# Heat and Mass Transfer Gravity Driven Fluid Flow over a Symmetrically-Vertical Plane through Neural Networks

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## Abstract

**:**

_{2}O

_{3}-water nanofluid flow containing electrified Al

_{2}O

_{3}-nanoparticles adjacent to a symmetrically-vertical plane wall. The proposed model becomes a set of nonlinear problems through similarity transformations. The nonlinear problem is solved using the bvp4c method. The results of the proposed model concerning heat and mass transfer with nanoparticle electrification and buoyancy parameters are depicted in the Figures and Tables. It was revealed that the electrification of nanoparticles enhances the heat and mass transfer capabilities of the Al

_{2}O

_{3}water nanoliquid. As a result, the electrification of nanoparticles could be an important mechanism to improve the transmission of heat and mass in the flow of Al

_{2}O

_{3}-water nanofluids. Furthermore, the numerical solutions of the nanofluid model of heat/mass transfer using the deep neural network (DNN) along with the procedure of Bayesian regularization scheme (BRS), DNN-BRS, was carried out. The DNN process is provided by taking eight and ten neurons in the first and second hidden layers along with the log-sigmoid function.

## 1. Introduction

_{2}O

_{3}nanoparticles in water. The enhanced thermal-physical properties of nanofluids have a substantial impact on their efficiency in convection operations. Researchers are now using these types of nanoparticles in a single base fluid to achieve promising heat transfer efficiency in the scientific domain. Thermophoresis and Brownian diffusion are defined by Buongiorno [2] for the perfection of heat transfer employing the thermal conductivity concept. He introduced the correlation between the mass, momentum, and heat transport of nanofluids. It should be noted that the nanoparticle electrification due to the Brownian diffusion of nanosized particles in a region of non-conductive fluid has been ignored. A nanofluid is a colloidal suspended nanoparticle in a base fluid. The botanists Brown [3] and Einstein [4] explained the idea of the incessant (random) motion of nanoparticles moving through molecules of a base fluid and colliding with each other by utilizing Brownian motion. Due to Brownian motion, the particles remain suspended in the base fluid, do not settle due to gravity, and are often electrically charged, Piazza, & Parola [5] and Oakley [6]. Burton [7], deBethune [8], Hunter [9], Shaw [10], Hemsley et al. [11], Gul et al. [12,13], and Mahian et al. [14] have investigated whether the colloidal particles, when immersed into a fluid, are electrically charged in nature, owing to the preferential adsorption of common ions in the base fluid onto the surface of the colloidal particles by Van der Waals forces. Accelerating these charged particles produces both an electric and a magnetic field. Loeb [15] and Soo [16] pointed out that the Brownian distribution of solid suspended particles within a non-conducting fluid region of a particulate system creates the collision of particles and particle-wall interface, thereby ensuing in fixed electrification of the solid particles. A slightly charged particle can affect the dynamics of a particulate system. Kang and Wang [17] have suggested an appliance for augmenting the thermal conductivity of nanofluids based on the cross-coupling among the thermal motion of nanoparticles and the electric field produced nearby every charged particle. Several investigations on the formation of buoyancy-driven nanofluid flow past a vertical flat plate have already been performed (Kuznetsov and Nield [18], Khan and Aziz [19], Gorla and Chamkha [20], Aziz and Khan [21], Ahmad et al. [22], Kayalvizhi et al. [23]). Several studies on the magnetohydrodynamic (MHD) buoyancy-driven nanofluid flow along a vertical plate with several physical characteristics have been examined in recent years. Bouselsa et al. [24] examined the effects of an Al

_{2}O

_{3}nanofluid flow inside a heat exchanger for the improvement of heat transfer. Slimani et al. [25] considered the inspiration of an Al

