Role of Induced Magnetic Field on Hydromagnetic Mixed Convection Flow in Vertical Microannulus in Existence of Radial Magnetic Field

  • Basant K. Jha
  • Babatunde AinaEmail author
Review Article


The present study presents the convective flow of an electrically conducting fluid in vertical microannulus formed by two concentric cylinders in the presence of imposed radial magnetic field and induced magnetic field. The governing equations of the motion are a set of simultaneous ordinary differential equations, and their exact solutions in dimensionless form have been obtained for the velocity field, the induced magnetic field and the temperature field. The expressions for the induced current density and skin friction have also been obtained. Results for various flows characteristic are presented through graphs and tables delineating the effect of various parameters characterizing the flow. Furthermore, it is observed that the reverse flow near the inner surface of outer cylinder is controlled by selecting suitable values of Knudsen number, fluid–wall interaction parameter and radius ratio.


Induced magnetic field Hydromagnetic Microannulus Radial magnetic field Velocity slip Temperature jump 

List of symbols

\( C_{\rho 0} \)

Specific heat at constant pressure

\( \ln \)

Fluid–wall interaction parameter, \( {{\beta_{t} } \mathord{\left/ {\vphantom {{\beta_{t} } {\beta_{\nu } }}} \right. \kern-0pt} {\beta_{\nu } }} \)

\( g \)

Gravitational acceleration


Grashof number

\( Gr/Re \)

Mixed convection parameter

\( k_{1} \)

Radius of the inner cylinder

\( k_{2} \)

Radius of the outer cylinder

\( Kn \)

Knudsen number, \( {\lambda \mathord{\left/ {\vphantom {\lambda w}} \right. \kern-0pt} w} \)

\( M \)

Magnetic field parameter

\( q \)

Volume flow rate

\( Q \)

Dimensionless volume flow rate

\( Pr \)

Prandtl number

\( r^{'}\)

Dimensional radial coordinate

\( R \)

Dimensionless radial coordinate

\( \mathop R\limits^{ \wedge } \)

Specific gas constant


Reynolds number

\( T \)

Temperature of fluid

\( T_{0} \)

Reference temperature

\( T_{1} \)

Temperature at outer surface of the inner cylinder

\( u \)

Axial velocity

\( u_{\text{m}} \)

Mean velocity


Dimensionless axial velocity

\( w \)

Dimensional gap between the cylinders

\( \sigma_{t} ,\sigma_{v} \)

Thermal and tangential momentum accommodation coefficients, respectively

\( {{{\text{d}}P^{'}} \mathord{\left/ {\vphantom {{{\text{d}}P^{'}} {{\text{d}}Z^{'}}}} \right. \kern-0pt} {{\text{d}}Z^{'}}} \)

Pressure gradient along the axis of the microannulus

\( {{{\text{d}}P} \mathord{\left/ {\vphantom {{{\text{d}}P} {{\text{d}}Z}}} \right. \kern-0pt} {{\text{d}}Z}} \)

Dimensionless pressure gradient along the axis of the microannulus

\({Z^{'}}, {r^{'}}\)

Axial and radial coordinates, respectively

\( Z,R \)

Dimensionless axial and radial coordinates, respectively

Greek letters

\( \alpha \)

Thermal diffusivity

\( \beta_{0} \)

Coefficient of thermal expansion

\( \beta_{t} ,\beta_{v} \)

Dimensionless variables


Ratio of specific heats

\( \mu_{0} \)

Dynamic viscosity

\( \theta \)

Dimensionless temperature



\( \nu \)

Fluid kinematic viscosity \( \left( {{{\mu_{0} } \mathord{\left/ {\vphantom {{\mu_{0} } {\rho_{0} }}} \right. \kern-0pt} {\rho_{0} }}} \right) \)


Ratio of radii \( \left( {{{k_{1} } \mathord{\left/ {\vphantom {{k_{1} } {k_{2} }}} \right. \kern-0pt} {k_{2} }}} \right) \)

\( \lambda \)

Molecular mean free path

\( k_{0} \)

Thermal conductivity


Skin friction


Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Weng HC, Chen CK (2009) Drag reduction and heat transfer enhancement over a heated wall of a vertical annular microchannel. Int J Heat Mass Transf 52:1075–1079CrossRefGoogle Scholar
  2. 2.
    Avci M, Aydin O (2009) Mixed convection in a vertical microannulus between two concentric microtubes. J Heat Transf Trans ASME 131:014502CrossRefGoogle Scholar
  3. 3.
    Jha BK, Aina B (2015) Mathematical modelling and exact solution of steady fully developed mixed convection flow in a vertical micro-porous-annulus. J Afr Mat 26:1199–1213MathSciNetCrossRefGoogle Scholar
  4. 4.
    Sadeghi M, Sadeghi A, Saidi MH (2014) Gaseous slip flow mixed convection in vertical microducts of constant but arbitrary geometry. AIAA J Thermophys Heat Transf 28(4):771–784CrossRefGoogle Scholar
  5. 5.
    Sadeghi M, Baghani MH, Saidi (2014) Gaseous slip flow mixed convection in vertical microducts with constant axial energy input. ASME J Heat Transf 136(3):032501CrossRefGoogle Scholar
  6. 6.
    Weng HC, Jian SJ (2012) Developing mixed convection in a vertical microchannel. Adv Sci Lett 5:1–6CrossRefGoogle Scholar
  7. 7.
    Jha BK, Aina B, Isa Sani (2015) MHD natural convection flow in a vertical micro-concentric-annuli in the presence of radial magnetic field: an exact solution. J Ain Shams Eng. CrossRefGoogle Scholar
  8. 8.
    Sheikholeslami M, Gorji-Bandpy M, Ganji DD (2014) Magnetohydrodynamic free convection of Al2O3–water nanofluid considering thermophoresis and Brownian motion effects. Comput Fluids 94:147–160MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sheikholeslami M, Gorji-Bandpy M (2014) Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field. Powder Technol 256:490–498CrossRefGoogle Scholar
  10. 10.
    Sheikholeslami M, Gorji-Bandpy M, Ganji DD (2014) Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid. Powder Technol 254:82–93CrossRefGoogle Scholar
  11. 11.
    Sarveshanand Singh AK (2015) Magnetohydrodynamic free convection between vertical parallel porous plates in the presence of induced magnetic field. SpringerPlus. CrossRefGoogle Scholar
  12. 12.
    Singh RK, Singh AK (2012) Effect of induced magnetic field on natural convection in vertical concentric annuli. Acta Mech Sin 28(2):315–323ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kumar A, Singh AK (2013) Unsteady MHD free convective flow past a semi-infinite vertical wall with induced magnetic field. Appl Math Comput 222:462–471MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dileep K, Singh AK (2016) Effects of heat source/sink and induced magnetic field on natural convective flow in vertical concentric annuli. Alex Eng J. CrossRefGoogle Scholar
  15. 15.
    Jha BK, Aina B (2016) Role of induced magnetic field on MHD natural convection flow in a vertical microchannel formed by two electrically non-conducting infinite vertical parallel plates. Alex Eng J. CrossRefGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Department of MathematicsAhmadu Bello UniversityZariaNigeria
  2. 2.Department of MathematicsBingham UniversityAbujaNigeria

Personalised recommendations