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New insights on fractional thermoelectric MHD theory

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Abstract

A mathematical model of thermoelectric MHD theory has been developed in the context of a study of heat conduction using a new fractal form of Green–Naghdi theory without energy dissipation (FGN-II). Theories of coupled (CT) and Green–Naghdi of type II (GN-II) thermoelectric fluid follow as limit cases. This model is applied to different problems of thermoelectric fluid bounded by an infinite plane surface with constant temperature in the presence of a uniform magnetic field. State space approach utilized for one-dimensional heat source situations. Techniques based on the Laplace transform are employed. The resulting formulation is applied to a problem for the entire space with heat sources distributed in a plane. Reflection approach tackles semi-space problem with heat source distribution and entire space solution. The numerical technique is used to achieve the Laplace transform inversion. The comparisons in the figures help assess how the fractional order parameter effects on the behavior of the field quantities in the new theory.

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Abbreviations

B :

Magnetic induction vector

q :

Heat flux vector

\(C_p\) :

Specific heat at constant pressure

\(F_i\) :

Lorentz force

H :

Magnetic field intensity vector

H o :

Constant component of magnetic field

J :

Conduction electric density vector

\(k\) :

Thermal conductivity

\(P_r\) :

Prandtl number

\(S\) :

Seebeck coefficient

\(k_o\) :

Seebeck coefficient at temperature \(T_o\)

t :

Time

T :

Temperature

\(T_w\) :

Temperature of the plate

\(T_\infty\) :

Temperature of the fluid away from the plate

\(T_o\) :

\(= T_w - T_\infty\), Reference temperature

\(U\) :

The standard speed, \([U] = {\text{m}}/{\text{s}}\)

Q :

Intensity of the applied heat source

M :

\(= \frac{\upsilon \sigma_o B_o^2 }{{\rho U^2 }}\), Magnetic number

\({{\varvec{u}}}\) :

Components of velocity vector \((u,v,w)\)

x :

Space coordinates (x, y, z)

\(\rho\) :

Density

\(\alpha\) :

Fractional derivative parameters

\(\tau_{ij}\) :

\(= \mu \left( {u_{i,j} + u_{j,i} } \right)\), Stress components

\(\mu_o\) :

Magnetic permeability

\(\mu\) :

Dynamic viscosity

\(\upsilon_o\) :

\(= \mu /\rho\) Kinematic viscosity

\(\sigma_o\) :

Electrical conductivity

\(\Pi\) :

Peltier Coefficient

\(\pi_o\) :

Peltier Coefficient at temperature \(T_0\)

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Acknowledgements

The authors gratefully acknowledge the approval and the support of this research study by the Grant No. SCIA-2023-12-2109 from the Deanship of Scientific Research in Northern Border University, Arar, KSA.

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Authors and Affiliations

  1. Department of Physics, Faculty of Science, Northern Border University, Arar, Saudi Arabia

    Amani S. Alruwaili

  2. Department of Mathematics, Faculty of Science, Northern Border University, Arar, Saudi Arabia

    Abaker A. Hassaballa & Mohamed H. Hendy

  3. Department of Mathematics, Faculty of Science, Al Arish University, Al Arish, Egypt

    Mohamed H. Hendy

  4. Department of Mathematics, Faculty of Science, Qassim University, 51452, Buraydah, Saudi Arabia

    Magdy A. Ezzat

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All authors whose names appear on the submission: * made substantial contributions to the conception or design of the work; or the acquisition, analysis or interpretation of data; or the creation of new software used in the work; * drafted the work or revised it critically for important intellectual content; * approved the version to be published and * agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

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Correspondence to Magdy A. Ezzat.

