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The effects of coupled heat and mass transfer in the fractional Jeffrey fluid over inclined plane

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Abstract

The flow of fractional Jeffrey fluid due to time-dependent motion of plate has been investigated. An inclined plate is considered, and thermal effects with mass diffusivity in flow are also taken into account. The set of governing equations is operated by integral transform namely Laplace transform technique. Caputo–Fabrizio time-fractional derivative is considered in energy and diffusion equations. Moreover, exact analytical solutions are achieved. In the limiting cases, the general solutions for accelerating and oscillating motion of Jeffrey fluid are obtained. Effects of involved physical parameters, fractional parameter \(\alpha\), inclination of plate \(\theta\), Prandtl number Pr and Schmidt number Sc on the flow, are graphically detected.

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Appendix

Appendix

  1. 1.
    $$\begin{array}{*{20}l} {\bar{C}\left( {y_{1} ,p;c,d} \right) = \frac{1}{p}\exp \left( { - y_{1} \sqrt {\frac{cp}{p + d}} } \right)} \hfill \\ {C\left( {y_{1} ,t;c,d} \right) = L^{ - 1} \left\{ {\bar{C}\left( {y_{1} ,p;c,d} \right)} \right\} = 1 - \frac{2c}{\pi }\int\limits_{0}^{\infty } {\frac{{\sin \left( {y_{1} x} \right)}}{{x\left( {x^{2} + c} \right)}}\exp } \left( { - \frac{{{\text{d}}tx^{2} }}{{x^{2} + c}}} \right){\text{d}}x} \hfill \\ \end{array}$$
  2. 2.
    $$\left\{ {\begin{array}{*{20}l} {\bar{D}\left( {y_{1} ,p;c,d,e} \right) = \frac{1}{p - e}\exp \left( { - y_{1} \sqrt {\frac{cp}{p + d}} } \right) = \bar{C}\left( {y_{1} ,p;c,d} \right) + \bar{E}\left( {y_{1} ,p;c,d,e} \right)} \hfill \\ {\bar{E}\left( {y_{1} ,p;c,d,e} \right) = \frac{1}{p - e}\bar{C}\left( {y_{1} ,p;c,d} \right)} \hfill \\ {E\left( {y_{1} ,t;c,d,e} \right) = L^{ - 1} \left\{ {\bar{E}\left( {y_{1} ,p;c,d,e} \right)} \right\}} \hfill \\ {L^{ - 1} \left\{ {\bar{E}\left( {y_{1} ,p;c,d,e} \right)} \right\} = \exp \left( {et - y_{1} \sqrt {\frac{ce}{d + e}} } \right) - 1 - \frac{2ce}{\pi }\int\limits_{0}^{\infty } {\frac{{\sin \left( {y_{1} x} \right)}}{{x\left[ {x^{2} \left( {d + e} \right) + ce} \right]}}\exp } \left( { - \frac{{{\text{d}}tx^{2} }}{{x^{2} + c}}} \right){\text{d}}x} \hfill \\ \end{array} } \right\}$$
  3. 3.
    $$\begin{array}{*{20}l} {\bar{G}\left( {y_{1} ,p;c,d} \right) = \frac{1}{{p^{2} }}\exp \left( { - y_{1} \sqrt {\frac{cp}{p + d}} } \right) = \frac{1}{p}\bar{C}\left( {y_{1} ,p;c,d} \right)} \hfill \\ {G\left( {y_{1} ,t;c,d} \right) = \int\limits_{o}^{t} {C\left( {y_{1} ,\tau ;c,d} \right){\text{d}}\tau } } \hfill \\ \end{array}$$
  4. 4.
    $$H\left( {y_{1} ,t;a_{1} ,b_{2} } \right) = L^{ - 1}\left[ \exp \left( { - y_{1} \sqrt {\frac{{a_{1} p}}{{p + b_{2} }}} } \right) \right] = \delta \left( t \right)e^{{ - y_{1} \sqrt {b_{1} } }} + \int\limits_{0}^{\infty } {\frac{{y_{1} }}{2u\sqrt \pi }\sqrt {\frac{{a_{1} b_{2} }}{t}} } { \exp }\left( { - \frac{{y_{1}^{2} }}{4u} - b_{2} t - a_{1} u} \right)I_{1} \left( {2\sqrt {a_{1} b_{2} ut} } \right){\text{d}}u$$

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Haq, S.U., Haq, E.U., Khan, M.A. et al. The effects of coupled heat and mass transfer in the fractional Jeffrey fluid over inclined plane. J Therm Anal Calorim 139, 1355–1365 (2020). https://doi.org/10.1007/s10973-019-08448-6

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Keywords

  • Fractional Jeffrey fluid
  • Inclined plane
  • Exact solutions
  • Heat and mass transfer