Numerical Analysis of Electrohydrodynamic Flow in a Circular Cylindrical Conduit by Using Neuro Evolutionary Technique

This paper analyzes the mathematical model of electrohydrodynamic (EHD) fluid flow in a circular cylindrical conduit with an ion drag configuration. The phenomenon was modelled as a nonlinear differential equation. Furthermore, an application of artificial neural networks (ANNs) with a generalized normal distribution optimization algorithm (GNDO) and sequential quadratic programming (SQP) were utilized to suggest approximate solutions for the velocity, displacements, and acceleration profiles of the fluid by varying the Hartmann electric number (Ha2) and the strength of nonlinearity (α). ANNs were used to model the fitness function for the governing equation in terms of mean square error (MSE), which was further optimized initially by GNDO to exploit the global search. Then SQP was implemented to complement its local convergence. Numerical solutions obtained by the design scheme were compared with RK-4, the least square method (LSM), and the orthonormal Bernstein collocation method (OBCM). Stability, convergence, and robustness of the proposed algorithm were endorsed by the statistics and analysis on results of absolute errors, mean absolute deviation (MAD), Theil’s inequality coefficient (TIC), and error in Nash Sutcliffe efficiency (ENSE).


Introduction
Electro fluid systems have attracted a large number of researchers in recent years due to their wide range of applications in engineering and industrial processes [1,2]. The influence of electric fields on fluids has been studied from a theoretical point of view in the recent past to create novel processes [3]. The interaction of hydrodynamics and electrodynamics in dielectric fluid flow in the presence of an electric field is referred to as electrohydrodynamics (EHD) [4,5]. It is the study of the motion of ionized fluid and controls transportation phenomena in fluid flow. EHD flows have a variety of applications in different fields such as electrostatic precipitators [6,7], designs of dielectric pumps [8], the flow of ions in diverter tokamak reactors [9], thermal management of microelectronics by using electro gas dynamical pumps [10], application of Taylor cone EHD jets for electrospray of liquid atomization [11], electro-fluid control of colloidal particles near electrodes [12], MEMS devices [13], inkjet devices [14], propulsion of small scale naval vessels [15], processing of nickel particle flows and their regulation in medical powder [16], etc.
McKee et al. [17] in 1997, for the first time, dealt with the electrohydrodynamic (EHD) flow of fluid in an ion drag configuration in a circular cylindrical conduit, which was governed by a nonlinear boundary value problem (BVP). They studied velocity profiles by obtaining perturbation solutions due to variations in small and large parameters governing the nonlinear problem. For large values of strength of nonlinearity (α), they used the combination of Ranga-Kutta methods and the finite difference method to calculate the numerical solutions for fluid velocity. Further, they concluded that for α 1 and α 1, the perturbation solutions were expanded to series solutions with O(1) and O(n α ), respectively. Paullet [18], in 1999, proved the existence and authentication of solutions to the EHD flow of fluid in the cylindrical conduit. The results obtained by [18] for a larger value of α were qualitatively different from those obtained by [17]. Statistics showed that for α 1 results by [17] diverged because the perturbation solutions were expanded to O(1/α) instead of O(1) [18]. Mastroberardino [19] used the homotopy perturbation method to find analytical solutions for the velocity profile of EHD flow by varying the values of the nonlinearity parameter and the Hartmann electrical number. Panday [20] used two semianalytical techniques based on an optimal homotopy analysis method and the homotopy asymptotic method to solve the EHD problem by varying different values of the parameters. Further, [21] developed a new homotopy perturbation method (NHPM) that depended on two components of homoptopy series to study the EHD flow equation. Moghtadaei [22] used the hybridization of the spectral homotopy analysis method (SHAM) and the spectral collocation method (SCM) to investigate the EHD problem in ion drag configuration. Pradip Roul [23,24] developed a new approximate method, namely the discrete Adomian decomposition method (DADM), to approximate the solution of a strongly nonlinear singular boundary value problem describing the electrohydrodynamic flow of a fluid in an iron drag configuration in a circular cylindrical conduit. Further, [25] used the differential transformation method to find an analytical solution for the velocity profile by varying the strength of nonlinearity (0 < α <1). This assumption helped overcome the limitation of nonlinearity and singularity. Various numerical and analytical techniques such as the pseudospectral collocation method [26], spectral collocation method [27], discrete optimized homotopy analysis method (DOHAM) [28], DTM-Pade' approximation [29], least square method [30], Galerkin Method, Collocation Method [31] and optimal B-spline collocation method [32,33] have been used. The electrohydrodynamic (EHD) flow of fluid has been studied by various numerical and analytical approaches but due to the singularity and nonlinearity in the nature of its mathematical model, it is difficult for traditional techniques to find its approximate solutions. In this paper, we have solved the EHD flow of fluid in a circular cylindrical conduit with an ion drag configuration numerically by applying a stochastic technique based on artificial neural networks (ANNs). In recent times, ANNs have been widely used to solve the number of problems arising in petroleum engineering [34], thermal engineering [35][36][37][38], civil engineering [39], biological models [40,41], and wire coating dynamics [42]. These applications of stochastic techniques are motivating factors for authors to analyze, explore, and investigate a nonlinear singular model of the EHD flow of fluid. In this paper, we designed an evolutionary soft computing technique to optimize the mathematical model of the electrohydrodynamic flow of fluid with an ion drag configuration by using the hybridization of the generalized normal distribution optimization (GNDO) algorithm and sequential quadratic programming (SQP). Contributions of the given study are highlighted through the following salient features: -This paper developed a neuroevolutionary soft computing paradigm to analyze the mathematical model of electrohydrodynamic (EHD) flow of fluid in the circular cylindrical conduit by exploiting the global search efficiency of GNDO and local search support of SQP. The design technique is named ANNs-GNDO-SQP. - The proposed algorithm was implemented on the EHD problem to study the effect of variations in the nonlinearity parameter and the Hartmann electrical number on w * (r * ), w * (r * ), and w * (r * ). The performance of the design paradigm was validated through statistics of the fitness function, absolute errors, mean absolute deviation, error in Nash Sutcliffe efficiency, and Theil's inequality coefficient.

