Heliyon 9 (2023) e14303
Available online 6 March 2023
2405-8440/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Research article
A design of neuro-computational approach for double-diffusive natural convection nanofluid flow
Muhammad Shoaib
a,b, Rafia Tabassum
a, Kottakkaran Sooppy Nisar
c,d,*, Muhammad Asif Zahoor Raja
e,**, Nahid Fatima
f, Nuha Al-Harbi
g, Abdel-Haleem Abdel-Aty
h,iaDepartment of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
bYuan Ze University, AI Center, Taoyuan, 320, Taiwan
cDepartment of Mathematics, College of Sciences and Humanities, Prince Sattam bin Abdulaziz University, Al Kharj, 16278, Saudi Arabia
dSchool of Technology, Woxsen University, Hyderabad, 502345, Telangana State, India
eFuture Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin, 64002, Taiwan, ROC
fDepartment of Mathematics & Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
gDepartment of Physics, Faculty of Applied Sciences, Umm Al-Qura University, Makkah, Saudi Arabia
hDepartment of Physics, College of Sciences, University of Bisha, P.O. Box 344, Bisha, 61922, Saudi Arabia
iPhysics Department, Faculty of Science, Al-Azhar University, Assiut, 71524, Egypt
A R T I C L E I N F O Keywords:
Intelligent computing Artificial neural networks Levenberg-Marquardt method Lobatto III-A technique Suction/injection Brownian motion Double-diffusive convection Nanofluid
Thermophoresis effect
A B S T R A C T
The artificial intelligence based neural networking with Back Propagated Levenberg-Marquardt method (NN-BPLMM) is developed to explore the modeling of double-diffusive free convection nanofluid flow considering suction/injection, Brownian motion and thermophoresis effects past an inclined permeable sheet implanted in a porous medium. By applying suitable transformations, the PDEs presenting the proposed problem are transformed into ordinary ones. A reference dataset of NN-BPLMM is fabricated for multiple influential variants of the model representing scenarios by applying Lobatto III-A numerical technique. The reference data is trained through testing, training and validation operations to optimize and compare the approximated solution with desired (standard) results. The reliability, steadiness, capability and robustness of NN- BPLMM is authenticated through MSE based fitness curves, error through histograms, regres- sion illustrations and absolute errors. The investigations suggest that the temperature enhances with the upsurge in thermophoresis impact during suction and decays for injection, whereas increasing Brownian effect decreases the temperature in the presence of wall suction and reverse behavior is seen for injection. The best measures of performance in form of mean square errors are attained as 7.1058×10−10,2.9262×10−10,1.1652×10−08,1.5657×10−10 and 5.5652×10−10 against 969, 824, 467, 277 and 650 iterations. The comparative study signifies the authenticity of proposed solver with the absolute errors about 10−7 to 10−3 for all influential parameters results.
* Corresponding author. Department of Mathematics, College of Sciences and Humanities, Prince Sattam bin Abdulaziz University, Al Kharj, 16278, Saudi Arabia
** Corresponding author.
E-mail addresses: [email protected] (M. Shoaib), [email protected] (R. Tabassum), [email protected], [email protected].
sa (K.S. Nisar), [email protected] (M.A.Z. Raja), [email protected] (N. Fatima), [email protected] (N. Al-Harbi), [email protected].
sa (A.-H. Abdel-Aty).
