Slippage phenomenon in hydromagnetic peristaltic rheology with hall current and viscous dissipation

Aamir Ali; Sana Mumraiz; Hafiz Junaid Anjum; Saleem Asghar; Muhammad Awais

doi:10.1515/ijnsns-2019-0226

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Slippage phenomenon in hydromagnetic peristaltic rheology with hall current and viscous dissipation

Aamir Ali , Sana Mumraiz , Hafiz Junaid Anjum , Saleem Asghar and Muhammad Awais

Published/Copyright: June 15, 2021

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 23 Issue 5

Abstract

The current research explores the slippage phenomenon in hydromagnetic peristaltic activity of a non-Newtonian fluid with porous media in an asymmetric channel. The analysis is performed under the influence of thermal radiation, Hall current, Joule heating and viscous dissipation. The problem is formulated with the assumption of small Reynolds number and large wavelength. Analytical solutions are achieved through perturbation technique and the impacts of involved influential parameters are examined through graphs. It is observed that the pressure gradient rises with fourth grade fluid parameter and decreases with increasing phase difference. The pressure rise increases in pumping regime and decreases in co-pumping regime for increasing magnetic field parameter, whereas it has opposite effects for hall parameter. It is also noted that the velocity drops in the middle of the channel, while it increases near the boundaries for growing slip parameter and magnetic field parameters and it has the opposite behavior for hall and permeability parameters. The slip parameter increases the temperature of the fluid and decreases the concentration. Also, in trapping phenomena, the bolus size reduces by enlarging Deborah parameter. The present research has profound use in biomedical science, polymer technology and artificial heart polishing.

Keywords: Hall current; MHD; peristaltic flow; porous medium; slip conditions; viscous dissipation

Corresponding author: Aamir Ali, Department of Mathematics, COMSATS University Islamabad, Attock Campus, Kamra Road, Attock 43600, Pakistan, E-mail: aamir_ali@cuiatk.edu.pk

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

ψ 0 = C 1 e y α α + C 2 e − y α α + C 3 + y C 4 ,

θ 0 = 3 Br e − 2 y α 4 3 + 4 Rd α 2 − C 2 2 1 + β − C 1 2 e 4 y α 1 + β + 8 C 2 e y α α C 4 + β + C 4 β − 8 C 1 e 3 y α α C 4 + β + C 4 β − 2 e 2 y α y 2 α − 2 C 1 C 2 1 + β + α C 4 2 + 1 + C 4 2 β + C 5 + y C 6 ,

ϕ 0 = 3 BrScSr e − 2 y α 4 3 + 4 Rd α 2 C 2 2 1 + β + C 1 2 e 4 y α 1 + β − 8 C 2 e y α α C 4 + β + C 4 β + 8 C 1 e 3 y α α C 4 + β + C 4 β + 2 e 2 y α y 2 α − 2 C 1 C 2 1 + β + α C 4 2 + 1 + C 4 2 β + C 7 + y C 8 ,

ψ 1 = e − 3 y α 4 α − C 2 3 − C 1 3 e 6 y α − 2 e 4 y α − 2 C 9 + 3 C 1 2 C 2 − 5 + 2 y α + 2 e 2 y α 2 C 10 + 3 C 1 C 2 2 5 + 2 y α + C 11 + y C 12 ,

θ 1 = Br e − 4 y α 32 3 + 4 Rd α 2 − 16 C 2 3 1 + C 4 e y α α β + 16 C 1 3 1 + C 4 e 7 y α α β + 96 e 3 y α 2 C 10 1 + C 4 + C 2 2 C 12 + 3 C 1 C 2 1 + C 4 7 + 2 y α α β + 96 e 5 y α − 2 C 9 1 + C 4 + C 1 − 2 C 12 + 3 C 1 C 2 1 + C 4 − 7 + 2 y α α β + 96 e 4 y α y 2 α C 1 C 10 α − β + 9 C 1 2 C 2 3 α − β + C 2 C 9 α − β + C 12 1 + C 4 α β + 12 C 1 e 6 y α − 4 C 9 α + β + C 1 2 C 2 × − 25 α + 12 y α 3 / 2 − 33 β + 12 y α β − 12 C 2 e 2 y α 4 C 10 α + β + C 1 C 2 2 25 α + 12 y α 3 / 2 + 33 β + 12 y α β + 3 C 2 4 5 α + 3 β + 3 C 1 4 e 8 y α 5 α + 3 β + C 13 + y C 14 ,

ϕ 1 = − BrScSr e − 4 y α 32 3 + 4 Rd α 2 − 16 C 2 3 1 + C 4 e y α α β + 16 C 1 3 1 + C 4 e 7 y α α β + 96 e 3 y α 2 C 10 1 + C 4 + C 2 2 C 12 + 3 C 1 C 2 1 + C 4 7 + 2 y α α β + 96 e 5 y α − 2 C 9 1 + C 4 + C 1 − 2 C 12 + 3 C 1 C 2 1 + C 4 − 7 + 2 y α α β + 96 e 4 y α y 2 α C 1 C 10 α − β + 9 C 1 2 C 2 3 α − β + C 2 C 9 α − β + C 12 1 + C 4 α β + 12 C 1 e 6 y α − 4 C 9 α + β + C 1 2 C 2 × − 25 α + 12 y α 3 / 2 − 33 β + 12 y α β − 12 C 2 e 2 y α 4 C 10 α + β + C 1 C 2 2 25 α + 12 y α 3 / 2 + 33 β + 12 y α β + 3 C 2 4 5 α + 3 β + 3 C 1 4 e 8 y α 5 α + 3 β + C 15 + y C 16 ,

where the values of the constants are:

C 1 = F 0 + h 1 − h 2 α e h 1 α 2 − h 1 − h 2 α 1 + α β 1 + e h 2 α − 2 + h 1 − h 2 α − 1 + α β 1 ,

C 2 = − e h 1 + h 2 α F 0 + h 1 − h 2 α e h 1 α 2 − h 1 − h 2 α 1 + α β 1 + e h 2 α − 2 + h 1 − h 2 α − 1 + α β 1 ,

C 3 = h 1 + h 2 e h 1 α 2 + F 0 α + α β 1 + e h 2 α − 2 + F 0 α − α β 1 2 e h 1 α 2 − h 1 − h 2 α 1 + α β 1 + e h 2 α − 2 + h 1 − h 2 α − 1 + α β 1 ,

C 4 = − e h 1 α 2 + F 0 α + α β 1 + e h 2 α − 2 + F 0 α − α β 1 e h 1 α 2 − h 1 − h 2 α 1 + α β 1 + e h 2 α − 2 + h 1 − h 2 α − 1 + α β 1 ,

C 5 = h 2 − β 2 h 1 − h 2 + 2 β 2 − A 1 4 3 + 4 Rd α 2 − A 3 3 + 4 Rd α − h 1 + β 2 h 1 − h 2 + 2 β 2 − 1 − A 2 4 3 + 4 Rd α 2 + A 4 3 + 4 Rd α ,

where

A 1 = 3 Br e − 2 h 1 α C 2 2 1 + β − 4 C 2 e h 1 α α 2 β + 2 C 4 1 + β + C 1 e h 1 α h 1 2 α 1 + β + e 2 h 1 α C 1 2 e 2 h 1 α 1 + β + 8 C 1 e h 1 α α C 4 + β + C 4 β + 2 h 1 2 α 2 β + 2 C 4 β + C 4 2 1 + β , A 2 = 3 Br e − 2 h 2 α C 2 2 1 + β − 4 C 2 e h 2 α α 2 β + 2 C 4 1 + β + C 1 e h 2 α h 2 2 α 1 + β + e 2 h 2 α C 1 2 e 2 h 2 α 1 + β + 8 C 1 e h 2 α α C 4 + β + C 4 β + 2 h 2 2 α 2 β + 2 C 4 β + C 4 2 1 + β , A 3 = 3 Br − 2 C 1 C 2 h 1 + C 4 2 h 1 α − 2 C 1 C 2 h 1 β + h 1 α β + 2 C 4 h 1 α β + C 4 2 h 1 α β − C 2 2 e − 2 h 1 α 1 + β 2 α + C 1 2 e 2 h 1 α 1 + β 2 α + 2 C 2 e − h 1 α C 4 + β + C 4 β + 2 C 1 e h 1 α C 4 + β + C 4 β β 2 , A 4 = 3 Br − 2 C 1 C 2 h 2 + C 4 2 h 2 α − 2 C 1 C 2 h 2 β + h 2 α β + 2 C 4 h 2 α β + C 4 2 h 2 α β − C 2 2 e − 2 h 2 α 1 + β 2 α + C 1 2 e 2 h 2 α 1 + β 2 α + 2 C 2 e − h 2 α C 4 + β + C 4 β + 2 C 1 e h 2 α C 4 + β + C 4 β β 2 ,

C 6 = e − 2 h 1 + h 2 α 4 3 + 4 Rd α 2 h 1 − h 2 + 2 β 2 − 4 e 2 h 1 + h 2 α 3 + 4 Rd α 2 − 3 Br − 2 e 2 h 1 + h 2 α h 1 + h 2 α 2 × C 4 2 + 1 + C 4 2 β h 1 − h 2 + 2 β 2 − 8 C 1 e 2 h 1 + h 2 α α C 4 + β + C 4 β e h 2 α − 1 + α β 2 + e h 1 α 1 + α β 2 + C 2 2 1 + β e 2 h 2 α − 1 + 2 α β 2 + e 2 h 1 α 1 + 2 α β 2 − C 1 2 1 + β × e 2 h 1 + 2 h 2 α − 1 + 2 α β 2 + e 2 2 h 1 + h 2 α 1 + 2 α β 2 − 4 C 2 − C 1 e 2 h 1 + h 2 α h 1 + h 2 α 1 + β h 1 − h 2 + 2 β 2 + 2 α β e h 1 + 2 h 2 α − 1 + α β 2 + e 2 h 1 + h 2 α 1 + α β 2 + 2 C 4 α 1 + β + e 2 h 1 + h 2 α 1 + α β 2 ,

