Magnetohydrodynamic transient flow in a circular cylinder

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Abstract

A transient flow problem of unidirectional electrically conducting fluid flow in a circular cylinder in the presence of the normal magnetic field has examined. The governing equation of fluid flow has presented by using Brinkman model and solved by separation of variables technique. The analytical solution of modelled equation is found in the form of Bessel and modified Bessel functions. Moreover, the influence of relevant parameters on the velocity of fluid is also analyzed by the graphical and tabular illustrations. The inertial and pressure forces enhance the velocity, while viscosity reduces it. In strong Magnetic field, the solution reduces to steady for shorter time scale and the steady solution can be obtained as limiting case of the solution, but the reverse nature of velocity has been obtained with Reynolds number.

Keywords

Unsteady flow Circular Magnetic field Hartmann number Bessel function 

List of symbols

\(\hbox {a}\)

Radius of cylinder

\({\hbox {B}}^{\prime }_0 \)

Magnetic field

\(\hbox {Br}\)

Brinkman number

\(\hbox {Eu}\)

Euler number

\(\hbox {G}\)

Negative applied pressure gradient \(\left( {{-\partial \hbox {p}}/{\partial \hbox {z}}} \right) \)

\(\hbox {Ha}\)

Hartmann number

\(\hbox {I}_\mathrm{n} \left( \hbox {r} \right) \)

Modified Bessel function of first kind of order n

\(\hbox {J}_\mathrm{n} \left( \hbox {r} \right) \)

Bessel function of first kind of order n

\(\hbox {Y}_\mathrm{n} \left( \hbox {r} \right) \)

Bessel function of second kind of order n

\(\hbox {p}\)

Pressure

\(\hbox {r}\)

Radial coordinate in non-dimensional form

\(\hbox {r}^{\prime }\)

Radial coordinate

\(\hbox {Re}\)

Reynolds number

\(\hbox {t}\)

Time

\(\hbox {t}^{\prime }\)

Non-dimensional time

\(\hbox {U}\)

Non-dimensional fluid velocity in axial direction

\(\hbox {u}_\mathrm{z}^*\)

Characteristics velocity

\(\hbox {U}_{\max } \)

Non-dimensional maximum velocity in axial direction

\(\hbox {U}_\mathrm{a} \)

Non-dimensional average velocity in axial direction

\({\hbox {u}}^{\prime }_{\mathrm{z}} \)

Fluid velocity in axial direction

\({\hbox {u}}^{\prime }_\mathrm{r} \)

Fluid velocity in radial direction

\(\hbox {u}^{\prime }_\theta \)

Fluid velocity in angular direction

\(\hbox {u}\)

Non-dimensional velocity defined in Eq. (6)

\(\hbox {z}\)

Dimensionless axial coordinate

\(\hbox {z}^{\prime }\)

Axial coordinate

Greek symbols

\(\upalpha _\mathrm{n} \)

Separation constant

\(\upmu \)

Fluid viscosity

\(\uprho \)

Fluid density

\(\upsigma \)

Fluid conductivity

\(\uptheta ^{\prime }\)

Angular coordinate

Notes

Acknowledgements

The author (SLY) would like to thank the Council of Scientific and Industrial Research, New Delhi, India, for financial support in the form of a Senior Research Fellowship.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Institute of ScienceBanaras Hindu UniversityVaranasiIndia

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