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Levenberg–Marquardt neural network for entropy optimization on Casson hybrid nanofluid flow with nonlinear thermal radiation: a comparative study

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Abstract

The purpose of this study is to investigate entropy optimization in the magneto-hydrodynamic and electro-magneto-hydrodynamic flow of a Casson hybrid nanofluid over a rotating disk with nonlinear thermal radiation. The governing dimensional partial differential equations were reduced to ordinary differential equations by using appropriate transforms and solved numerically. The effects of several physical factors on the velocity, temperature, entropy generation, Bejan number, Nusselt number, and skin friction coefficient in comparison to the nanofluid and hybrid nanofluid scenarios over a rotating disk are explored both tabularly and graphically. The constructed artificial neural network is the most appropriate for predicting the skin friction coefficient and Nusselt number over a rotating disk. As the magnetic field strength increased, the velocity profiles decreased in the nanofluid and hybrid nanofluid scenarios. When the thermal radiation increased, the amount of entropy generated for the nanofluids and hybrid nanofluids also increased. We built the artificial neural networking model using 51 sample values of the skin friction coefficient and Nusselt number as outputs. This section provides various dimensionless parameters, which are all inputs. We utilized 70% of the data for training, and 15% for validation and testing. The Levenberg–Marquardt algorithm and back-propagation were used to train the neural network. The best validation performance for skin friction and the Nusselt number for the Casson hybrid nanofluid across a rotating disk are 6652e-07 at epoch 138 and 2.7094e-05 at epoch 7. Additionally, the training, validation, testing, and performance of the ANN model were closer to unity.

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Abbreviations

\((u^{ * } ,v^{ * } ,w^{ * } )\) :

Velocity components

\((r^{ * } ,\phi^{ * } ,z^{ * } )\) :

Directions (m s−1)

Be:

Bejan number

\(\mu\) :

Dynamic viscosity \(\left( {{\text{kg}}\;{\text{m}}^{ - 1} \;{\text{s}}^{ - 1} } \right)\)

\(\upsilon\) :

Kinematic viscosity \(({\text{m}}^{2} \;{\text{s}}^{ - 1} )\)

\(\sigma\) :

Electric conductivity \(({\text{S}}\;{\text{m}}^{ - 1} )\)

\(k\) :

Thermal conductivity \(({\text{W}}\;{\text{m}}^{ - 1} \;{\text{K}}^{ - 1} )\)

\(\phi_{1} ,\;\phi_{2}\) :

Nanoparticles volume fraction

\(\beta_{f}\) :

Thermal expansion \({\text{K}}^{ - 1}\)

\(\rho\) :

Density \(\left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right)\)

\(c_{p}\) :

Heat capacity \(({\text{J}}\;{\text{kg}}^{ - 1} \;{\text{K}}^{ - 1} )\)

\(C_{f} \;{\text{Re}}^{1/2}\) :

Skin friction coefficient

\(\alpha_{f}\) :

Thermal diffusivity of the base fluid, (m2 s−1)

\(T_{w}\) :

Surface temperature, \({\text{K}}\)

\(T\) :

Temperature of the fluid, \({\text{K}}\)

\(\sigma^{*}\) :

Stefan Boltzmann constant

\(\beta\) :

Casson fluid parameter

\(T_{f}\) :

Temperature of heated fluid, \({\text{K}}\)

\(T_{\infty }\) :

Ambient fluid temperature, \({\text{K}}\)

\(F^{\prime}\left( \eta \right)\) :

Radial velocity

\(\Omega\) :

Constant angular velocity

\(k^{*}\) :

Mean absorption coefficient

\(p\) :

Pressure

\(h_{f}\) :

Heat transfer coefficient, \({\text{W}}\;{\text{m}}^{ - 2} \;{\text{k}}\)

\({\text{Ec}} = \frac{{r^{2} \Omega^{2} }}{{\left( {C_{p} } \right)_{f} \left( {T_{f} - T_{\infty } } \right)}}\) :

Eckert number

\({\text{Br}} = \frac{{\mu_{f} \Omega^{2} R^{2} }}{{k_{f} \Delta T}}\) :

Rotational Brinkman number

\(\theta_{f} = \frac{{T_{f} }}{{T_{\infty } }}\) :

Temperature ratio parameter

\({\text{Gr}} = \frac{{g\beta_{f} T_{\infty } (T_{f} - 1)r^{3} }}{{\upsilon_{f}^{2} }}\) :