C 7 = h 2 − β 3 h 1 − h 2 + 2 β 2 A 5 4 3 + 4 Rd α 2 + A 7 3 + 4 Rd α − h 1 + β 3 h 1 − h 2 + 2 β 2 − 1 + A 6 4 3 + 4 Rd α 2 − A 8 3 + 4 Rd α ,

where

A 5 = 3 Br e − 2 h 1 α ScSr C 2 2 1 + β − 4 C 2 e h 1 α α 2 β + 2 C 4 1 + β + C 1 e h 1 α h 1 2 α 1 + β + e 2 h 1 α C 1 2 e 2 h 1 α 1 + β + 8 C 1 e h 1 α α C 4 + β + C 4 β + 2 h 1 2 α 2 β + 2 C 4 β + C 4 2 1 + β ,

A 6 = 3 Br e − 2 h 2 α ScSr C 2 2 1 + β − 4 C 2 e h 2 α α 2 β + 2 C 4 1 + β + C 1 e h 2 α h 2 2 α 1 + β + e 2 h 2 α C 1 2 e 2 h 2 α 1 + β + 8 C 1 e h 2 α α C 4 + β + C 4 β + 2 h 2 2 α 2 β + 2 C 4 β + C 4 2 1 + β ,

A 7 = 3 BrScSr − 2 C 1 C 2 h 1 + C 4 2 h 1 α − 2 C 1 C 2 h 1 β + h 1 α β + 2 C 4 h 1 α β + C 4 2 h 1 α β − C 2 2 e − 2 h 1 α 1 + β 2 α + C 1 2 e 2 h 1 α 1 + β 2 α + 2 C 2 e − h 1 α C 4 + β + C 4 β + 2 C 1 e h 1 α C 4 + β + C 4 β β 3 ,

A 8 = 3 BrScSr − 2 C 1 C 2 h 2 + C 4 2 h 2 α − 2 C 1 C 2 h 2 β + h 2 α β + 2 C 4 h 2 α β + C 4 2 h 2 α β − C 2 2 e − 2 h 2 α 1 + β 2 α + C 1 2 e 2 h 2 α 1 + β 2 α + 2 C 2 e − h 2 α C 4 + β + C 4 β + 2 C 1 e h 2 α C 4 + β + C 4 β β 3 ,

C 8 = e − 2 h 1 + h 2 α 4 3 + 4 Rd α 2 h 1 − h 2 + 2 β 3 − 4 e 2 h 1 + h 2 α 3 + 4 Rd α 2 + 3 BrScSr − 2 e 2 h 1 + h 2 α h 1 + h 2 α 2 × C 4 2 + 1 + C 4 2 β h 1 − h 2 + 2 β 3 − 8 C 1 e 2 h 1 + h 2 α α C 4 + β + C 4 β e h 2 α − 1 + α β 3 + e h 1 α 1 + α β 3 + C 2 2 1 + β e 2 h 2 α − 1 + 2 α β 3 + e 2 h 1 α 1 + 2 α β 3 − C 1 2 1 + β e 2 h 1 + 2 h 2 α − 1 + 2 α β 3 + e 2 2 h 1 + h 2 α 1 + 2 α β 3 − 4 C 2 − C 1 e 2 h 1 + h 2 α × h 1 + h 2 α 1 + β h 1 − h 2 + 2 β 3 + 2 α β e h 1 + 2 h 2 α − 1 + α β 3 + e 2 h 1 + h 2 α 1 + α β 3 + 2 C 4 α 1 + β e h 1 + 2 h 2 α − 1 + α β 3 + e 2 h 1 + h 2 α 1 + α β 3 .

C 9 = − e − 4 h 1 + h 2 α 4 e h 1 α 1 + α β 1 + e h 2 α − 1 + α β 1 e h 1 α 2 − h 1 − h 2 α 1 + α β 1 + e h 2 α − 2 + h 1 − h 2 α − 1 + α β 1 × 4 e 4 h 1 + 5 h 2 α F 1 α − 1 + α β 1 − 4 e 5 h 1 + 4 h 2 α F 1 α 1 + α β 1 − 12 e 5 h 1 + 3 h 2 α × + e 3 h 1 + 5 h 2 α + e 4 h 1 + h 2 α − 2 + h 1 − h 2 2 α − 1 + α β 1 2 + C 2 3 − 2 e 5 h 1 + h 2 α × − 2 e h 1 + 5 h 2 α + e 2 h 1 + 2 h 2 α 2 + α − 3 h 1 + 3 h 2 − 2 1 + h 1 − h 2 α β 1 + h 1 − h 2 α β 1 2 + e 2 2 h 1 + h 2 α 2 + α 3 h 1 − 3 h 2 + 2 1 − h 1 − h 2 α β 1 − h 1 − h 2 α β 1 2 + 6 C 1 2 C 2 4 e 5 h 1 + h 2 α − 4 + h 1 + h 2 α + e 2 2 h 1 + 3 h 2 α 8 + α − 10 β 1 − h 1 − 1 + α β 1 × 3 − 2 h 2 α − 5 α β 1 + 2 h 2 α β 1 + h 2 − 1 + α β 1 7 − 2 h 2 α − 5 α β 1 + 2 h 2 α β 1 + e 2 3 h 1 + 2 h 2 α 8 + α 10 β 1 + 2 α h 1 + h 1 α β 1 2 + h 2 3 + 8 α β 1 + 5 α β 1 2 − h 1 1 + α β 1 7 + 5 α β 1 + 2 h 2 α + α β 1 + 2 C 1 3 e 6 h 1 + h 2 α 4 cosh h 1 − h 2 α + 4 − 1 + h 1 − h 2 α β 1 cosh 2 h 1 − h 2 α + α 3 h 1 − 3 h 2 − 2 β 1 + h 1 − h 2 α β 1 2 sinh 2 h 1 − h 2 α

