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Influence of Hall and Slip on MHD Reiner-Rivlin blood flow through a porous medium in a cylindrical tube

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Abstract

The current study aims to scrutinize the peristalsis of magnetohydrodynamics Reiner-Rivlin fluid model in a cylindrical tube through porous medium. The basic equations, including momentum, heat, and concentration are tackled in the valuable presence of Hall current, Joule heating, viscous-dissipation, and Soret effects. Moreover, slip effect is also entertained. The considered system is simplified by approximating with long wavelengths and very low Reynolds numbers. Perturbation technique is chosen to obtain the closed form analytical solutions. The impact of influential parameters is presented through plots and physically discussed in detail. The main conclusions of this work are that the velocity and temperature fields exhibit opposite behavior for the Hartman number and Hall parameter. The slip parameter has minimizing impact on velocity distribution. The fluid velocity tends to increase when Reiner-Rivlin fluid parameter is incremented. The Darcy number has upgrading impact on fluid temperature. The solute concentration minifies when slip parameter is incremented. It can be visualized that size of trapped bolus becomes larger on increasing the value of Hall and slip parameter. This research investigation basically examine the blood flow through an artery under strong electric and magnetic field effects. Therefore, this study laid the ground work for scientists, engineers, and medical practitioners working in physiological field.

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The data used to support the findings of this study are included within the article.

Abbreviations

\(\overline{\user2{V}} = \left( {u\left( {r,z,t} \right),0,w\left( {r,z,t} \right)} \right)\) :

Velocity vector

\({\varvec{J}} = \frac{A}{{m^{2} }}\) :

Current density

\(\sigma = \frac{S}{m}\) :

Electric conductivity

\(m = \left( {\frac{{\sigma B_{0} }}{{en_{e} }}} \right)\) :

Hall effect

\(\tau_{ij}\) :

Cauchy stress tensor

\(\theta\) :

Temperature in dimensionless form

(\(T_{0} ,T_{1} )\) :

Temperature at lower and upper wall, respectively

\(Q\) :

Volume flow rate

\(\delta_{ij}\) :

Kronecker delta

\(e_{ij}\) :

Strain tensor rate

\(\mu_{c}\) :

Cross viscosity parameter

\(k\) :

Permeability parameter

\(\rho = \frac{{{\text{kg}}}}{{{\text{m}}^{3} }}\) :

Fluid density

\({\text{Sr}} = \left( {\frac{{\rho \left( {T_{1} - T_{0} } \right)DK_{T} }}{{\mu T_{m} \left( {C_{1} - C_{0} } \right)}}} \right)\) :

Soret number

\(\mu\) :

Fluid viscosity

\(\rho C_{{\text{p}}}\) :

Specific heat constant

\(D_{{\text{m}}}\) :

Diffusion constant

\(T_{{\text{m}}}\) :

Mean temperature

\(\kappa = \frac{{\text{W}}}{{{\text{m}}.{\text{k}}}}\) :

Thermal conductivity

\(\left( {\beta ,\beta_{1} ,\beta_{2} } \right)\) :

Slip parameters

\(\phi\) :

Concentration in dimensionless form

\(\left( {C_{0} ,C_{1} } \right)\) :

Concentration at lower and upper wall, respectively

\(B = \left( {0,0,B_{0} } \right)\) :

Magnetic field

\(\delta = \left( {\frac{d}{\lambda }} \right)\) :

Wave number

\({\text{Re}} = \frac{\rho cd}{\mu }\) :

Reynolds number

\(\lambda = \frac{{\overline{\mu }_{{\text{c}}} c}}{\mu d}\) :

Fluid parameter

\(M^{2} = \frac{{\sigma B_{0}^{2} d^{2} }}{\mu }\) :

Hartmann number

\(D_{{\text{a}}} = \frac{{k_{0} }}{{d^{2} }}\) :

Darcy resistance parameter

\({\text{Pr}} = \left( {\frac{{\mu C_{{\text{p}}} }}{\kappa }} \right)\) :

Prandtl number

\({\text{Sc}} = \left( {\frac{\mu }{\rho D}} \right)\) :

Schmidt number

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The authors received no financial support for this research and for publication of this article.