Thermal Grashof number

\(K = \frac{{\upsilon_{f} }}{{\Omega K^{ * } }}\) :

Porosity parameter

\(\delta = \frac{a}{\Omega }\) :

Stretching-strength parameter

\({\text{Re}} = r\left( {\frac{r\Omega }{{\upsilon_{f} }}} \right)\) :

Rotational Reynolds number

\(F^{*} = \left( {\frac{{C_{d} }}{{rK^{{*^{1/2} }} }}} \right)\) :

Non-uniform inertia coefficient

\({\text{Fr}} = \frac{{C_{d} }}{{\sqrt {K^{ * } } }}\) :

Forchheimer number

\(C_{d}\) :

Drag coefficient

\(\Delta T = T_{f} - T_{\infty }\) :

Temperature difference

\(\alpha = \frac{\Delta T}{{T_{\infty } }}\) :

Temperature ratio parameter

\(D = \frac{r}{R}\) :

Dimensionless radial coordinate

\({\text{Bi}} = \frac{{h_{f} }}{{k_{f} }}\sqrt {\frac{{\upsilon_{f} }}{2\Omega }}\) :

Biot number

\(E_{1} = \frac{E}{B\Omega r}\) :

Electric field parameter

\(M = \frac{{\sigma_{f} B_{0}^{2} }}{{\Omega \rho_{f} }}\) :

Magnetic interaction parameter

\(N_{G} = \frac{{S^{\prime\prime\prime}_{gen} }}{{2\left( {k_{f} \Delta T\Omega /T_{w} \upsilon_{f} } \right)}}\) :

Dimensionless entropy generation rate

\({\text{NuRe}}^{ - 1/2}\) :

Nusselt number

\(R_{d} = \frac{{4\sigma^{*} T^{3} }}{{k^{*} k_{f} }}\) :

Radiation parameter

\(Q = \frac{{Q_{0} }}{{\Omega \left( {\rho C_{p} } \right)_{f} }}\) :

Heat absorption/generation coefficient

\(S = \frac{W}{{\sqrt {2\Omega \upsilon_{f} } }}\) :

Suction parameter

\({\text{Pr}} = \frac{{\upsilon_{f} }}{{\alpha_{f} }}\) :

Prandtl number

\({\text{Re}} = \frac{{\Omega r^{2} }}{{\upsilon_{f} }}\) :

Rotational Reynolds number

\(\alpha_{f}\) :

Thermal diffusivity of the base fluid, \({\text{m}}^{2} \;{\text{s}}^{ - 1}\)

\(f\) :

Base fluid

\(nf\) :

Nanofluid

\(hnf\) :

Hybrid nanofluid

\(s_{1}\) :

First solid nanoparticle

\(s_{2}\) :

Second solid nanoparticle

EMHD:

Electro-magneto-hydrodynamic

ANN:

Artificial neural networks

NLTR:

Non linear thermal radiation

BP:

Backpropagation

PDE:

Partial differential equations

ODE:

Ordinary differential equations

LMFFBOA:

Levenberg–Marquardt feedforward backpropagation optimization approach

LMA:

Levenberg–Marquardt algorithm

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Authors and Affiliations

  1. School of Computer Science and Engineering, Vellore Institute of Technology (VIT), Vellore, 632014, India

    Kakelli Anil Kumar

  2. Department of Mathematics, St. Peters Engineering College, Medchal, Dhulapally, Hyderabad, Telangana, 500100, India

    K. Sakkaravarthi

  3. Department of Mathematics, S.A.S. Vellore Institute of Technology (VIT), Vellore, 632014, India

    P. Bala Anki Reddy

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Correspondence to P. Bala Anki Reddy.

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Authors have no actual or potential conflict of interest including financial, personal, or other relationships with other people or organizations. This work has been carried out by the authors: Kakelli Anil Kumar, Sakkaravarthi K and P. Bala Anki Reddy.

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Kumar, K.A., Sakkaravarthi, K. & Bala Anki Reddy, P. Levenberg–Marquardt neural network for entropy optimization on Casson hybrid nanofluid flow with nonlinear thermal radiation: a comparative study. Eur. Phys. J. Plus 139, 555 (2024). https://doi.org/10.1140/epjp/s13360-024-05359-w

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