Hall effect on MHD Jeffrey fluid flow with Cattaneo–Christov heat flux model: an application of stochastic neural computing

Exploration and exploitation of intelligent computing infrastructures are becoming of great interest for the research community to investigate different fields of science and engineering offering new improved versions of problem-solving soft computing-based methodologies. The current investigation presents a novel artificial neural network-based solution methodology for the presented problem addressing the properties of Hall current on magneto hydrodynamics (MHD) flow with Jeffery fluid towards a nonlinear stretchable sheet with thickness variation. Generalized heat flux characteristics employing Cattaneo–Christov heat flux model (CCHFM) along with modified Ohms law have been studied. The modelled PDEs are reduced into a dimensionless set of ODEs by introducing appropriate transformations. The temperature and velocity profiles of the fluid are examined numerically with the help of the Adam Bashforth method for different values of physical parameters to study the Hall current with Jeffrey fluid and CCHFM. The examination of the nonlinear input–output with neural network for numerical results is also conducted for the obtained dataset of the system by using Levenberg Marquardt backpropagated networks. The value of Skin friction coefficient, Reynold number, Deborah number, Nusselt number, local wall friction factors and local heat flux are calculated and interpreted for different parameters to have better insight into flow dynamics. The precision level is examined exhaustively by mean square error, error histograms, training states information, regression and fitting plots. Moreover, the performance of the designed solver is certified by mean square error-based learning curves, regression metrics and error histogram analysis. Several significant results for Deborah number, Hall parameters and magnetic field parameters have been presented in graphical and tabular form.