C 10 = e − 2 h 1 + h 2 α 4 e h 1 α 1 + α β 1 + e h 2 α − 1 + α β 1 e h 1 α 2 − h 1 − h 2 α 1 + α β 1 + e h 2 α − 2 + h 1 − h 2 α − 1 + α β 1 × − 4 e 3 h 1 + h 2 α F 1 α e h 1 α 1 + α β 1 + e h 2 α − 1 + α β 1 + 12 C 1 2 C 2 e 5 h 1 + 3 h 2 α + e 3 h 1 + 5 h 2 α + e 4 h 1 + h 2 α − 2 + h 1 − h 2 2 α − 1 + α β 1 2 + C 2 3 − 4 e 3 h 1 + h 2 α − 4 e h 1 + 3 h 2 α + e 4 h 2 α 4 + α 3 h 1 − 3 h 2 − 2 β 1 + 4 − h 1 + h 2 α β 1 + h 1 − h 2 α β 1 2 + e 4 h 1 α 4 + α − 3 h 1 + 3 h 2 + 2 β 1 + 4 − h 1 + h 2 α β 1 + − h 1 + h 2 α β 1 2 + 6 C 1 C 2 2 4 e 3 h 1 + h 2 α 4 + h 1 + h 2 α + e 2 h 1 + 2 h 2 α × − 8 + α 3 h 2 + h 1 − 7 + 2 − h 1 + h 2 α + 10 β 1 + 4 3 h 1 − 2 h 2 + h 1 h 1 − h 2 α α β 1 − h 1 − h 2 5 + 2 h 1 α α β 1 2 + e 2 2 h 1 + h 2 α − 8 + α − 10 β 1 + h 1 1 + α β 1 × 3 + 5 α β 1 + 2 h 2 α + α β 1 − h 2 1 + α β 1 7 + 5 α β 1 + 2 h 2 α + α β 1 + 2 C 1 3 e 5 h 1 + h 2 α 2 − 1 + h 1 − h 2 α β 1 cosh h 1 − h 2 α + 2 cosh 2 h 1 − h 2 α + α − 3 h 1 + 3 h 2 − 2 β 1 + h 1 − h 2 α β 1 2 sinh h 1 − h 2 α

C 11 = e − 3 h 1 + h 2 α 2 α e h 1 α 1 + α β 1 + e h 2 α − 1 + α β 1 e h 1 α 2 − h 1 − h 2 α 1 + α β 1 + e h 2 α − 2 + h 1 − h 2 α − 1 + α β 1 × C 2 3 e 2 h 1 α − e 2 h 2 α e h 1 + 2 h 2 α 1 + 3 h 1 α − h 1 α β 1 + e 3 h 2 α 1 − h 2 α + h 2 α β 1 − e 2 h 1 + h 2 α 1 + 3 h 2 α + h 2 α β 1 + e 3 h 1 α − 1 + h 1 α + h 1 α β 1 + 6 C 1 2 C 2 e 3 h 1 + 2 h 2 α − 1 − h 1 α + α β 1 − e 3 2 h 1 + h 2 α 1 + h 2 α + α β 1 + e 5 h 1 + 4 h 2 α 1 + α 2 h 2 + h 1 − 1 + 2 h 1 − h 2 α 1 + α β 1 + e 4 h 1 + 5 h 2 α 1 + α − h 2 − 1 + 2 h 2 α − 1 + α β 1 + 2 h 1 1 − h 2 α + h 2 α β 1 + 6 C 1 C 2 2 e 2 h 1 + 5 h 2 α − 1 + h 2 α − α β 1 + e 5 h 1 + 2 h 2 α − 1 + h 1 α + α β 1 + e 3 h 1 + 4 h 2 α 1 + α − 2 h 2 − h 1 1 + 2 h 1 − h 2 α − 1 + α β 1 + e 4 h 1 + 3 h 2 α 1 + α h 2 1 + 2 h 2 α 1 + α β 1 − 2 h 1 1 + h 2 α + h 2 α β 1 + 2 e 4 h 1 + h 2 α F 1 h 1 − h 2 α 3 / 2 2 α β 1 cosh h 1 − h 2 α + 1 + α β 1 2 sinh h 1 − h 2 α + C 1 3 e 3 h 1 + 8 h 2 α − 1 − h 1 α + h 1 α β 1