Contents lists available at ScienceDirect
Heliyon
journal homepage: www.cell.com/heliyon
https://doi.org/10.1016/j.heliyon.2023.e14303
Received 13 September 2022; Received in revised form 24 February 2023; Accepted 28 February 2023
Heliyon 9 (2023) e14303 1. Introduction
In recent years, the study of nanofluid has gained the great attention of scientists in engineering and industrial fields. The appli- cations of nanofluid include microelectronics, fuel cells, nanofluid detergents, geothermal extraction, nano-drugs delivery and numerous others. Nanofluid is composed of nanoparticles and base liquids e.g. water, kerosene and ethylene glycol mixture. The nanoparticles, consist of Cu, Al2O3, TiO2, SiC and graphite, supports base fluid to boost the thermal conductivity. Originally, Choi [1]
encompassed the nanotechnology in order to improve the heat transfer process of fluid. Later on, many researchers investigated nanofluid technology over inclined surfaces. Kausar et al. [2] deliberated MHD Casson nanofluid with radiation and chemical reaction effects past an inclined porous sheet. Soret-Dufour effect along with thermophoresis on non-Newtonian nanofluid stream past an inclined plane is investigated by Idowu and Falodun [3]. Goud et al. [4] presented the joule heat and thermal radiation, chemical reaction effects on MHD Casson nanoliquid flow. Khademi et al. [5] numerically scrutinized conjugate-mixed convection stream of nanofluid with transverse magnetic field past a permeable inclined flat surface. The analysis of convective thin film nanofluid flow considering magnetic impact over an inclined sheet is presented by Saeed et al. [6].
Many studies have been conducted on nanofluid flow considering various surfaces implanted in porous medium [7]. Gandhi and Sharma [8] investigated the heat sink/source and joule heating influences on mixed convective stream considering stretchable vertical sheet implanted in porous medium. Arifuzzaman et al. [9] conferred the chemically reactive MHD fluid flow considering the impacts of radiation and heat absorption past an oscillating porous sheet. Khan et al. [10] inspected the influence of irregular heat source/sink on nanofluid subjected to radiative mixed convection stream past porous cylinder engrossed in porous medium. The linear stability examination of hybrid nanofluid flow in porous medium with low suction/injection and magnetic effects is conferred by Kapen et al.
[11]. Nisar et al. [12] examined the heat transfer effect on MHD laminar hybrid nanofluid past a stretchable surface subjected to porous media. The influence of a magnetic field on free convection hybrid nanoliquid flow embedded in an inclined porous media is delib- erated by Izadi et al. [13]. Solar radiation impact in Darcy-Forchheimer porous medium for hybrid nanofluid stream over a flat surface is presented by Alzahrani et al. [14].
For many years, double diffusion (DD) phenomenon has gained undeniable importance in different scientific disciplines such as astrophysics, geosciences, oceanography and chemical processes (Beghein et al. [15]). The combined effect of temperature gradients with concentration gradients creates double diffusive convection phenomenon in porous medium. Abundant investigations on DD have been conducted by researchers [16]. Pal et al. [17] conferred the Soret-Dufour DD convection considering heat and mass transmission impact on micropolar liquid in a porous medium. Alvarez et al. [18] inspected a DD free convective problem by ´ considering viscous nanofluid in a porous medium. Aly et al. [19] investigated the DD free convection phenomenon on nanofluid stream in porous cavity with passive control nanoparticles. The numerical analysis in asymmetric microchannel of double diffusion micropolor nanofluid flow by employing thermal radiation and magnetic field effects is deliberated by Tripathi et al. [20]. Parkash et al. [21] presented heat transfer effect with DD convection flow phenomenon on EMHD ionic nanofluid. Nandeppanavar et al. [22]
examined the non-linear thermal radiation impact on DD free convective Casson nanoliquid flow over a vertically moveable sheet.
Parvin et al. [23] exemplified the DD convection effect on Maxwell nanofluid subjected to an inclined shrinking surface.
There are many studies related to the effects of suction/injection on convective boundary layer flow. Raza et al. [24] explored the dual solution of convective boundary layer micropolar liquid flow with suction and injection impact. The influence of suction/injection on boundary layer viscous nanofluid stream past moving flat sheet is conferred by Ferdows et al. [25]. The suction and injection impacts on convective boundary layer flow past a vertical porous sheet are explored by Jha and Samaila [26]. The suction and injection effects on Ag-water nanoliquid flow in a porous flat medium by taking viscous-Ohmic dissipation are conferred by Pandey et al. [27].