Introduction
The significant influence of Hall current utilizing the Ohms law in hydro-magnetic (MHD) flow of non-Newtonian fluid is a recent inclination. Hall current is noticeable when the magnetic field is strong, or the density of the fluid is low because electrons carry an excited current which moves faster as compared to ions and produces an isotropic conductivity. Dynamics of fluids with Hall current effect on MHD have an extensive and broad use in the field of engineering and industries like geophysical, astrophysical space, bio-fluids, nuclear power reactor, fluid engineering and has many practical applications such as the construction of turbines, Hall sensor, Hall accelerator, centrifugal machines, MHD energy generators, control of crystal growth systems, lubrication restraint of high accelerated spinning machines, magneto astronautical flows, etc. Effect of MHD flow and Hall current inside rotating plates is reported by Shah et al. [1]. Kumar et al. [2] investigated MHD fluid flow between vertical conducting walls in the presence of the Hall effect. Opanuga et al. [3] investigated the Influence of Hall current for entropy generation of radiative MHD convective Casson fluid flow model. Akbar et al. [4] Hall current and ion slip effect on hydromagnetic biologically inspired hybrid nanofluid flow model. Awan et al. [5] examined the effect of Hall current along with electrical MHD on micropolar nanofluid. Recently, bio-heat transfer in the human body gains the reflection of numerous analysts because of its wide applications in human thermal standards which includes heat convection because of the progression of blood from the pores of tissues in a human body, radiation process among surfaces and conduction process in tissues, etc. The impact of Hall current on MHD with heat and mass transmission in a porous medium with thermal radiations was investigated by Shah et al. [6]. Chu et al. [7] examined the influence of heat transfer and radiative heat flux on Rabinowitsch. Hayat et al. [8] observed the heat transfer impacts in magneto-hydrodynamic (MHD) axisymmetric stream of third-grade liquid between the extending sheets. Riaz et al. [9] used the HAM along with a Genetic algorithm for the investigation of peristaltic transport of Jeffry fluid in a porous medium. The importance and usage of non-Newtonian fluid in the modern world of science has tremendous application in technology and industrial areas because all the rheological properties of fluid do not describe by Navier-Stokes equations. In the classification of non-Newtonian fluid Jeffery fluid is a rate type material which means it has a time derivative rather convective derivative. The Jeffrey fluid has numerous industrial and technological applications including wire coating, dying, polymer productions, food dispensation, geophysics, chemical and petroleum, plastic manufacturing, biological fluid, etc. Sreelakshami et al. [10] present the relation of Jeffrey fluid for non-Newtonian fluids. The power-law fluid with the effect of MHD and entropy generation was studied by Khan et al. [11]. Noreen et al. [12] examined Dufour and Soret effects on Jeffrey fluid. The MHD boundary layer with Jeffrey fluid was examined by Shahzad et al. [13]. Researchers investigate a Jeffery fluid under different circumstances. Patel and Maher [14] work on the Jeffery-Hemal flow with a magnetic field. Vaidya et al. [15] look at the peristaltic Jeffery liquid with heat movement in an opposite permeable layer. Nazeer et al. [16] give the impact of non-linear thermal radioactivity on the 3D Jeffery fluid over shrinking/stretching surface in the occurrence of heterogeneous-homogeneous reactions, and injection/suction. Some potential studies about Jeffery fluid are found in these Refs. [17][18][19][20]. Asha and Sunitha [21] studied the effect of heat transfer and hall current on peristaltic blood flow on MHD with Jeffery fluid in a permeable channel. More features of Jeffery fluid with Hall current over 3D are investigated by Sinha et al. [22]. In literature, a lot of research have been done on the transportation of heat and mass theories such as enhancing the heat transfer rates and pressure loss reduction using compact heat exchanger [23]. Cattaneo model was further modified by Christov by changing time-derivative with Oldroyd-B variant [24,25] and stability is reported in [26]. Further relevant studies on Cattaneo-Christov heat flux model (CCHFM) can be seen in [27][28][29]. Alamir et al. [30] work on the perspective of CCHFM. Shah et al. [31] used this model for micropolar ferrofluid on a stretched sheet. Cattaneo-Cristov heat flux model incorporated with slip condition is studied by Ahmad et al. [32]. Khan et al. [33] worked on numerical and analytical solutions of Maxwell fluid on a stretched cylinder with CCHFM. The objective of the present research is to explore the effects of Hall current on MHD with Jeffery fluid over a nonlinear stretchable sheet. We examined the applications of Hall current on MHD with a different perspective. We analyze the characteristics of CCHFM over the variable stretchable sheet with varied thickness. In this regards, we consider the influences of heat transfer, temperature, velocity, stretching sheet with variable thickness, effects of the electric field, induced magnetic field, Hall current parameter, Jeffrey parameters, Deborah number, Nusselt number, skin friction coefficient, shear stress are computed. Mathematical modeling will be presented to construct the nonlinear coupled ordinary differential equations. Similarity transformations are applied and transformed governing equations using Adams Bash-forth method. Further, the experimental data will be analyzed using the Artificial Neural Network model with the Levenberg Marquardt method (ANN-LMM). Artificial intelligence techniques-based stochastic approaches are based on machine learning mechanism which works on the pattern of human behavior to find the stiff and valuable solutions to various types of important problems related to face identification, device management system, radar assembling, cancer diagnostic mechanism, and virus deification. The main components in the intelligent system work with the setting and adjustments of neurons and layers which play a vital and significant role for the best modeling of the designed networks and for their optimizations through different local and global heuristics. system etc. Robbins and Monro [34] analyzed intelligent computing infrastructure for the mathematical system. Mehmood et al. [35] examined the thermal transfer through a fluid flow via the design of a stochastic intelligent computing system. The ANN technique for heat transfer rate are analyzed by Sheikholeslami et al. [36]. An exclusive description made via investigators on this regime consists of [37][38][39]. Transformed governing equations are analyzed numerically using Adams Bashforth method [40][41][42][43].

Description of the fluid flow system
Consider the electrically conducted, unsteady Jeffrey fluid which passing over a stretching surface with fluctuating thickness. The stretching is by the side of the axial direction (x-axis) and y-axis which is perpendicular to the stretchable surface. The applied magnetic field B [0,B 0 ,0] is taken along y axis. The low magnetic Reynold number is taken so that the induced magnetic field is negligible. The modified ohm's law by adding Hall's current effect are taken into account. The expressions of the Jeffrey model for the non-Newtonian fluids are given as: where P denotes the pressure, τ be the Cauchy stress tensor, S is the stress tensor, R 1 is the Rivlin-Ericken tensor, λ 1 stands for the ratio of relaxation to the retardation times while the λ 2 is a retardation time (see Fig. 1). Governing equations after boundary layer approximation are reduced to [5,10,13,18,28]: with associated conditions In the above expression, u, v, w and x, y represent the velocity components and Cartesian coordinates, respectively. ν represents the kinematic viscosity, μ is the dynamic viscosity and ρ is the density. The stretching rate U (x) U 0 (x + b) n , with U 0 be the reference velocity, b represents the relative stretching parameter and n denotes the velocity exponent. The sheet is non-flat, and its surface is taken at y A(x + b) 0.5(1−n) where A is the stretching coefficient while the quantities are assumed to be constant along z-axis.
Modified Ohm's law with Hall's current effect is defined as: Here J denotes a current density, σ denotes electric conductivity parameter, p e is called electronic pressure parameter and B 0 denotes the magnetic field parameter. The components of J can be given as follows: (8) where m is the Hall parameter i-e m σ B 2 0 en e . The heat equation of steady viscous flow is defined as: where ρ is the density, T the temperature, c p stands for the specific heat while q be the heat flux. Cattaneo-Christov heat flux law is defined as: Here κ represents the thermal conductivity, λ be the thermal relaxation factor. For incompressible flow From Eqs. (9) and (10) we have Conditions for fluid temperature are The expression α κ ρc p , T W , T ∞ defined respective thermal diffusivity parameter, surface temperature and ambient temperature.