+ e 6 h 1 + 5 h 2 α cosh h 1 − h 2 α 1 + h 1 α − h 1 α β 1 − 8 h 2 α sinh h 1 − h 2 α + sinh h 1 − h 2 α − 3 + 5 h 1 α + 3 h 1 α β 1 − 4 1 − h 2 α + h 2 α β 1 sinh h 1 − h 2 α

C 12 = e − 7 2 h 1 + h 2 α 4 α − h 1 + h 2 α cosh 1 2 h 1 − h 2 α + 2 + − h 1 + h 2 α β 1 sinh 1 2 h 1 − h 2 α × C 2 3 − e h 1 α + e h 2 α 3 e h 1 α + e h 2 α + 6 C 1 C 2 2 − e 2 2 h 1 + h 2 α + e 2 h 1 + 2 h 2 α + e 3 h 1 + h 2 α h 1 − h 2 α + 6 C 1 2 C 2 e 5 h 1 + 3 h 2 α − e 3 h 1 + 5 h 2 α − 2 e 4 h 1 + h 2 α h 1 − h 2 α + e 3 h 1 + h 2 α C 1 3 e h 1 α − e h 2 α 3 e h 1 α + e h 2 α + 2 F 1 α e h 2 α − 1 + α β 1 − e h 1 α 1 + α β 1

C 13 = − 1 h 1 − h 2 + 2 β 2 − Br h 2 − β 2 32 3 + 4 Rd α − 16 C 2 3 1 + C 4 e − 3 h 1 α β α + 16 C 1 3 1 + C 4 e 3 h 1 α β α + 96 e − h 1 α 2 C 10 1 + C 4 + C 2 2 C 12 + 3 C 1 C 2 1 + C 4 7 + 2 h 1 α β α + 3 C 2 4 e − 4 h 1 α 5 α + 3 β α + 96 e h 1 α − 2 C 1 C 12 − 2 C 9 1 + C 4 + 3 C 1 2 C 2 1 + C 4 − 7 + 2 h 1 α β α + 3 C 1 4 e 4 h 1 α 5 α + 3 β α + 12 C 1 e 2 h 1 α − 4 C 9 α + β + C 1 2 C 2 − 25 α + 12 h 1 α 3 / 2 − 33 β + 12 h 1 α β α − 12 C 2 e − 2 h 1 α 4 C 10 α + β + C 1 C 2 2 25 α + 12 h 1 α 3 / 2 + 33 β + 12 h 1 α β α + 96 h 1 2 C 1 C 10 α − β + 9 C 1 2 C 2 2 α − β + C 2 C 9 α − β + C 12 1 + C 4 α β − 3 Br h 2 − β 2 e − 4 h 1 α 8 3 + 4 Rd α 3 / 2 C 2 4 5 α + 3 β + 24 C 1 C 2 2 e 3 h 1 α α 6 C 1 e h 1 α h 1 α − β + 1 + C 4 × 5 + 2 h 1 α β − 2 C 2 3 e h 1 α 2 1 + C 4 α β + C 1 e h 1 α 19 α + 12 h 1 α 3 / 2 + 27 β + 12 h 1 α β + e 3 h 1 α − 4 C 1 3 1 + C 4 e 4 h 1 α α β + 16 1 + C 4 α C 10 + C 9 e 2 h 1 α + C 12 e h 1 α h 1 α β − C 1 4 e 5 h 1 α 5 α + 3 β + 8 C 1 e h 1 α C 9 e 2 h 1 α α + β + 2 α C 10 h 1 α − β + C 12 e h 1 α β − 2 C 2 e 2 h 1 α 4 C 10 α + β + e h 1 α 12 C 1 2 1 + C 4 e 2 h 1 α − 5 + 2 h 1 α α β − 8 α C 9 e h 1 α h 1 α − β + C 12 β + C 1 3 e 3 h 1 α − 19 α + 12 h 1 α 3 / 2 − 27 β + 12 h 1 α β β 2 + Br h 1 + β 2 32 3 + 4 Rd α − 16 C 2 3 1 + C 4 e − 3 h 2 α β α + 16 C 1 3 1 + C 4 e 3 h 2 α β α + 96 e − h 2 α 2 C 10 1 + C 4 + C 2 2 C 12 + 3 C 1 C 2 1 + C 4 7 + 2 h 2 α β α + 3 C 2 4 e − 4 h 2 α 5 α + 3 β α + 96 e h 2 α − 2 C 1 C 12 − 2 C 9 1 + C 4 + 3 C 1 2 C 2 1 + C 4 − 7 + 2 h 2 α β α + 3 C 1 4 e 4 h 2 α 5 α + 3 β α

