key: cord-0077936-88sleh69 authors: Shoaib, Muhammad; Rafia, Tabassum; Raja, Muhammad Asif Zahoor; Khan, Waqar Azeem; Waqas, Muhammad title: Further analysis of double-diffusive flow of nanofluid through a porous medium situated on an inclined plane: AI-based Levenberg–Marquardt scheme with backpropagated neural network date: 2022-05-10 journal: J Braz DOI: 10.1007/s40430-022-03451-9 sha: ac6c17a522317c8b7694562068987518a037b5ee doc_id: 77936 cord_uid: 88sleh69 The present article exploits a novel application of AI-based Levenberg–Marquardt scheme with backpropagated neural network (LMS–BPNN) to analyze the double-diffusive free convection nanofluid flow model (DDFC-NFM) over an inclined plate in the existence of Brownian motion and thermophoresis properties embedded in a porous medium. The governing PDEs representing DDFC-NFM are transformed into system of nonlinear ODEs by applying suitable transformation. The reference data set is generated from Lobatto III-A numerical solver by variation of magnetic field parameter (M), thermal Grashof number (Gr), angle of inclination (α), Brownian motion parameter (Nb), Dufour-solutal Lewis number (Ld), modified Dufour parameter (Nd) and thermophoresis parameter (Nt) for all scenarios of the designed LMS–BPNN. The approximate solution and its comparison with standard solution are analyzed by execution of training, testing and validation procedure of the designed LMS–BPNN. The effectiveness and reliable performance of LMS–BPNN are endorsed with MSE-based fitness curve, regression analysis, error histogram analysis and correlation index. Results reveal that velocity increases with the rise in Gr, whereas reverse trend has been noticed for angle of inclination and magnetic field parameter and the temperature profile increases with the increase in Nb, Nd and Nt. The solutal concentration profile increases with the increment in Ld, while an increase in Nd causes a decrease in it. When Nt increases, the enhancement in the nanoparticle volume frictions occurs, but an opposite behavior is depicted for Brownian motion parameter. In recent time, nanofluid and its findings have received significance importance over the past few years. This is due to their diverse and massive applications such as nano-drug delivery, food processing, power plant, geothermal extraction, nanoliquid detergent and numerous others. Firstly, the word 'nanofluid' was invented by Choi [1] which is a liquid composed of nano-scaled particle suspended in base fluid. This nanometer-sized material has unique chemical and physical characteristics. Usually, the base fluid has low thermal conductivity such as water, engine oil and ethylene glycol used for this purpose, and the nanoparticles consist of Cu, AlN, SiC, Al 2 O 3 and graphite. It supports base fluid as due to high thermal conductivity it enhances the heat transfer process which saves cost and times. Several authors investigated nanofluid flow past an inclined plate under different effects. Khan et al. [2] studied viscosity of MHD flow of mixed convection Eyring-Powell nanofluid past an inclined plate. Zeeshan et al. [3] numerically investigated bi-phase coupled stress nanofluid past an inclined surface with Hafnium and metallic nanoparticles. Laminar conjugate mixed convection flow of nanofluid with transverse magnetic field past an inclined flat surface embedded in a porous medium is discussed by Khademi et al. [4] . Mass and energy transport boundary-layer flow of nanofluid in the presence of Soret-Dufour effect past an inclined plate is presented by Rafique et al. [5] . Idowu and Falodun [6] analyzed the heat and mass transfer MHD flow of nanofluid past an inclined surface in the existence of thermophoresis and Soret-Dufour effect. Recently, the researchers also investigated the combination of three different nanoparticles with base liquid, known as ternary hybrid fluid. The viscosity and thermal conductivity of these fluids depend upon three volume fraction parameters. Animasaun et al. [7] exemplified the threedimensional ternary nanofluids considering suction effect and bi-stretching surface. Yook et al. [8] investigated the ternary fluid past convectively heated sheet considering the effect of heat sink/source and magnetic flux density. Many authors investigated the non-Newtonian fluid and its findings over a porous medium [9] [10] [11] . Rashad et al. [12] exemplified natural convective non-Newtonian nanofluid flow in a porous medium past a radiative plate. Slip motion of MHD nanofluid over a stretching surface in porous media is examined by Kumar et al. [13] . Megahed [14] studied within a porous medium the convective heat transfer effect past a Maxwell fluid flow past a stretching surface. Rasheed et al. [15] discussed mixed convection flow of tangent hyperbolic fluid implanted in a porous medium with chemically reactive and magnetic field effect. Heat transfer effect on peristaltic