Similarity Conversion system
The best suitable transformation system for the presented model [44] is proved as follows: Using these transformations, we have: With the transformed boundary conditions: In which differentiation is with respect to ξ . We further assume the following [44]: where ξ − α η, and α A n+1 With the transformed boundary conditions: Here M denotes the magnetic field parameter, γ is the thermal relaxation parameter, β is the Deborah number, Pr is the Prandtl number, α is the thermal diffusivity and m is the Hall parameter. In boundary conditions α is the wall thickness parameter, η corresponds to the surface of the sheet which non-dimensional similarity variable, f is the dimensionless stream function, T is the temperature of the fluid, T W is the surface temperature, T ∞ is the ambient temperature. 8 is the dimensionless temperature. where prime denotes the derivative with respect to η. Parameters involved in the nondimensional equations are: represent the magnetic field parameter, Pr ρc p ν k, represents Prandtl number γ λU 0 (x + b) n−1 is the thermal relaxation parameter, α A (n + 1)U 0 2ν is wall thickness parameter where A y(x + b) n and β λ 2 A 1 is the Deborah number where A 1 γ /λ.

Skin friction factor and Nusselt number
The skin friction coefficient at the stretched surface is written as: Here τ w x , τ w z are shear-stress of the surface in the horizontal direction and shear stress is perpendicular to the where The heat transfer rate relations are written as follows: Here q w is the surface heat flux. Non-dimensional form is: where Nu x represents the local Nusselt Number.

Structure of the designed intelligent network
Stat of the art Adams Bash-forth numerical method is incorporated with the assistance of ND-Solve command exploited through Mathematica software. The considered numerical is the best suitable numerical computing technique for the generation of the dataset for further designing of the artificial neural networks. The diagram of the designed intelligent network is described in Fig. 2.
The above-mentioned network is a mathematical system inspired by biological neural networks, which is dependent upon the collection of neurons. Neurons are the integral and important component of the designed soft computingbased intelligent network that transformed the data obtained through any deterministic-based method like Adam's numerical method and then gives the result in the output layer. Data traveled from the input layer to the layer connected with the output setting of the network. Different layers are also incorporated into the designed networks for finding the best possible outcomes. The total data set is classified into 70% training, 15% validation, 15% testing. The present article carries out the Hall current with Jeffrey fluid and CCHFM. In this regards the experimental data will be analyzed by using the Artificial Neural Network model with Leven-berg Marquardt method (ANN-LMM).

Numerical results and discussion
In a current research article, reference numerical result and ANN is applied for the estimate of the Hall effect on MHD flow with Jeffrey fluid and heat transfer with CCHFM. Numerical solutions with the help of ND-Solve command and Artificial Neural Network (ANNs) are investigated. Table 1 is constructed for all variants of the presented MHD flow of the Jeffrey fluid system under the impact of heat. Velocity profile f (η) is represented through case study 1, and another velocity profile g(η) is shown via case study 2, whereas case study 3 represents the temperature profile θ (η). Scenarios 1, 2, 3 denotes the variables. Scenarios 1, 2 and 3 of case study 1 represent the Hall current parameter (m), wall thickness parameter (α), ratio of relaxation to the retardation time (λ 1 ). Case study 2 is about Hall current parameter (m), Deborah number (β). Case study 3 represents the thermal relaxation parameter (γ ), Prandtl number (Pr.), velocity exponent parameter (n).

