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Ferromagnetic Chaos in thermal convection of fluid through fractal–fractional differentiations

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Abstract

Thermal convection suppresses the thermal stability and instability during the interaction between the magnetic fields because thermal convection is the most significant driver of time-dependent patterns of motion within magnetized and non-magnetized chaotic. In this manuscript, a mathematical modeling is proposed subject to the magnetohydrodynamic conductive fluid lying on an infinite horizontal layer subject to heat from below with gravity. The mathematical model is based on nonlinear ordinary differential equations and such model has been investigated by means of the Boussinesq approximation and Darcy's law. The newly defined techniques of fractal–fractional differential operators, namely Atangana–Baleanu and Caputo–Fabrizio fractal–fractional differentiations, have been imposed on the governing equations. The mathematical analysis based on the equilibrium points and stability criteria is investigated to examine the dynamic responses of a magnetized and non-magnetized conductive fluid model. The numerical simulations have been performed by Adams methods, which is so-called the explicit scheme of the Adams–Bashforth method. Our results suggest that the comparative evolution of trajectories between magnetized and non-magnetized chaotic behaviors has strong effects due to Lorentz force that showed the resistivity in chaotic phenomenon.

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Abbreviations

ρ0 :

Reference value of density

V :

Velocity

B :

Applied magnetic field

T :

Temperature

Φ:

Electric potential

p :

Pressure of fluid

Pr:

Prandtl number

\(\aleph_{1} ,\aleph_{2} ,\aleph_{3} ,\aleph_{4}\) :

Letting parameters

\(E_{{{\rm x}_{1} ,{\rm y}_{1} ,{\rm z}_{1} }}^{1} ,E_{{{\rm x}_{1} ,{\rm y}_{1} ,{\rm z}_{1} }}^{2} ,E_{{{\rm x}_{1} ,{\rm y}_{1} ,{\rm z}_{1} }}^{3}\) :

Equilibrium points

\(\tau_{1} ,\vartheta_{1}\) :

Fractal and fractional parameters

\(\frac{{{\rm d}^{{\tau_{1} ,\vartheta_{1} }} }}{{{\rm d}t^{{\tau_{1} ,\vartheta_{1} }} }}\) :

Fractal–fractional differential operator of Caputo–Fabrizio

\(\frac{{{\rm d}^{{\tau_{2} ,\vartheta_{2} }} }}{{{\rm d}t^{{\tau_{2} ,\vartheta_{2} }} }}\) :

Fractal–fractional differential operator of Atangana–Baleanu

β :

Thermal expansion coefficient

j :

Electric current density

σ:

Electric conductivity

t :

Time

αe :

Thermal diffusivity

ν:

Density of fluid

Ha:

Hartman number

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Acknowledgements

The author 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. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

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Authors and Affiliations

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

    Kashif Ali Abro & Abdon Atangana

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

    Kashif Ali Abro

  3. CONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México

    J. F. Gómez-Aguilar

  4. Department of Medical Research, China Medical University Hospital, China, Medical University, Taichung, Taiwan

    Abdon Atangana

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  1. Kashif Ali Abro
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Contributions

Kashif Ali Abro was involved in conceptualization, methodology, resources, formal analysis, writing-original draft, supervision; Abdon Atangana performed conceptualization, methodology, software, writing—original draft preparation; J.F. Gómez-Aguilar contributed to conceptualization, methodology, writing—review editing, validation, final draft preparation, and supervision.

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Correspondence to Kashif Ali Abro.

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Abro, K.A., Atangana, A. & Gómez-Aguilar, J.F. Ferromagnetic Chaos in thermal convection of fluid through fractal–fractional differentiations. J Therm Anal Calorim 147, 8461–8473 (2022). https://doi.org/10.1007/s10973-021-11179-2

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