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A fractional and analytic investigation of thermo-diffusion process on free convection flow: an application to surface modification technology

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It is worth noted that soft and hard inchromizing entirely depends upon high temperature on materials. Due to this fact, thermal diffusion process can enhance the life expectancy of tools based on surface modification technology. The main objective of this investigation is to explore the thermo-diffusion effects on unsteady-free convection flow in the presence of magnetic field. For developing the governing equations of thermal diffusion process in terms of fractional differentiation, a modern approach of Atangana–Baleanu differential operator is invoked for knowing the memory effects on thermal diffusion process. The temperature, velocity, and concentration are obtained through analytical calculation via Laplace and Fourier sine transform methods. A parametric study is focused for hidden phenomenon of thermo-diffusion process which exhibits typical and rheological properties such as optimal temperature ranges, temperature resistance, increase or decrease of temperature, and few others. Finally, the characteristics of thermal diffusion process are presented graphically based on some physical parameters such as heat transfer (Grashof number), heat capacity (Prandtl number), enthalpy (Dufour number), momentum and mass diffusivity (Schmidt number), magnetization (magnetic field), and few other embedded parameters.

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\( u\left( {y,t} \right) \) :

Velocity field

\( T\left( {y,t} \right) \) :

Temperature distribution

\( R \) :

Radiative parameter

\( {\text{Du}} \) :

Dufour number

\( k \) :

Chemical reaction

\( K \) :

Magnetic lines of force

\( {\mathbf{E}}_{\alpha } \) :

Mittag–Leffler function

\( \xi \) :

Fourier sine transform variable

\( g \) :

Acceleration due to gravity

\( q_{\text{r}} \) :

Radiative heat fluxes in the y-direction

\( k \) :

Thermal conductivity of the fluid

\( \beta \) :

Volumetric coefficient of thermal expansion

\( \mu \) :

Coefficient of viscosity

\( \nu \) :

Kinematic viscosity

\( C\left( {y,t} \right) \) :

Mass concentration

\( P_{r} \) :

Prandtal number

\( G_{\text{m}} \) :

Mass Grashof number

\( S_{c} \) :

Schmidt number

\( G_{r} \) :

Grashof number

\( \alpha \) :

Fractional parameter

\( M \) :

Magnetic parameter

\( p \) :

Laplace variable

\( R \) :

Radiative parameter

\( \sigma \) :

Electric conductivity

\( \rho \) :

Density of the fluid

\( \beta^{*} \) :

Volumetric coefficient of expansion with concentration

\( w \) :

Conditions on the wall

\( \gamma_{o} - \gamma_{12} \) :

Letting parameters


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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.

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Abro, K.A. A fractional and analytic investigation of thermo-diffusion process on free convection flow: an application to surface modification technology. Eur. Phys. J. Plus 135, 31 (2020). https://doi.org/10.1140/epjp/s13360-019-00046-7

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