Performance analysis of numerical solution:
We obtained the non-dimensional velocities and temperature profiles for emerging parameters. Using ND-solve command in MATHEMATICA software with ADAM BASHFORTH method, we obtained solutions for profiles f (η), g(η) and θ (η) for various cases.     Fig. 6. When we increase the value of Hall current parameter m, the velocity profile also increases. Figure 4 shows the impact of the magnetic field parameter (M). When we increase the value of M the velocity component f (η) shows a reduction. This is due to the fact that magnetic field M induces  the resistive force which is also called a Lorentz force while the velocity profile reduces because when the Lorentz force becomes weaker, the motion of the fluid reduces and the fluid become to rest. It is because of the fact the magnetic field acts as retarding/controlling agent and has the ability to control the fluids velocity upto desired value. Figure 5 influences the ratio of relaxation to the retardation time λ 1 on the velocity profile f (η) along with the boundary layer because the physical ratio of relaxation to the retardation time depends upon the retardation time. As λ 1 increases the relaxation time and   it reduces the retardation time. Jeffrey fluid parameter was the reason for the variation of the momentum boundary layer, so the velocity reduces in the variable sheet. Figure 7 exhibit the effect of Deborah number β on the velocity profile g(η).
As it was observed from Fig. 7 that for the larger value of the Deborah number β, velocity profile g(η) reduces. Physically, Deborah number depends upon retardation time so with the enhancement of the retardation time Deborah number decreases but it increases for the gradient of the velocity profile of the Jeffrey fluid. Figures 8, 9 and 10 exhibit the influence of γ , Pr, n with the ranges 0.2 ≤ γ ≤ 0.8, 0.71 ≤ Pr. ≤ 2.0, 0.1 ≤ n ≤ 0.9 respectively on the temperature profile θ (η). Figure 8 shows that an increment in the value of thermal relaxation parameter γ results into a reduction of the temperature profile θ (η) because the temperature of variable sheet decreases with the enhancement of the thermal relaxation γ . In case when the thermal relaxation parameter reduces to zero i-e (γ 0) the CCHFM becomes the classical Fourier law of heat conduction. Figure 9 presents the impact of Prandtl number Pr. On the temperature profile θ (η).it is observed from the figure that the temperature profile θ (η) reduces for the larger values of Pr. Number. Physically, the Prandtl number depends upon the thermal diffusivity and thermal diffusivity becomes lower with the enhancement of the Prandtl fluid because of the fluid with high pr. number shows less conduction. Due to this reason when we increase the value of pr. number thermal diffusivity reduced and temperature profile reduces. Figure 10 demonstrated the influence of the velocity exponent parameter (n) on θ (η). For the larger n, the profile θ (η) increases. The positive value of n i-e (n > 0) shows that the variable sheet is stretching. For the transverse velocity distributions, it shows the same behavior. Variation for M, Pr., α, λ 1 , m, γ , β of and skin friction coefficient and Nusselt number are presented in Table 2.

Performance analyses on outcomes of the networks
The elaborative numerical solution of transformed system of ODEs by ANN is presented for various parameters of profiles f (η),g(η) and θ (η). Solution by ANN with Levenberg Marquardt method (ANN-LMM) interpreted through error histogram, plot fit, training states, performance and regression. Performances of three scenarios of all the cases of case study 1, 2 and 3 are presented. Result will be analyzed by comparison.

Case study 1
The   Fig. 21a-c displayed the error histograms of scenario 3 for three cases. The data set points with an error close to zero having less errors for cases 1 and 3 whereas bit more errors for case 2. The sub- Fig. 22a-c present the fitness plots which show that the error is close to zero i.e., the predicted value suits to experimental data values. The error for all the cases lies in the range of (−2 × 10 −4 to 2 × 10 −4 , −1 × 10 −5 to 1 × 10 −5 , −2 × 10 −4 to 2 × 10 −4 ). Figure 23 demonstrated the performance analysis graphically After training the data set the least value of MSE is obtained. The best validation performance at epochs (284, 293, 249) is given as (1.6407e −9 , 22.23755e −11 , 8.87428e −10 ) which shows that the data set is well trained. Further, the training states give Mu, gradient, validation checks observed in Fig. 24a-c. The gradient is (9.96e −8 , 9.97e −8 , 9.83e −8 ) for the three cases of scenario 3 which gives consistently viable results. Additionally, the value of Mu (1e −9 , 1e −9 , 1e −9 ) is found close to zero for scenario 3 in each case. Sub- Figure 25a-c also show the regression analysis of the predicted and target value of scenario 3 with regression index R 1 that presents the rationality of the accurate performance of the ANN network model.