_{2}O

_{3}nanofluid for energy transfer, using a conical encloser. Anwar et al. [26] scrutinized the impact of heat source/heat sink and radiation on the flux of unstable nanofluids driven by MHD buoyancy over a moving vertical porous plate. Chandel et al. [27] explored the impact of the thermal performance of the nanofluid flow beyond a vertical surface. Arulmozhi et al. [28] looked at the effects of radiation and chemical reaction on the nanofluid flow propelled by MHD on an infinite moving vertical plate. Several researchers have studied the modeling of the buoyancy of magnetohydrodynamic (MHD)-induced nanofluids beyond a vertical plate with various physical aspects [24,25,26,27,28]. In all those previous MHD, the base fluid was considered electrically conducting. However, as suggested by refs. [5,6,7,8,9,10,11,12,13,14,15,16,17], hardly any attention has been given to the nanoparticle electrification appliance in a non-conducting base fluid when modeling the nanofluid flow. Again, research is still ongoing into increasing the thermal performance of nanofluids, which is understood from the latest available literature by Mishra et al. [29]. In polymer technology, the magnetic field strength and heat/mass transfer into the flow over a symmetrically heated surface are important theoretically and practically. This includes geothermal reservoirs, thermal insulation, porous solids drying, enhanced oil recovery, thermal insulation, nuclear reactor cooling, subwoofer catalytic reactors, enhanced oil recovery, and the rest of it. These all benefit from the joint convection device and heat flow on the vertical symmetric surface.

_{2}O

_{3}-water nanofluid flow past a vertical wall, which are limited in the existing literature. It is expected that the nanoparticle electrification mechanism would not only be a new strategy for heat transport enhancement in nanofluids but would also be of great importance in designing numerous industrial applications for nanofluids. This would be useful to determine the quality of finished products with desired properties, which highly rely on the heat transfer rate or cooling rate between the solid surface and the fluid.

- A DNN learning process was used to solve the model problem.
- The DNN training procedure is presented by taking ten and twenty neurons in the first and second hidden layers as well as the log-sigmoid activation function.
- The correctness of the procedure was observed by using the comparison between the obtained and reference results, while the negligible absolute error (AE) performances for solving the fluid model were used to enhance the worth of the scheme.

## 2. Materials and Methods

_{2}O

_{3}-water nanofluid containing electrified alumina nanoparticles past a vertical plane wall was studied. The surface wall was taken vertically along the x-axis. $Tw$ and $Cw$ present the temperature and concentration of the nanofluids at the wall surface. Where ${C}_{\infty}\mathrm{and}{T}_{\infty}$ are the free stream components of C and T. $Tw-{T}_{\infty}>0$ and $Cw-{C}_{\infty}>0$ imply that the temperature and wall concentrations exceed the temperature and ambient concentration. The physical scheme is depicted in Figure 1. According to the above norms, the governing equations of the flow field in a two-dimensional cartesian coordinate system are derived from Equation (4) to Equation (8) and after boundary layer simplification following Ullah et al. [30], and Abbas et al. [31], and Pati et al. [35], are respectively expressed as.

_{2}O

_{3}-water nanofluid thermophysical properties.

## 3. Results

#### Neural Networks

## 4. Conclusions

_{2}O

_{3}-water nanofluid flow containing electrified Al

_{2}O

_{3}-nanoparticles adjacent to a symmetrically vertical plane wall, the impacts of Al

_{2}O

_{3}nanoparticle electrification parameter M and buoyancy ratio $Nr$ on non-dimensional concentration, temperature, and velocity profiles, as well as dimensionless coefficients of heat and mass transfer have been thoroughly examined. The concluding results show that the heat transfer rate increased with the inclusion of the Al

_{2}O

_{3}nanoparticles in the base fluid of water. The significant results are highlighted herewith.

- An increase in M causes the velocity of the nanofluid in the boundary layer region to increase, whereas an increase in Nr causes the velocity to decrease.
- An increase in $Nr$ raises the temperature of the nanofluid, whereas an increase in M lowers the temperature near the plane wall.
- In the vicinity of the plane wall, increasing $Nr$ enhances the concentration of nanoparticles, while increasing M reduces the concentration.
- The dimensionless heat and mass transfer coefficients of the nanofluid rise alongside M, while both reduce with $Nr$.
- The process of DNN is presented in two hidden layers, with 8 and 10 neurons, to solve the model.
- The log-sigmoid function is used as an activation function for both hidden layers to solve the mathematical model.
- The reliability and exactness are observed through the negligible values of the absolute error for each variation of the model.
- The numerical achieved observations of the fluid dynamical system have been computed to reduce the mean square error performances, which were performed in negligible values for the testing and training.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