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Appendices

Appendix A

$$ \begin{gathered} a_o = \frac{k_1^2 \cosh k_2 y - k_2^2 \cosh k_1 y}{{k_1^2 - k_2^2 }},\quad a_1 = \frac{k_1^3 \sinh k_2 y - k_2^3 \sinh k_1 y}{{k\_1 \ k_2 (k_1^2 - k_2^2 )}}, \hfill \\ a_2 = \frac{\cosh k_1 y - \cosh k_2 y}{{k_1^2 - k_2^2 }},\quad a_3 = \frac{k_2 \sinh k_1 y - k_1 \sinh k_2 y}{{k_1 k_2 (k_1^2 - k_2^2 )}}. \hfill \\ \end{gathered} $$

Appendix B

$$ \begin{aligned} \ell_{11} & = \frac{(k_1^2 - m)\cosh k_2 y - (k_2^2 - m)\cosh k_1 y}{{k_1^2 - k_2^2 }}, \\ \ell_{12} & = - \ n(s + M)\left[ {\frac{k_2 \sinh k_1 y - k_1 \sinh \ k_2 y}{{k_1 k_2 \left( {k_1^2 - k_2^2 } \right)}}} \right]{,} \\ \ell_{13} & = \frac{{k_2 \left( {k_1^2 - s - M} \right)\sinh k_1 y - k_1 \left( {k_2^2 - s - M} \right)\sinh k_2 y}}{{k_1 k_2 \left( {k_1^2 - k_2^2 } \right)}}{,} \\ \ell_{14} & = - n\left[ {\frac{{{\text{cosh}}k_1 y - \cosh k_2 y}}{k_1^2 - k_2^2 }} \right]{,}\quad \ell_{21} = m\ K_o \left[ {\frac{k_2 \sinh k_1 y - k_1 \sinh \ k_2 y}{{k_1 k_2 \left( {k_1^2 - k_2^2 } \right)}}} \right]{,} \\ \ell_{22} & = \frac{{\left( {k_1^2 - s - M} \right)\cosh k_2 y - \left( {k_2^2 - s - M} \right)\cosh k_1 y}}{k_1^2 - k_2^2 }{,} \\ \ell_{23} & = K_o \left[ {\frac{{{\text{cosh}}k_1 y - \cosh k_2 y}}{k_1^2 - k_2^2 }} \right]{,} \\ \ell_{24} & = \frac{{k_2 \left( {k_1^2 - m} \right)\sinh k_1 y - k_1 \left( {k_2^2 - m} \right)\sinh k_2 y}}{{k_1 k_2 \left( {k_1^2 - k_2^2 } \right)}}\,{,} \\ \ell_{31} & = m\left[ {\frac{{k_2 \left( {k_1^2 - s - M} \right)\sinh k_1 y - k_1 \left( {k_2^2 - s - M} \right)\sinh k_2 y}}{{k_1 k_2 \left( {k_1^2 - k_2^2 } \right)}}} \right]{,} \\ \ell_{32} & = - n(s + M)\left[ {\frac{{{\text{cosh}}k_1 y - \cosh k_2 y}}{k_1^2 - k_2^2 }} \right]{,} \\ \ell_{33} & = \frac{{\left( {k_1^2 - s - M} \right)\cosh k_1 y - \left( {k_2^2 - s - M} \right)\cosh k_2 y}}{k_1^2 - k_2^2 }{,} \\ \ell_{34} & = - n\left[ {\frac{k_1 \sinh k_1 y - k_2 \sinh k_2 y}{{k_1^2 - k_2^2 }}} \right]{,}\quad \ell_{41} = mK_o \left[ {\frac{{{\text{cosh}}k_1 y - \cosh k_2 y}}{k_1^2 - k_2^2 }} \right]\,{,} \\ \ell_{42} & = (s + M)\left[ {\frac{{k_2 \left( {k_1^2 - m} \right)\sinh k_1 y - k_1 \left( {k_2^2 - m} \right)\sinh k_2 y}}{{k_1 k_2 \left( {k_1^2 - k_2^2 } \right)}}} \right]{,} \\ \ell_{43} & = K_o \left[ {\frac{k_1 \sinh k_1 y - k_2 \sinh k_2 y}{{k_1^2 - k_2^2 }}} \right]{,} \\ \ell_{44} & = \frac{(k_1^2 - \, m)\cosh k_1 y - (k_2^2 - m)\cosh k_2 y}{{k_1^2 - k_2^2 }}{.} \\ \end{aligned} $$

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Alruwaili, A.S., Hassaballa, A.A., Hendy, M.H. et al. New insights on fractional thermoelectric MHD theory. Arch Appl Mech (2024). https://doi.org/10.1007/s00419-024-02597-3

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