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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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Abstract

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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Abbreviations

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

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

References

  1. 1.

    M. Stupnisek, B. Stefotic, Development of the diffusion process for carbide coatings on steels, in Proceedings of YUSTOM 1986, Plitvice Lakes (1986), pp. 323–330

  2. 2.

    T. Arai, Development of carbide and nitride coating by thermo-reactive deposition and diffusion, in Surface Modification Technologies (1990), p. 587

  3. 3.

    K.A. Abro, I. Khan, Analysis of heat and mass transfer in MHD flow of generalized casson fluid in a porous space via non-integer order derivative without singular kernel. Chin. J. Phys. 55(4), 1583–1595 (2017)

  4. 4.

    A.A. Kashif, H.S. Shaikh, M. Norzieha, K. Ilyas, T. Asifa, A mathematical study of magnetohydrodynamic Casson fluid via special functions with heat and mass transfer embedded in porous plate. Malays. J. Fundam. Appl. Sci. 14(1), 20–38 (2018)

  5. 5.

    M.M. Dur, A.A. Kashif, A.S. Muhammad, Application of modern approach of Caputo–Fabrizio fractional derivative to MHD second grade fluid through oscillating porous plate with heat and mass transfer. Int. J. Adv. Appl. Sci. 5(10), 97–105 (2018)

  6. 6.

    K. Nakanishi, H. Takeda, H. Tachikawa, T. Arai, Fluidized bed carbide coating process-development and application, in The 8th International Congress on Heat Treatment of Materials, Kyoto, Japan (1992), p. 507

  7. 7.

    R.R. Srinivasa, G.R. Jithender, J.R. Anand, M.M. Rashidi, Thermal diffusion and diffusion thermo effects on an unsteady heat and mass transfer magnetohydrodynamic natural convection Couette flow using FEM. J. Comput. Des. Eng. 3(4), 349–362 (2016)

  8. 8.

    A. Hobinya, I.A. Abbas, Analytical solutions of photo-thermo-elastic waves in a non-homogenous semiconducting material. Results Phys. 10, 385–390 (2018)

  9. 9.

    S. Venkateswarlu, S.V.K. Varma, K. Kiran, Thermo-diffusion and non-uniform heat source/sink effects on hydromagnetic flow of Cu and TiO2 water-based nanofluid partially filled with a porous medium. Inform. Med. Unlocked 13, 51–61 (2018)

  10. 10.

    A.A. Kashif, K. Ilyas, J.F. Gomez-Aguilar, Thermal effects of magnetohydrodynamic micropolar luid embedded in porous medium with Fourier sine transform technique. J. Braz. Soc. Mech. Sci. Eng. 41, 174–181 (2019). https://doi.org/10.1007/s40430-019-1671-5

  11. 11.

    A. Derya, Y. Aylin, Cauchy and source problems for an advection-diffusion equation with Atangana–Baleanu derivative on the real line Chaos. Solitons Fractals 118, 361–365 (2019)

  12. 12.

    A.A. Kashif, J.F. Gomez-Aguilar, A comparison of heat and mass transfer on a Walter’s-B fluid via Caputo–Fabrizio versus Atangana–Baleanu fractional derivatives using the Fox-H function. Eur. Phys. J. Plus 134, 101 (2019). https://doi.org/10.1140/epjp/i2019-12507-4

  13. 13.

    K.A. Abro, A.M. Anwer, H.A. Shahid, K. Ilyas, I. Tlili, Enhancement of heat transfer rate of solar energy via rotating Jeffrey nanofluids using Caputo–Fabrizio fractional operator: an application to solar energy. Energy Rep. 5, 41–49 (2019). https://doi.org/10.1016/j.egyr.2018.09.009

  14. 14.

    S. Ambreen, A.A. Kashif, A.S. Muhammad, Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: applications to thermal science. J. Therm. Anal. Calorim. (2018). https://doi.org/10.1007/s10973-018-7897-0

  15. 15.

    A.A. Kashif, A.A. Irfan, M.A. Sikandar, K. Ilyas, On the thermal analysis of magnetohydrodynamic Jeffery fluid via modern non integer order derivative. J. King Saud Univ. Sci. (2018). https://doi.org/10.1016/j.jksus.2018.07.012

  16. 16.

