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A Comparative Study for Solidification of Nanoparticles Suspended in Nanofluids through Non-Local Kernel Approach

  • Research Article-mechanical Engineering
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Abstract

The nanoparticles reduce heat dissipation during the solidification and combustion process, this is because the dispersion of nanoparticles accelerates the solidification processes. In this paper, the comparative study of an unsteady free convection MHD flow of nanofluids through an accelerated long vertical plate placed in a porous medium with embedded nanoparticles is proposed for solidification trends. The flow is confined to a moving plate under the inclined magnetic field and affected by chemical reactions, Newtonian heating, and thermal radiation. The governing equations for concentration, temperature, and velocity are reduced in dimensionless and fractionalized with innovative fractional derivatives of Caputo-Fabrizio and Atangana-Baleanu. The Laplace transform is employed on governing equations based on water as base fluid with nanoparticles. The numerical inversion technique, namely the Zakian method, is used. The results suggest the decaying behavior of velocity and temperature with both fractional models based on the memory of the flow at a specific time.

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Data Availability Statement

The data that support the findings of this study are publicly available and the corresponding author can provide upon request.

Abbreviations

\({w}^{^{\prime}}\) :

Velocity field [\(m{s}^{-1}\)]

\(g\) :

Gravitational acceleration [\(m{s}^{-2}\)]

\({\beta }_{T}\) :

Volumetric coefficient of thermal extension [\({K}^{-1}\)]

\({T}^{^{\prime}}\) :

Temperature [\(K\)]

\({T}_{\infty }^{^{\prime}}\) :

Fluid temperature away from the plate [\(K\)]

\({T}_{w}\) :

The temperature of the fluid at the plate [\(K\)]

\({C}^{^{\prime}}\) :

The concentration of the fluid [\(kg{m}^{-3}\)]

\({C}_{\infty }^{^{\prime}}\) :

Fluid concentration away from the plate [\(kg{m}^{-3}]\)

\({C}_{w}\) :

The concentration of the fluid at the plate, \(\left(y=0\right)\) [\(kg{m}^{-3}]\)

\({C}_{P}\) :

Specific heat at constant pressure [\(J{Kg}^{-1}{K}^{-1}\)]

\(\mu \) :

Dynamic viscosity [\(Kg{m}^{-1}{s}^{-1}\)]

\(R\) :

Radiation parameter

\(\upsilon \) :

Kinematic viscosity [\({m}^{2}{s}^{-1}\)]

\(\rho \) :

The density of fluid [\(Kg{m}^{-3}\)]

\(D\) :

Mass diffusion coefficient [\({m}^{2}{s}^{-1}\)]

\(\tau \) :

Shear stress

\({B}_{0}\) :

Magnetic fluid parameter

\(Gr\) :

Thermal Grashof number

\(Gm\) :

Mass Grashof number

\(Pr\) :

Prandtl number

\(Sc\) :

Schmidt number

\(\sigma \) :

Electrical conductivity of the fluid

\({q}_{r}\) :

Thermal flux

\({S}_{1}\) :

Dimensionless heat absorption or generation coefficient

\({\lambda }_{1}\) :

Chemical reaction parameter

\(M\) :

Magnetic parameter

\({\mu }_{nf}\) :

Dynamic viscosity of nanofluid

\({\rho }_{nf}\) :

Density of nanofluid

\({K}_{nf}\) :

Thermal conductivity of nanofluid

\({\sigma }_{nf}\) :

Electrical conductivity of nanofluid

\({\left(\rho {C}_{P}\right)}_{nf}\) :

Heat capacity of nanofluid

\(\varphi \) :

The volume fraction of nanofluid

CF:

Caputo-Fabrizio time-fractional derivative

AB:

Atangana-Baleanu time-fractional derivative

NFs:

Nanofluids

HNFs:

Hybrid nanofluids

\(\left(\mathrm{\alpha },\upbeta ,\upgamma \right)\) :

Fractional parameters

MHD:

Magnetohydrodynamics

\({\sigma }^{*}\) :

Stephan Boltzmann's constant

\({\kappa }^{*}\) :

The mean absorption coefficient

q :

Laplace transform variable

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Acknowledgements

Dr. Kashif Ali Abro is highly thankful and grateful to Mehran University of Engineering and Technology, Jamshoro, Pakistan, for generous support and facilities of this research work.

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Authors and Affiliations

  1. Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan

    Samia Riaz, Muhammad Amir & Qasim Ali

  2. Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan

    Imran Qasim Memon & Kashif Ali Abro

  3. Institute of Ground Water Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa

    Kashif Ali Abro

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  1. Samia Riaz
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Riaz, S., Amir, M., Memon, I.Q. et al. A Comparative Study for Solidification of Nanoparticles Suspended in Nanofluids through Non-Local Kernel Approach. Arab J Sci Eng 48, 11645–11663 (2023). https://doi.org/10.1007/s13369-022-07493-y

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