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Functional application of Fourier sine transform in radiating gas flow with non-singular and non-local kernel

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Abstract

Several industrial manufacturing processes mostly depend upon measurement of gases for controlling the product quality, process control, environmental compliance and production efficiencies. We propose here unsteady natural convection radiating flow as an application of gas flow measurement through mathematical modeling based on governing fractional differential equations. The mathematical model of unsteady natural convection radiating flow is analyzed by Fourier sine and Laplace transform with novel analytical calculations and results. From the analysis of mathematical point of view, the main novelty of considered technique is fractional approach of Atangana–Baleanu which provides non-singular effect of gas with Mittag–Leffler kernel, promising straightforward convergence of imposed initial and boundary conditions that is not assumed for perturbation and discretization. The analytical solutions are obtained for the temperature distribution and velocity field of gas flow based on sine and cosine sinusoidal waves, and these are expressed in terms of elementary functions. Finally, in order to meet the physical aspects of the problem, the multiple variations and differential parametric analysis have been presented through graphical illustrations.

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References

  1. Koca I, Atangana A (2016) Solutions of Cattaneo–Hristov model of elastic heat diffusion with Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Therm Sci 21:2299–2305

    Article  Google Scholar 

  2. Khan U, Khan SI, Ahmed N, Bano S, Mohyudin ST (2016) Heat transfer analysis for squeezing flow of a Casson fluid between parallel plates. Ain Shams Eng J 7:497–504

    Article  Google Scholar 

  3. Khan I, Gul A, Shafie S (2017) Effects of magnetic field on molybdenum disulfide nanofluids in mixed convection flow inside a channel filled with a saturated porous medium. J Porous Media 20:435–448

    Article  Google Scholar 

  4. Sheikh NA, Ali F, Khan I, Saqib M, Khan A (2017) MHD flow of micropolar fluid over an oscillating vertical plate embedded in porous media with constant temperature and concentration. Math Probl Eng. https://doi.org/10.1155/2017/9402964

    Article  MathSciNet  Google Scholar 

  5. Kashif AA, Solangi MA, Laghari MH (2017) 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

    MathSciNet  MATH  Google Scholar 

  6. Muzaffar HL, Abro KA, Shaikh AA (2017) Helical flows of viscoelastic fluid in circular pipe. Int J Adv Appl Sci 4(10):97–105

    Article  Google Scholar 

  7. Saqib M, Ali F, Khan I, Sheikh NA, Jan SAA (2018) Exact solutions for free convection flow of generalized Jeffrey fluid: a Caputo–Fabrizio fractional model. Alex Eng J 57:1849–1858

    Article  Google Scholar 

  8. Saqib M, Ali F, Khan I, Sheikh NA (2016) Application of Caputo–Fabrizio derivatives to MHD free convection flow of generalized Walter’s-B fluid model. Eur Phys J Plus 131:377

    Article  Google Scholar 

  9. Nadeem SA, Ali F, Khan I, Saqib M (2018) A modern approach of Caputo–Fabrizio time-fractional derivative to MHD free convection flow of generalized second-grade fluid in a porous medium. Neural Comput Appl. https://doi.org/10.1007/s00521-016-2815-5

    Article  Google Scholar 

  10. Ilyas K, Shah NA, Mahsud Y, Vieru D (2017) Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo–Fabrizio derivatives. Eur Phys J Plus 132:194. https://doi.org/10.1140/epjp/i2017-11456-2

    Article  Google Scholar 

  11. Pakdemirli M, Hayat T, Yürüsoy M, Abbasbandyd S, Asghar S (2011) Perturbation analysis of a modified second grade fluid over a porous plate. Nonlinear Anal Real World Appl 12:1774–1785

    Article  MathSciNet  MATH  Google Scholar 

  12. Yépez-Martínez H, Gómez-Aguilar JF (2019) A new modified definition of Caputo–Fabrizio fractional-order derivative and their applications to the multi-step Homotopy analysis method (MHAM). J Comput Appl Math 346:247–260

    Article  MathSciNet  MATH  Google Scholar 

  13. Arshad K, Abro KA, Tassaddiq A, Khan I (2017) 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):279

    Article  Google Scholar 

  14. Hong CP, Asako Y, Morini GL, Reshman D (2019) Data reduction of average friction factor of gas flow through adiabatic micro-channels. Int J Heat Mass Transf 129:427–431

    Article  Google Scholar 

  15. Lutsenko NA, Fetsov SS (2019) Influence of gas compressibility on gas flow through bed of granular phase change material. Int J Heat Mass Transf 130:693–699

    Article  Google Scholar 

  16. Avdiaj S, Šetina J (2017) Flowmeter for very low gas flows of inert gases. Measurement 111:384–388

    Article  Google Scholar 

  17. Abro KA, Ilyas K, Gómez-Aguilar JF (2018) A mathematical analysis of a circular pipe in rate type fluid via Hankel transform. Eur Phys J Plus 133:397

