Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Analysis of magnetohydrodynamics transient flow in a horizontal annular duct

  • 9 Accesses

Abstract

The aim of this research is to investigate the key role of radial magnetic field with the transient flow of a viscous, incompressible, electrically-conducting fluid between two horizontal concentric cylinders. In this case, motion of such fluid creates a viscous force within the annulus region. The governing equation of the problem has been presented by using Brinkman model and solution is done by the separable variable’s method. The analytical solution of the problem is established in the form of Bessel, modified Bessel functions and eigenfunctions. Further, we have discussed the results in detail by using graphs and tables, to show the impacts of various parameters on the velocity profile. Moreover, the variation of velocity, skin-frictions, mass flux and the effect on ratio of radii i.e. outer radius to inner radius are thoroughly described with some non-dimensional parameters as Hartmann number, Reynolds number, Euler number and time. As a result, for higher value of Hartmann number, the solution reduces from unsteady to steady for shorter time scale, the steady solution can be obtained as limiting case of the solution. For some lower values of time, velocity rapidly enhances but after certain interval of time, the velocity does not change.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

Abbreviations

\( a^{{\prime }} \) :

Non-dimensional inner radius

\( b^{{\prime }} \) :

Non-dimensional outer radius

\( B_{0}^{{\prime }} \) :

Magnetic field

\( B{\kern 1pt} r \) :

Brinkman number

\( Eu \) :

Euler number

\( G \) :

Negative applied pressure gradient \( \left( {{{ - \partial p} \mathord{\left/ {\vphantom {{ - \partial p} {\partial z}}} \right. \kern-0pt} {\partial z}}} \right) \)

\( Ha \) :

Hartmann number

\( I_{n} \left( r \right) \) :

Modified Bessel function of first kind of order n

\( J_{n} \left( r \right) \) :

Bessel function of first kind of order n

\( Y_{n} \left( r \right) \) :

Bessel function of second kind of order n

\( K_{n} \left( r \right) \) :

Modified Bessel function of second kind of order n

p :

Pressure

\( \lambda \) :

Ratio of radii of cylinder Outer and Inner

r :

Radial coordinate in non-dimensional form

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

Radial coordinate

\( \text{Re} \) :

Reynolds number

t :

Time

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

Non-dimensional time

\( U \) :

Non-dimensional fluid velocity in axial direction

\( u_{z}^{*} \) :

Characteristics velocity

\( U_{\hbox{max} } \) :

Non-dimensional maximum velocity in axial direction

\( U_{a} \) :

Non-dimensional average velocity in axial direction

\( u_{z}^{{\prime }} \) :

Fluid velocity in axial direction

\( u_{r}^{{\prime }} \) :

Fluid velocity in radial direction

\( u_{\theta }^{{\prime }} \) :

Fluid velocity in angular direction

\( u \) :

Non-dimensional velocity defined in Eq. (7)

\( z \) :

Dimensionless axial coordinate

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

Axial coordinate

\( \alpha_{n} \) :

Separation constant

\( \mu \) :

Fluid viscosity

\( \rho \) :

Fluid density

\( \sigma \) :

Fluid conductivity

\( \theta^{{\prime }} \) :

Angular coordinate

References

  1. 1.

    Couette MM (1890) Effect of viscous incompressible fluid flow in concentric annuli. Annales de Chimie et de Physique (Ann Chem Phys) 21(6):433–510

  2. 2.

    Ramamoorthy P (1961) Flow between two concentric rotating cylinders with a radial magnetic field. Phys Fluid 4:1444–1445

  3. 3.

    Hartmann J, Lazarus F (1937) Mercury dynamics II experimental investigations on the flow of mercury in a homogeneous magnetic field. Series: Mathematisk-fysiske meddelelser / det Kgl. Danske videnskabernes selskab 15(7):1–45

  4. 4.

    Gold R (1962) Magnetohydrodynamics pipe flow. Part I J Fluid Mech 13:505–512

  5. 5.

    Goldstein RJ, Briggs DG (1964) Transient free convection about vertical plates and circular cylinders. J Heat Transf 86(4):490–500

  6. 6.

    Tsui YT, Tremblay B (1984) On transient natural convection heat transfer in the annulus between concentric, horizontal cylinders with isothermal surfaces. Int J Heat Mass Transf 27(1):103–111

  7. 7.

    Grebeth C, Thess A, Marty P (1990) Theoretical study of the MHD flow around a cylinder in crossed electric and magnetic field. Eur J Mech Fluids 9(3):239–257

  8. 8.

    Velusamy K, Garg VK (1992) Transient natural convection over a heat generating vertical cylinder. Int J Heat Mass Transf 35(5):1293–1306

  9. 9.

    Singh SK, Jha BK, Singh AK (1997) Natural convection in vertical concentric annuli under a radial magnetic field. Heat Mass Transf 32:399–401

  10. 10.