propulsion of Jeffrey nanofluid within a porous rectangular medium is presented by Riaz et al. [16] . Yadav [17] analyzed the effect of Darcy number and viscosity on the arrival of convective motion in a couple-stress fluid in a porous medium. The effect of concentration with temperature gradient referring to buoyancy-driven flows is known as doublediffusive convection. The double-diffusive convention has a variety of applications in scientific field such as oceanography, geophysics applications, chemical reaction, petroleum reservoirs, aerospace defense, solar collector, food processing, energy storage and numerous others. Firstly, Pera and Gebhart [18] investigated numerically this phenomenon for vertical laminar fluid motions. After this, numerous studies have been investigated on double diffusion [19] [20] [21] . Nag and Molla [22] studied the double-diffusive natural convention effect on non-Newtonian nanofluid in square cavity. Double-diffusion effect over square cavity on natural convention flow with entropy generation is investigated by Said et al. [23] . The suspension of nano-encapsulated phase change materials in the presence of double-diffusion nanofluid flow in rotating porous cavity is examined by Raizah and Aly [24] . Prasad et al. [25] analyzed the double-diffusion natural convection Casson nanoliquid flow over an inclined surface within a Darcian porous medium. MHD Casson fluid past a vertically inclined plane in a porous medium with double-D = diffusion convective flow is discussed by Sailaja et al. [26] . Double-diffusion effect on mixed convection flow in the existence of static magnetic field within rectangular domain is investigated by Moolya and Satheesh [27] . The natural convective boundary-layer nanofluid flow in the presence of Brownian and thermophoresis effect is taking into consideration by many researchers [28, 29] . The suspension of randomly moving particles in gas or liquid which enhance the collision of molecules is known as Brownian motion, whereas thermophoresis is the movement of tiny particles toward decreasing thermal gradient. Heat transfer free convection nanofluid flow with Brownian and thermophoresis effect is studied by Haddad et al. [30] . Ganji et al. [31] examined the Brownian and thermophoresis effect on free convection MHD Al 2 O 3 -H 2 O nanofluid flow. Heat transfer natural convection flow on nanofluid with Brownian and thermophoresis effect in L-shaped enclosure is studied by Rana et al. [32] . The effect of Brownian motion and thermophoresis in wavy porous cavity on free convection nanofluid flow is presented by Pop et al. [33] . Sometimes, it is not possible to find an exact solution of a problem analytically for this purpose; researcher uses different numerical and semi-numerical techniques to solve the problem. Some techniques are homotopy perturbation method [34] , Keller-Box method [35] , spectral relaxation method [36] , Galerkin finite element method [21] and many others [37] [38] [39] . All the above-mentioned cited studies on different nanofluidic systems are solved by using different numerical and semi-numerical methods, but AI-based numerical computing paradigms are important to exploit double-diffusive free convection nanofluid flow model (DDFC-NFM) due to their worthiness and efficiency. Some authors already applied these stochastic numerical techniques in different fields such as thermodynamic [40] , plasma physics [41] , astrophysics [42] , finance [43] , nanofluid model [44] , Emden-Fowler system [45, 46] , HIV infection model [47] , nonlinear corneal shape model [48] , mosquito dispersal model [49] and COVID-19 models [50, 51] . All these inspiring factors motivate the researchers to exploit consistent and precise AI algorithmbased numerical computational paradigm for numerical analysis of mathematical model for double-diffusive free convection nanofluid by conducting graphical and numerical studies to explore the effect of all variants on velocity, nanoparticle volume fraction, solutal concentration and temperature profile. MATLAB is used for this purpose. The innovative contributions of the present study are as follows: • The novel application of Levenberg-Marquardt scheme with backpropagated neural network (LMS-BPNN) is adopted to analyze the double-diffusive free convection nanofluid flow model (DDFC-NFM) over an inclined plane in the existence of thermophoresis and Brownian motion properties implanted within a porous medium. • The mathematical modeling is presenting for the proposed problem, i.e., DDFC-NFM. • The reference data set is generated from Lobatto III-A numerical solver by the variation of and for all seven scenarios of the designed LMS-BPNN. • The solver LMS-BPNN is designed with the help of procedure based on testing, training and validation to find the approximate solutions of DDFC-NFM. Furthermore, the comparative study is conducted through MSE in order