Problem Formulation
In this section, we present the mathematical overview of the governing equation. Consider a circular cylindrical conduit in an axisymmetric coordinate system of radius (r) with insulating conduit walls, as illustrated though Figure 1. An electric field → E 0 is produced in fluid by applying voltage (V). Current density in a dielectric medium for a single ionized particle is defined as [20,43] where K is the ion mobility, and ρ f denotes the charge density of ions in fluid. The velocity of the fluid in generalized form is given as where, the pressure gradient (∂p/∂z) is assumed to be constant. By using Equation (2), the Naiver Stokes equation reduces to where µ denotes the viscosity (dynamic) of the EHD fluid. The velocity profiles of fluid depend on time scales such as U and L which determine the coupling between ions and fluid.
where, ε 0 denotes the permittivity constant of free space, t c is the relation time of charge, t f is the time of fluid transport, and ρ 0 denotes the charge density at the inlet screen. From Equation (1), assuming that l(z = 0) = l 0 the charge density can be defined as In order to calculate the velocity profile of the EHD fluid, Equations (3)-(5) are integrated to obtain a nonlinear differential equation which is given as with the boundary conditions defined at the center and wall of a conduit as w (r) = 0 at r = 0; w = 0 at r = a.

Design Methodology
In this section, the structure of the design methodology is presented to solve the EHD problem with ion drag configuration. The methodology consisted of two steps. Initially, an approximate solution model for differential equations based on mean square error was constructed. Secondly, an optimization process of the fitness function was carried out by utilizing the hybridization of GNDO and SQP.

ANN Modeling
In this section, mathematical modeling of electrohydrodynamic (EHD) fluid flow in an ion drag configuration is emphasized by using artificial neural networks (ANNs). Feedforward ANNs were utilized to model approximate solutionsŵ * (r * ) for Equation (8) aŝ since, the nth order continuous derivative of Equation (10) exists, therefore, the first and second-order derivatives are defined as where, W = (α, ζ, β) = (α 1 , α 2 , . . . , α m , ζ 1 , ζ 2 , . . . , ζ m , β 1 , β 2 , . . . , β m ) are weights in ANNs architecture, m represents the number of neurons and f is a connection function or activation function. For instance, we consider the log-sigmoid function. Application of the log-sigmoid function will model Equations (10)-(12) aŝ Energies 2021, 14, 7774 ANNs architecture for the EHD flow of fluid in terms of input, hidden, and output layers is shown in Figure 2.

Formulation of the Fitness Function
In this section, an objective function known as the fitness function is formulated in a mean square error sense for the nonlinear EHD problem.
where, Θ 1 and Θ 2 are error functions corresponding to the nonlinear differential equation and boundary conditions, which are defined as and where, N = 1 h , and h represents the mesh points.