List of symbols | Greek symbols |

$u$, $v$—Components of velocity $\left(\mathrm{m}{\mathrm{s}}^{-1}\right).$ | ${\mu}_{nf}$—The viscosity of nanofluid $\left(\mathrm{m}\mathrm{P}\mathrm{a}\right).$ |

$Sc$—Schmidt number | $\varphi $—Nanoparticle volume fraction |

${D}_{B}$—Brownian diffusion coefficient $\left({\mathrm{m}}^{2}/\mathrm{s}\right).$ | $\eta $—Dimensionless transform variable |

$({E}_{x},{E}_{y})$—Electric intensity $\left(\mathrm{N}/\mathrm{C}\right).$ | $c{p}_{f}$—Heat capacitance $\left(\mathrm{J}/\mathrm{k}\mathrm{g}\xb7\mathrm{K}\right).$ |

$Nr$—Buoyancy ratio. | ${k}_{nf}$—Thermal conductivity $\left(\mathrm{W}/\mathrm{m}\mathrm{K}\right).$ |

$T$—Temperature of fluid $\left(\mathrm{K}\right).$ | $\alpha $—Stretching shrinking parameter |

$\mathrm{Pr}$—Prandtl number. | $c{p}_{nf}$—Heat capacitance of nanofluid |

${D}_{T}$—Thermophoretic coefficient $\left({\mathrm{m}}^{2}{\mathrm{s}}^{-1}\right).$ | ${\rho}_{f}$—Base fluid density $\left(\mathrm{K}\mathrm{g}{\mathrm{m}}^{-3}\right).$ |

$Nb$—Brownian motion parameter. | ${\mu}_{f}$—Viscosity of the base fluid $\left(\mathrm{m}\mathrm{P}\mathrm{a}\right).$ |

$g$—Gravitational acceleration $\left(\mathrm{m}{\mathrm{s}}^{-2}\right).$ | ${\sigma}_{nf}$ Electrical conductivity |

$Nc$—Nanoparticle ionization parameter | ${\rho}_{nf}$—Hybrid nanofluid density $\left(\mathrm{K}\mathrm{g}{\mathrm{m}}^{-3}\right).$ |

${T}_{w},{T}_{\infty}$—Lower, Upper wall temperature $\left(\mathrm{K}\right).$ | ${\beta}_{T}$—Thermal expansion $\left({\mathrm{K}}^{-1}\right).$ |

${N}_{\mathrm{Re}}$—Momentum transfer number. | $\mathsf{\Theta},f$—Dimensional thermal, velocity fields |

