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Heat transfer in a non-uniform channel on MHD peristaltic flow of a fractional Jeffrey model via porous medium

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Abstract

Peristalsis has recently been a hot topic in biomedical engineering and biological sciences because of its importance. This paper provides a rudimentary insight into the peristaltic transport of an MHD fractional fluid over a permeable medium in a non-uniform channel that is the aim of the research. In mathematical modeling, fluid fills the porous region according to an altered Darcy's law. The derived equations were solved analytically via the long-wavelength hypothesis reliant on the small Reynolds number hypothesis, to solve the leading equations. Temperature, velocity, pressure gradient, friction forces, and pressure rise are all solved using confined solutions. The obtained results were validated with the state-of-threat literature reports. It was claimed that our systematic approach. The numerical results are computed, discussed numerically, and graphs are used to present them. The effects of relevant parameters on the previously mentioned quantities are investigated by plotting graphs based on the computational results. The results show that the parameter effect is extremely powerful. As a limiting case of the problem considered, an adequate comparison with prior results in the literature has been made.

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Acknowledgements

The authors are grateful for the financial support provided by Taif University Researchers Supporting Project (TURSP-2020/96), Taif University, Taif, Saudi Arabia.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

    A. M. Abd-Alla & Esraa N. Thabet

  2. Department of Mathematics, Faculty of Science, Aswan University, Aswan, 81528, Egypt

    Esraa N. Thabet

  3. Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif, 21944, Saudi Arabia

    F. S. Bayones & Abdullah M. Alsharif

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  1. A. M. Abd-Alla
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  2. Esraa N. Thabet
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  3. F. S. Bayones
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  4. Abdullah M. Alsharif
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Correspondence to Esraa N. Thabet.