Optimization Process
In this section, an optimization process of ANNs based fitness function is incorporated by hybridization of the generalized normal distribution optimization algorithm and sequential quadratic programming. The detailed flow chart of the ANNs-GNDO-SQP algorithm is shown in Figure 3.

Generalized Normal Distribution Optimization Algorithm
The GNDO algorithm is a novel metaheuristic algorithm inspired by the classical generalized normal distribution theory [44]. It is a global search technique that is widely used for parameter extraction; unlike other metaheuristics, the GNDO algorithm is easy to implement requiring an initial population size and essential termination conditions. The working procedure of GNDO is based on two processes, which are explained as -Exploitation Exploitation refers to the process of finding the best solution around the search space consisting of the current positions of all individuals. A distribution model for optimization based on a relationship between the distribution of individuals and their normal distribution is given as where, Ω t i ,P i ,Φ i andĤ, are the trial vector, standard variance, mean position (generalized), and penalty factor respectively. These parameters are defined aŝ here, M denotes the mean position, a, b, h, and h 1 are random numbers from 0 to 1 andΦ is the current position of the individual. x t i is the ith individual, and x t Best is the current best individual.

-Exploration
A process of searching entire population space for finding the global best solution is called exploration. In GNDO, exploration of search space is based on randomly selected individuals, which can be expressed as Global information sharing Local information sharing (24) here, β denotes the adjustment parameter, h 1 and h 2 are constants between 0 and 1, and v 1 and v 2 are trial vectors that are defined as where, a1, a2, and a3 are integers. An overview of the working procedure of GNDO is provided in pseudocode as Algorithm 1. Recently, the GNDO algorithm has been applied to study the parameter extraction photovoltaic models [44] and heat transfer in temperature fins [45].

Sequential Quadratic Programming
The SQP algorithm is a nonlinear programming technique that the research community has extensively used to solve a number of real world problems arising in different fields of engineering and applied mathematics. In this paper, the SQP algorithm was implemented to refine the process and weights obtained by the GNDO during the global search phase. Currently, SQP algorithms are used to find approximate solutions for two-block nonconvex constrained optimization problems [46], real power loss minimization in radial distribution systems [47], dynamic realtime optimization framework for cycling energy systems [48], measurement error in invariance testing [49], and maximum likelihood-based measurement noise covariance estimation [50].
In the present study, we exploited the global search ability of the GNDO algorithm with the sequential quadratic programming with the "fmincon" environment of MATLAB to calculate the approximate solutions using an ANN structure for the EHD flow of fluid. The detailed working procedure of the design algorithm is shown in Figure 3. The ANN-GNDO-SQP algorithm has a simple structure and is easy to implement. GNDO updates the position of an individual using generalized normal distribution formula, and SQP complements its local convergence. The experimental analysis showed that the proposed algorithm converged to the best solutions for a number of real-world problems by training the weights from the interval [−5, 5]. The convergence of design scheme was slightly affected by increasing the domain. The parameter settings for execution of the proposed algorithm are shown through Table 1.

