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Effects of thermal and concentration convection and induced magnetic field on peristaltic flow of Williamson nanofluid in inclined uniform channel

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Abstract

Noteworthiness of double-diffusive convection with nanofluids by induced magnetic field and diverse wave shapes for peristaltic stream of Williamson fluid in a uniform channel has been theoretically examined. Mathematical formulation of two-directional and two-dimensional flow of a Williamson fluid model is explained in detail. The exact solutions of nanoparticles are described. Perturbation solutions of nonlinear PDEs are also calculated. Numerical and graphical results are displaced to see the conduct of temperature, concentration, nanoparticle volume fraction, pressure rise, pressure gradient, magnetic force function and stream functions. It is concluded that the profile of temperature increases and profile of concentration decreases by increasing values of Brownian motion parameter \( N_{\text{b}} \), thermophoresis parameter \( N_{\text{t}} \), Dufour parameter \( N_{TC} \) and Soret parameter \( N_{CT} \).

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Author information

Authors and Affiliations

  1. MCS, National University of Sciences and Technology, Islamabad, Pakistan

    Safia Akram

  2. Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

    Qamar Afzal

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  1. Safia Akram
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  2. Qamar Afzal
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Correspondence to Safia Akram.

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Appendix

Appendix

$$ \begin{aligned} \kappa_{1} &= \frac{{N_{b} + N_{t} }}{{h\left( {1 - N_{CT} N_{TC} } \right)}},\quad \kappa_{2} = \frac{{\kappa_{1} e^{{\kappa_{1} h}} \left( {N_{b} \left( {G_{rt} - N_{CT} G_{rc} } \right) + G_{rF} N_{t} } \right)}}{{N_{b} \left( {e^{{\kappa_{1} h}} - 1} \right)}}, \\ \kappa_{3} &= \frac{{N_{b} \left( {N_{CT} + 1} \right)G_{rc} - G_{rF} \left( {N_{b} + N_{t} } \right)}}{{hN_{b} }},\quad \kappa_{4} = 2C_{1}^{2} M^{6} + 2C_{2}^{2} M^{6} , \\ \kappa_{5} &= \frac{{2\kappa_{2} M^{2} \left( {C_{2} \left( {\kappa_{1}^{2} + M^{2} } \right) - 2C_{1} \kappa_{1} M} \right)}}{{M^{2} - \kappa_{1}^{2} }},\quad \kappa_{6} = \frac{{2\kappa_{1}^{4} \kappa_{2} \kappa_{3} }}{{M^{2} \left( {\kappa_{1}^{2} M^{2} - \kappa_{1}^{4} } \right)}}, \\ \kappa_{7} &= \frac{{4\kappa_{1}^{6} \kappa_{2}^{2} }}{{\left( {\kappa_{1}^{2} M^{2} - \kappa_{1}^{4} } \right)^{2} }},\quad \kappa_{8} = \left( {2\kappa_{2} M^{4} \cosh (hM) + 2\kappa_{1} \kappa_{2} M^{3} \sinh (hM)} \right), \\ \kappa_{9} &= \kappa_{1}^{2} \kappa_{3} \left( {\kappa_{1}^{2} - M^{2} } \right)\left( {h^{2} M^{2} - 2hM\sinh (hM) + 2\cosh (hM) - 2} \right), \\ \kappa_{10} &= \kappa_{1}^{2} \left( {M^{3} \sinh (hM) - \kappa_{2} (\cosh (hM) - 1)} \right) - M\kappa_{1}^{4} \sinh (hM), \\ \end{aligned} $$