## References

- Choi, U.S. Enhancing Thermal Conductivity of Fluids with Nanoparticles; AS-MEFED; Argonne National Lab. (ANL): Argonne, IL, USA, 1995; Volume 231, pp. 99–105. [Google Scholar]
- Boungiorno, J. Convective transport in nanofluids. ASME J. Heat Transfer
**2006**, 128, 240–250. [Google Scholar] [CrossRef] - Brown, R. XXVII. A brief account of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Philos. Mag.
**1828**, 4, 161–173. [Google Scholar] [CrossRef] [Green Version] - Einstein, A. Investigations on the Theory of the Brownian Movement; Courier Corporation: Chelmsford, MA, USA, 1956. [Google Scholar]
- Piazza, R.; Parola, A. Thermophoresis in colloidal suspensions. J. Phys. Condens. Matter
**2008**, 20, 153102. [Google Scholar] [CrossRef] - Oakley, H.B. The Origin of the Charge on Colloidal Particles. J. Phys. Chem.
**1926**, 30, 902–916. [Google Scholar] [CrossRef] - Burton, E.F. The Physical Properties of Colloidal Solutions; Longmans Green: London, UK, 1921. [Google Scholar]
- deBethune, A.J.; Licht, T.S.; Swendeman, N. The temperature coefficients of electrode potentials. J. Electrochem. Soc.
**1959**, 106, 616. [Google Scholar] [CrossRef] - Hunter, R.J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: Cambridge, MA, USA, 2013. [Google Scholar]
- Shaw, D.J. Introduction to Colloid and Surface Chemistry; Butterworths: Oxford, UK, 1980. [Google Scholar]
- Hemsley, A.R.; Collinson, M.E.; Kovach, W.L.; Vincent, B.; Williams, T. The role of self-assembly in biological systems: Evidence from iridescent colloidal sporopollenin in Selaginella megaspore walls. Philos. Trans. R. Soc. Lond. B Biol. Sci.
**1994**, 345, 163–173. [Google Scholar] - Gul, T.; Khan, M.A.; Noman, W.; Khan, I.; Abdullah Alkanhal, T.; Tlili, I. Fractional order forced convection carbon nanotube nanofluid flow passing over a thin needle. Symmetry
**2019**, 11, 312. [Google Scholar] [CrossRef] [Green Version] - Gul, T.; Anwar, H.; Khan, M.A.; Khan, I.; Kumam, P. Integer and non-integer order study of the GO-W/GO-EG nanofluids flow by means of Marangoni convection. Symmetry
**2019**, 11, 640. [Google Scholar] [CrossRef] [Green Version] - Mahian, O.; Kolsi, L.; Amani, M.; Estellé, P.; Ahmadi, G.; Kleinstreuer, C.; Marshall, J.S.; Siavashi, M.; Taylor, R.A.; Niazmand, H.; et al. Recent advances in modeling and simulation of nanofluid flows-Part I: Fundamentals and theory. Phys. Rep.
**2019**, 790, 1–48. [Google Scholar] [CrossRef] - Loeb, L.B. The generation of static charges by processes involving ionization of gases. In Static Electrification; Springer: Berlin/Heidelberg, Germany, 1958; pp. 201–224. [Google Scholar]
- Soo, S.L. Effect of electrification on the dynamics of a particulate system. Ind. Eng. Chem. Fundam.
**1964**, 3, 64–68. [Google Scholar] [CrossRef] - Kang, Z.; Wang, L. Effect of thermal-electric cross coupling on heat transport in nanofluids. Energies
**2017**, 10, 123. [Google Scholar] [CrossRef] [Green Version] - Kuznetsov, A.V.; Nield, D.A. Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci.
**2010**, 49, 243–247. [Google Scholar] [CrossRef] - Khan, W.A.; Aziz, A. Natural convection flow of a nanofluid over a vertical plate with uniform surface heat flux. Int. J. Therm. Sci.
**2011**, 50, 1207–1214. [Google Scholar] [CrossRef] - Gorla, R.S.R.; Chamkha, A. Natural convective boundary layer flow over a nonisothermal vertical plate embedded in a porous medium saturated with a nanofluid. Nanoscale Microscale Thermophys. Eng.
**2011**, 15, 81–94. [Google Scholar] [CrossRef] [Green Version] - Aziz, A.; Khan, W.A. Natural convective boundary layer flow of a nanofluid past a convectively heated vertical plate. Int. J. Therm. Sci.
**2012**, 52, 83–90. [Google Scholar] [CrossRef] - Ahmad, R.; Mustafa, M.; Turkyilmazoglu, M. Buoyancy effects on nanofluid flow past a convectively heated vertical Riga-plate: A numerical study. Int. J. Heat Mass Transf.
**2017**, 111, 827–835. [Google Scholar] [CrossRef] - Kayalvizhi, J.; Vijaya Kumar, A.G.; Öztop, H.F.; Sene, N.; Abu-Hamdeh, N.H. Heat Transfer Enhancement through Thermodynamical Activity of H2O/Clay Nanofluid Flow over an Infinite Upright Plate with Caputo Fractional-Order Derivative. Energies
**2022**, 15, 6082. [Google Scholar] [CrossRef] - Bouselsal, M.; Mebarek-Oudina, F.; Biswas, N.; Ismail, A.A.I. Heat Transfer Enhancement Using Al2O3-MWCNT Hybrid-Nanofluid inside a Tube/Shell Heat Exchanger with Different Tube Shapes. Micromachines
**2023**, 14, 1072. [Google Scholar] [CrossRef] - Slimani, R.; Aissa, A.; Mebarek-Oudina, F.; Khan, U.; Sahnoun, M.; Chamkha, A.J.; Medebber, M.A. Natural convection analysis flow of Al2O3-Cu/water hybrid nanofluid in a porous conical enclosure subjected to the magnetic field. Eur. Phys. J. Appl. Phys.