Simulation and Discussion
In this section, the design algorithm was implemented on the EHD flow to calculate the approximate solutions for the velocity, displacements, and acceleration profiles of the fluid by varying the Hartmann electric number Ha 2 and the strength of nonlinearity (α). To demonstrate the accuracy of the proposed algorithm, the results were compared with the orthogonal Bernstein collocation method (OBCM) [20] the least square method (LSM) [30], and numerical solutions obtained by an implicit Runge-Kutta method (bvp5c), using MATLAB. Velocity profiles for different values of the Hartmann electric number and nonlinearity parameter are illustrated in Figures 4 and 5 respectively. From Figure 4, it can be seen that for fixed values of the nonlinearity parameter (α = 0.5, 1.0, 1.5, 2.0) an increase in the Hartmann electric number Ha 2 causes an increase in the velocity profile, while Figure 5 shows that an increase in the nonlinearity parameter with a fixed Hartmann electric number Ha 2 = 1, 2, 6, 10 causes a decrease in the velocity profile of the EHD flow of fluid. The strength of nonlinearity has the adverse effect on the velocity profile against the Hartmann electric number. Moreover, it can be seen that a plug flow profile appears for small values of α with significantly large values of Ha 2 . Statistics of w * (r * ), w * (r * ) and w * (r * ) for different values of Ha 2 and α are dictated in Tables 2 and 3. The results for w * (0) obtained by the ANN-GNDO-SQP algorithm were compared with the orthogonal Bernstein collocation method (OBCM) and numerical solutions for α = 0.5, 1 and 2 with Ha 2 = 1, 2, 6, 10, 20, and 50 as shown in Table 4. Absolute errors in our solutions are plotted in Figure 6. The design scheme overlapped the analytical solution and had minimum absolute errors in comparison with the least square method (LSM) and OBCM as shown in Table 5. In order to show the superiority of ANN-GNDO-SQP, we investigated the effect of large values of Ha 2 and α on the velocity of the fluid. Paullet [18] showed that solutions of Equation (8) Table 2. The values of w * (r * ), w * (r * ), and w * (r * ) for different variations in the Hartmann electric number and nonlinearity parameter.  Table 3. The values of w * (r * ), w * (r * ), and w * (r * ) for different variations in the Hartmann electric number and nonlinearity parameter.
where, w(r i ) denotes the analytical solution, whileŵ * r * i represents the approximate solution by the ANN-GNDO-SQP algorithm. Global values of fitness function and performance indices can be calculated as To study the optimization behavior of the objective function, 100 independent executions of the ANN-GNDO-SQP algorithm were carried out. Convergence of fitness function and TIC for Equation (17) Figure 10. Normal probability curves are plotted in Figure 11 for the values of ENSE. Furthermore, Table 6 depicts the stability analysis of the design algorithm. For 100 independent runs, the results in each run were in good agreement with the exact solutions. The accuracy of the design algorithm can be seen from values of ENSE that are approximately equal to zero for each case. Statistical analysis of the values of fitness function and performance indices are shown in Table 7. Global values of these indicators lay around 10 −5 to 10 −7 which shows the robustness and accuracy of the design scheme. Table 6. Stability analysis in terms of fitness value, mean absolute deviations, Theil's inequality coefficient, and error in Nash Sutcliffe efficiency of the design algorithm when executed for 100 multiple runs.      Figure 11.
(a-f) shows the normal probability curves for ENSE. Computational complexity analysis (CCA) was evaluated for the proposed algorithm based on the average time taken to calculate unknown neurons in the ANN structure using the GNDO-SQP algorithm. The execution time of the ANN-GNDO-SQP algorithm for different values of the nonlinearity parameter and Hartmann electric number in the EHD flow problem in terms of minimum, mean, and standard deviations are shown in Table 8. The results showed the consistency and efficiency of the proposed algorithm. All calculations and evaluation for this research were performed on an HP laptop EliteBook 840 G2 with intel (R) Core (TM) i5-5300 CPU @ 2.30 GHz, 8.00 GB RAM, 64 bits, operating in Microsoft Windows 10 Education edition, running R2018a version of MATLAB.

Conclusions
In this paper, we analyzed the mathematical model of the electro hydrodynamic (EHD) flow of fluid in a circular cylindrical conduit with an ion drag configuration. The problem was modelled by the boundary value problem with strong nonlinearity due to the rational function form of the nonlinearity of the EHD flow's equation. The solution of such a nonlinear differential equation is always a difficult task for any traditional, as well as modern, numerical techniques. Therefore, to calculate the velocity profile of the EHD flow we designed a neuro soft computing technique based on artificial neural networks.
Our key contributions are summarized as follows:

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We developed a neuroevolutionary soft computing paradigm to analyze the mathematical model of the electrohydrodynamic (EHD) flow of fluid in the circular cylindrical conduit by exploiting the global search efficiency of GNDO algorithm and local search support of SQP.

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The ANN-GNDO-SQP algorithm is an unsupervised learning mechanism that has a simple structure and is easy to implement. It does not require any prior information about the problem aside from the essential initial and terminal setting of parameters in GNDO-SQP for the execution.

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The design scheme known as the ANN-GNDO-SQP algorithm was implemented to study the effect of variations in the nonlinearity parameter and Hartmann electrical number on w * (r * ), w * (r * ), and w * (r * ) of the fluid.

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The results concluded that an increase in Hartmann electric number Ha 2 with fixed values of nonlinearity parameter caused an increase in the velocity profile, while an increase in the nonlinearity parameter with a fixed Hartmann electric number possessed an inverse relation with the velocity of the fluid.

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The results obtained by the proposed algorithm were compared with the least square method and orthogonal Bernstein collocation method. Statistics showed that the ANN-GNDO-SQP algorithm overlapped the exact solutions and analytical solution with minimum absolute errors.

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The results of performance measures show that the proposed algorithm is accurate, reliable, and efficient in solving real-world problems. Informed Consent Statement: Not Applicable.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.