Intelligent Backpropagated Neural Networks for Numerical Computations for MHD Squeezing Fluid Suspended by Nanoparticles between Two Parallel Plates


 In the present research, artificial intelligence based backpropagated neural networks with Levenberg-Marquardt algorithm (BNN-LMA) are utilized to interpret the numerical computation for squeezing 2D magneto-hydrodynamic (MHD) nanofluid flow between two parallel plates. The non-linear system of ODEs represents the magneto-hydrodynamic, squeezing nanofluidic flow model (MHD-SNFM). A reference dataset for BNN-LMA is formulated by utilizing Adam numerical solver for different scenarios of MHD-SNFM by variation of squeezing number, Hartmann Number and heat source parameter. The validation, testing and training processes of BNN-LMA are exploited to analyze the approximate solution of MHD-SNFM for different scenarios and correctness of proposed BNN-LMA is verified by comparison of reference outcomes. The performance of BNN-LMA to solve the MHD-SNFM is validated through regression analysis, histogram studies and mean square error (MSE).


Introduction:
The squeezing flow, first explored by Josef Stefen, is the form of flow in which flow is squeezed, deformed or pressed out between two parallel plates or disks. Moreover, it demonstrates the movement of fluid particles, its contact with the surface of plates and effects of other parameters such as temperature, viscosity and heat source parameter etc. In this research we are dealing with nanofluid squeezed between two parallel particles, nanofluid comprises of nanoparticles mixed base fluids. These nanoparticles are metal oxides, metals or carbon nanotubes.
In the recent years, most of the literature has been done on study of nano-particles and nanofluids. Nanofluids, first named by Choi [1], are basically nanoparticles suspended in base fluids. T Hayat et. al [2] have studied the behavior of parameters on velocity, temperature and concentration profiles, also computed and analyzed the heat and mass transfer. The fluid flow and heat transfer characteristics are analyzed by Mehmood et al. [3]. Siddique et al. [4] analyses the MHD squeezing flow between two parallel surfaces. Shoaib et al. [5] investigated the heat and mass transfer in 3-D MHD radiative flow of of hybrid nanofluid. If one nanoparticle is mixed with base fluid, it is mono nanofluid while if two or more nanoparticles are added then it is hybrid nanofluid. Ali Imran et al. [6] theoretically investigate the heat transfer of nanofluid flow in ciliated channel.
Azimi et al. [7] have discussed MHD squeezing flow of nanofluid between parallel plates and compared the analytical and numerical results. The heat transfer in unsteady nanofluid flow between two moveable parallel plates is investigated by Ganji DD et al. [8], and analyzed the effects of various parameters. The chemical reactions, velocity slip, thermal radiation and Brownian motion in 3-D flow of Casson nanofluid has been studied by M Umar et al. [9]. Babazadeh et al. [10] have studied the effects of thermophoresis, magnetic forces on nanoparticles squeezed between two plates. Noor et al. [11][12] have analyzed the unsteady MHD squeezing flow of Jeffery fluid in a porous medium and effects of viscous dissipation and chemical reaction on MHD squeezing flow of Casson nanofluid between parallel plates in a porous medium. Many of other researchers [13][14][15][16][17][18][19][20][21] have contributed their work on squeezing flow of nanofluids including hybrid nanofluids and Casson nanofluid.
In most of the engineering problems, the system of PDEs representing mathematical relations of the problem are transformed into ODEs. In many cases, the solution to scientific problems does not admitted analytically, this leads to the equation to be solved by using special techniques. One of the methods are reconstruction of variational iteration method [ [38][39], fluid dynamics [40][41] and nanotechnology [42][43].
Although above cited literature on nanofluid flows containing nanoparticles for different fluidic systems, mostly on squeezing flows by using various traditional numerical and analytical methods; but stochastic numerical techniques are required to exploit for squeezing flow problems due to their worth, effectiveness and robustness. The stochastic numerical methods are already implemented for various research problems by the research workers [44][45][46][47][48][49]. Some most recent artificial intelligence based techniques are Emden-Fowler Model [50], nonlinear unipolar electro hydrodynamic pump flow model [51], non-linear corneal shape model [52] and COVID-19 Models [53][54]. These soft computing infrastructures are inspiring factors for the authors source of motivation to exploit an accurate and reliable alternate framework based on soft computing infrastructure for the solution of heat generation in mixed convected Williamson fluid stretching flow problem by conducting a parametric study to examine the effects or various physical quantities on the velocity, concentration and temperature profiles. Mathematica and MATLAB software are utilized for numerical treatment.
The experimental insights of computing simulation are highlighted as follows:  Intelligent computation is presented by Levenberg-Marquardt algorithm based backpropagated neural networks to study the MHD squeezing nanofluid flow between two parallel plates.

Problem Formulation of MHD-SNFM:
In this research, numerical computation for the analysis of 2D unsteady, incompressible, squeezing nanofluid flow between two parallel plates having distance = ± √1 − = ±ℎ( ) has been interpreted, where is the distance between the plates at = 0, and is squeezed parameter, when > 0 the plates are squeezed but when < 0 and = 1 the plates are separated. The magnetic field's variation with time is defined as = The governing equations of conservation of momentum and energy are given as [55]: Where u and v are velocity components, T is temperature, pressure, is function of heat source, ℎ effective density, ( ) ℎ effective heat capacity and ℎ is electrical conductivity of nanofluid. The boundary conditions are: Now by using some parameters given below: The system of PDEs are transformed into system of ODEs as follows: And the boundary conditions are: Where S is squeezing number, Ha is Hartmann Number, Pr is Prandtl number and Hs is heat source parameter and 1 , 2 , 3 , 4 and 5 are dimensional constants.

Solution Methodology:
The nftool, an effective algorithm in artificial based neural networks (NNs) toolbox in MATLAB software package is utilized to execute the designed backpropagated neural network with Levenberg Marquardt Algorithm (BNN-LMA). The solution methodology consists of essential description for dataset and implementation procedure for implementation of designed BNN-LMA. The designed neural network for BNN-LMA is shown in Figure 2 and flow chart of methodology presented in Figure 3.
The reference dataset of designed BNN-LMA is created for inputs between 0 and 1 with time interval of 0.01 by utilizing Adam numerical solver through "NDSolve" in Mathematica software package by variation of squeezing number, Hartmann Number and heat source parameter in MHD-SNFM as listed below in the Table 1. 3. Interpretation of Results: The possible outcomes of numerical computation for the designed neural networks backpropagated with Levenberg Marquardt algorithm is exploited for the proposed MHD-SNFM as presented in equations (7)(8)(9). The three scenarios of MHD-SNFM by variation of squeezing number S, Hartmann Number Ha and heat source parameter Hs are formulated for four different cases for both velocity and temperature parameters of MHD squeezing nanofluidic flow model as mentioned in Table 1. The efficiency of results of BNN-LMA is observed with matching outcomes of Adam numerical solver for all the three scenarios of MHD-SNFM as shown in Figure 6 which is further endorsed by error plots. The investigation through regression analysis is carried out by co-relation studies. Figure 7 shows the results of regression outcomes of respective three variants of MHD-SNFM.
One may see that the value of correlation R close to unity indicates perfect modeling, in terms of training, testing and validation certified the correctness of BNN-LMA for the designed MHD-SNFM.
Additionally, the corresponding numerical values listed in Tables 2-4