Raslen et al. [28] examined the impacts of suction/injection on KKL-model of MHD stagnation point convective nanofluid stream with stretching surface in the presence of nonlinear thermal radiation. Ramudu et al. [29] calculated the radiation influence on MHD shear thickness Casson nanoliquid over vertical stretchable sheet considering suction and injection. Upreti et al. [30] examined the influence of suction/injection on MHD natural convective stream of Ag-kerosene oil fluid in the existence of thermophoresis effect.
The majority of above-mentioned literature consists of traditional numerical/semi numerical techniques such as Homotopy analysis method [31], Keller box method [32], RKF45 [33] and spectral relaxation method [34] and shooting method [35] etc., but the AI based numerical computing frameworks due to their worthiness and effectiveness are significant. Recently, some authors imple- mented these computing intelligence based paradigms in many disciplines such as particle percolation problem under vibration [36], vibrating screen model [37], MXene-ionic liquids model [38], biogas powered dual-fuel model [39], heat transfer in renewable energy system [40], COVID-19l model [41], Thomas-fermi system [42], nonlinear reactive transport model [43], predictor prey model [44], Lassa fever transmission model [45], double diffusion convection model [46], irreversibility model [47] and CNTs fluid [48], Wil- liamson nanofluid model [49]. These recent and valuable researches motivated authors to exploit an effective and reliable intelligent computing methodology of neural networking with back propagated Levenberg-Marquardt method (NN-BPLMM) for solving the model of double-diffusive free convection nanofluid flow (DDNFM) with suction/injection.
The objectives of the innovative work performed in this study based on artificial NN-BPLMM are briefly highlighted as follows:
•To explore model of double-diffusive free convection nanofluid flow (DDNFM) in the presence of suction/injection, thermophoresis and Brownian motion effects past an inclined permeable plate implanted in porous medium the AI based NN-BPLMM is introduced.
•By utilizing the appropriate transformation, the PDEs exhibiting DDNFM are changed into ordinary ones.
•By varying parameters for all scenarios of designed NN-BPLMM, the reference data set is created from Lobatto III-A numerical solver.
M. Shoaib et al.
•Utilizing training, testing and validation operations, the said data is trained through constructing an MSE based fitness merit function.
•The merit function is optimized to compute and compare the approximated solution with desired (standard) results.
•To authenticate the reliability, steadiness, capability and robustness of NN-BPLMM, the MSE based fitness curves, error histograms, regression illustrations and absolute errors are analyzed.
•The behaviors and effects of multiple influential variants on temperature, solutal concentration and nanoparticle volume friction profiles are also investigated.
2. Mathematical modeling
Consider steady, laminar free convection flow of nanoliquid past an inclined permeable plate embedded in a porous medium with suction/injection. The thermophoresis and Brownian motion phenomena are also deliberated. The y-axis is normal to the plate and x- axis is aligned vertically upward making acute angle α (shown in the geometry of the problem Fig. 1). The further explanation regarding mathematical modeling is given in [50,51].
After solving the governing PDEs system (given in Sarkar and Kundu et al. [51]) by similarity transformation following ODEs with boundary conditions acquired:
d3f dη3+
3 4fd2f
dη2− 1 2
(df dη
)2
− Kdf
dη+φGmcosα− γGncosα+θGrcosα=0, (1)
Fig. 1. Flow geometry.
Fig. 2. The neural network for DDNFM.
Heliyon 9 (2023) e14303
d2θ dη2+
d2φ dη2Nd+
(dθ dη
)2 Nt+3
4Prfdθ dη+
dγ dη
dθ
dηNb=0, (2)
d2φ dη2+
d2θ dη2Ld+
3 4fdφ
dηLe=0, (3)
d2γ dη2+
d2θ dη2
Nt Nb+3
4fdγ
dηLn=0, (4)
f(0) =fw,f ’(0) =0,θ(0) =1,φ(0) =1,Ntθ’(0) +Nbγ’(0) =0,f′(η)→ 0,γ(η)→ 0,φ(η)→ 0,θ(η)→ 0,as η→ ∞. (5) The description of the parameters used in Eqs. (1)–(5) are expressed in nomenclature.