$$ \begin{aligned} \kappa_{11} &= \left( {2\kappa_{1} - M} \right)\left( {M^{2} - \kappa_{1}^{2} } \right)\left( {\kappa_{1} \left( {2M - \kappa_{1} } \right)\left( {3C_{1} \kappa_{3} M^{2} - 2\kappa_{4} } \right) + 6\kappa_{5} M^{2} } \right), \\ \kappa_{12} &= \kappa_{6} \left( {h\kappa_{1} + 1} \right)\left( {M - \kappa_{1} } \right)\left( {2M - \kappa_{1} } \right)\left( {2\kappa_{1} + M} \right)\left( {M - 2\kappa_{1} } \right), \\ \kappa_{13} &= C_{1} \kappa_{3} \left( {\left( {h^{2} M^{2} + 2} \right)\cosh (hM) - 2hM\sinh (hM) - 2} \right), \\ \kappa_{14} &= \left( {8\kappa_{4} \sinh^{4} \left( {\frac{hM}{2}} \right)\left( {2hM\coth \left( {\frac{hM}{2}} \right) - 1} \right) + 12M^{2} \kappa_{13} } \right), \\ \kappa_{15} &= - \kappa_{1}^{2} \left( {4\kappa_{5} + 3\left( {4\kappa_{6} + \kappa_{7} } \right)} \right) + 3\left( {4\kappa_{6} + \kappa_{7} } \right)M^{2} + \kappa_{1} \left( { - 8\kappa_{5} + 4\kappa_{6} + \kappa_{7} } \right)M, \\ \kappa_{16} &= \left( {M - \kappa_{1} } \right)\,\left( {4\kappa_{1}^{2} - M^{2} } \right)\left( {2\kappa_{7} M^{4} - \kappa_{1}^{4} \left( {4\kappa_{5} + \kappa_{7} } \right)} \right), \\ \end{aligned} $$
$$ \begin{aligned} \kappa_{17} &= \left( {M - 2\kappa_{1} } \right)\left( {M - \kappa_{1} } \right)^{2} \left( {2\kappa_{1} + M} \right)\left( {\kappa_{1} + M} \right)(hM\cosh (hM) - \sinh (hM)), \\ \kappa_{18} &= \kappa_{4} - 3M^{2} \left( {\frac{{\kappa_{6} }}{{M^{2} - \kappa_{1}^{2} }} + \frac{{\kappa_{7} }}{{M^{2} - 4\kappa_{1}^{2} }} + \frac{{\kappa_{5} }}{{2\kappa_{1} M - \kappa_{1}^{2} }}} \right), \\ \kappa_{19} &= \frac{{\kappa_{7} \left( {2\kappa_{1} \sinh (hM) + M\cosh (hM)} \right)}}{{M^{2} - 4\kappa_{1}^{2} }} - \frac{{4\kappa_{1} \kappa_{5} e^{{h\kappa_{1} }} \left( {\kappa_{1} e^{hM} \sinh (hM) + M} \right)}}{{\left( {M - \kappa_{1} } \right)^{2} \left( {\kappa_{1} - 2M} \right)}}, \\ \kappa_{20} &= \frac{{2\left( {\kappa_{4} - 30C_{1} \kappa_{3} M^{2} } \right)}}{{M^{4} }} - 24C_{5} - \frac{{6\kappa_{7} }}{{\kappa_{1}^{2} M^{2} - 4\kappa_{1}^{4} }} + \frac{{24\kappa_{5} M}}{{\kappa_{1} \left( {2M - \kappa_{1} } \right)\left( {\kappa_{1} - M} \right)^{3} }}, \\ \kappa_{21} &= \frac{{\sinh (hM)\left( {12\kappa_{3} M^{2} \left( {7C_{1} - 2C_{2} hM} \right) + 24C_{5} M^{4} } \right)}}{{M^{5} }} - \frac{{\kappa_{4} \sinh (2hM)}}{{M^{5} }}, \\ \kappa_{22} &= \frac{{\cosh (hM)\left( {12\kappa_{3} M^{2} \left( {7C_{2} - 2C_{1} hM} \right) + 24C_{6} M^{4} } \right)}}{{M^{5} }} - \frac{{3\kappa_{7} e^{{ - 2h\kappa_{1} }} }}{{\kappa_{1}^{3} M^{2} - 