# 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.

### References

- 1.Subramanya K (2003) Theory and applications of fluid mechanics, 11th edn. Tata McGraw-Hill, New DelhiGoogle Scholar
- 2.Gold R (1962) Magnetohydrodynamics pipe flow, part-I. J Fluid Mech 13:505–512MathSciNetCrossRefMATHGoogle Scholar
- 3.Agrawal AK, Kishor B, Raptis A (1989) Effects of MHD free convection and mass transfer on the flow past a vibrating infinite vertical circular cylinder. J Heat Mass Transf 24(4):243–250Google Scholar
- 4.Singh SK, Jha BK, Singh AK (1997) Natural convection in vertical concentric annuli under a radial magnetic field. Heat Mass Transf 32:399–401CrossRefGoogle Scholar
- 5.Ganesan PG, Rani HP (2000) Unsteady free convection MHD flow past a vertical cylinder with heat and mass transfer. Int J Therm Sci 39:265–272CrossRefGoogle Scholar
- 6.Kumar H, Rajathy R (2006) Numerical Study of MHD flow past a circular cylinder at low and moderate Reynolds numbers. Int J Comput Methods Eng Sci Mech 7:461–473CrossRefMATHGoogle Scholar
- 7.Sharma BR, Hemanta K (2015) MHD flow, heat and mass transfer due to axially moving cylinder in presence of thermal diffusion, radiation and chemical reactions in a binary fluid mixture. Int J Comput Appl 110:52–59Google Scholar
- 8.Singh RK, Singh AK (2012) Effect of induced magnetic field on natural convection in vertical concentric annuli. Acta Mech Sin 28(2):315–323MathSciNetCrossRefMATHGoogle Scholar
- 9.Kumar D, Singh AK (2015) Effect of induced magnetic field on natural convection with a Newtonian heating/cooling in vertical concentric annuli. Proc Eng 127:568–574CrossRefGoogle Scholar
- 10.Kumar A, Singh AK (2013) Effect of induced magnetic field on natural convection in vertical concentric annuli heated/cooled asymmetrically. J Appl Fluid Mech 6(1):15–26Google Scholar
- 11.Singh B, Lal J (1984) Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity. Int J Numer Methods Fluids 4(3):291–302CrossRefMATHGoogle Scholar
- 12.Elgazery NS, Hassan MA (2007) Numerical study of radiation effect on MHD transient mixed-convection flow over a moving vertical cylinder with constant heat flux. Int J Numer Methods Biomed Eng 24(11):1183–1202MathSciNetMATHGoogle Scholar
- 13.Reddy MG, Reddy NB (2009) Thermal Radiation and mass transfer effects on MHD free convective flow past a vertical cylinder with variable surface temperature and concentration. J Navel Archit Eng 6(1):1–24MathSciNetMATHGoogle Scholar
- 14.Loganathan P, Kannan M, Ganesan P (2011) Thermal radiation effects on MHD flow over a moving semi-infinite vertical cylinder. Int J Math Anal 5(6):257–274MATHGoogle Scholar
- 15.Boricic AZ, Jovanovic MM, Boricic BZ (2012) Magnetohydrodynamic effects on unsteady dynamic, thermal and diffusion boundary layer flow over a horizontal circular cylinder. Therm Sci 16(2):S311–S321CrossRefGoogle Scholar
- 16.Deka RK, Paul A (2013) Transient free convective MHD flow past an infinite vertical cylinder. Theor Appl Mech 40(3):385–402MathSciNetCrossRefMATHGoogle Scholar
- 17.Machireddy GR (2013) Chemically reactive species and radiation effects on MHD convective flow past a moving vertical cylinder. Ain Shams Eng J 4:879–888CrossRefGoogle Scholar
- 18.Niu J, Zheng L, Zhang X (2014) Unsteady MHD convection heat transfer along an accelerating/decelerating cylinder with variable fluid properties. Eur Phys J Plus 129:1–15CrossRefGoogle Scholar
- 19.Sharma BR, Dutta N (2016) Chemical reaction and thermal radiation effects on unsteady MHD flow of an incompressible viscous fluid past a moving vertical cylinder. Int J Eng Sci Res Technol 5(8):536–544Google Scholar
- 20.Poornima T, Sreenivasulu P, Reddy NB (2016) Chemical reaction effects on an unsteady MHD mixed convective and radiative boundary layer flow over a circular cylinder. J Appl Fluid Mech 9(6):2877–2885Google Scholar
- 21.Vanita V, Kumar A (2016) Effect of radial magnetic field on free convective flow over ramped velocity moving vertical cylinder with ramped type temperature and concentration. J Appl Fluid Mech 9(6):2855–2864Google Scholar
- 22.Rajesh V, Bég OA, Sridevi C (2016) Finite difference analysis of unsteady MHD free convective flow over moving semi-infinite vertical cylinder with chemical reaction and temperature oscillation effects. J Appl Fluid Mech 9(1):157–167CrossRefGoogle Scholar
- 23.Yadav SL, Singh AK (2016) Analysis of entropy generation in annular porous duct. Transp Porous Media 111:425–440MathSciNetCrossRefGoogle Scholar
- 24.Jha BK, Apere CA (2012) Magnetohydrodynamic transient free-convective flow in a vertical annulus with thermal boundary condition of the second kind. J Heat Transf 134:1–8CrossRefGoogle Scholar
- 25.Jha BK, Yusuf TS (2017) MHD transient free convection flow in vertical concentric annulus with isothermal and adiabatic boundaries. Int J Eng Technol 11:40–52Google Scholar