3. Solution methodology
This section states the essential descriptions regarding the methodology of DDNFM by employing the designed AI based neural networking with back propagated Levenberg-Marquardt method (NN-BPLMM).
The proposed methodology comprised two phases. Firstly, the reference dataset of NN-BPLMM is produced by Lobatto-IIIA nu- Fig. 3. Process flow architecture of Proposed NN-BPLMM for DDNFM.
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Table 1
Scenarios and cases for all influential parameters of DDNFM.
Scenario Cases Parameters
fw Nb Nt Ld Le Ln
1 1 1.4 0.04 0.3 1.0 0.5 1.0
2 1.4 0.09 0.3 1.0 0.5 1.0
3 1.4 0.2 0.3 1.0 0.5 1.0
4 −1.2 0.04 0.3 1.0 0.5 1.0
5 −1.2 0.09 0.3 1.0 0.5 1.0
6 −1.2 0.2 0.3 1.0 0.5 1.0
2 1 1.4 0.2 0.1 1.0 0.5 1.0
2 1.4 0.2 0.4 1.0 0.5 1.0
3 1.4 0.2 0.7 1.0 0.5 1.0
4 −1.2 0.2 0.1 1.0 0.5 1.0
5 −1.2 0.2 0.4 1.0 0.5 1.0
6 −1.2 0.2 0.7 1.0 0.5 1.0
3 1 1.4 0.2 0.3 0.2 0.5 1.0
2 1.4 0.2 0.3 0.4 0.5 1.0
3 1.4 0.2 0.3 0.6 0.5 1.0
4 −1.2 0.2 0.3 0.2 0.5 1.0
5 −1.2 0.2 0.3 0.4 0.5 1.0
6 −1.2 0.2 0.3 0.6 0.5 1.0
4 1 1.4 0.2 0.3 1.0 1.5 1.0
2 1.4 0.2 0.3 1.0 2.3 1.0
3 1.4 0.2 0.3 1.0 3.0 1.0
4 −1.2 0.2 0.3 1.0 1.5 1.0
5 −1.2 0.2 0.3 1.0 2.3 1.0
6 −1.2 0.2 0.3 1.0 3.0 1.0
5 1 1.4 0.2 0.3 1.0 0.5 0.2
2 1.4 0.2 0.3 1.0 0.5 0.4
3 1.4 0.2 0.3 1.0 0.5 0.6
4 −1.2 0.2 0.3 1.0 0.5 0.2
5 −1.2 0.2 0.3 1.0 0.5 0.4
6 −1.2 0.2 0.3 1.0 0.5 0.6
Table 2
Evaluation of results of NN-BPLMM for designed DDNFM.