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Appendix I

Appendix I

$$ k = \frac{ - \beta }{{1 + R}},\;\;f_{1} = \left( {1 + \lambda_{1}^{{\alpha_{1} }} \partial_{t}^{{\alpha_{1} }} } \right)\,,\,\,\,\,\,\,\,\,\,\,f_{2} = \,\,\left( {1 + \lambda_{1}^{{\beta_{1} }} \partial_{t}^{{\beta_{1} }} } \right), $$
$$ c_{1} = \frac{ - 1}{{2\left( {h + \gamma } \right)}},\;\;c_{2} = \frac{{\left( {1 - kh\left( {2\gamma + h} \right)} \right)}}{2},\;\;A = f_{2} + f_{1} M^{2} K, $$
$$ a_{1} = \frac{{e^{{\frac{ - \sqrt A y}{{\sqrt K \sqrt {f_{2} } }}}} }}{{2\left( {1 + e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)A^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} }}, $$
$$ a_{2} = e^{{\frac{2h\sqrt A y}{{\sqrt K \sqrt {f_{2} } }}}} \sqrt {f_{2} } \sqrt K - e^{{\frac{2\sqrt A y}{{\sqrt K \sqrt {f_{2} } }}}} \sqrt {f_{2} } \sqrt K - e^{{\frac{h\sqrt A y}{{\sqrt K \sqrt {f_{2} } }}}} h\sqrt A , $$
$$ a_{3} = - e^{{\frac{{\sqrt A \left( {h + 2y} \right)}}{{\sqrt K \sqrt {f_{2} } }}}} h\sqrt A + e^{{\frac{\sqrt A y}{{\sqrt K \sqrt {f_{2} } }}}} y\sqrt A + e^{{\frac{{\sqrt A \left( {2h + y} \right)}}{{\sqrt K \sqrt {f_{2} } }}}} y\sqrt A , $$
$$ \begin{gathered} a_{4} = e^{{\frac{\sqrt A h}{{\sqrt K \sqrt {f_{2} } }}}} \left( {2f_{2}^{2} - f_{1} f_{2} K\left( { - 2\frac{dp}{{dx}} + 2c_{2} Gr + kGr\left( {h^{2} + 2K} \right) - 4M^{2} } \right) + f_{1}^{2} K^{2} M^{2} \left( {2\frac{dp}{{dx}} - 2c_{2} Gr - kGrh^{2} + 2M^{2} } \right)} \right), \hfill \\ a_{5} = e^{{\frac{{\sqrt A \left( {h + 2y} \right)}}{{\sqrt K \sqrt {f_{2} } }}}} \left( {2f_{2}^{2} - f_{1} f_{2} K\left( { - 2\frac{dp}{{dx}} + 2c_{2} Gr + kGr\left( {h^{2} + 2K} \right) - 4M^{2} } \right) + f_{1}^{2} K^{2} M^{2} \left( {2\frac{dp}{{dx}} - 2c_{2} Gr - kGrh^{2} + 2M^{2} } \right)} \right), \hfill \\ a_{6} = e^{{\frac{\sqrt A y}{{\sqrt K \sqrt {f_{2} } }}}} \left( { - 2f_{2}^{2} + f_{1}^{2} K^{2} M^{2} \left( { - 2\left( {\frac{dp}{{dx}} - 2c_{2} Gr + M^{2} } \right) + kGry^{2} } \right) + f_{1} f_{2} K\left( { - \frac{dp}{{dx}} + 2c_{2} Gr - 4M^{2} + kGr\left( {2K + y^{2} } \right)} \right)} \right), \hfill \\ a_{7} = e^{{\frac{{\sqrt A \left( {2h + y} \right)}}{{\sqrt K \sqrt {f_{2} } }}}} \left( { - 2f_{2}^{2} + f_{1}^{2} K^{2} M^{2} \left( { - 2\left( {\frac{dp}{{dx}} - 2c_{2} Gr + M^{2} } \right) + kGry^{2} } \right) + f_{1} f_{2} K\left( { - \frac{dp}{{dx}} + 2c_{2} Gr - 4M^{2} + kGr\left( {2K + y^{2} } \right)} \right)} \right), \hfill \\ \end{gathered} $$
$$ b_{1} = 2\left( {1 + e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)A^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} , $$
$$ b_{2} = - 2\left( {1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1} f_{2}^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} K^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} + 2\left( { - 1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} \sqrt {f_{2} } K^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}}} M^{2} , $$
$$ b_{3} = 2\left( { - 1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} hK^{2} M^{2} \sqrt A , $$
$$ b_{4} = - 4e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} f_{1} f_{2} hK\sqrt A \cosh \left( {\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}} \right), $$
$$ b_{5} = - 2\left( { - 1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{2}^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} \sqrt K + \left( {1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1} f_{2}^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} GrK^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} \left( {2c_{2} + c_{1} h + kh^{2} + 2kK} \right), $$
$$ b_{6} = - 4\left( { - 1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1} f_{2}^{{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} K^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} M^{2} - 2c_{2} \left( { - 1 + e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} \sqrt {f_{2} } GrK^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} M^{2} , $$
$$ b_{7} = \left( { - 1 + e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} \sqrt {f_{2} } Grh\left( {2c_{1} + kh} \right)K^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}}} M^{2} + 2\left( { - 1 + e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} \sqrt {f_{2} } K^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}}} M^{4} , $$
$$ b_{8} = - 2\left( {1 + e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{2}^{2} \sqrt A h - 4c_{1} e^{{\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} f_{1} f_{2} GrK^{2} \sqrt A - 2c_{2} \left( { - 1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} GrK^{2} M^{2} \sqrt A , $$
$$ \begin{gathered} b_{9} = - \frac{1}{3}\left( { - 1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} Grh^{2} \left( {3c_{1} + kh} \right)K^{2} M^{2} \sqrt A + 2\left( { - 1 - e^{{\frac{2h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} } \right)f_{1}^{2} hK^{2} M^{4} \sqrt A + \hfill \\ \,\,\,\,\,\,\,\,\,\,\frac{1}{3}e^{{\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} f_{1} f_{2} Grh\left( {6c_{2} + 3c_{1} h + kh^{2} } \right)K\sqrt A \cosh \left( {\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}} \right), \hfill \\ \end{gathered} $$
$$ b_{10} = 4e^{{\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} f_{1} f_{2} Gr\left( {c_{1} + kh} \right)K^{2} \sqrt A \cosh \left( {\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}} \right) - 8e^{{\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}}} f_{1} f_{2} hKM^{2} \sqrt A \cosh \left( {\frac{h\sqrt A }{{\sqrt K \sqrt {f_{2} } }}} \right). $$

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Abd-Alla, A.M., Thabet, E.N., Bayones, F.S. et al. Heat transfer in a non-uniform channel on MHD peristaltic flow of a fractional Jeffrey model via porous medium. Indian J Phys 97, 1799–1809 (2023). https://doi.org/10.1007/s12648-022-02554-2

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