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Pulsating hydromagnetic flow and heat transfer of Jeffrey ferro-nanofluid in a porous channel: a dynamics of blood

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Abstract

An analytical investigation has been carried out to study the influence of MHD pulsatile flow of Jeffrey ferro-nanofluid in a porous channel with the effect of Joule heating, viscous dissipation and heat source/sink. Blood is considered as Jeffrey fluid (base fluid) and \(\mathrm{Fe}_3\mathrm{O}_4\) (magnetite) taken as nanoparticles. The Maxwell Garnett model for thermal conductivity of nanofluid is utilized. Flow is prompted by the pulsatile pressure gradient. The governing flow equations are solved analytically using the perturbation procedure. The influences of different parameters on velocity, temperature and rate of heat transfer have been analysed. The results depicts that the velocity of nanofluid is heightened with a rise in the frequency parameter while declining with increasing magnetic field and volume fraction. The temperature of nanofluid is increased by increasing viscous dissipation. The rate of heat transfer rises with an increase in nanoparticle volume fraction.

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Abbreviations

\(x^{*},y^{*}\) :

Dimensional Cartesian coordinates

x, y :

Dimensionless Cartesian coordinates

\(u^{*}\) :

Dimensional velocity component in \(x^*\) direction (m/s)

u :

Dimensionless velocity component in x direction

\(P^{*}\) :

Dimensional pressure (Kg m\(^{-1}\) S\(^{-1}\))

P :

Dimensionless pressure

\(t^{*}\) :

Dimensional time (s)

t :

Dimensionless time

\(\mu \) :

Dynamic viscosity

\(\sigma ^{*}\) :

Stefan–Boltzmann constant (W m\(^{-2}\) K\(^{-4}\))

\(\sigma \) :

Electrical conductivity \((\Omega \mathrm{m})^{-1}\)

\(k^{*}\) :

Roseland mean absorption \((\mathrm{m}^{-1})\)

\(B_{0}\) :

A strength of an applied magnetic field

\(q_{r}\) :

Radiative heat flux (W m\(^{-2}\))

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

Temperatures at top and bottom walls

\(\rho \) :

Density (Kg/m\(^{3}\))

\(C_p\) :

Specific heat (J/Kg K)

\(\rho C_P\) :

Effective specific heat capacity

K :

Thermal conductivity (W/m K)

\(\phi \) :

Nanoparticle volume fraction

A :

Known constant

\(\epsilon (<<1)\) :

Positive quantity

\(T^{*}\) :

Temperature of the nanofluid (K)

h :

Distance between the walls (m)

\(\omega \) :

Frequency

H :

Frequency parameter

R :

Cross-flow Reynolds number

M :

Hartmann number

Pr:

Prandtl number

Ec:

Eckert number

Rd:

Radiation parameter

f :

Base fluid

s :

Nanoparticle

nf :

Nanofluid

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Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, 632 014, India

    T. Thamizharasan & A. Subramanyam Reddy

Authors
  1. T. Thamizharasan
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Appendix

