curvilinear coordinate system is employed in which R is oriented along radial direction while X is along the flow direction. The relation between curvilinear coordinate system and Cartesian coordinate system Ẍ ,Ÿ configured at O is given by the transformations 55  Formulated nonlinear governing model in laboratory frame is given as [55][56][57] : The expressions for ∇ 2 and in above equations are: (1) (2) (3) R = b 1 + b 2 cos 2π (X − ct) .
H 1 (X, t) = b 1 + b 2 cos 2π (X − ct) , H 2 (X, t) = −b 1 − b 2 cos 2π (X − ct) .  where β 1 and β 2 denote the first and second order slip parameters, h is convective heat coefficient while T 0 expresses temperature at upper and lower walls and P is the pressure. If (R, X, V, U) and (r, x, v, u) denote the coordinates and velocities in the laboratory and wave frame. From laboratory frame to wave frame transformations for steady problem 10,56 are: Introducing stream function ψ along with dimensionless variables in laboratory frame 57-59 as follows: In above equations, r, x, v,u, p and θ are dimensionless coordinates, velocities, pressure and temperature with δ and k as wave number and radius of curvature. Non dimensional governing model for the above-mentioned quantities along with small wave number and low Reynolds number approach is: By eliminating pressure, Eq. (12) simplified in the form: Corresponding boundary conditions are written as: .

Methodology
The methodology adopted in this study is presented here in terms of reference numerical solutions, neural networks modeling and training with Levenberg-Marquardt algorithm as graphically depicted in Fig. 2. The methodology is systematically represented with necessary detail in terms of process block structure consist of problem, modeling optimization and storage steps.
The reference numerical solutions are determined by employing the Adams predictor corrector method 60-62 to solve the system of Eqs. (13)(14)(15) by using boundary conditions as given in Eq. (16). In this numerical approach, predictor solution is predicted first and then corrector is used to calculate the accurate solution by using already predicted solution. Considering the Eqs. (13)(14)(15) for velocity u and temperature θ as: To derive two-step predictor formula for first equation in (18), integration gives 62 :   The dataset generated by Adams predictor corrector method is used in neural networks structure in terms of represented with three layers; input layer, hidden layer and output layer based on a supervised learning approach with a back propagation with Levenberg-Marquart algorithm, while the number of neurons ranging can be taken between 50 and 80 with structure as shown in in Fig. 3.
Training data is acquired from governing model using Adams Bash forth method. Reference data is randomly divided into three groups i.e., one set of example can set training (90%), validation (5%) and testing (5%). Input and the corresponding target provided to the ANN model is received by neurons which combine them, perform a nonlinear operation on the result in the hidden layer, and then outputs appear in output layer. Weighted summation of inputs is added with biases in the hidden layer which is transferred then using hyperbolic tangent sigmoid function as an activation function and can be calculated mathematically as 63 : In which, w jk represents the weights of jth neuron in previous layer to the kth neurons and kth neuron is denoted by n k . Levenberg-Marquardt training algorithm is used to fine tune the weights and biases in the networks to minimize the error, and obtain a high performance of accurate solution. The procedural step of Levenberg-Marquardt training is presented in Fig. 2 optimization block. Finally, the output activation function is a linear function mathematically represented as: Learning/retrain operation is continuous execute as the desire set level of error or fitness is achieved, and neural network performances are then evaluated to predict the model efficacy. One set of the architecture analysis of a neural network model is represented in Fig. 3.
Different error and accuracy definitions are proposed to assess neural network model. The performance operators are given as follows 63,64 where p i is the predicted value and y i is the corresponding targeted value, y i represents the average of targeted value. Values of R closer to 1 and lower MSE values are representative of more reliable and accurate prediction.
The software packages used in implementation of the design methodology consist of two different programs. Firstly, we used 'NDSolve' routine with algorithm/method 'adams' with automatics settings of stoppage and

Results and discussion
In this section, results of proposed study ANN-LMM along with the reference solutions as per procedure provided in the last section are presented for fluidic model of peristaltic flow through a curved channel to predict the flow and heat transfer characteristics.
In the presented simulated study, neural networks are exploited to peristaltic flow through a curved channel to predict the flow and heat transfer characteristics. Numerical solutions, as well as, neural network estimated results are figured out and explained. The physical parameter settings of different variation are tabulated in Table 1. Case study 1 represents the variable of velocity profile with three scenarios. Scenarios 1, 2 and 3 considered variation in values of volume fraction of gold nanoparticles ϕ 2 , first order slip parameter β 1 and second order slip parameter β 2, respectively, which, each scenario has four corresponding cases as mentioned in Table 1. Similarly, case study 2, defines for fluid temperature with two scenarios in which variation in magnitudes of ϕ 2 and Bi are considered with four different cases.