    Article  Google Scholar 

  18. Ying Y, Lian Y, Tang S, Liu WK (2017) High-order central difference scheme for Caputo fractional derivative. Comput Methods Appl Mech Eng 317:42–54

    Article  MathSciNet  Google Scholar 

  19. Abro KA, Yildirim A (2019) Fractional treatment of vibration equation through modern analogy of fractional differentiations using integral transforms. Iran J Sci Technol Trans A Sci 43:1–8

    Article  MathSciNet  Google Scholar 

  20. Dassios IK, Baleanu D (2018) Caputo and related fractional derivatives in singular systems. Appl Math Comput 337:591–606

    MathSciNet  Google Scholar 

  21. Abro KA, Gómez-Aguilar JF (2019) 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. https://doi.org/10.1007/978-3-030-11662-0_5

  22. Wang YM, Ren L (2019) A high-order L2-compact difference method for Caputo type time-fractional sub-diffusion equations with variable coefficients. Appl Math Comput 342:71–93

    MathSciNet  Google Scholar 

  23. Abro KA, Memon AA, Abro SH, Khan I, Tlili I (2019) 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

    Article  Google Scholar 

  24. Alkahtani BST, Atangana A (2016) Controlling the wave movement on the surface of shallow water with the Caputo–Fabrizio derivative with fractional order. Chaos, Solitons Fractals 89:539–546

    Article  MathSciNet  MATH  Google Scholar 

  25. Abro KA, Memon AA, Memon AA (2018) Functionality of circuit via modern fractional differentiations. Analog Integr Circuits Signal Process 1:1–11

    Google Scholar 

  26. Atangana A, Koca I (2017) Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos, Solitons Fractals 89:447–454

    Article  MathSciNet  MATH  Google Scholar 

  27. Siyal A, Abro KA, Solangi MA (2018) Thermodynamics of magnetohydrodynamic Brinkman fluid in porous medium: applications to thermal science. J Therm Anal Calorim 1:1–9

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Owolabi KM (2018) Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives. Chaos, Solitons Fractals 115:160–169

    Article  MathSciNet  MATH  Google Scholar 

  30. Abro KA, Chandio AD, Abro IA, Khan I (2018) 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 1:1–11

    Google Scholar 

  31. Abro KA, Khan I, Tassadiqq A (2018) 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 13:1–9

    Article  MathSciNet  MATH  Google Scholar 

  32. Saad KM (2018) Comparing the Caputo, Caputo–Fabrizio and Atangana–Baleanu derivative with fractional order: fractional cubic isothermal auto-catalytic chemical system. Eur Phys J Plus 133(3):94

    Article  Google Scholar 

  33. Abro KA, Memon AA, Uqaili MA (2018) A comparative mathematical analysis of RL and RC electrical circuits via Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. Eur Phys J Plus 133:113

    Article  Google Scholar 

  34. Abro KA, Rashidi MM, Khan I, Abro IA, Tassadiq A (2018) Analysis of stokes’ second problem for nanofluids using modern fractional derivatives. J Nanofluids 7:738–747

    Article  Google Scholar 

  35. Owolabi KM, Atangana A (2017) Analysis and application of new fractional Adams–Bashforth scheme with Caputo–Fabrizio derivative. Chaos, Solitons Fractals 105:111–119

    Article  MathSciNet  MATH  Google Scholar 

  36. Abro KA, Mirza MH, Baig MM (2017) An analytic study of molybdenum disulfide nanofluids using modern approach of Atangana-Baleanu fractional derivatives. Eur Phys J Plus 132:439

    Article  Google Scholar 

  37. Kashif AA, Mirza MH, Baig MM (2018) A mathematical analysis of magnetohydrodynamic generalized Burger fluid for permeable oscillating plate. Punjab Univ J Math 50(2):97–111

    MathSciNet  Google Scholar 

  38. Kashif AA, Solangi MA (2017) Heat transfer in magnetohydrodynamic second grade fluid with porous impacts using Caputo–Fabrizo fractional derivatives. Punjab Univ J Math 49(2):113–125

    MathSciNet  MATH  Google Scholar 

  39. Kumar D, Singh J, Tanwar K, Baleanu D (2019) A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag–Leffler laws. Int J Heat Mass Transf 138:1222–1227

    Article  Google Scholar 

  40. Goswami A, Singh J, Kumar D, Sushila A (2019) An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Phys A Stat Mech Appl 524:563–575. https://doi.org/10.1016/j.physa.2019.04.058

    Article  MathSciNet  Google Scholar 

  41. Kumar D, Singh J, Baleanu D, Rathore S (2018) Analysis of a fractional model of the Ambartsumian equation. Eur Phys J Plus 133:259

    Article  Google Scholar 

  42. Singh J, Kumar D, Baleanu D (2018) On the analysis of fractional diabetes model with exponential law. Adv Differ Equ 2018:231

    Article  MathSciNet  MATH  Google Scholar 

  43. Singh J, Kumar D, Baleanu D, Rathore S (2018) An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation. Appl Math Comput 335:12–24