Scenario Case MSE Time Gradient Value Mu Parameter Epochs
Training Validation Testing
1 1 7.90E-09 9.18E-09 7.57E-09 3m27s 9.15E-07 1E-09 1000
2 2.15E-10 2.47E-10 2.02E-10 1m32s 9.86E-08 1E-10 842
3 8.28E-10 7.11E-10 1.20E-09 1m4s 9.98E-08 1E-09 969
4 8.05E-09 1.20E-08 1.21E-08 1m41s 7.93E-07 1E-08 1000
5 3.92E-09 4.20E-09 4.55E-09 1m24s 2.17E-06 1E-09 1000
6 2.14E-09 2.10E-09 3.39E-09 1m6s 9.98E-08 1E-09 747
2 1 5.02E-09 6.46E-09 6.76E-09 28s 9.94E-08 1E-08 349
2 1.01E-09 1.48E-09 1.25E-09 32s 9.95E-08 1E-09 908
3 3.14E-10 2.93E-10 2.58E-10 1m8s 9.91E-08 1E-10 824
4 1.83E-09 8.03E-09 2.97E-09 24s 9.96E-08 1E-09 436
5 2.08E-09 1.77E-09 2.76E-09 1m5s 6.24E-07 1E-09 1000
6 3.61E-09 3.70E-09 2.52E-09 1m28s 3.17E-06 1E-09 1000
3 1 6.92E-09 5.91E-09 5.55E-09 31s 9.98E-08 1E-08 327
2 6.34E-09 8.26E-09 5.09E-09 39s 9.95E-08 1E-08 466
3 7.63E-09 1.17E-08 1.77E-08 38s 9.96E-08 1E-08 467
4 1.27E-09 1.44E-09 1.20E-09 1m39s 3.52E-07 1E-09 1000
5 1.88E-09 2.35E-09 2.19E-09 1m15s 9.99E-08 1E-09 784
6 1.96E-09 1.79E-09 2.15E-09 1m26s 9.99E-08 1E-09 798
4 1 1.35E-09 1.63E-09 1.70E-09 27s 9.95E-08 1E-09 321
2 2.95E-09 2.98E-09 3.02E-09 41s 9.84E-08 1E-09 199
3 1.64E-09 1.57E-09 1.46E-09 55s 9.85E-08 1E-09 277
4 1.47E-09 1.58E-09 1.11E-09 50s 9.40E-08 1E-09 438
5 2.54E-09 2.23E-09 2.06E-09 41s 3.57E-07 1E-09 1000
6 5.98E-09 6.84E-09 5.07E-09 1m36s 9.90E-08 1E-10 882
5 1 6.55E-09 6.96E-09 6.58E-09 10s 9.93E-08 1E-08 179
2 5.85E-09 5.67E-09 5.10E-09 10s 5.85E-09 1E-08 194
3 5.62E-10 5.57E-10 5.97E-10 34s 9.83E-08 1E-09 650
4 8.70E-10 8.60E-10 7.11E-10 30s 9.97E-08 1E-09 584
5 8.97E-10 8.58E-10 7.25E-10 27s 9.99E-08 1E-09 527
6 1.44E-09 1.79E-09 1.66E-09 47s 9.99E-08 1E-09 617
Heliyon 9 (2023) e14303
merical solver in MATLAB software using ‘bvp4c’ package taking input between 0 and 8 interval and varying Nb,Nt,Ld,Le and Ln.
Later on, ‘nftool’ operation is performed to train data by dividing it into different proportions as validation data (10%), training data (80%) and testing data (10%). The trained data is used to find the approximated solution through MSE based fitness merit function.
The designed NN-BPLMM in Fig. 2 is constructed for a single neuron and the flow chart of complete structure of the study is depicted in Fig. 3. The reliability, steadiness, capability and fitness of NN-BPLMM are authenticated through MSE based fitness curves, error through histograms, transition statistics (Mu and gradients) and regression illustrations. The numerical Lobatto-IIIA solution and NN-BPLMM outcomes are compared and the corresponding absolute errors are calculated for θ(η),φ(η)and γ(η).
4. Result interpretations and discussion
The designed NN-BPLMM is executed to solve all scenarios each with six cases, the first three cases for suction (fw=1.4)and the last three cases for injection (fw= − 1.2), by the variation of Nb,Nt,Ld,Le and Ln for DDNFM. Table 1 depicts the variation of parameters
Fig. 4. Mean square error illustrations of NN-BPLMM for case 3 of each scenario of DDNFM.
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setting of all the scenarios and cases of DDNFM which the remainder of the work utilizes. The MSE convergence for testing, training and validation with epochs, gradient, time and back-propagated activator i-e., Mu are illustrated in Table 2.