Before providing the results of proposed algorithm ANN-LMM, first the necessary elaborative information/ criteria/justification for selection of appropriate inputs samples, hidden neurons as well as training, testing and validations percentages is presented.
The dataset is formulated with the help of Adams numerical solver for all four variations of each scenario of both case studies of the systems model for inputs between − 1.4 and 1.4 with step size 0.015, i.e., 561 data points or sample for each case. The dataset constructed for each case is used for finding the approximate solution of the problem, if we increase the data points the accuracy of the algorithm increase but then you need more hidden neurons in the model to accurate neural networks modeling. So, increase the number of data points will increase the accuracy but at the cast of more computational requirement. After exhaustive experimentation and keeping in view of reasonable compromise between the accuracy of the model and complexity, 561 data points are selected for each case of the system model. The selection of the appropriate neurons for neural network structure is always bit complex procedure. Normally, the number of the neurons in the neural network structure are determined on the basis of the tradeoff between accuracy of model on the training data points. i.e., biased inputs without prior knowledge of original/exact target and level of precision on the testing data inputs, i.e., unbiased inputs with no information of targets. We have extensively performed the simulation study with setting of the neurons between 10, 20 or 30, a relatively a low level of accuracy in training, testing and validation data points is achieved by the proposed methodology with the no noticeable difference in performance for training, testing and validation inputs. However, similarly if increasing the neurons around 300 or more, then results are better for training data inputs while no improvement for testing data inputs. Additionally, with the increase of hidden neurons in neural network structure the computational complexity also increases considerably with some benefit of precision. Therefore, considering both options (accuracy and complexity) after extensive simulation studies, we set 60 number of hidden neurons for our numerical experimentation for solving the model presented in Eqs. (14)(15)(16).
We have conducted the experimentations with different combination of the testing, validation and training arbitrary selected data samples, i.e., 70%, 50% and 90% training, 15%, 25% and 5% testing, and 15%, 25% and 5% validation. observations/finding/remarks. The 70% training, 15% testing and 15% validation, results are consistently obtained and convergent but with low level of the accuracy on the basis of MSE, mean square deviation from the reference numerical solution, in the range of 10 -06 to 10 -07 and 10 -05 to 10 -06 for testing and validations inputs, while in case of 50% training, 25% testing and 25% validation, results of the training are in 10 -05 to 10 -06 but validation and testing accuracy decrease considerable between10 -01 to 10 -02 , and accordingly, in case of 90% training, 5% testing and 5%. The accuracy of testing inputs is found in the range of 10 -09 to 10 -10 , while testing and validations also lies mostly in the range of 10 -08 to 10 -09 . The results provided here are based on 60 hidden neurons, while small change of number of neurons in neural network structure have no bit impact on the performance.
Keeping in view of all these simulations studies, rest of the study is presented selecting 60 hidden neurons, 561 input or target instances/samples, 90% training and 5% testing and validation, for proposed computing scheme.  Fig. 4, effect of gold nanoparticle's volume fraction on fluid velocity is examined. It is evident from graph that maximum value of velocity occurs near the center of the channel. The velocity is increasing near the upper wall whereas an opposite trend is seen closed to the lower wall. This trend may return to an increasing shear stress caused by addition of more gold nanoparticles in lower half. Velocity of fluid shows a reduction in the central part of channel while it enhances near the walls towards variation in β 1 as displayed in Fig. 5.
Increment in values of β 1 represents more slippage at the boundary of surface which decrease the resistive forces and fluid velocity is unaffected by surface motion. Moreover, variation in values of β 2 leads to accelerate the velocity in the vicinity of the upper channel and an opposite trend is observed for lower half. This fact is depicted in Fig. 6. Variational trend of temperature of hybrid nanofluid against φ 2 and Bi is portrayed in Figs. 7 and 8.