**2020**, 92, 10904. [Google Scholar] [CrossRef] - Anwar, T.; Kumam, P.; Shah, Z.; Watthayu, W.; Thounthong, P. Unsteady radiative natural convective MHD nanofluid flow past a porous moving vertical plate with heat source/sink. Molecules
**2020**, 25, 854. [Google Scholar] [CrossRef] [Green Version] - Chandel, S.; Sood, S. Numerical analysis of Williamson-micropolar nanofluid flow through porous rotatory surface with slip boundary conditions. Int. J. Appl. Comput. Math.
**2022**, 8, 134. [Google Scholar] [CrossRef] - Arulmozhi, S.; Sukkiramathi, K.; Santra, S.S.; Edwan, R.; Fernandez-Gamiz, U.; Noeiaghdam, S. Heat and mass transfer analysis of radiative and chemical reactive effects on MHD nanofluid over an infinite moving vertical plate. Results Eng.
**2022**, 14, 100394. [Google Scholar] [CrossRef] - Mishra, S.; Nayak, M.; Misra, A. Thermal conductivity of nanofluids—A comprehensive review. Int. J. Thermofluid Sci. Technol.
**2020**, 7, 070301. [Google Scholar] [CrossRef] - Ullah, Z.; Bilal, M.; Sarris, I.E.; Hussanan, A. MHD and Thermal Slip Effects on Viscous Fluid over Symmetrically Vertical Heated Plate in Porous Medium: Keller Box Analysis. Symmetry
**2022**, 14, 2421. [Google Scholar] [CrossRef] - Abbas, A.; Ashraf, M.; Sarris, I.E.; Ghachem, K.; Labidi, T.; Kolsi, L.; Ahmad, H. Numerical Simulation of the Effects of Reduced Gravity, Radiation and Magnetic Field on Heat Transfer Past a Solid Sphere Using Finite Difference Method. Symmetry
**2023**, 15, 772. [Google Scholar] [CrossRef] - Sabir, Z. Neuron analysis through the swarming procedures for the singular two-point boundary value problems arising in the theory of thermal explosion. Eur. Phys. J. Plus
**2022**, 137, 638. [Google Scholar] [CrossRef] - Shoaib, M.; Naz, S.; Nisar, K.S.; Raja, M.A.Z.; Aslam, S.; Ahmad, I. MHD Casson Nanofluid in Darcy-Forchheimer Porous Medium in the Presence of Heat Source and Arrhenious Activation Energy: Applications of Neural Networks. Int. J. Model. Simul.
**2022**, 43, 1–24. [Google Scholar] [CrossRef] - Raja, M.A.Z.; Shoaib, M.; Khan, Z.; Zuhra, S.; Saleel, C.A.; Nisar, K.S. Supervised neural networks learning algorithm for three dimensional hybrid nanofluid flow with radiative heat and mass fluxes. Ain Shams Eng. J.
**2022**, 13, 101573. [Google Scholar] [CrossRef] - Ramanuja, M.; Kavitha, J.; Sudhakar, A.; Radhika, N. Study of MHD SWCNT-Blood Nanofluid Flow in Presence of Viscous Dissipation and Radiation Effects through Porous Medium. J. Niger. Soc. Phys. Sci.
**2023**, 5, 1054. [Google Scholar] [CrossRef] - Mahrukh, M.; Kumar, A.; Gu, S.; Kamnis, S.; Gozali, E. Modeling the effects of concentration of solid nanoparticles in liquid feedstock injection on high-velocity suspension flame spray process. Ind. Eng. Chem. Res.
**2016**, 55, 2556–2573. [Google Scholar] [CrossRef] [Green Version] - Shoaib, M.; Kainat, R.; Ijaz Khan, M.; Prasanna Kumara, B.C.; Naveen Kumar, R.; Zahoor Raja, M.A. Darcy-Forchheimer entropy-based hybrid nanofluid flow over a stretchable surface: Intelligent computing approach. Waves Random Complex Media
**2022**, 1–24. [Google Scholar] [CrossRef] - Raja, M.A.Z.; Shoaib, M.; Tabassum, R.; Khan, N.M.; Kehili, S.; Bafakeeh, O.T. Stochastic numerical computing for entropy optimized of Darcy-Forchheimer nanofluid flow: Levenberg Marquardt Algorithm. Chem. Phys. Lett.
**2022**, 807, 140070. [Google Scholar] [CrossRef] - Shoaib, M.; Tabassum, R.; Nisar, K.S.; Raja, M.A.Z.; Fatima, N.; Al-Harbi, N.; Abdel-Aty, A.H. A design of neuro-computational approach for double-diffusive natural convection nanofluid flow. Heliyon
**2022**, 9, e14303. [Google Scholar] [CrossRef] [PubMed] - Shoaib, M.; Zubair, G.; Nisar, K.S.; Raja, M.A.Z.; Khan, M.I.; Gowda, R.P.; Prasannakumara, B.C. Ohmic heating effects and entropy generation for nanofluidic system of Ree-Eyring fluid: Intelligent computing paradigm. Int. Commun. Heat Mass Transf.
**2021**, 129, 105683. [Google Scholar] [CrossRef] - Ali, I.; Gul, T.; Khan, A. Unsteady Hydromagnetic Flow over an Inclined Rotating Disk through Neural Networking Approach. Mathematics
**2023**, 11, 1893. [Google Scholar] [CrossRef]