4\kappa_{1}^{5} }} + \kappa_{21} , \\ \kappa_{23} &= - \frac{{6C_{1} M\sinh (hM) + 6C_{2} M\cosh (hM) + h^{3} \kappa_{3} }}{{M^{2} }} + \frac{{6\kappa_{2} e^{{ - h\kappa_{1} }} }}{{\kappa_{1}^{3} M^{2} - \kappa_{1}^{5} }}, \\ \kappa_{24} &= \frac{{2\left( {\kappa_{4} - 30C_{1} \kappa_{3} M^{2} } \right)}}{{M^{4} }} - \frac{{6\kappa_{7} }}{{\kappa_{1}^{2} M^{2} - 4\kappa_{1}^{4} }} + \frac{{24\kappa_{5} M}}{{\kappa_{1} \left( {2M - \kappa_{1} } \right)\left( {\kappa_{1} - M} \right)^{3} }}, \\ \kappa_{25} &= \frac{{24e^{{ - h\kappa_{1} }} \left( {\kappa_{1}^{2} \kappa_{5} \left( {\kappa_{1} + M} \right)e^{hM} - \kappa_{6} \left( {2M - \kappa_{1} } \right)\left( {M - \kappa_{1} } \right)^{2} } \right)}}{{\kappa_{1}^{3} \left( {2M - \kappa_{1} } \right)\left( {\kappa_{1} + M} \right)\left( {\kappa_{1} - M} \right)^{3} }}, \\ \end{aligned} $$
$$ \begin{aligned} C_{1} &= - \frac{{\kappa_{3} }}{{M^{4} }} - \frac{{\kappa_{2} }}{{M^{4} - \kappa_{1}^{2} M^{2} }}, \\ C_{2} &= \frac{{2M^{2} \left( {\kappa_{1}^{2} \left( {\kappa_{2} (hM\sinh (hM) - \cosh (hM) + 1) - hM^{4} } \right) + h\kappa_{1}^{4} M^{2} - \kappa_{2} M^{2} } \right) + \kappa_{9} }}{{2\kappa_{1}^{2} M^{4} \left( {M^{2} - \kappa_{1}^{2} } \right)(hM\cosh (hM) - \sinh (hM))}} \\ & \quad + \frac{{F_{0} \left( {2\kappa_{1}^{2} M^{6} - 2\kappa_{1}^{4} M^{4} } \right) + e^{{ - \kappa_{1} h}} \left( { - 2h\kappa_{1} \kappa_{2} M^{4} - 2\kappa_{2} M^{4} } \right)}}{{2\kappa_{1}^{2} M^{4} \left( {\kappa_{1}^{2} - M^{2} } \right)(hM\cosh (hM) - \sinh (hM))}}, \\ C_{3} &= \frac{{\kappa_{1}^{2} \kappa_{3} - \kappa_{2} M^{2} }}{{M^{4} \kappa_{1}^{2} }}, \\ \end{aligned} $$
$$ \begin{aligned} C_{4} &= \frac{{F_{0} \left( {2\kappa_{1}^{4} M^{4} \cosh (hM) - 2\kappa_{1}^{2} M^{6} \cosh (hM)} \right) + e^{{ - \kappa_{1} h}} \kappa_{8} - 2M^{2} (\kappa_{10} + \kappa_{2} M^{2} \cosh (hM)}}{{2\kappa_{1}^{2} M^{3} \left( {\kappa_{1}^{2} - M^{2} } \right)(hM\cosh (hM) - \sinh (hM))}} \\ & \quad + \frac{{\kappa_{1}^{2} \kappa_{3} \left( {M^{2} - \kappa_{1}^{2} } \right)\left( {\left( {h^{2} M^{2} + 2} \right)\cosh (hM) - 2hM\sinh (hM) - 2} \right)}}{{2\kappa_{1}^{2} M^{3} \left( {\kappa_{1}^{2} - M^{2} } \right)(hM\cosh (hM) - \sinh (hM))}}, \\ \end{aligned} $$
$$ \begin{aligned} C_{5} &= \frac{{\kappa_{11} }}{{6\kappa_{1} M^{4} \left( {M - 2\kappa_{1} } \right)\left( {M - \kappa_{1} } \right)\left( {2M - \kappa_{1} } \right)\left( {\kappa_{1} + M} \right)}} + \frac{{\kappa_{6} \left( {2\kappa_{1} - M} \right)}}{{M^{2} \left( {M - 2\kappa_{1} } \right)\left( {M - \kappa_{1} } \right)\left( {\kappa_{1} + M} \right)}} \\ & \quad - \frac{{\kappa_{7} }}{{M^{2} \left( {M - 2\kappa_{1} } \right)\left( {2\kappa_{1} + M} \right)}}, \\ \end{aligned} $$