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Institute of Space Technology 2750, Islamabad, 44000, Pakistan

    M. Yasin & R. Naz

  2. Department of Mathematical Sciences, Fatima Jinnah Women University, Rawalpindi, 46000, Pakistan

    S. Hina

Authors
  1. M. Yasin
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  2. S. Hina
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  3. R. Naz
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Correspondence to S. Hina.

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Appendix

Appendix

$$ H = \left( {\frac{{M^{2} }}{{1 + m^{2} + \frac{1}{{{\text{Da}}}}}}} \right); L = H^{2} \eta \left( {{\text{BesselY}}\left[ {1,H\eta } \right] + H\beta {\text{BesselY}}\left[ {0,H\eta } \right]} \right), {\text{L}}_{1} = - {\text{BesselJ}}\left[ {1,H\eta } \right] $$
$$ L_{2} = H\beta BesselJ\left[ {0,H\eta } \right], L_{3} = \left( {BesselY\left[ {1,H\eta } \right] + H\beta BesselY\left[ {0,H\eta } \right]} \right), $$
$$ L_{4} = \eta BesselY\left[ {1,H\eta } \right] - H\beta \eta BesselY\left[ {0,H\eta } \right], L_{5} = L_{3} H\eta^{3} BesselJ\left[ {2,H\eta } \right], $$
$$ \frac{{\partial p_{0} }}{\partial z} = - \frac{{QL - L_{5} }}{{\eta L_{4} }}, L{\prime} = H^{3} \beta \eta \left( { - HBesselY\left[ {1,H\eta } \right] + BesselY\left[ {0,H\eta } \right]\left( {\beta + \eta } \right)} \right)\eta^{\prime}, $$
$$ L_{1}{\prime} = \frac{1}{2}H\eta^{\prime}\left( { - BesselJ\left[ {0,H\eta } \right] + BesselJ\left[ {2,H\eta } \right]} \right), L_{2}{\prime} = - H^{2} \beta \eta^{\prime}BesselJ\left[ {1,H\eta } \right], $$
$$ L_{3}{\prime} = - \frac{1}{2}H\left( { - BesselY\left[ {0,H\eta } \right] + BesselY\left[ {2,H\eta } \right] + 2H\beta \eta^{\prime}BesselY\left[ {1,H\eta } \right]} \right), $$
$$ L_{4}{\prime} = H\left( {HBesselY\left[ {1,H\eta } \right]\beta \eta + BesselY\left[ {0,H\eta } \right]\left( { - \beta + \eta } \right)} \right)\eta^{\prime}, $$
$$ L_{5}{\prime} = H\eta^{2} \left( {BesselJ\left[ {2,H\eta } \right]\eta L_{3}{\prime} + L_{3} \left( {BesselJ\left[ {2,H\eta } \right] + H\eta BesselJ\left[ {1,H\eta } \right]} \right)\eta^{\prime}} \right), $$
$$ L_{7} = H^{2} \left( { - BesselJ\left[ {0,H\eta } \right] + H\beta BesselJ\left[ {1,H\eta } \right]} \right), L_{8} = BesselJ\left[ {0,H\eta } \right] - H\beta BesselJ\left[ {1,H\eta } \right], $$
$$ L_{9} = \left( { - 1 + H^{2} \eta \beta } \right), L_{10} = - BesselJ\left[ {0,H\eta } \right] + H\beta BesselJ\left[ {1,H\eta } \right], $$