+ 12 C 1 e 2 h 2 α − 4 C 9 α + β + C 1 2 C 2 − 25 α + 12 h 2 α 3 / 2 − 33 β + 12 h 2 α β α − 12 C 2 e − 2 h 2 α 4 C 10 α + β + C 1 C 2 2 25 α + 12 h 2 α 3 / 2 + 33 β + 12 h 2 α β α + 96 h 2 2 C 1 C 10 α − β + 9 C 1 2 C 2 2 α − β + C 2 C 9 α − β + C 12 1 + C 4 α β + 3 Br h 2 − β 2 e − 4 h 2 α 8 3 + 4 Rd α 3 / 2 C 2 4 5 α + 3 β + 24 C 1 C 2 2 e 3 h 2 α α 6 C 1 e h 2 α h 2 α − β + 1 + C 4 5 + 2 h 2 α β − 2 C 2 3 e h 2 α 2 1 + C 4 α β + C 1 e h 2 α 19 α + 12 h 2 α 3 / 2 + 27 β + 12 h 2 α β + e 3 h 2 α − 4 C 1 3 1 + C 4 e 4 h 2 α α β + 16 1 + C 4 α C 10 + C 9 e 2 h 2 α + C 12 e h 2 α h 2 α β − C 1 4 e 5 h 2 α 5 α + 3 β + 8 C 1 e h 2 α C 9 e 2 h 2 α α + β + 2 α C 10 h 2 α − β + C 12 e h 2 α β − 2 C 2 e 2 h 2 α 4 C 10 α + β + e h 2 α 12 C 1 2 1 + C 4 e 2 h 2 α − 5 + 2 h 1 α α β − 8 α × C 9 e h 2 α h 2 α − β + C 12 β + C 1 3 e 3 h 2 α − 19 α + 12 h 2 α 3 / 2 − 27 β + 12 h 2 α β β 2

C 14 = − Br e − 4 h 1 + h 2 α 32 3 + 4 Rd α 2 h 1 − h 2 + 2 β 2 16 C 1 3 1 + C 4 e 4 h 1 + h 2 α α β e 3 h 1 α 1 + 3 α β 2 + e 3 h 2 α − 1 + 3 α β 2 − 3 C 2 4 5 α + 3 β e 4 h 1 α 1 + 4 α β 2 + e 4 h 2 α − 1 + 4 α β 2 + 3 C 1 4 5 α + 3 β e 4 h 1 + 2 h 2 α − 1 + 4 α β 2 + e 4 2 h 1 + h 2 α 1 + 4 α β 2 − 48 C 1 e 4 h 1 + h 2 α × 2 C 10 h 1 + h 2 α α − β h 1 − h 2 + 2 β 2 + 4 C 12 α β e h 1 α 1 + α β 2 + e h 2 α − 1 + α β 2 + C 9 α + β e 2 h 1 α 1 + 2 α β 2 + e 2 h 2 α − 1 + 2 α β 2 − 96 1 + C 4 α β 2 C 10 e 3 h 1 + h 2 α × e h 1 α 1 + α β 2 + e h 2 α − 1 + α β 2 + e 4 h 1 + h 2 α C 12 h 1 + h 2 α 3 / 2 h 1 − h 2 + 2 β 2 + 2 C 9 e h 1 α 1 + α β 2 + e h 2 α − 1 + α β 2 − 288 C 1 C 2 2 α 3 C 1 e 4 h 1 + h 2 α h 1 + h 2 × α α − β h 1 − h 2 + 2 β 2 + 1 + C 4 e 3 h 1 + h 2 α β e h 1 α 7 + 5 α β 2 + 2 h 2 α + α β 2 + e h 2 α − 7 + 5 α β 2 − 2 h 2 α + α β 2 + 12 C 2 8 C 9 e 4 h 1 + h 2 α h 1 2 α 2 h 2 2 − h 1 2 − 16 C 12 α β e 3 h 1 + 4 h 2 α − e 4 h 1 + 3 h 2 α + 8 C 9 e 4 h 1 + h 2 α α β h 1 2 − h 2 2 − 16 C 9 e 4 h 1 + h 2 α α 2 β h 1 + h 2 + 16 C 12 α β β 2 e 3 h 1 + 4 h 2 α − e 4 h 1 + 3 h 2 α + 16 C 9 α β β 2 e 4 h 1 + h 2 α h 1 + h 2 + 4 C 10 α + β e 2 h 1 + 2 h 2 α − 1 + 2 α β 2 + e 2 2 h 1 + h 2 α × 1 + 2 α β 2 + 24 C 1 2 1 + C 4 e 4 h 1 + h 2 α α β e h 1 α − 7 − 5 α β 2 + 2 h 1 α + α β 2 + e h 2 α 7 − 5 α β 2 − 2 h 2 α − α β 2 + C 1 3 e 2 3 h 1 + 2 h 2 α − 33 β + 2 α 3 / 2 6 h 1 − 19 β 2 + 6 α β 2 h 1 − 9 β 2 + 24 h 1 α 2 β 2 + α − 25 + 24 h 1 β β 2 + e 2 2 h 1 + 3 h 2 α 33 β + 24 h 2 α 2 β 2 − 6 α β 2 h 2 + 9 β 2 − 2 α 3 / 2 6 h 2 + 19 β 2 + α 25 + 24 h 1 β β 2 + 4 C 2 3 4 1 + C 4 α β × e h 1 + 4 h 2 α − 1 + 3 α β 2 + e 4 h 1 + h 2 α 1 + 3 α β 2 + 3 C 1 e 2 h 1 + h 2 α e 2 h 2 α − 33 β + 24 h 1 α 2 β 2 + 6 α β − 2 h 1 + 9 β 2 + α 3 / 2 − 12 h 1 + 38 β 2 + α − 25 + 24 h 1 β β 2 + e 2 h 1 α 33 β + 24 h 2 α 2 β 2 + 6 α β 2 h 2 + 9 β 2 + 2 α 3 / 2 6 h 2 + 19 β 2 + α 25 + 24 h 2 β β 2