    A.A. Kashif, D.C. Ali, A.A. Irfan, K. Ilyas, Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo–Fabrizio and Atangana–Baleanu fractional derivatives embedded in porous medium. J. Therm. Anal. Calorim. 135, 1–11 (2018). https://doi.org/10.1007/s10973-018-7302-z

  17. 17.

    A. Coronel-Escamilla, J.F. Gómez-Aguilar, L. Torres, R.F. Escobar-Jiménez, M. Valtierra-Rodríguez, Synchronization of chaotic systems involving fractional operators of Liouville-Caputo type with variable-order. Physica A 487(1), 1–21 (2017)

  18. 18.

    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)

  19. 19.

    H.L. Muzaffar, A.A. Kashif, A.S. Asif, Helical flows of fractional viscoelastic fluid in a circular pipe. Int. J. Adv. Appl. Sci. 4(10), 97–105 (2017)

  20. 20.

    A.A. Kashif, H. Mukarrum, M.M. Baig, Slippage of fractionalized oldroyd-B fluid with magnetic field in porous medium. Prog. Fract. Differ. Appl. Int. J. 3(1), 69–80 (2017)

  21. 21.

    K.A. Abro, K. Ilyas, J.F. Gomez-Aguilar, A mathematical analysis of a circular pipe in rate type fluid via Hankel transform. Eur. Phys. J. Plus 133, 397 (2018). https://doi.org/10.1140/epjp/i2018-12186-7

  22. 22.

    M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 73–85 (2015)

  23. 23.

    Q. Al-Mdallal, K.A. Abro, I. Khan, Analytical solutions of fractional Walter’s-B fluid with applications, Complexity (2018), Article ID 8918541

  24. 24.

    K.A. Abro, A.M. Anwar, A.U. Muhammad, A comparative mathematical analysis of RL and RC electrical circuits via Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. Eur. Phys. J. Plus 133, 113 (2018). https://doi.org/10.1140/epjp/i2018-11953-8

  25. 25.

    A.A. Kashif, A.M. Ali, A.M. Anwer, Functionality of circuit via modern fractional differentiations. Analog Integr. Circuits Signal Process. Int. J. 99(1), 11–21 (2019). https://doi.org/10.1007/s10470-018-1371-6

  26. 26.

    A. Abdon, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. (2016). https://doi.org/10.2298/TSCI160111018A

  27. 27.

    A.A. Kashif, Y. Ahmet, Fractional treatment of vibration equation through modern analogy of fractional differentiations using integral transforms. Iran. J. Sci. Technol. Trans. A Sci. 43, 1–8 (2019). https://doi.org/10.1007/s40995-019-00687-4

  28. 28.

    F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative. Chaos Solitons Fractals 117, 16–20 (2018)

  29. 29.

    A.A. Kashif, J.F. Gomez-Aguilar, Dual fractional analysis of blood alcohol model via non-integer order derivatives, in Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol. 194, ed. by J.F. Gómez, L. Torres, R.F. Escobar (Springer, New York, 2019). https://doi.org/10.1007/978-3-030-11662-0_5

  30. 30.

    K.M. Owolabi, Behavioural study of symbiosis dynamics via the Caputo and Atangana–Baleanu fractional derivatives. Chaos Solitons Fractals 122, 89–101 (2019)

  31. 31.

    Sümeyra Uçar, Esmehan Uçar, Necati Özdemir, Zakia Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative. Chaos Solitons Fractals 118, 300–306 (2019)

  32. 32.

    K. Arshad, A.A. Kashif, T. Asifa, K. Ilyas, Atangana–Baleanu and Caputo Fabrizio analysis of fractional derivatives for heat and mass transfer of second grade fluids over a vertical plate: a comparative study. Entropy 19(8), 1–12 (2017)

  33. 33.

    A. Abdon, J.F. Gómez-Aguilar, Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws. Chaos Solitons Fractals (2017). https://doi.org/10.1016/j.chaos.2017.03.022

  34. 34.

    M.A. Imran, F. Miraj, I. Khan, I. Tlili, MHD fractional Jeffrey’s fluid flow in the presence of thermo-diffusion, thermal radiation effects with first order chemical reaction and uniform heat flux. Results Phys. 10, 10–17 (2018)

  35. 35.

    K.A. Ali, A.S. Muhammad, Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo–Fabrizoi fractional derivatives. Punjab Univ. J. Math. 49(2), 113–125 (2017)

  36. 36.