to validate the consistent accuracy. • The effectiveness and reliable performance of LMS-BPNN are endorsed with MSE-based fitness curve, regression analysis, error histogram analysis and correlation index. Consider time-independent 2D incompressible natural convection flow of nanofluid past an inclined impermeable plate implanted in the existence of thermophoresis and Brownian motion effect within a porous medium. Figure 1 represents the geometry of the present fluid flow system, where y-axis is perpendicular to an inclined plate and x-axis makes an acute angle with vertical part. A uniform transverse magnetic field of strength B 0 is applied normal toward the direction of nanofluid flow. The influence of induced magnetic field and magnetic Reynolds number is negligible. Further, we consider the temperature T w , the solute concentration C w and nanoparticle volume fraction N w have constant values. Ambient values of T, C and N are T ∞ , C ∞ and N ∞ , respectively. Assume C w > C ∞ and N w < N ∞ , so that as a consequence thermal and nanoparticle volume fraction buoyancy effect of an upward fluid is encouraged. The system of PDEs with BCs after considering Soret and Dufour effect is as follows [52] [53] [54] : Consider dimensionless variables [55] : Here, c is the empirical constant. The following equations can be attained by substituting the above variables: Mathematical expressions for physical parameters used in Eqs. (8)- (12) are: Involving stream function we have the following form of Eqs. (9)-(13) as: The further information regarding problem formulation can be seen in [55] . The process of finding the solution of the above PDEs is complex and tough. Lie group transformations suggested by Mukhopadhyay et al. [56, 57] and Islam et al. [58] are applied to solve the system of PDEs. This implies the following are corresponding ODEs with boundary conditions. The solution methodology is comprised of two steps: Firstly, the reference data set of LMS-BPNN is generated by solving transformed ODEs system presented in Eqs. parameter (Nd) and thermophoresis parameter (Nt). Later on, the designed AI-based Levenberg-Marquardt scheme with backpropagated neural network (LMS-BPNN) is implemented with the help of MATLAB command 'ftool' which is an artificial neural network (ANN) toolbox. The 'nftool' command is utilized to examine the MSE results, histogram studies and regression analysis that validates the performance of LMS-BPNN of the proposed DDFC-NFM. The solution for f � ( ), ( ), ( ) and ( ) for input 0 to 10 is randomly dispersed, and the reference data set is segmented to generate a set for training data (80%), validation data (10%) and testing data (10%) to operate the designed LMS-BPNN. The proposed LMS-BPNN is presented as an artificial neural network in Fig. 2 , and a flowchart of methodology is depicted in Fig. 3 . The designed LMS-BPNN is operated under the influence of parameters of interest Gr, , M , Ld, Nb, Nd and Nt for DDFC-NFM. There are seven scenarios each with four cases. Table 1 shows the numerical values of parameters of interest associated with DDFC-NFM which are used in the rest of work. Figures 4 and 5 show the performance and transition state of the proposed LMS-BPNN, while Figs. 6, 7 and 8 show the fitness curve with error analysis for scenarios 1-3, 4-6 and 7, respectively. Figures 9 and 10 show the regression of DDFC-NFM for scenarios 1-4 and 5-7 by LMS-BPNN, respectively. Furthermore, the MSE convergence for performance of training, testing and validation, performance, epochs, backpropagated operator, i.e., Mu, and time taken are depicted in Table 2 . The convergence curves of MSE for the second case of all seven scenarios of DDFC-NFM are represented in Fig. 4 (I)-(VII) for training, testing and validation. The excellent or best curves are achieved at 654, 234, 111, 166, 219, 215 and 141 epochs, while MSE is almost 10 −9 , 10 −9 → 10 −7 , 10 −10 → 10 −9 , 10 −10 → 10 −9 , 10 −10 → Moreover, the total numerical analysis for all seven scenarios is shown in Table 2 . The performance measures of LMS-BPNN for all cases of scenario lie in the range 10 −9 , 10 −10 → 10 −9 for all corresponding cases of scenario 2. Moreover, for scenarios 3-7 the performances are 10 −10 → 10 −8 , 10 −10 → 10 −9 , 10 −10 → 10 −9 , 10 −10 → 10 −9 and 10 −9 , respectively, for the designed DDFC-NFM. In Table 2 , the precise and accurate performance of LMS-BPNN is certified by numerical illustrations for solving each variant of DDFC-NFM. The MATLAB software is used to analyze the results of LMS-BPNN for investigating the effects of variation of thermal Grashof number (Gr), angle of inclination ( ) and magnetic field parameter (M) of velocity profile f � ( ) with absolute errors as depicted in Fig. 11 . Figure 11a depicts the impact of thermal Grashof number on f � ( ) with an absolute error about 10 −6 → 10 −3 as shown in Fig. 11b , while Fig. 11c