It is noticed that an increment in volume fraction of gold nanoparticles for fixed concentration of MWCNTs causes temperature of fluid to upgrade. Clearly, the presence of nanoparticles near cancerous tissues produces   Case Study 1. Performance analysis for 4 cases of the all 3 scenarios namely ϕ 2, β 1 and β 2 , for a reliable prediction of velocity profile is displayed in Figs. 9, 10, 11, 12, 13 and 14. Figure 9a-d displays the network performance in terms of mean square error against epochs curve. It is depicted that the trained data is very accurately validated for all cases of scenario 1 with the best performance at epochs 190, 257, 343 and 211 corresponding to least mean square error of order e −10 , respectively. Figure 10a-d     Fig. 11a-d for different cases of scenario 1 represent that the error between the target and output of network is very close to zero. Moreover, the positive difference is comparably lesser than negative difference for case 1 and case 4, while contrary trend is shown for cases 2 and 3. Figure 12a-d represents regression plots for scenario 1. Since, R be the correlation for the outputs with targets and should be closed to R = 1 and avoid random scenario of R = 0; regression plots of all four cases show that data is highly correlated and concentrated i.e., R = 1. It is also noteworthy that regression plot for case 1 of scenario 1 is highly efficient due to small empty space. The empty space is because of missing values in data.
Numerical simulations are further summarized by plotting time series response for target and output data in Fig. 13a-d. In present study, 'ton data' time series format is used to arrange the data according to standard network cell array form. There is a specific interval for each case of scenario 1 which contains both output and target and the error for each time step of data is presented up to total time steps i.e., 561. This means that the analysis predicts the proposed results for training and testing data with certain accuracy, i.e., reasonable precision, but error performance is relatively poor for testing from training. Further, the results are validated with auto-correlation plots in Fig. 14a-d. It is depicted that error autocorrelation is smaller for case 3 of scenario 1. The similar trends for neural network results for different cases of scenario 2 and scenario 3 are observed.
Case study 2. The analysis for neural network to analyze the impact of temperature on hybrid nanofluid for the two scenarios with several cases is provided in Table 1. Performance of the network is plotted in Fig. 15. In case of MSE versus epochs and it is shown that best validation performance is achieved for increasing number of iterations for both scenarios. Moreover, validated results for case 2 of scenario 1 are more efficient than case 1 with performance at 2.7656e −10 at epochs 324, while case 3 of scenario 2 is more accurate than case 1 with validation performance 4.3012e −11 at epochs 284. In case of training state plots, it can be inferred that gradient parameter and mu factor are reducing with increase in number of iterations. These states are further befitted by accurate displaying of error histogram. Additionally, Most of the errors against components of neural networks i.e. training, validation and testing phases approach zero. Eventually, for given time steps of dataset, results are  Table 2. Mathematical relations for thermophysical properties of hybrid nanomaterials are also represented in Table 3.
Moreover, the details values of the parameters while execution on neural networks in terms of selected number of neurons in the architecture, observed epochs, values of MSE, R value, errors for gradient index and mu parameter for respective cases of each scenario of the two case studies are tabulated in Tables 4 and 5, respectively. This tabular representation is useful in verification and validation of results of the proposed neural network based stochastic solver. It is noted that R value in each case is 1 which is an indicative of good fit. Except this, numerical results found by Adams Bashforth technique for stream function, velocity and temperature for corresponding interval [− 1.4, 1.4] with step size 0.1 are represented in Table 6 for fixed values of ϕ 2 = 0.02, β 1 = β 2 = 0.01, k = 5.0, F = − 1.4 and Bi = 0.5, which satisfy the results.

Conclusions
A novel application of intelligent computing framework mainly based on neural networks is portrayed for solving a model arises in nanotechnology field with bio-medical and geometrical features. Accuracy is validated through achievement of MSE of the order of 10 −10 or 10 −11 . An effective correlation observed for target with outputs having value close to R = 1, a decreasing trend in gradient index with error near zero. In addition, time series response is shows consistent overlapping of results with reference solutions, and the error remained close to 0 error line for each scenario of the system model. Experimental observations show the accuracy, robustness and stability of the proposed computing frameworks, i.e., the neural network paradigm trained with Levenberg-Marquart algorithm. It is also proved that neural network is a powerful tool for analysis and network outputs show a reasonable accuracy for flow variables against different cases of each scenario of the system model. Moreover, the problem is physically imperative concerning its geometrical features as several physiological vessels are curved in nature. Present attempt is appropriate in medical field to estimate and get the best possible outcomes for the effectiveness of nanomaterials, and generally to identify patterns in different scientific divisions by employing neural network algorithm. This is evidently a developing field and future attempts with fractional evolutionary/swarming based optimization algorithms 65-70 look promising for better outcomes and to enhance the neural network performance considerably.

Properties Hybrid nanofluid
Density Heat capacity Thermal conductivity Thermal expansion coefficient