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Thermal stratification of rotational second-grade fluid through fractional differential operators

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Abstract

This manuscript predicts the change in temperature at a different epilimnion to the change in temperature at a different hypolimnion. The fractional analysis on rotational second-grade fluid with sinusoidal boundary conditions is performed for knowing thermal stratification. The mathematical modeling is also proposed by means of modern fractional differential operators, namely Caputo–Fabrizio and Atangana–Baleanu derivatives, for rotational second-grade fluid. Most of authors have proposed the classical solutions of rotational second-grade fluid, which are obtained by the Laplace transform only. Our fractionalized mathematical model of rotational second-grade fluid has been solved via Fourier sine and Laplace transform techniques simultaneously. The solutions of velocity and temperature have been investigated and expressed in the format of Mittag–Leffler and Fox-H functions. Both fractional solutions are presented for comparison of velocity and temperature through Caputo–Fabrizio and Atangana–Baleanu derivatives. Finally, our results showed that the fractional solutions investigated for the velocity and temperature via Fourier sine and Laplace transform methods are stable and rapid than classical solutions.

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Acknowledgements

The authors are 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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Authors and Affiliations

  1. Institute of Ground Water Studies, Faculty of Natural and Agricultural Sciences, 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 & Ambreen Siyal

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  1. Kashif Ali Abro
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Appendix

Appendix

$${\mathcal{L}}^{ - 1} \left( {\frac{{s^{2} }}{{(s + a)(s^{2} + \omega^{2} )}}} \right) = \frac{{a^{2} e^{{ - {\text{at}}}} }}{{(a^{2} + \omega^{2} )}} - \frac{\omega }{{\sqrt {a^{2} + \omega^{2} } }}\sin \left( {\omega t + \tan^{ - 1} \left( {\frac{\omega }{a}} \right)} \right),$$
(28)
$${\mathcal{L}}^{ - 1} \left( {\frac{{s^{2} + a_{1} s + a_{0} }}{{s^{2} (s + b)}}} \right) = \frac{{b^{2} - a_{1} b + a_{0} }}{{b^{2} }}e^{{ - {\text{bt}}}} + \frac{{a_{1} b - a_{0} }}{{b^{2} }} + \frac{{a_{0} }}{b}t,$$
(29)
$${\mathcal{L}}^{ - 1} \left( {\frac{\sqrt s }{{s - a^{2} }}} \right) = \frac{1}{{\sqrt {\pi t} }} + ae^{{\text {a}^{ 2} {\text{t}}}} {\text{erf}}\left( {a\sqrt t } \right),$$
(30)
$${\mathcal{L}}^{ - 1} \left( {\frac{{s^{\alpha } }}{{s\left( {s^{\alpha } + \beta } \right)}}} \right) = E_{\alpha } \left( { - \beta t^{\alpha } } \right),$$
(31)
$${\mathcal{L}}^{ - 1} \left( {\frac{1}{{\left( {s^{\alpha } - \beta } \right)}}} \right) = t^{\alpha - 1} {\mathbf{E}}_{\alpha ,\alpha } \left( { - \beta t^{\alpha } } \right).$$
(32)

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Abro, K.A., Siyal, A. & Atangana, A. Thermal stratification of rotational second-grade fluid through fractional differential operators. J Therm Anal Calorim 143, 3667–3676 (2021). https://doi.org/10.1007/s10973-020-09312-8

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