Case study 2
The analysis of networks for velocity profile g(η) of different scenarios for various emerging parameters of case study 2 (CS2) are illustrated in Figs. 26, 27, 28, 29, 30, 31, 32, 33, 34 and 35. Figure 26a-c depicts accurately our model through the prediction of the data set after training. Higher positive error exhibits for case 1 and negative error for cases 2 and 3. Figure 27a-c demonstrates the fitting plots graphically for scenario 1 of three cases. Absolute errors for the presented fluid system show good performance analysis as its most of the values lie in the ranges (−5 × 10 −5 to 5 × 10 −5 , −2 × 10 −4 to 2 × 10 −4 , −1 × 10 −4 to 1 × 10 −4 ) respectively with better precision. The sub- Fig. 28a-c presents the performance analysis with the best validation performance of the given data set for three cases of scenario 1. The minimum value of MSE are (1.57918e −10 , 1.41400e −10 , 2.98581e −10 ) at the epoch (12,21,39) with good validation. The sub- Fig. 29a-c depict the training states of all variants associated with scenario 1. The magnitudes of the gradient (6.14e −8 , 9.60e −8 , 9.35e −8 ) at the epoch (12,21,39). Further, the value of Mu for all the cases of scenario 1 is (1e −12 , 1e −11 , 1e −11 ) which shows the convergence of the ANN   Fig. 31a-c depicts that most of the values give reasonable accuracy for case 1, whereas higher negative zero error for case 2 and 3. Figure 32 shows the plot fitness of all variants associated with scenario 1. The error was found close to (−2 × 10 −4 to 2 × 10 −4 ) for all the cases. Figure 33a-c shows the mean square error and the best validation performance at epoch (7,9,9) are given (6.29335e −9 , 1.96818e −9 , 2.22200e −9 ), respectively, which shows that the data set is well trained. Moreover, . Figure 35a-c shows the best regression analysis (R 1) plot for variants associated with scenario 2 with good accuracy between the target and output values.

Case study 3
The network is designed to plot the temperature profile θ(η) for two scenarios (γ , Pr.) for all variants associated with the Jeffrey fluid flow system and are presented graphically in Figs. 36, 37, 38, 39, 40, 41, 42, 43, 44 and 45. Figure 36a-c shows the histogram analysis of the ANN model of scenario 1 for three cases. The zero error line close to zero    Figure 44a-c shows the training states of all variants of scenario 2. As it is observed from Fig. 44 training states depend upon Mu, gradient, validation checks. The value of the gradient for three cases are (9.98e −8 , 9.97e −8 , 9.99e −8 ) at the corresponding epoch (318. 374, 187), respectively. The magnitude of Mu for three values are (1e −10 , 1e −8 , 1e −13 ) which is close to zero with good accuracy. Figure 45 shows the best regression analysis of scenario 2 for all variants with good error fitness of ANN showing the closeness of output and target values.

Tabular description for case studies 1-3
The results presented in Tables 3, 4 Table 6).

Concluding remarks
In the presented investigation, a Hall current effect on Magnetohydrodynamics flow with Jeffrey fluid and Heat transfer with CCHFM over a stretchable sheet with varied thickness. The results are effectively analyzed through designed ANN-LMM using error histogram, plot fit, performance, training states, regression plot. Major outcomes of the present study are summarized below: 1. Both velocity components f (η) and g(η) along with the skin friction coefficient in the horizontal as well as in and z-axis direction are accelerated with the increase in Hall current parameter (m). Actually, it happens due to the controlling mechanism of electric conductivity for the fluid system, which accelerates molecular movement. 2. Magnetic field parameter reduces the thickness of momentum boundary layer along x-axis, while an increment in M will tend to reduce fluid velocity as magnetic field parameter is the ratio of electromagnetic force to the viscous force and due to this fact drag force is enhanced resulting in the increment in the skin friction coefficients along with x and z axes directions, respectively.     Whereas along x-axis and z-axis the skin friction coefficient increases for M. 3. The temperature profile for θ (η) shows a reduction with an increment of Pr as is the ratio of momentum diffusivity to the thermal diffusivity as due to the large value of Pr, the thermal diffusivity becomes low which declines the temperature profile. 4. The velocity component f (η) tends to increase with an increment in Deborah number β, while the opposite behavior is found for g(η). 5. Velocity profile f (η) decreases for larger value of ratios of relaxation to the retardation time (λ 1 ) while g(η) shows opposite behavior.
6. With an increment in relaxation time of the heat flux γ tend to decrease temperature profile θ(η) . 7. Local Nusselt number Nu x increase with increment in the Pr, α, m, β and decreases with increase in M, λ 1 , γ /As the larger value of the Nu x corresponds to more effective convection in the fluid flow system. 8. In the Artificial neural network, the error between the target and output value after training are analyzed by an error histogram. The Regression (R) of the trained data set for all the cases is 1.i-e (R 1).