    MathSciNet  Google Scholar 

  44. Singh J, Kumar D, Baleanu D (2019) New aspects of fractional Biswa-Milovic model with Mittag-Leffler law. Math Model Nat Phenom 14:303

    Article  MathSciNet  MATH  Google Scholar 

  45. Kashif AA, Gomez-Aguilar JF (2019) 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. https://doi.org/10.1140/epjp/i2019-12507-4

    Article  Google Scholar 

  46. Abro KA, Khan I, Gomez-Aguilar JF (2019) Thermal effects of magnetohydrodynamic micropolar fluid embedded in porous medium with Fourier sine transform technique. J Braz Soc Mech Sci Eng 41:174–181. https://doi.org/10.1007/s40430-019-1671-5

    Article  Google Scholar 

  47. Shah Z, Islam S, Ayaz H, Khan S (2019) Radiative heat and mass transfer analysis of micropolar nanofluid flow of Casson fluid between two rotating parallel plates with effects of hall current. ASME J Heat Transf. https://doi.org/10.1115/1.4040415

    Article  Google Scholar 

  48. Shah Z, Tassaddiq A, Islam S, Alklaibi A, Khan I (2019) Cattaneo–Christov heat flux model for three-dimensional rotating flow of SWCNT and MWCNT nanofluid with Darcy-Forchheimer porous medium induced by a linearly stretchable surface. Symmetry 11:331

    Article  Google Scholar 

  49. Shah Z, Bonyah E, Islam S, Gul T (2019) Impact of thermal radiation on electrical MHD rotating flow of carbon nanotubes over a stretching sheet. AIP Adv 9:015115

    Article  Google Scholar 

  50. Shah Z, Gul T, Islam S, Khan A (2017) Effects of hall current on steady three dimensional non-Newtonian nanofluid in a rotating frame with Brownian motion and thermophoresis effects. J Eng Technol 6:280–296

    Google Scholar 

  51. Sheikholeslami M, Shah Z, Ahafi A A, Khan I, Tilili I (2019) Uniform magnetic force impact on water based nanofluid thermal behavior in a porous enclosure with ellipse shaped obstacle. Sci Rep. https://doi.org/10.1038/s41598-018-37964-y

    Article  Google Scholar 

  52. Sheikholeslami M, Shah Z, Tassaddiq A, Shafee A, Khan I (2019) Application of electric field for augmentation of ferrofluid heat transfer in an enclosure including double moving walls. IEEE Access 7:21048–21056

    Article  Google Scholar 

  53. Ahmad B, Shah SIA, Haq SU, Shah NA (2017) Analysis of unsteady natural convective radiating gas flow in a vertical channel by employing the Caputo time- fractional derivative. Eur Phys J Plus 132:380

    Article  Google Scholar 

  54. Cogley AC, Vincenti WC, Gilles SE (1968) Differential approximation for radiative transfer in a non-grey gas near equilibrium. AIAA J 6(3):551–553

    Article  Google Scholar 

  55. Kashif AA, Shaikh AA, Dehraj S (2016) Exact solutions on the oscillating plate of Maxwell fluids. Mehran Univ Res J Eng Technol 35(1):157–162

    Google Scholar 

  56. Khan A, Abro KA, Tassaddiq A, Khan I (2017) 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

    Article  Google Scholar 

  57. Abro KA, Khan I (2017) 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

    Article  MathSciNet  Google Scholar 

  58. Mugheri DM, Abro KA, Solangi MA (2018) 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

    Article  Google Scholar 

  59. Abro KA, Saeed SH, Mustapha N, Khan I, Tassadiq A (2018) 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

    Google Scholar 

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

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

    Kashif Ali Abro & Muhammad Nawaz Mirbhar

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

    J. F. Gómez-Aguilar

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  1. Kashif Ali Abro
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  2. Muhammad Nawaz Mirbhar
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Correspondence to Kashif Ali Abro.

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Appendix

Appendix

$$ {\mathcal{L}}^{ - 1} \left( {\frac{1}{{s^{\alpha } + a}}} \right) = p^{\alpha - 1} t^{\varUpsilon - 1} {\mathbf{\mathcal{E}}}_{\alpha ,\alpha } \left( { - at^{\alpha } } \right), $$
(18)
$$ {\mathcal{L}}^{ - 1} \left( {\frac{1}{{s^{n + 1} }}} \right) = \frac{{t^{n} }}{{\varGamma \left( {n + 1} \right)}}, $$
(19)
$$ {\mathcal{L}}^{ - 1} \left\{ {f\left( s \right)*g\left( s \right)} \right\} = \mathop \smallint \limits_{0}^{t} f\left( t \right)g\left( {t - z} \right){\text{d}}z. $$
(20)

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Abro, K.A., Mirbhar, M.N. & Gómez-Aguilar, J.F. Functional application of Fourier sine transform in radiating gas flow with non-singular and non-local kernel. J Braz. Soc. Mech. Sci. Eng. 41, 400 (2019). https://doi.org/10.1007/s40430-019-1899-0

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