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A Novel Formulation for MHD Slip Flow of Elastico-Viscous Fluid Induced by Peristaltic Waves with Heat/Mass Transfer Effects

  • Research Article-Mechanical Engineering
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Abstract

Prime motive of this research is to formulate and analyze the elastico-viscous fluid flow triggered by sinusoidal waves progressing across the walls of symmetric channel. The research is based on the Walters-B model that depicts normal stress effects attributed to the fluid elasticity. Here, we endeavor to derive correct form of equations representing peristalsis of Walters-B liquid. The said model is investigated by employing generic Robin-type boundary conditions signifying partial slip effects. The case of hydrodynamic flow has been separately analyzed. Series expansions are considered about a parameter \(\delta\), measuring the ratio of wavelength of peristaltic wave to the channel width. Emphasis is paid to the influence of new attributes of the problem, namely viscoelasticity and wall slip on the pressure rise, axial flow, temperature and concentration of species. Heat transfer coefficient is computed graphically across the channel wall for a variety of controlling flow parameters. Even at small Reynolds number, a marked axial velocity is predicted, provided that elastico-viscous fluid parameter is sufficiently large. Moreover, heat transfer rate at channel walls is lowered due to the inclusion of viscoelasticity.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Fatima Jinnah Women University, The Mall, Rawalpindi, 46000, Pakistan

    Javeriah Rani & S. Hina

  2. School of Natural Sciences, National University of Sciences and Technology, Islamabad, 44000, Pakistan

    Meraj Mustafa

Authors
  1. Javeriah Rani
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  2. S. Hina
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  3. Meraj Mustafa
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Correspondence to Meraj Mustafa.

Appendix

Appendix

Here, we provide expressions of the coefficients appearing in the solutions (given in Sect. 3):