Appendix

$$\begin{aligned} B_{1}= & {} \frac{A_{2} }{A_{1} } \frac{1}{1+\lambda _{1} }, B_{2} =R, B_{3} =\frac{A_{5} }{A_{1} } M^{2} ,\\ B_{4}= & {} \frac{B_{2} +\sqrt{B_{2}^{2} +4B_{1} B_{3} } }{2B_{1} }, B_{5} =\frac{B_{2} -\sqrt{B_{2}^{2} +4B_{1} B_{3} } }{2B_{1} },\\ B_{6}= & {} \frac{H^{2} }{A_{1} B_{3} } , B_{7} =\frac{-B_{6} (e^{B_{4} } -1)}{e^{B_{4} } -e^{B_{5} } } , B_{8} =\frac{B_{6} (e^{B_{4} } -1)}{e^{B_{4} } -e^{B_{5} } } -B_{6} ,\\ B_{9}= & {} B_{1} +\frac{A_{2} }{A_{1} } \frac{\lambda i}{1+\lambda _{i} } ,\\ B_{10}= & {} \frac{A_{5} }{A_{1} } M^{2} +iH^{2} , B_{11} =\frac{B_{2} +\sqrt{B_{2}^{2} +4B_{9} B_{10} } }{2B_{9} },\\ B_{12}= & {} \frac{B_{2} -\sqrt{B_{2}^{2} +4B_{9} B_{10} } }{2B_{9} } , B_{13} =\frac{H^{2} }{A_{1} B_{10} } ,\\ B_{14}= & {} \frac{-B_{13} (e^{B11} -1)}{e^{B_{11} } -e^{B_{12} } }, B_{15} =\frac{B_{13} (e^{B_{11} } -1)}{e^{B_{11} } -e^{B_{12} } } -B_{13},\\ B_{16}= & {} \left( \frac{A_{4} }{A_{3} } +\frac{4}{3} \frac{Rd}{A_{3} } \right) \frac{1}{\Pr }, \\ B_{17}= & {} \frac{Q}{A_{3} } , B_{18} =\frac{B_{2} +\sqrt{B_{2}^{2} -4B_{16} B_{17} } }{2B_{16} } ,\\ B_{19}= & {} \frac{B_{2} -\sqrt{B_{2}^{2} -4B_{16} B_{17} } }{2B_{16} } ,\\ B_{20}= & {} \left( \frac{A_{2} }{A_{3} } \frac{1}{1+\lambda _{1} } EcB_{8}^{2} B_{4}^{2} +\frac{A_{5} }{A_{3} } M^{2} EcB_{8}^{2} \right) ,\\ B_{21}= & {} \left( \frac{A_{2} }{A_{3} } \frac{1}{1+\lambda _{1} } EcB_{7}^{2} B_{5}^{2} +\frac{A_{5} }{A_{3} } M^{2} EcB_{7}^{2} \right) ,\\ B_{22}= & {} \left( \frac{A_{2} }{A_{3} } \frac{1}{1+\lambda _{1} } 2EcB_{8} B_{4} B_{7} B_{5} +\frac{A_{5} }{A_{3} } 2M^{2} EcB_{8} B_{7} \right) ,\\ B_{23}= & {} \frac{A_{5} }{A_{3} } 2M^{2} EcB_{8} B_{6} ,\\ B_{24}= & {} \frac{A_{5} }{A_{3} } 2M^{2} EcB_{7} B_{6} , B_{25} =\frac{A_{5} }{A_{3} } M^{2} EcB_{6}^{2} , \\ B_{26}= & {} \frac{-B_{20} }{4B_{4}^{2} B_{16} -2B_{4} B_{2} +B_{17} } ,\\ B_{27}= & {} \frac{-B_{21} }{4B_{5}^{2} B_{16} -2B_{5} B_{2} +B_{17} } ,\\ B_{28}= & {} \frac{-B_{22} }{\left( B_{4} +B_{5} \right) ^{2} B_{16} -\left( B_{4} +B_{5} \right) B_{2} +B_{17} } , \\ B_{29}= & {} \frac{-B_{23} }{B_{4}^{2} B_{16} -B_{4} B_{2} +B_{17} } , B_{30} =\frac{-B_{24} }{B_{5}^{2} B_{16} -B_{5} B_{2} +B_{17} } ,\\ B_{31}= & {} \frac{-B_{25} }{B_{17} } , \\ B_{32}= & {} \frac{1}{\left( e^{B_{19} } -e^{B_{18} } \right) }\\&\left( \begin{array}{l} {1+B_{26} \left( e^{B_{18}} -e^{2B_{4} } \right) +B_{27} \left( e^{B_{18} } -e^{2B_{5} } \right) } \\ {+B_{28} \left( e^{B_{18} } -e^{(B_{4} +B_{5} )} \right) +B_{29} \left( e^{B_{18} } -e^{B_{4} } \right) } \\ {+B_{30} \left( e^{B_{18}} -e^{B_{5} } \right) +B_{31} \left( e^{B_{18}} -1\right) } \end{array}\right) ,\\ B_{33}= & {} -\left( B_{26} +B_{27} +B_{28} +B_{29} +B_{30} +B_{31} +B_{32} \right) , \\ B_{34}= & {} B_{17} -iH^{2} ,\\ B_{35}= & {} \frac{B_{2} +\sqrt{B_{2}^{2} -4B_{16} B_{34} } }{2B_{16} } , \\ B_{36}= & {} \frac{B_{2} -\sqrt{B_{2}^{2} -4B_{16} B_{34} } }{2B_{16} } , \\ B_{37}= & {} \left( \frac{A_{2} }{A_{3} } \frac{1}{1+\lambda _{1} } 2EcB_{8} B_{4} B_{15} B_{11} +\frac{A_{5} }{A_{3} } 2M^{2} EcB_{8} B_{15} \right) ,\\ B_{38}= & {} \left( \frac{A_{2} }{A_{3} } \frac{1}{1+\lambda _{1} } 2EcB_{8} B_{4} B_{14} B_{12} +\frac{A_{5} }{A_{3} } 2M^{2} EcB_{8} B_{14} \right) ,\\ B_{39}= & {} \left( \frac{A_{2} }{A_{3} } \frac{1}{1+\lambda _{1} } 2EcB_{7} B_{5} B_{15} B_{11} +\frac{A_{5} }{A_{3} } 2M^{2} EcB_{7} B_{15} \right) , \\ B_{40}= & {} \left( \frac{A_{2} }{A_{3} } \frac{1}{1+\lambda _{1} } 2EcB_{7} B_{5} B_{14} B_{12} +\frac{A_{5} }{A_{3} } 2M^{2} EcB_{7} B_{14} \right) ,\\ B_{41}= & {} \frac{A_{5} }{A_{3} } 2M^{2} EcB_{8} B_{13} , B_{42} =\frac{A_{5} }{A_{3} } 2M^{2} EcB_{7} B_{13} , \\ B_{43}= & {} \frac{A_{5} }{A_{3} } 2M^{2} EcB_{6} B_{15} , \\ B_{44}= & {} \frac{A_{5} }{A_{3} } 2M^{2} EcB_{6} B_{14} , B_{45} =\frac{A_{5} }{A_{3} } 2M^{2} EcB_{6} B_{13} , \\ B_{46}= & {} \frac{-B_{37} }{\left( B_{4} +B_{11} \right) ^{2} B_{16} -\left( B_{4} +B_{11} \right) B_{2} +B_{34} } , \\ B_{47}= & {} \frac{-B_{38} }{\left( B_{4} +B_{12} \right) ^{2} B_{16} -\left( B_{4} +B_{12} \right) B_{2} +B_{34} } , \\ B_{48}= & {} \frac{-B_{39} }{\left( B_{5} +B_{11} \right) ^{2} B_{16} -\left( B_{5} +B_{11} \right) B_{2} +B_{34} } ,\\ B_{49}= & {} \frac{-B_{40} }{\left( B_{5} +B_{12} \right) ^{2} B_{16} -\left( B_{5} +B_{12} \right) B_{2} +B_{34} } ,\\ B_{50}= & {} \frac{-B_{41} }{B_{4} {}^{2} B_{16} -B_{4} B_{2} +B_{34} } , \\ B_{51}= & {} \frac{-B_{42} }{B_{5} {}^{2} B_{16} -B_{5} B_{2} +B_{34} },\\ B_{52}= & {} \frac{-B_{43} }{B_{11} {}^{2} B_{16} -B_{11} B_{2} +B_{34} } ,\\ B_{53}= & {} \frac{-B_{44} }{B_{12} {}^{2} B_{16} -B_{12} B_{2} +B_{34} } , \\ B_{54}= & {} \frac{-B_{45} }{B_{34} } , \\ B_{55}= & {} \frac{1}{\left( e^{B_{36} } -e^{B_{35} } \right) }\\&\left( \begin{array}{l} {B_{46} \left( e^{B_{35} } -e^{\left( B_{4} +B_{11} \right) } \right) +B_{47} \left( e^{B_{35} } -e^{\left( B_{4} +B_{12} \right) } \right) } \\ {+B_{48} \left( e^{B_{35} } -e^{\left( B_{5} +B_{11} \right) } \right) +B_{49} \left( e^{B_{35} } -e^{\left( B_{5} +B_{12} \right) } \right) } \\ {+B_{50} \left( e^{B_{35} } -e^{B_{4} } \right) +B_{51} \left( e^{B_{35} } -e^{B_{5} } \right) } \\ {+B_{52} \left( e^{B_{35} } -e^{B11} \right) +B_{53} \left( e^{B_{35} } -e^{B_{12} } \right) } \\ {+B_{54} \left( e^{B_{35} } -1\right) } \end{array}\right) ,\\ B_{56}= & {} -\left( B_{46} +B_{47} +B_{48} +B_{49} +B_{50} +B_{51} +B_{52} +B_{53}\right. \\&\left. +B_{54} +B_{55} \right) . \end{aligned}$$

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Thamizharasan, T., Reddy, A.S. Pulsating hydromagnetic flow and heat transfer of Jeffrey ferro-nanofluid in a porous channel: a dynamics of blood. Eur. Phys. J. Spec. Top. 231, 1205–1214 (2022). https://doi.org/10.1140/epjs/s11734-022-00528-3

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