$$ \begin{aligned} C_{6} &= \frac{{C_{2} \left( {2C_{1} M^{4} (2hM\cosh (2hM) - \sinh (2hM)) + 3\kappa_{3} \left( {\left( {2h^{2} M^{2} + 5} \right)\sinh (hM) - 5hM\cosh (hM)} \right)} \right)}}{{6M^{2} (hM\cosh (hM) - \sinh (hM))}} \\ & \quad - \frac{{\kappa_{1} \kappa_{5} e^{{h\left( {M - \kappa_{1} } \right)}} \left( {4h\kappa_{1}^{4} + M^{3} (hM - 1) + 4\kappa_{1}^{3} - \kappa_{1} M^{2} } \right) + e^{{ - \kappa_{1} h}} (\kappa_{5} \kappa_{1}^{3} Me^{hM} (5hM - 4) + \kappa_{12} )}}{{\kappa_{17} \kappa_{1}^{2} \left( {2M - \kappa_{1} } \right)}} \\ & \quad + \frac{{\kappa_{6} \left( {\kappa_{1}^{2} \left( {hM\left( {\kappa_{1}^{2} + 2M^{2} - 3\kappa_{1} M} \right)\sinh (hM) - \left( {\kappa_{1}^{2} + 2M^{2} - 3\kappa_{1} M} \right)\cosh (hM) + \kappa_{1}^{2} } \right) - 2M^{4} } \right)}}{{\kappa_{1}^{2} M^{2} \left( {M - \kappa_{1} } \right)^{2} \left( {2M - \kappa_{1} } \right)\left( {\kappa_{1} + M} \right)(hM\cosh (hM) - \sinh (hM))}} \\ & \quad + \frac{{\kappa_{1}^{2} \left( {M^{2} - 4\kappa_{1}^{2} } \right)\,\kappa_{14} + 3M^{4} \kappa_{7} e^{{ - 2\kappa_{1} h}} \left( {2h\kappa_{1} + 1} \right)}}{{12\kappa_{1}^{2} \left( {M^{2} - 4\kappa_{1}^{2} } \right)\,M^{4} (hM\cosh (hM) - \sinh (hM))}} + \frac{{\left( {\kappa_{1} M\left( {M - 2\kappa_{1} } \right)\,\left( {2\kappa_{1} + M} \right)\kappa_{15} + \kappa_{16} } \right)}}{{4\kappa_{1}^{2} M^{2} \left( {2M - \kappa_{1} } \right)\kappa_{17} }} \\ & \quad + \frac{{\left( {\kappa_{5} \left( {M^{2} - 4\kappa_{1}^{2} } \right) + \kappa_{1} \kappa_{7} \left( {2M - \kappa_{1} } \right)} \right)(Mh\sinh (hM) - \cosh (hM))}}{{\kappa_{1} M^{2} \left( {M - 2\kappa_{1} } \right)\left( {2M - \kappa_{1} } \right)\left( {2\kappa_{1} + M} \right)(hM\cosh (hM) - \sinh (hM))}} \\ & \quad + \frac{{F_{1} }}{\sinh (hM) - hM\cosh (hM)}, \\ \end{aligned} $$
$$ C_{7} = - \frac{{2C_{1} \kappa_{3} }}{{M^{2} }} - \frac{{\left( {4\left( {\kappa_{5} + \kappa_{6} } \right) + \kappa_{7} } \right)M^{2} + \kappa_{4} \left( {M - \kappa_{1} } \right)^{2} }}{{4M^{4} \left( {M - \kappa_{1} } \right)^{2} }} - \frac{{\left( {4\kappa_{6} + \kappa_{7} } \right)\left( {M - 2\kappa_{1} } \right)}}{{4\kappa_{1}^{2} M\left( {M - \kappa_{1} } \right)^{2} }}, $$
$$ \begin{aligned} C_{8} &= - \frac{{e^{{ - \kappa_{1} h}} \kappa_{6} \left( {\kappa_{1} \sinh (hM) + M\cosh (hM)} \right)}}{{\kappa_{1}^{2} \left( {M^{2} - \kappa_{1}^{2} } \right)(hM\cosh (hM) - \sinh (hM))}} - \frac{{\left( {4\kappa_{6} + \kappa_{7} } \right)\left( {6\kappa_{1} M^{3} - 3M^{4} } \right)\cosh (hM)}}{{12\kappa_{1}^{2} M^{3} \left( {M - \kappa_{1} } \right)^{2} (hM\cosh (hM) - \sinh (hM))}} \\ & \quad + \frac{{3M^{2} \left( {F_{1} M^{2} \cosh (hM) - 4C_{1} \kappa_{3} \sinh^{4} \left( {\tfrac{hM}{2}} \right)} \right) + \kappa_{3} C_{2} (2hM - \sinh (2hM)) - 2\kappa_{18} }}{{6M^{3} \left( {hM\cosh (hM) - \sinh (hM)} \right)}} \\ & \quad + \frac{{2M\kappa_{1} \kappa_{4} \cosh (hM)(\cosh (2hM) - 5) - \kappa_{1}^{2} \kappa_{4} \cosh (hM)(\cosh (2hM) - 5)}}{{12M^{3} \left( {M - \kappa_{1} } \right)^{2} (hM\cosh (hM) - \sinh (hM))}} \\ & \quad - \frac{{3e^{{ - 2h\kappa_{1} }} \kappa_{19} + 64\kappa_{1}^{2} C_{2} C_{1} M^{3} \sinh^{3} \left( {\tfrac{hM}{2}} \right)\cosh^{3} \left( {\tfrac{hM}{2}} \right)}}{{12\kappa_{1}^{2} (hM\cosh (hM) - \sinh (hM))}} \\ & \quad - \frac{{\cosh (hM)\left( {\kappa_{4} (\cosh (2hM) - 5) - 3\left( {4\left( {\kappa_{5} + \kappa_{6} } \right) + \kappa_{7} } \right)} \right)}}{{12M\left( {M - \kappa_{1} } \right)^{2} (hM\cosh (hM) - \sinh (hM))}} \\ & \quad - \frac{{\left( {4\kappa_{6} + \kappa_{7} } \right)\left( {6\kappa_{1} M^{3} - 3M^{4} } \right)\cosh (hM)}}{{12\kappa_{1}^{2} M^{3} \left( {M - \kappa_{1} } \right)^{2} (hM\cosh (hM) - \sinh (hM))}}, \\ \end{aligned} $$
$$ \begin{aligned} C_{9} &= \frac{1}{6}hR_{m} \left( { - 6C_{1} - \frac{{6\kappa_{1} \kappa_{2} }}{{\kappa_{1}^{3} M^{2} - \kappa_{1}^{5} }}} \right) - \frac{1}{6}R_{m} \left( {3h^{2} \left( {A - C_{4} } \right) + \kappa_{23} } \right), \\ C_{10} &= - \frac{1}{6}R_{m} \left( { - 6C_{1} - \frac{{6\kappa_{1} \kappa_{2} }}{{\kappa_{1}^{3} M^{2} - \kappa_{1}^{5} }}} \right), \\ C_{11} &= \frac{{R_{m} }}{12}\left( { - \frac{{24e^{{ - \kappa_{1} h}} \left( {\kappa_{1}^{2} \kappa_{5} \left( {\kappa_{1} + M} \right)e^{hM} - \kappa_{6} \left( {2M - \kappa_{1} } \right)\left( {M - \kappa_{1} } \right)^{2} } \right)}}{{\kappa_{1}^{3} \left( {2M - \kappa_{1} } \right)\left( {\kappa_{1} + M} \right)\left( {\kappa_{1} - M} \right)^{3} }}} \right. \\ & \quad + \kappa_{22} + 12C_{8} h^{2} - 4C_{1} C_{2} M\cosh (2hM) \\ & \quad + h\left. {\left( {\kappa_{20} + \frac{{24\left( {\kappa_{6} \left( {\kappa_{1} - 2M} \right)\left( {M - \kappa_{1} } \right)^{2} + \kappa_{5} \kappa_{1}^{2} \left( {\kappa_{1} + M} \right)} \right)}}{{\kappa_{1}^{2} \left( {\kappa_{1} + M} \right)\left( {\kappa_{1} - 2M} \right)\,\left( {\kappa_{1} - M} \right)^{3} }}} \right)} \right), \\ C_{12} &= - \frac{1}{24}R_{m} \left( {\frac{{24(\kappa_{6} \left( {M - \kappa_{1} } \right)^{2} \left( {\kappa_{1} - 2M} \right) + \kappa_{5} \kappa_{1}^{2} \left( {\kappa_{1} + M} \right))}}{{\kappa_{1}^{2} \left( {\kappa_{1} + M} \right)\left( {\kappa_{1} - 2M} \right)\left( {\kappa_{1} - M} \right)^{3} }} + \kappa_{24} - 24C_{5} } \right). \\ \end{aligned} $$

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Akram, S., Afzal, Q. Effects of thermal and concentration convection and induced magnetic field on peristaltic flow of Williamson nanofluid in inclined uniform channel. Eur. Phys. J. Plus 135, 857 (2020). https://doi.org/10.1140/epjp/s13360-020-00869-9

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