    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. Int J Heat Mass Transf 24(4):243–250

  11. 11.

    Ganesan PG, Rani HP (2000) Unsteady free convection MHD flow past a vertical cylinder with heat and mass transfer. Int Therm Sci 39:265–272

  12. 12.

    Chamkha AJ (2000) Unsteady laminar hydro magnetic fluid-particle flow and heat transfer in channels and circular pipes. Int J Heat Fluid Flow 21:740–746

  13. 13.

    Kumar A, Singh AK (2010) Transient magnetohydrodynamic Couette flow with ramped velocity. Int J Fluid Mech Res 37(5):435–446

  14. 14.

    Di Prima RC, Swinney HL (2005) Instabilities and transition in flow between concentric rotating cylinders. Hydrodyn Instab Transit Turbul 45:139–180

  15. 15.

    Singh AK, Sacheti NC, Chandran P (1994) Transient effects on magnetohydrodynamic Couette flow with rotation. Int J Eng Sci 32:133–139

  16. 16.

    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 7:461–473

  17. 17.

    Sankar M, Venkatachalappa M, Shivakumara IS (2006) Effect of magnetic field on natural convection in a vertical cylindrical annulus. Int J Eng Sci 44:1556–1570

  18. 18.

    Bakalis PA, Hatzikonstantinou PM (2011) MHD and thermal flow between isothermal vertical concentric cylinders with the rotation of the inner cylinder. Numer Heat Transf Part A 59:836–856

  19. 19.

    Singh RK, Singh AK (2012) Effect of induced magnetic field on natural convection in vertical concentric annuli. Acta Mech Sin 28(2):315–323

  20. 20.

    Machireddy GR (2013) Chemically reactive species and radiation effects on MHD convective flow past a moving vertical cylinder. Ain Shams Eng J 4:879–888

  21. 21.

    Deka RK, Paul A (2013) Transient free convective MHD flow past an infinite vertical cylinder. Theor Appl Mech 40(3):385–402

  22. 22.

    Rajesh V, Beg OA (2014) MHD transient nano-fluid flow and heat transfer from a moving vertical cylinder with temperature oscillation. Comput Thermal Sci Int J 6(5):439–450

  23. 23.

    Jha BK, Aina B, Isa S (2015) Transient magnetohydrodynamic free convective flow in vertical micro-concentric annuli. Proc Inst Mech Eng Part N J Nanomater Nanoeng Nanosyst 230(4):188–199

  24. 24.

    Kumar D, Singh AK (2015) Effect of induced magnetic field on natural convection with Newtonian heating/cooling in vertical concentric annuli. Proc Eng 127:568–574

  25. 25.

    Deka RK, Paul A, Chaliha A (2017) Transient free convection flow past vertical cylinder with constant heat flux and mass transfer. Ain Shams Eng J 8:643–651

  26. 26.

    Jha BK, Oni MO (2018) Transient natural convection flow between vertical concentric cylinders heated/cooled asymmetrically. Proc Inst Mech Eng Part A: J Power Energy 232(7):926–939

  27. 27.

    Maurya JP, Yadav SL, Singh AK (2018) Magnetohydrodynamic transient flow in a circular cylinder. Int J Dyn Control 6(4):1477–1483

  28. 28.

    Chandrasekhar S, Elbert D (1954) The roots of Yn (λη) Jn (λ) − Jn (λη) Yn(λ) = 0. Math Proc Camb Philos Soc 50(2):266–268

Download references

Acknowledgments

The Author (JPM) would like to acknowledge the financial support received from the University Grants Commission, New Delhi, India.

Author information

Correspondence to Shyam Lal Yadav.

Appendix

Appendix

These constants which are mentioned below, we have used in the mathematical expressions:

$$ S = s_{1} + s_{3} + s_{5} - s_{2} - s_{4} - s_{6} , $$
$$ P = p_{1} - p_{2} - p_{3} , $$
$$ s_{1} = \frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{Y_{0} (\alpha_{n} \lambda )}}{{\alpha_{n} }}\left\{ {\lambda \;J_{1} (\alpha_{n} \lambda ) - J_{1} (\alpha_{n} )} \right\}, $$
$$ s_{2} = \frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{J_{0} (\alpha_{n} \lambda )}}{{\alpha_{n} }}\left\{ {\lambda \;Y_{1} (\alpha_{n} \lambda ) - Y_{1} (\alpha_{n} )} \right\}, $$
$$ s_{3} = L.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{Y_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{1} , $$
$$ s_{4} = L.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{J_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{2} , $$
$$ s_{5} = M.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{Y_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{3} , $$
$$ s_{6} = M.\frac{{Eu.\text{Re} }}{{Ha^{2} }}.\frac{{J_{0} (\alpha_{n} \lambda )}}{{Ha^{2} + \alpha_{n}^{2} }}I_{4} , $$
$$ I_{1} = \left\{ \begin{aligned} \lambda Ha.\;J_{0} (\alpha_{n} \lambda ).I_{1} (\lambda Ha) + \alpha_{n} \lambda J_{1} (\alpha_{n} \lambda ).I_{0} (\lambda Ha) \hfill \\ - Ha.J_{0} (\alpha_{n} ).I_{1} (Ha) - \alpha_{n} .J_{1} (\alpha_{n} )I_{0} (Ha) \hfill \\ \end{aligned} \right\}, $$
$$ I_{2} = \left\{ \begin{aligned} \lambda Ha.\;Y_{0} (\alpha_{n} \lambda ).I_{1} (\lambda Ha) + \alpha_{n} \lambda .Y_{1} (\alpha_{n} \lambda ).I_{0} (\lambda Ha) \hfill \\ - Ha.Y_{0} (\alpha_{n} ).I_{1} (Ha) - \alpha_{n} .Y_{1} (\alpha_{n} )I_{0} (Ha) \hfill \\ \end{aligned} \right\}, $$
$$ I_{3} = \left\{ \begin{aligned} \alpha_{n} \lambda .K_{0} (\lambda Ha).\;J_{1} (\alpha_{n} \lambda ) - \lambda Ha\;J_{0} (\alpha_{n} \lambda ).K_{1} (\lambda Ha) \hfill \\ - \alpha_{n} .K_{0} (Ha).\;J_{1} (\alpha_{n} ) + Ha\;J_{0} (\alpha_{n} ).K_{1} (Ha) \hfill \\ \end{aligned} \right\}, $$
$$ I_{4} = \left\{ \begin{aligned} \alpha_{n} \lambda .K_{0} (\lambda Ha).\;Y_{1} (\alpha_{n} \lambda ) - \lambda Ha\;Y_{0} (\alpha_{n} \lambda ).K_{1} (\lambda Ha) \hfill \\ - \alpha_{n} .K_{0} (Ha).\;Y_{1} (\alpha_{n} ) + Ha\;Y_{0} (\alpha_{n} ).K_{1} (Ha) \hfill \\ \end{aligned} \right\}, $$
$$ p_{1} = \frac{{\lambda^{2} }}{2}\left\{ {Y_{0} (\alpha_{n} \lambda ).\;J_{1} (\alpha_{n} \lambda ) - \;J_{0} (\alpha_{n} \lambda )Y_{1} (\alpha_{n} \lambda )} \right\}^{2} , $$
$$ p_{2} = \frac{1}{2}\left\{ {Y_{0} (\alpha_{n} \lambda ).\;J_{0} (\alpha_{n} ) - \;J_{0} (\alpha_{n} \lambda )Y_{0} (\alpha_{n} )} \right\}^{2} , $$
$$ p_{3} = \frac{1}{2}\left\{ {Y_{0} (\alpha_{n} \lambda ).\;J_{1} (\alpha_{n} ) - \;J_{0} (\alpha_{n} \lambda )Y_{1} (\alpha_{n} )} \right\}^{2} , $$
$$ \xi_{1} = L.Ha.I_{1} (Ha) - M.Ha.K_{1} (Ha), $$
$$ \eta_{1} = Y_{0} (\alpha_{n} \lambda ).\alpha_{n} \;J_{1} (\alpha_{n} ) - J_{0} (\alpha_{n} \lambda ).\alpha_{n} Y_{1} (\alpha_{n} ), $$
$$ \xi_{2} = L.Ha.I_{1} (\lambda Ha) - M.Ha.K_{1} (\lambda Ha), $$
$$ \eta_{2} = Y_{0} (\alpha_{n} \lambda ).\alpha_{n} \;J_{1} (\alpha_{n} \lambda ) - J_{0} (\alpha_{n} \lambda ).\alpha_{n} Y_{1} (\alpha_{n} \lambda ), $$
$$ \xi_{3} = \lambda I_{1} (\lambda Ha) - I_{1} (Ha), $$
$$ \eta_{3} = \lambda K_{1} (\lambda Ha) - K_{1} (Ha), $$
$$ \xi_{4} = \lambda J_{1} (\alpha_{n} \lambda ) - J_{1} (\alpha_{n} ), $$
$$ \eta_{4} = \lambda Y_{1} (\alpha_{n} \lambda ) - Y_{1} (\alpha_{n} ). $$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Maurya, J.P., Yadav, S.L. & Singh, A.K. Analysis of magnetohydrodynamics transient flow in a horizontal annular duct. Int. J. Dynam. Control (2020). https://doi.org/10.1007/s40435-020-00611-4

Download citation

Keywords

  • Unsteady flow
  • Annulus
  • Magnetic field
  • Hartmann number
  • Modified Bessel function and eigenfunctions