C 15 = − 1 h 1 − h 2 + 2 β 3 − BrScSr h 2 − β 3 32 3 + 4 Rd α − 16 C 2 3 1 + C 4 e − 3 h 1 α β α + 16 C 1 3 1 + C 4 e 3 h 1 α β α + 96 e − h 1 α 2 C 10 1 + C 4 + C 2 2 C 12 + 3 C 1 C 2 1 + C 4 7 + 2 h 1 α β α + 3 C 2 4 e − 4 h 1 α 5 α + 3 β α + 96 e h 1 α − 2 C 1 C 12 − 2 C 9 1 + C 4 + 3 C 1 2 C 2 1 + C 4 − 7 + 2 h 1 α β α + 3 C 1 4 e 4 h 1 α 5 α + 3 β α + 12 C 1 e 2 h 1 α − 4 C 9 α + β + C 1 2 C 2 − 25 α + 12 h 1 α 3 / 2 − 33 β + 12 h 1 α β α − 12 C 2 e − 2 h 1 α 4 C 10 α + β + C 1 C 2 2 25 α + 12 h 1 α 3 / 2 + 33 β + 12 h 1 α β α + 96 h 1 2 C 1 C 10 α − β + 9 C 1 2 C 2 2 α − β + C 2 C 9 α − β + C 12 1 + C 4 α β + 3 BrScSr h 2 − β 3 e − 4 h 1 α 8 3 + 4 Rd α 3 / 2 C 2 4 5 α + 3 β + 24 C 1 C 2 2 e 3 h 1 α α 6 C 1 e h 1 α h 1 α − β + 1 + C 4 5 + 2 h 1 α β − 2 C 2 3 e h 1 α 2 1 + C 4 α β + C 1 e h 1 α 19 α + 12 h 1 α 3 / 2 + 27 β + 12 h 1 α β + e 3 h 1 α − 4 C 1 3 1 + C 4 e 4 h 1 α α β + 16 1 + C 4 α C 10 + C 9 e 2 h 1 α + C 12 e h 1 α h 1 α β − C 1 4 e 5 h 1 α 5 α + 3 β + 8 C 1 e h 1 α C 9 e 2 h 1 α α + β + 2 α C 10 h 1 α − β + C 12 e h 1 α β − 2 C 2 e 2 h 1 α 4 C 10 α + β + e h 1 α 12 C 1 2 1 + C 4 e 2 h 1 α − 5 + 2 h 1 α α β − 8 α C 9 e h 1 α h 1 α − β + C 12 β + C 1 3 e 3 h 1 α − 19 α + 12 h 1 α 3 / 2 − 27 β + 12 h 1 α β β 3 − BrScSr h 1 + β 3 32 3 + 4 Rd α − 16 C 2 3 1 + C 4 e − 3 h 2 α β α + 16 C 1 3 1 + C 4 e 3 h 2 α β α + 96 e − h 2 α 2 C 10 1 + C 4 + C 2 2 C 12 + 3 C 1 C 2 1 + C 4 7 + 2 h 2 α β α + 3 C 2 4 e − 4 h 2 α 5 α + 3 β α + 96 e h 2 α − 2 C 1 C 12 − 2 C 9 1 + C 4 + 3 C 1 2 C 2 1 + C 4 − 7 + 2 h 2 α β α + 3 C 1 4 e 4 h 2 α 5 α + 3 β α

+ 12 C 1 e 2 h 2 α − 4 C 9 α + β + C 1 2 C 2 − 25 α + 12 h 2 α 3 / 2 − 33 β + 12 h 2 α β α − 12 C 2 e − 2 h 2 α 4 C 10 α + β + C 1 C 2 2 25 α + 12 h 2 α 3 / 2 + 33 β + 12 h 2 α β α + 96 h 2 2 C 1 C 10 α − β + 9 C 1 2 C 2 2 α − β + C 2 C 9 α − β + C 12 1 + C 4 α β − 3 BrScSr h 2 − β 3 e − 4 h 2 α 8 3 + 4 Rd α 3 / 2 C 2 4 5 α + 3 β + 24 C 1 C 2 2 e 3 h 2 α α 6 C 1 e h 2 α h 2 α − β + 1 + C 4 5 + 2 h 2 α β − 2 C 2 3 e h 2 α 2 1 + C 4 α β + C 1 e h 2 α 19 α + 12 h 2 α 3 / 2 + 27 β + 12 h 2 α β + e 3 h 2 α − 4 C 1 3 1 + C 4 e 4 h 2 α α β + 16 1 + C 4 α C 10 + C 9 e 2 h 2 α + C 12 e h 2 α h 2 α β − C 1 4 e 5 h 2 α 5 α + 3 β + 8 C 1 e h 2 α C 9 e 2 h 2 α α + β + 2 α C 10 h 2 α − β + C 12 e h 2 α β − 2 C 2 e 2 h 2 α 4 C 10 α + β + e h 2 α 12 C 1 2 1 + C 4 e 2 h 2 α − 5 + 2 h 1 α α β − 8 α C 9 e h 2 α h 2 α − β + C 12 β + C 1 3 e 3 h 2 α − 19 α + 12 h 2 α 3 / 2 − 27 β + 12 h 2 α β β 3

