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An evolutionary-based neural network approach to investigate heat and mass transportation by using non-Fourier double-diffusion theories for Prandtl nanofluid under Hall and ion slip effects

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Abstract

Achieving precise predictions and classifications with artificial neural networks (ANNs) while minimizing the consumption of computational resources and time continues to be a substantial objective in the realm of scientific inquiry. The primary objective of this research is to create ANNs that achieve a harmonious equilibrium between precision and computational speed, thereby enabling their implementation in a wide array of contexts. This study aims to examine the characteristics of Prandtl nanofluids using ANN-HCS-PNF, an innovative computational framework that integrates artificial neural networks and a hybridized cuckoo search method. The aforementioned framework is utilized in conjunction with the Cattaneo–Christov double heat flow model, with Hall and ion slip effects also taken into account. The central argument of this manuscript pertains to the examination of concentration, velocity, and temperature profiles in the flow of Prandtl nanofluid using the Cattaneo–Christov double heat flux model. This investigation focuses specifically on the consequences of Hall and ion slip effects. In order to explore the complexities of flow dynamics, scientists utilize an unsupervised ANN approach. The process entails converting the partial differential equations that regulate the flow of Prandtl nanofluids into a set of ordinary differential equations. Following this, the transformed equations are utilized in the construction of a reference dataset that supports the evolutionary methodology. The complex nanofluid flow velocities demonstrate a clear and direct relationship with non-dimensional parameters, such as the flexible number and Prandtl fluid parameter, in addition to the Hall and ion slip parameters. Temperature profiles demonstrate a strong correlation with the Brownian parameter and ion slip parameter as they increase, while the Prandtl number and thermal parameter exhibit an inverse correlation. In a similar fashion, the concentration profile exhibits an inverse correlation with the Hall parameter, Schmidt number, and concentration relaxation parameters, but a direct correlation with the Hartmann number. This study enhances the advancement of computationally efficient ANN models utilized in the analysis of complex fluid dynamics, thereby creating novel opportunities for their implementation across diverse domains.

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Data Availability Statement

No data associated in the manuscript.

Abbreviations

\(\tilde{u},\tilde{v},\tilde{w}\,\left[ {{\text{m}}/{\text{s}}} \right]\) :

Velocity components

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{z} \,\left[ {\text{m}} \right]\) :

Cartesian coordinates

\(\tilde{T},\tilde{T}_{\infty } ,\tilde{T}_{w}\) :

Fluid, ambient, and surface fluid temperature, respectively

\(\tilde{C},\tilde{C}_{\infty } ,\tilde{C}_{w}\) :

Fluid, ambient, and surface fluid concentration, respectively

\(\begin{gathered} k\left[ {{\text{W}}/\left( {{\text{m}} \cdot {\text{K}}} \right)} \right],\rho^{*} \left[ {{\text{kg/m}}^{3} } \right],\hfill \\ C_{p} \left[ {J/\left( {{\text{kg}} \cdot {\text{K}}} \right)} \right], q* = - k\nabla \tilde{T}\left[ {{\text{W}}/{\text{m}}^{2} } \right],J^{*} = \rho u. \end{gathered}\) :

Thermal conductivity, density, specific heat, and heat, mass flux and magnetic field, respectively

\(K^{ * * * } ,\,D^{ * * * }\) :

Thermal conductivity and molecular diffusivity, respectively

\(\nu ,\,\rho ,\,\mu\) :

Kinematic viscosity, fluid density, and dynamics viscosity, respectively

\(\alpha_{i} ,\,\alpha_{h} ,\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\sigma }\) :

Ion slip and Hall and electrical conductivity parameters, respectively

\(A,C,B = \left( {0,0,B_{0} } \right)\left[ {{\text{tesla }}\left( T \right)} \right]\) :

Material constants and magnetic field, respectively

\(\alpha * = K/\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\rho } \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{c}_{p} } \right)_{f} ,\,K\left[ {{\text{m}}^{2} /{\text{s}}} \right]\) :

Thermal diffusivity and thermal conductivity, respectively

\(\left( {\rho C} \right)_{p} ,\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{D}_{\gamma } ,\,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{D}_{T}\) :

Heat capacitance, Brownian, and thermophoretic diffusion coefficients, respectively

\(\beta_{1} = \frac{1}{\mu Ac},\,\beta_{2} = \frac{{bU_{w} }}{{2c^{2} v}}\) :

Prandtl fluid parameter, flexible number, respectively

\(({\text{Ha}})_{2} = \frac{{\sigma B_{0}^{2} }}{{a\rho_{f} }},\,\gamma = \frac{{h_{f} }}{k}\sqrt{\frac{v}{a}}\) :

Hartmann parameter, Biot number, respectively

\(\Pr = \frac{v}{{a^{ * } }},\,({\text{Ha}})_{2} = \frac{{\sigma B_{0}^{2} }}{{a\rho_{f} }},\) :

Prandtl number, Hartmann parameter, respectively

\({\text{Nb}} = \frac{{(\rho c)_{p} D_{B} C_{\infty } }}{{(\rho c)_{f} \nu }},\,{\text{Sc}} = \frac{v}{{D_{B} }},\alpha = \frac{b}{a}\) :

Brownian parameter, Schmidt number, ratio parameter

\({\text{Nt}} = \frac{{(\rho c)_{p} D_{T} (T_{f} - T_{\infty } )}}{{(\rho c)_{f} \nu T_{\infty } }},\) :

Thermophoretic parameter

\(\lambda_{1} = \gamma_{1}^{ * } a, \, \lambda_{2} = \gamma_{2}^{ * } a,\) :

Thermal and concentration relaxation parameters, respectively

\(\nabla ,\,\gamma_{1}^{ * } \left[ {\text{s}} \right],\,\gamma_{2}^{ * } \left[ {\text{s}} \right].\) :

Laplace operator, thermal and solutal relaxation times

\(f^{\prime } ,\,g^{\prime }\) :

Velocities in \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{y}\) space coordinates, respectively

\(\theta ,\phi\) :

Dimensionless temperature and concentration, respectively

CS, HCS:

Cuckoo search, hybridized cuckoo search

PNF, HM:

Prandtl nanofluid, Hydromagnetic, respectively

ANNs, NNF:

Artificial neural networks, non-Newtonian fluid

CC-model, H–I effect:

Cattaneo–Christov model, Hall and ion effect, respectively

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Authors and Affiliations

  1. Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Khyber Pakhtunkhwa, 23200, Pakistan

    Shazia Habib, Saeed Islam & Zeeshan Khan

  2. Department of Natural Sciences, University of Engineering and Technology Mardan, Mardan, Khyber Pakhtoonkhwa, 23200, Pakistan

    Shazia Habib

  3. School of Mechanical Engineering, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Waseem

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  1. Shazia Habib
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Correspondence to Waseem.

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Habib, S., Islam, S., Khan, Z. et al. An evolutionary-based neural network approach to investigate heat and mass transportation by using non-Fourier double-diffusion theories for Prandtl nanofluid under Hall and ion slip effects. Eur. Phys. J. Plus 138, 1122 (2023). https://doi.org/10.1140/epjp/s13360-023-04740-5

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