$$C_{2} = \frac{{F_{0} }}{2\eta } - \frac{1}{{2\eta^{2} }}\left( {\frac{{\left( {2\eta + F_{0} } \right)\sinh M\eta }}{{ - M\cosh M\eta + \sinh M\eta \left( {\frac{1}{\eta } - \beta_{1} M^{2} } \right)}}} \right),$$
$$C_{4} = \frac{1}{2\eta }\left( {\frac{{2\eta + F_{0} }}{{ - M\cosh M\eta + \sinh M\eta \left( {\frac{1}{\eta } - \beta_{1} M^{2} } \right)}}} \right),$$
$$D_{1} = \frac{1}{2} - \frac{{{\text{Br}}M^{4} C_{4}^{2} \eta }}{4}\left( {\frac{\eta }{2} + \beta_{2} } \right) + \frac{{{\text{Br}}M^{2} C_{4}^{2} }}{4}\left( {\frac{1}{2}\cosh 2M\eta + \beta_{2} M\sinh 2M\eta } \right),$$
$$E_{1} = \frac{1}{2} + \frac{{ {\text{SrBr}}M^{4} C_{4}^{2} \eta }}{8}\left( {\frac{\eta }{2} + \beta_{3} } \right) - \frac{{ {\text{SrBr}}M^{2} C_{4}^{2} }}{4}\left( {\frac{1}{2}\cosh 2M\eta + \beta_{3} M\sinh 2M\eta } \right),$$
$$\begin{aligned} & E_{2} = \frac{1}{{2\left( {\eta + \beta_{3} } \right)}},\,l_{1} = M^{2} C_{2} C_{4x} \left( {\text{Re} + kM^{2} } \right),\, \\ & l_{2} = - M^{3} C_{2x} C_{4} \left( {\text{Re} + kM^{2} } \right),\, D_{2} = \frac{1}{{2\left( {\eta + \beta_{2} } \right)}}, \\ \end{aligned}$$
$$A_{2} = \frac{{F_{1} }}{2\eta } - \frac{{A_{4} }}{\eta }\sinh M\eta - \frac{{l_{2} }}{{4M^{3} }}\eta \sinh M\eta - \cosh M\eta \left( {\frac{{l_{1} }}{{2M^{3} }} - \frac{{5l_{2} }}{{4M^{4} }}} \right),$$
$$\begin{aligned} & A_{4} = \frac{1}{{2M\cosh M\eta + \sinh M\eta \left( {2\beta_{1} M^{2} - \frac{2}{\eta }} \right)}} \\ & \left[ { - \frac{{F_{1} }}{\eta } + \eta \sinh M\eta \left( {\frac{{2l_{2} }}{{M^{3} }} - \frac{{l_{1} }}{{M^{2} }}} \right) - \frac{{l_{2} \eta^{2} }}{{2M^{2} }}\cosh M\eta } \right. \\ & \quad - \beta_{1} \left\{ {\eta \cosh M\eta \left( {\frac{{l_{1} }}{M} - \frac{{l_{2} }}{{2M^{2} }} + 2kM^{3} C_{2x} C_{4} } \right)} \right. \\ & \quad + \sinh M\eta \left( {\frac{{2l_{1} }}{{M^{2} }} + \frac{{l_{2} \eta^{2} }}{2M} - \frac{{4l_{2} }}{{M^{3} }} - 4kM^{2} C_{2x} C_{4} - 2kM^{2} C_{2} C_{4x} } \right) \\ & \quad \left. {\left. { - \,4kM^{3} C_{4x} C_{4} \cosh M\eta \sinh M\eta } \right\}} \right], \\ \end{aligned}$$
$$p_{1} = \Pr \text{Re} C_{2} D_{1x} , p_{2} = \Pr \text{Re} \left( {C_{2} D_{2x} - C_{2x} D_{2} } \right), p_{3} = \frac{{\Pr \text{Re} {\text{Br}}C_{4} M^{4} }}{4}\left( {C_{2} C_{4x} - C_{4} C_{2x} } \right),$$
$$p_{4} = \Pr \text{Re} MC_{4} D_{1x} , p_{5} = \Pr \text{Re} MC_{4} D_{2x} , p_{6} = \frac{{\Pr \text{Re} {\text{Br}}M^{5} C_{4}^{2} C_{4x} }}{4},$$
$$p_{7} = - \frac{{\Pr \text{Re} {\text{Br}}M^{2} C_{2} C_{4} C_{4x} }}{4}, p_{8} = - \frac{{\Pr \text{Re} {\text{Br}}M^{3} C_{4}^{2} C_{4x} }}{4}, p_{9} = - \Pr \text{Re} D_{2} C_{4x} ,$$
$$p_{10} = - \frac{{\Pr \text{Re} {\text{Br}}M^{4} C_{4}^{2} C_{4x} }}{4}, p_{11} = \frac{{\Pr \text{Re} {\text{Br}}M^{3} C_{4}^{2} C_{4x} }}{4},$$
$$p_{12} = - {\text{Br}}C_{4} \left( {2M^{4} A_{4} + 2l_{1} - \frac{{4l_{2} }}{M} - kM^{4} C_{2} C_{4x} } \right), p_{13} = - \frac{{{\text{Br}}Ml_{2} C_{4} }}{2},$$
$$p_{14} = \frac{{\Pr \text{Re} {\text{Br}}M^{3} C_{4}^{2} C_{2x} }}{4} + \frac{{{\text{Br}}C_{4} }}{2}\left( {\frac{{l_{2} }}{2} - Ml_{1} - kM^{5} C_{4} C_{2x} } \right),$$