    A.A. Kashif, H. Mukarrum, M.B. Mirza, An analytic study of molybdenum disulfide nanofluids using modern approach of Atangana–Baleanu fractional derivatives. Eur. Phys. J. Plus. 132, 439 (2017). https://doi.org/10.1140/epjp/i2017-11689-y

  37. 37.

    M. Alegría-Zamudio, R.F. Escobar-Jiménez, J.F. Gómez-Aguilar, Fault tolerant system based on non-integers order observers: application in a heat exchanger. ISA Trans. 80, 286–296 (2018). https://doi.org/10.1016/j.isatra.2018.06.007

  38. 38.

    A.A. Kashif, K. Ilyas, S.N. Kottakkaran, Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit. Chaos Solitons Fractals 129, 40–45 (2019). https://doi.org/10.1016/j.chaos.2019.08.001

  39. 39.

    N.A. Sheikh, F. Ali, M. Saqib, I. Khan, S.A.A. Jan, A.S. Alshomrani, M.S. Alghamdi, Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results Phys. 7, 789–800 (2017)

  40. 40.

    J. Hristov, Transient heat diffusion with a non-singular fading memory. Therm. Sci. 20, 757–769 (2016)

  41. 41.

    A.A. Kashif, M.R. Mohammad, K. Ilyas, A.A. Irfan, T. Asifa, Analysis of Stokes’ second problem for nanofluids using modern fractional derivatives. J. Nanofluids 7, 738–747 (2018)

  42. 42.

    A.K. Muhammad, M.F. Saifullah, A new fractional model for tuberculosis with relapse via Atangana–Baleanu derivative. Chaos Solitons Fractals 116, 227–238 (2018)

  43. 43.

    I. Koca, A. Atangana, Solutions of Cattaneo-Hristov model of elastic heat diffusion with Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Therm. Sci. (2017). https://doi.org/10.2298/TSCI160102102M

  44. 44.

    A.K. Ali, K. Ilyas, T. Asifa, Application of Atangana–Baleanu fractional derivative to convection flow of MHD Maxwell fluid in a porous medium over a vertical plate. Math. Model. Nat. Phenom. 13, 1 (2018). https://doi.org/10.1051/mmnp/2018007

  45. 45.

    Y. Abdullahi, I. Mustafa, I.A. Aliyu, B. Dumitru, Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations. Chaos Solitons Fractals 116, 220–226 (2018)

  46. 46.

    Z. Hammouch, T. Mekkaoui, Control of a new chaotic fractional-order system using Mittag-Leffler stability. Nonlinear Stud. 22, 565–577 (2015)

  47. 47.

    A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)

  48. 48.

    A. Yusuf, S. Qureshi, M. Inc, A.I. Aliyu, D. Baleanu, A.A. Shaikh, Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel. Chaos Interdiscipl. J. Nonlinear Sci. AIP 28(12), 1–11 (2018)

  49. 49.

    S. Qureshi, A. Yusuf, Modeling chickenpox disease with fractional derivatives: from Caputo to Atangana–Baleanu. Chaos Solitons Fractals 122, 111–118 (2019). https://doi.org/10.1016/j.chaos.2019.03.020

  50. 50.

    A.K. Abro, H. Mukarrum, M.B. Mirza, Analytical solution of MHD generalized Burger’s fluid embedded with porosity. Int. J. Adv. Appl. Sci. 4(7), 80–89 (2017)

  51. 51.

    A.A. Kashif, A.S. Muhammad, H.L. Muzaffar, Influence of slippage in heat and mass transfer for fractionalized MHD flows in porous medium. Int. J. Adv. Appl. Math. Mech. 4(4), 5–14 (2017)

  52. 52.

    J. Muhammad, A.A. Kashif, A.K. Najeeb, Helices of fractionalized Maxwell fluid. Nonlinear Eng. 4(4), 191–201 (2015)

  53. 53.

    K.A. Abro, M. Hussain, B.M. Mirza, A mathematical analysis of magnetohydrodynamic generalized burger fluid for permeable oscillating plate. Punjab Univ. J. Math. 50(2), 97–111 (2018)

  54. 54.

    K.A. Ali, A.S. Asif, D. Sanuallah, Exact solutions on the oscillating plate of maxwell fluids. Mehran Univ. Res. J. Eng. Technol. 35(1), 157–162 (2016)

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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.

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Correspondence to Kashif Ali Abro.

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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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