represents the impact of angle of inclination on f � ( ) with an absolute error about 10 −7 → 10 −2 as shown in Fig. 11d. Similarly, Fig. 11e depicts the influence of M on f � ( ) with an absolute error about 10 −7 → 10 −3 as shown in Fig. 11f . One may notice that • The velocity enhances swiftly near the surface with the rise in Gr and then gradually decreases and approaches to zero as → ∞ . This is due to the fact that the Gr is the ratio of thermal buoyancy force to viscous hydrodynamic force in the boundary layer. So, the increment in the Gr causes an increment in thermal buoyancy force in the system, which weakens the bonds between fluids and consequently lowers the internal friction pressure and increases the gravity [59] . • The velocity f � ( ) decays with the increase in M due to the fact that a retarding force is exerted by magnetic field on free convection flow. • The velocity f � ( ) declines with the increase in . As angle of inclination increases, the effect of buoyancy force decreases due to thermal diffusion. The MATLAB software is used to analyze the results of LMS-BPNN for investigating the impact of change in Brownian motion parameter (Nb), modified Dufour parameter (Nd) and thermophoresis parameter (Nt) of temperature profile ( ) with absolute errors as illustrated in Fig. 12. Figure 12a depicts the Brownian motion parameter impact on ( ) with an absolute error about 10 −7 → 10 −3 (Fig. 12b) , while the effect of the modified Dufour parameter on ( ) with an absolute error about 10 −7 → 10 −4 is shown in Fig. 12c, d. Similarly, Fig. 12e represents the impact of thermophoresis parameter on ( ) with an absolute error about 10 −7 → 10 −3 as shown in Fig. 12f . One may notice that When the Nt, Nb and Nd increase, the thermal boundary layer expands. In a recent meta-analysis, it was concluded that due to Brownian motion of nanoparticles internal pressure of nanoparticles increases and as a result temperature increases [60] . Another study [61] reveals that the different reactions to a temperature gradient's force are adequate to improve the temperature profile due to increased thermophoresis. The MATLAB software is used to analyze the results of LMS-BPNN for investigating the impact of variation of Dufour-solutal Lewis number (Ld) and modified Dufour parameter (Nd) on ( ) with absolute errors as depicted in Fig. 13 . Figure 13a shows the effect of Ld on ( ) with an absolute error about 10 −7 → 10 −4 as shown in Fig. 13b , while the impact of Nd on ( ) with an absolute error about 10 −7 → 10 −3 is shown in Fig. 13c The MATLAB software is used to analyze the results of LMS-BPNN for investigating the impact of variation of thermophoresis parameter (Nt) and Brownian motion parameter Nb on ( ) with absolute errors as depicted in Fig. 14 . Figure 14a shows the impact of Nb on ( ) with an absolute error about 10 −7 → 10 −3 as shown in Fig. 14b , while Fig. 14c represents the effect of Nt on ( ) with an absolute error about 10 −7 → 10 −3 as shown in Fig. 14d . One may reveal that: • Nanoparticle volume friction profile decreases, that is, boundary-layer thickness decreases with the enhancement of Nb. • The prominent and opposite effect is seen; as the value of Nt increases, the enhancement of nanoparticle volume frictions occurs. This is due to increasing diffusivity as increase in thermophoresis because particles move from hot to cold so there is an enhancement of density of nanoparticles within boundary layer. The technique of AI-based Levenberg-Marquardt scheme with backpropagated neural network (LMS-BPNN) had been used to analyze the problem of double-diffusive flow of nanofluid (DDFC-NFM) due to free convection over an inclined plane when Brownian motion and thermophoresis of tiny particles as the fluid flows through a porous medium are significant. The governing PDEs representing DDFC-NFM are transformed into system of nonlinear ODEs by applying suitable transformation. The reference data set is generated from Lobatto III-A numerical solver by variation of magnetic field parameter (M), thermal Grashof number (Gr), angle of inclination ( ), Brownian motion parameter (Nb), Dufour-solutal Lewis number (Ld), modified Dufour parameter (Nd) and thermophoresis parameter (Nt). The data sets as reference results are executed for training data (80%) validation data (10%) and testing data (10%) by operating LMS-BPNN solver. The proposed and reference outcomes verify the correctness of technique and are further endorsed through numerical and graphical illustration of mean square In future, one may work on different nanofluidic models [62] [63] [64] [65] to solve a problem through backpropagated neural networks. 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convection nanofluidic model Funding No funding is available for the present research. Conflict of interest All the authors of the manuscript declare that there is no conflict of interest.