$$\begin{aligned} & G_{0} = - \frac{1}{2}\left[ {\left( {p_{1} - \frac{{p_{12} }}{2}} \right)\left( {\eta \beta_{2} + \frac{{\eta^{2} }}{2}} \right) + \left( {p_{3} - \frac{{p_{13} }}{{8M^{2} }}} \right)\left( {\frac{{2\eta^{3} \beta_{2} }}{3} + \frac{{\eta^{4} }}{6}} \right)} \right. \\ & \quad + \cosh M\eta \left( {\frac{{2p_{4} }}{{M^{2} }} + \frac{{p_{8} }}{{M^{2} }} + \frac{{2p_{6} }}{{M^{2} }}\left( {\eta^{2} + \frac{6}{{M^{2} }}} \right) - \frac{{4p_{10} }}{{M^{3} }} - \frac{{p_{11} }}{{M^{2} }} + \beta_{2} \left( { - \frac{{4p_{6} \eta }}{{M^{2} }} + \frac{{2p_{10} \eta }}{M}} \right)} \right) \\ & \quad + \sinh M\eta \left( { - \frac{{8p_{6} \eta }}{{M^{3} }} + \frac{{2p_{10} \eta }}{{M^{2} }} + \beta_{2} \left( {\frac{{2p_{4} }}{M} + \frac{{2p_{6} }}{M}\left( {\eta^{2} + \frac{2}{{M^{2} }}} \right) + \frac{{p_{8} }}{M} - \frac{{2p_{10} }}{{M^{2} }} - \frac{{p_{11} }}{M}} \right)} \right) \\ & \quad + \sinh 2M\eta \left( {\frac{{p_{14} \eta }}{{2M^{2} }} - \frac{{p_{13} }}{{2M^{3} }}\eta + \beta_{2} \left( {\frac{{p_{7} }}{M} + \frac{{p_{12} }}{2M} + \frac{{p_{13} }}{2M}\left( {\eta^{2} + \frac{1}{{2M^{2} }}} \right) - \frac{{p_{14} }}{{2M^{2} }}} \right)} \right) \\ & \quad + \cosh 2M\eta \left( {\frac{{p_{7} }}{{2M^{2} }} + \frac{{p_{12} }}{{4M^{2} }} + \frac{{p_{13} }}{{4M^{2} }}\left( {\eta^{2} + \frac{3}{{2M^{2} }}} \right) - \frac{{p_{14} }}{{2M^{3} }} + \beta_{2} \left( {\frac{{p_{14} \eta }}{M} - \frac{{p_{13} \eta }}{{2M^{2} }}} \right)} \right) \\ & \quad \left. { + \cosh 3M\eta \left( {\frac{{p_{8} }}{{9M^{2} }} + \frac{{p_{11} }}{{9M^{2} }}} \right) + \sinh 3M\eta \left( {\beta_{2} \left( {\frac{{p_{8} }}{3M} + \frac{{p_{11} }}{3M}} \right)} \right)} \right], \\ \end{aligned}$$
$$\begin{aligned} & G_{1} = \frac{1}{{2\left( {\eta + \beta_{2} } \right)}} \\ & \left[ { - p_{2} \eta^{2} \left( {\beta_{2} + \frac{\eta }{3}} \right) + \cosh M\eta \left( { - \frac{{2p_{5} \eta }}{{M^{2} }} + \frac{{2p_{5} \beta_{2} }}{{M^{2} }} - \frac{{2p_{9} \beta_{2} }}{M}} \right)} \right. \\ & \quad \left. { + \sinh M\eta \left( {\frac{{4p_{5} }}{{M^{3} }} - \frac{{2p_{9} }}{{M^{2} }} - \frac{{2p_{5} \beta_{2} \eta }}{M}} \right)} \right], \\ \end{aligned}$$
$$q_{1} = \text{Re} {\text{Sc}}C_{2} E_{1x} - \frac{{ {\text{Sr}}}}{2 }\left( {p_{1} - \frac{{p_{12} }}{2}} \right),q_{2} = \text{Re} {\text{Sc}}\left( {C_{2} E_{2x} - C_{2x} E_{2} } \right) - {\text{Sr}}p_{2} ,$$
$$q_{3} = \frac{{\text{Re} {\text{Sc SrBr}}M^{4} C_{4} }}{8 }\left( {C_{4} C_{2x} - C_{2} C_{4x} } \right) - {\text{Sr}}\left( {p_{3} - \frac{{p_{13} }}{{8M^{2} }}} \right),$$
$$q_{4} = \text{Re} {\text{Sc}}MC_{4} E_{1x} - {\text{Sc}}\left( {p_{4} + \frac{{p_{8} }}{2} - \frac{{p_{11} }}{2}} \right),q_{5} = \text{Re} {\text{Sc}}MC_{4} E_{2x} - {\text{Sr}}p_{5} ,$$
$$q_{6} = - \frac{{\text{Re} {\text{Sc }}N_{t} {\text{Br}}M^{5} C_{4}^{2} C_{4x} }}{{8 N_{b} }} - {\text{Sr}}p_{6} ,q_{7} = \frac{{\text{Re} {\text{Sc SrBr}}M^{2} C_{2} C_{4} C_{4x} }}{4} - {\text{Sr}}\left( {p_{7} + \frac{{p_{12} }}{2}} \right),$$