C 16 = B r S c S r e − 4 h 1 + h 2 α 32 3 + 4 Rd α 2 h 1 − h 2 + 2 β 3 16 C 1 3 1 + C 4 e 4 h 1 + h 2 α α β e 3 h 1 α 1 + 3 α β 3 + e 3 h 2 α − 1 + 3 α β 3 − 3 C 2 4 5 α + 3 β e 4 h 1 α 1 + 4 α β 3 + e 4 h 2 α − 1 + 4 α β 3 + 3 C 1 4 5 α + 3 β e 4 h 1 + 2 h 2 α − 1 + 4 α β 3 + e 4 2 h 1 + h 2 α 1 + 4 α β 3 − 48 C 1 e 4 h 1 + h 2 α 2 C 10 h 1 + h 2 α α − β h 1 − h 2 + 2 β 3 + 4 C 12 α β e h 1 α 1 + α β 3 + e h 2 α − 1 + α β 3 + C 9 α + β e 2 h 1 α 1 + 2 α β 3 + e 2 h 2 α − 1 + 2 α β 3 − 96 1 + C 4 α β 2 C 10 e 3 h 1 + h 2 α e h 1 α 1 + α β 3 + e h 2 α − 1 + α β 3 + e 4 h 1 + h 2 α C 12 h 1 + h 2 α 3 / 2 h 1 − h 2 + 2 β 3 + 2 C 9 e h 1 α 1 + α β 3 + e h 2 α − 1 + α β 3 − 288 C 1 C 2 2 α 3 C 1 e 4 h 1 + h 2 α h 1 + h 2 α α − β h 1 − h 2 + 2 β 3 + 1 + C 4 e 3 h 1 + h 2 α β e h 1 α 7 + 5 α β 3 + 2 h 2 α + α β 3 + e h 2 α − 7 + 5 α β 3 − 2 h 2 α + α β 3 + 12 C 2 8 C 9 e 4 h 1 + h 2 α h 1 2 α 2 h 2 2 − h 1 2 − 16 C 12 α β e 3 h 1 + 4 h 2 α − e 4 h 1 + 3 h 2 α + 8 C 9 e 4 h 1 + h 2 α α β h 1 2 − h 2 2 − 16 C 9 e 4 h 1 + h 2 α α 2 β h 1 + h 2 + 16 C 12 α β β 3 e 3 h 1 + 4 h 2 α − e 4 h 1 + 3 h 2 α + 16 C 9 α β β 3 e 4 h 1 + h 2 α h 1 + h 2 + 4 C 10 α + β e 2 h 1 + 2 h 2 α − 1 + 2 α β 3 + e 2 2 h 1 + h 2 α 1 + 2 α β 3 + 24 C 1 2 1 + C 4 e 4 h 1 + h 2 α α β e h 1 α − 7 − 5 α β 3 + 2 h 1 α + α β 3 + e h 2 α 7 − 5 α β 3 − 2 h 2 α − α β 3 + C 1 3 e 2 3 h 1 + 2 h 2 α − 33 β + 2 α 3 / 2 6 h 1 − 19 β 3 + 6 α β 2 h 1 − 9 β 3 + 24 h 1 α 2 β 3 + α − 25 + 24 h 1 β β 3 + e 2 2 h 1 + 3 h 2 α 33 β + 24 h 2 α 2 β 3 − 6 α β 2 h 2 + 9 β 3 − 2 α 3 / 2 6 h 2 + 19 β 3 + α 25 + 24 h 1 β β 3 + 4 C 2 3 4 1 + C 4 α β × e h 1 + 4 h 2 α − 1 + 3 α β 3 + e 4 h 1 + h 2 α 1 + 3 α β 3 + 3 C 1 e 2 h 1 + h 2 α e 2 h 2 α − 33 β + 24 h 1 α 2 β 3 + 6 α β − 2 h 1 + 9 β 3 + α 3 / 2 − 12 h 1 + 38 β 3 + α − 25 + 24 h 1 β β 3 + e 2 h 1 α 33 β + 24 h 2 α 2 β 3 + 6 α β 2 h 2 + 9 β 3 + 2 α 3 / 2 6 h 2 + 19 β 3 + α 25 + 24 h 2 β β 3

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Received: 2019-09-11

Revised: 2021-04-11

Accepted: 2021-05-12

Published Online: 2021-06-15

Published in Print: 2022-08-26

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2019-0226

Keywords for this article

Hall current; MHD; peristaltic flow; porous medium; slip conditions; viscous dissipation