$$ \frac{{\partial^{2} p}}{{\partial z^{2} }} = \frac{{\left( {QL - L_{5} } \right)\eta L_{4}{\prime} + L_{4} *\left( {\eta *\left( { - QL^{\prime} + L_{5}{\prime} } \right) + \left( {QL - L_{5} } \right)\eta^{\prime}} \right)}}{{ L_{4}^{2} *\eta^{2} }},\frac{{\partial p_{1} }}{\partial z} = \frac{{2QH L_{7} - L_{11} }}{{ L_{12} }}, $$
$$ L_{11} = 2A_{1} \eta \left( {H L_{8} \eta + 2 L_{9} BesselJ\left[ {1,H\eta } \right]} \right), L_{12} = \eta \left( {H L_{10} \eta + 2BesselJ\left[ {1,H\eta } \right]} \right), $$
$$ A_{1} = \frac{1}{{L^{3} }}Hr\left( {H^{2} L_{3} \eta BesselJ\left[ {0,Hr} \right] + BesselY\left[ {0,Hr} \right]\left( {H^{2} \left( { L_{1} - L_{2} } \right)\eta + \frac{{\partial p_{0} }}{\partial z}} \right)} \right) \times $$
$$ \left( {H^{2} rBesselJ\left[ {1,Hr} \right]\left( {L L_{3} \eta^{\prime} - L_{3} \eta L^{\prime} + L\eta L_{3}{\prime} } \right) + L_{4} L^{\prime}\frac{{\partial p_{0} }}{\partial z} - L L_{4}{\prime} \frac{{\partial p_{0} }}{\partial z} - L L_{4} \frac{{\partial^{2} p}}{{\partial z^{2} }} + rBesselY\left[ {1,Hr} \right]\left( { - L^{\prime}\left( {H^{2} \left( { L_{1} - L_{2} } \right)\eta + \frac{{\partial p_{0} }}{\partial z}} \right) + L\left( {H^{2} \left( { L_{1} \eta^{\prime} - L_{2} \eta^{\prime} + \eta L_{1}{\prime} - \eta L_{2}{\prime} } \right) + \frac{{\partial^{2} p}}{{\partial z^{2} }}} \right)} \right)} \right), $$
$$ B_{1} = \frac{Br}{{L^{2} \left( {1 + m^{2} } \right)}}\left( {H^{2} Mr\eta \left( { L_{3} BesselJ\left[ {1,Hr} \right] + \left( { L_{1} - L_{2} } \right)BesselY\left[ {1,Hr} \right]} \right) + M\left( { - L_{4} + rBesselY\left[ {1,Hr} \right]} \right)\frac{{\partial p_{0} }}{\partial z}} \right)^{2} , $$
$$ + \frac{Br}{{L^{2} }}\left( {H^{3} L_{3} r\eta BesselJ\left[ {0,Hr} \right] + HrBesselY\left[ {0,Hr} \right]\left( {H^{2} \left( { L_{1} - L_{2} } \right)\eta + \frac{{\partial p_{0} }}{\partial z}} \right)} \right)^{2} $$
$$ B_{11} = - \frac{2}{{L* L_{7}^{3} \left( {1 + m^{2} } \right)}}Br^{2} Hr(H^{2} L_{3} \eta BesselJ\left[ {0,Hr\left] { + BesselY} \right[0,Hr} \right](H^{2} \left( { L_{1} - L_{2} } \right)\eta $$
$$ + \frac{{\partial p_{0} }}{\partial z}))\left( {2M\left( { L_{8} + L_{9} BesselJ\left[ {0,Hr} \right]} \right)A_{1} + M\left( { L_{10} + BesselJ\left[ {0,Hr} \right]} \right)\frac{{\partial p_{1} }}{\partial z}} \right)^{2} \left( { - L_{7} A_{1} + HBesselJ\left[ {1,Hr} \right]\left( {2 L_{9} A_{1} + \frac{{\partial p_{1} }}{\partial z}} \right)} \right). $$

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Yasin, M., Hina, S. & Naz, R. Influence of Hall and Slip on MHD Reiner-Rivlin blood flow through a porous medium in a cylindrical tube. Soft Comput 28, 2799–2810 (2024). https://doi.org/10.1007/s00500-023-09538-2

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