The mean square errors for case-3 of all influential parameters (scenarios) of DDNFM are described in Fig. 4(a–e) in terms of testing, training and validation. The finest performance is accomplished at 969, 824, 467, 277 and 650 epochs while MSE is 7.1058×10−10, 2.9262×10−10,1.1652×10−08,1.5657×10−10 and 5.5652×10−10, respectively. Since MSE show the difference between target solutions with desired results, so smallest MSE gives efficient results.
The statistical values for NN-BPLMM are shown in Fig. 5(a–e). The gradient values are [9.98×10−8,9.91×10−8,9.96× 10−8, 9.85×10−8,9.83×10−8] and Mu are [10−09,10−10,10−08,10−09,10−09] for case 3 of all scenarios. Moreover, the performance attitude of NN-BPLMM is around 10−10 to 10−9 of scenario 1 for each case, 10−10 to 10−9 of scenario 2 for each case, 10−9 of scenario 3 for each case, 10−10 of scenario 4 for each case, and 10−10 to 10−9 of scenario 5 for each case of designed DDNFM in Table 2. The efficient convergence rate of NN-BPLMM for each case of DDNFM has been verified by the results. With the smallest values of gradient and Mu, the better performance of convergence is achieved.
In Fig. 6 (a–e), the fitness overlapping graphs of target and output consequences of NN-BPLMM are illustrated for training, testing and validation as well as the error through histograms for each instance. The overlapping of fitness curves demonstrates the precision
Fig. 5. Statistical analysis of NN-BPLMM for case 3 of each scenario of DDNFM.
Heliyon 9 (2023) e14303
of the solution. There are 20 bins in a histogram with zero errors which divide negative and positive errors equally. The zero error of histograms is found around 5.2E− 06,6.86E− 06,4.21E− 05,5.34E− 05 and 1.08E− 06 for case 3 of each scenario, respectively.
These graphs reveal the consistency of the proposed solver.
The information about regression illustrations is explained in Fig. 7(a–c) for three scenarios of proposed model. The linearity between desired and achieved outputs is seen in regression graphs. The graphs consist of all data, trained data, validation data and testing data. Regression indicates the data fitting with curve/line, where regression R=1 specifies the best fitting [52].
4.1. Impact on temperature profile θ(η)
The comparative illustrations of upshot of Nb and Nt for temperature profile θ(η)with absolute error (AE) are described in Fig. 8.
Fig. 6. Histogram of Error and Fitness interpretation for NN-BPLMM for case 3 of each scenario of DDNFM.
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Fig. 8(a) shows the impact of Brownian motion parameter for suction and injection of θ(η)with AE about 10−8 to 10−4 as presented in Fig. 8(b) while Fig. 8(c) represents the impact of thermophoresis parameter for suction and injection of θ(η)with an absolute error about 10−7 to 10−4 as shown in Fig. 8(d).
One may reveals that:
•θ(η)decays in boundary layer area with an upsurge in Nb for suction, while for injection, temperature enhances with the enhancement of Nb across boundary layer region. This is because Nb accelerates the Brownian movement of nanoparticles, which raises their kinetic energy at both the molecular and nanoparticle levels and raises the temperature of nanofluids.
•Due to a rise in Nt, an increase in temperature is seen within the boundary layer region for suction, reaching a maximum at the plate surface. The phenomenon shows that the thermophoretic force causes a swift flow away from the surface as a result of the tem- perature gradient. Consequently, heated fluid began to flow away from the surface, raising the temperature. The behavior is reversed in case of injection.
4.2. Impact on solutal concentration profile φ(η)
The comparative illustrations of the upshot of Ld and Le for φ(η)with absolute error (AE) are described in Fig. 9. Fig. 9(a) shows the effect of Dufour-Solutal Lewis number for φ(η)with AE around 10−7 to 10−4 as revealed in Fig. 9(b). Fig. 9(c) signifies the behavior of Le for φ(η)with AE around 10−8 to 10−3 as shown in Fig. 9(d).