Property | Solid (Alumina) | Fluid (Water) |
---|---|---|

${\mathrm{c}}_{\mathrm{p}}\left(\mathrm{J}/\mathrm{kgk}\right)$ | 765 | 4179 |

$\mathrm{k}\left(\mathrm{W}/\mathrm{mk}\right)$ | 40 | 0.613 |

$\mathsf{\rho}\left({\mathrm{kg}/\mathrm{m}}^{3}\right)$ | 3970 | 997.1 |

Parameters | ${\mathit{\theta}}^{\prime}\left(0\right)$ | ${\mathit{S}}^{\prime}\left(0\right)$ | |||
---|---|---|---|---|---|

$Nt$ | $Nc$ | $Nb$ | $Sc$ | ||

0.1 | 0.01 | 0.1 | 0.1 | 0.38356 | 0.370880 |

0.3 | 0.418340 | 0.3781709 | |||

0.5 | 0.434021 | 0.381703 | |||

0.02 | 0.43194 | 0.370880 | |||

0.03 | 0.478149 | 0.370880 | |||

0.2 | 0.3973194 | 0.3642471 | |||

0.3 | 0.408781 | 0.3513482 | |||

0.2 | 0.38356 | 0.421032 | |||

0.3 | 0.38356 | 0.4421872 |

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**MDPI and ACS Style**

Awwad, F.A.; Ismail, E.A.A.; Gul, T.
Heat and Mass Transfer Gravity Driven Fluid Flow over a Symmetrically-Vertical Plane through Neural Networks. *Symmetry* **2023**, *15*, 1288.
https://doi.org/10.3390/sym15061288

**AMA Style**

Awwad FA, Ismail EAA, Gul T.
Heat and Mass Transfer Gravity Driven Fluid Flow over a Symmetrically-Vertical Plane through Neural Networks. *Symmetry*. 2023; 15(6):1288.
https://doi.org/10.3390/sym15061288

**Chicago/Turabian Style**

Awwad, Fuad A., Emad A. A. Ismail, and Taza Gul.
2023. "Heat and Mass Transfer Gravity Driven Fluid Flow over a Symmetrically-Vertical Plane through Neural Networks" *Symmetry* 15, no. 6: 1288.
https://doi.org/10.3390/sym15061288