$$q_{8} = - \frac{\text{Sr}}{2 }p_{13} ,q_{9} = \frac{{\text{Re} {\text{Sc SrBr}}M^{3} C_{4}^{2} C_{4x} }}{4 },q_{10} = - \text{Re} {\text{Sc}}E_{2} C_{4x} - {\text{Sr}}p_{9} ,$$
$$q_{11} = \frac{{\text{Re} {\text{Sc}} N_{t} {\text{Br}}M^{4} C_{4}^{2} C_{4x} }}{{8 N_{b} }} - {\text{Sr}}p_{10} , q_{12} = - \frac{{\text{Re} {\text{Sc SrBr}}M^{3} C_{4}^{2} C_{2x} }}{4} - {\text{Sr}}p_{14} ,$$
$$q_{13} = - \frac{{\text{Re} {\text{Sc SrBr}}M^{3} C_{4}^{2} C_{4x} }}{{4 {\text{Sr}}}},\,q_{14} = - \frac{\text{Sr}}{ 2}\left( {p_{8} + p_{11} } \right),$$
$$\begin{aligned} & H_{0} = - \frac{1}{2}\left[ {q_{1} \eta \left( {\beta_{3} + \frac{\eta }{2}} \right) + \frac{{q_{3} \eta^{3} }}{3}\left( {2\beta_{3} + \frac{\eta }{2}} \right)} \right. \\ & \quad + \cosh M\eta \left( {\frac{{2q_{4} }}{{M^{2} }} + \frac{{q_{9} }}{{M^{2} }} + \frac{{2q_{6} }}{{M^{2} }}\left( {\eta^{2} + \frac{6}{{M^{2} }}} \right) - \frac{{4q_{11} }}{{M^{3} }} - \frac{{q_{13} }}{{M^{2} }} + \beta_{3} \left( { - \frac{{4q_{6} \eta }}{{M^{2} }} + \frac{{2q_{11} \eta }}{M}} \right)} \right) \\ & \quad + \sinh M\eta \left( { - \frac{{8q_{6} \eta }}{{M^{3} }} + \frac{{2q_{11} \eta }}{{M^{2} }} + \beta_{3} \left( {\frac{{2q_{4} }}{M} + \frac{{2q_{6} }}{M}\left( {\eta^{2} + \frac{2}{{M^{2} }}} \right) + \frac{{q_{9} }}{M} - \frac{{2q_{11} }}{{M^{2} }} - \frac{{q_{13} }}{M}} \right)} \right) \\ & \quad + \cosh 2M\eta \left( {\frac{{q_{7} }}{{2M^{2} }} + \frac{{q_{8} }}{{2M^{2} }}\left( {\eta^{2} + \frac{3}{{2M^{2} }}} \right) - \frac{{q_{12} }}{{2M^{3} }} + \beta_{3} \left( {\frac{{q_{12} \eta }}{M} - \frac{{q_{8} \eta }}{{M^{2} }}} \right)} \right) \\ & \quad + \sinh 2M\eta \left( {\frac{{q_{12} \eta }}{{2M^{2} }} - \frac{{q_{8} \eta }}{{M^{3} }} + \beta_{3} \left( {\frac{{q_{7} }}{M} - \frac{{q_{12} }}{{2M^{2} }} + \frac{{q_{8} }}{M}\left( {\eta^{2} + \frac{1}{{2M^{2} }}} \right)} \right)} \right) \\ & \quad \left. { + \cosh 3M\eta \left( {\frac{{q_{9} }}{{9M^{2} }} + \frac{{q_{13} }}{{9M^{2} }} + \frac{{2q_{14} }}{{9M^{2} }}} \right) + \sinh 3M\eta \left( {\beta_{3} \left( {\frac{{q_{9} }}{3M} + \frac{{q_{13} }}{3M} + \frac{{2q_{14} }}{3M}} \right)} \right)} \right], \\ \end{aligned}$$
$$\begin{aligned} & H_{1} = \frac{1}{{2\left( {\eta + \beta_{3} } \right)}} \\ & \left[ { - q_{2} \eta^{2} \left( {\beta_{3} + \frac{\eta }{3}} \right) + \cosh M\eta \left( { - \frac{{2q_{5} \eta }}{{M^{2} }} - \beta_{3} \left( { - \frac{{2q_{5} }}{{M^{2} }} + \frac{{2q_{10} }}{M}} \right)} \right)} \right. \\ & \quad \left. { + \,\sinh M\eta \left( {\frac{{4q_{5} }}{{M^{3} }} - \frac{{2q_{10} }}{{M^{2} }} - \frac{{2q_{5} \beta_{3} \eta }}{M}} \right)} \right]. \\ \end{aligned}$$

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Rani, J., Hina, S. & Mustafa, M. A Novel Formulation for MHD Slip Flow of Elastico-Viscous Fluid Induced by Peristaltic Waves with Heat/Mass Transfer Effects. Arab J Sci Eng 45, 9213–9225 (2020). https://doi.org/10.1007/s13369-020-04722-0

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