One may reveals that:
•The solutal concentration rises with the escalation in Ld for suction and injection. Physically, the rising values of Ld lower the drift in boundary layer caused by porous medium.
•The upsurge in Le cause decrease in solutal concentration profile for both suction i.e fw>0 and injection i.e fw<0.
Fig. 6. (continued).
Heliyon 9 (2023) e14303
Fig. 7. NN-BPLMM based regression graphs for case 3 of each scenario of DDNFM.
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Fig. 8. Comparative outcomes of NN-BPLMM with numerical results of Lobatto III-A for θ(η)of DDNFM.
Heliyon 9 (2023) e14303
4.3. Impact on nanoparticle volume fraction γ(η)
The comparative illustrations of the upshot of Nt and Ln for γ(η)with absolute error (AE) are labeled in Fig. 10. Fig. 10(a) shows the results of Nt for γ(η)with AE about 10−7 to 10−3 as revealed in Fig. 10(b). Fig. 10(c) embodies the influence of nano-liquid Lewis Number on γ(η)with AE about 10−7 to 10−3 as shown in Fig. 10(d).
One may reveals that:
•During suction, an upswing in Nt rises the thermophoretic diffusion which enhances nanoparticle volume friction in the boundary layer area while in the presence of injection the prominent but reverse effect is noticed.
•As the value of nano-liquid Lewis Number enhances there is a decrease in nanoparticle volume frictions occurs in the case of wall suction this is due to poorer species diffusivity as increment in Ln whereas γ(η)decays for 0≤η<2 (approx.) and afterward in- creases slightly and reaches to zero for injection.
5. Conclusion
The analysis of DDNFM past an inclined permeable plate implanted in a porous medium considering suction/injection effect is presented through a computational based solver i.e., NN-BPLMM by varying parameters. The PDEs presenting DDNFM are transformed into ordinary ones. The dataset of NN-BPLMM is fabricated by applying Lobatto III-A numerical technique. The reference data is processed in testing (10% of data), training (80% of data) and validation (10% of data) operations to compute and compare the approximated solution with desired (standard) results. The following conclusions are drawn from the study.
•The temperature increases with an increase in Nt for suction and decreases for injection while for Nb there is a decrease in tem- perature in the presence of wall suction and reverse behavior is seen for injection.
•The solutal concentration profile upsurges with the escalation in the value of Ld for both suction and injection while the reverse trend is noticed for Le.
•During suction, nanoparticle volume friction profile rises with the upsurge in Nb and decreases in Ln while in the existence of injection the prominent but reverse impact is noticed.
Fig. 9.Comparative outcomes of NN-BPLMM with numerical results of Lobatto III-A for φ(η)of DDNFM.
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•The results of absolute errors signify the authenticity of NN-BPLMM with accuracy of order up to 3-7 decimal places reliable for each scenario of DDNFM.
In future, one may work via AI based neural networking with back propagated Levenberg-Marquardt method to solve problems on different nano-fluidic models such as Ree-Eyring fluid [53], micropolar nanofluid [54] and mixed convection nanofluid [55].
Moreover, one may apply unsupervised algorithms on the proposed problems such as used in Entropy generated system [56], MHD nanofluid model [57] and thermal radiative system [58].
Declarations
Author contribution statement
Muhammad Shoaib, Rafia Tabassum, Kottakkaran Sooppy Nisar: Conceived and designed the experiments; Performed the ex- periments; Wrote the paper.
Muhammad Asif Zahoor Raja: Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.
Nahid Fatima; Nuha Al-Harbi; Abdel-Haleem Abdel-Aty: Analyzed and interpreted the data; Wrote the paper.
Funding statement
This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).
Data availability statement
No data was used for the research described in the article.
Fig. 10.Comparative outcomes of NN-BPLMM with numerical results of Lobatto III-A for γ(η)of DDNFM.
Heliyon 9 (2023) e14303 Declaration of interest’s statement
The authors declare no competing interests.
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