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Heat transfer in an unsteady MHD flow through an infinite annulus with radiation
 Nazibuddin Ahmed^{1}Email author and
 Manas Dutta^{2}
https://doi.org/10.1186/s136610140279z
© Ahmed and Dutta; licensee Springer 2015
 Received: 7 July 2014
 Accepted: 30 December 2014
 Published: 16 January 2015
Abstract
An analytical study of a laminar unsteady magnetohydrodynamic flow of a viscous incompressible and electrically conducting Newtonian nonGray optically thin fluid between two infinite concentric vertical cylinders influenced by time dependent periodic pressure gradient subjected to a magnetic field applied in azimuthal direction and in the presence of appreciable thermal radiation and periodic wall temperature is presented. The governing equations of motion and energy are transformed into ordinary differential equations which are solved in closed form in terms of the modified Bessel functions (of first and second kind) of order zero. The induced magnetic field is neglected, assuming the magnetic Reynolds number to be considerably small. A parametric study accounting for the effects of various physical parameters on the velocity and temperature fields and on the coefficient of skin friction, the rate of heat transfer at the surface of the cylinders, and mass flux across a normal section of the annulus is conducted and the results are discussed graphically.
Keywords
 annular region
 thermal radiation
 periodic pressure gradient
 periodic wall temperature
 heat transfer
MSC
 76W05
1 Introduction
The magnetohydrodynamic flow and heat transfer problems in an annular region have assumed considerable industrial significance in the light of advancements in hydraulics and nuclear technology. Several authors have studied the magnetohydrodynamic (MHD) flows and heat transfer in channels and circular pipes. Chamkha [1] studied the unsteady laminar MHD flow and heat transfer in channels and circular pipes under the influence of two different applied pressure gradients (oscillating and ramp). Singh [2] analyzed the problem of MHD mixed convection periodic flow in a rotating vertical channel with heat radiation and presented an exact solution. MHD heat transfer problems with periodic wall temperature are also frequently encountered. IsraelCookey et al. [3] investigated the problem of MHD free convection and oscillating flow of an optically thin fluid bounded by two horizontal porous parallel walls with a periodic wall temperature. Reddy et al. [4] also introduced a periodic wall temperature into their study. Marin and Marinescu [5] proceeded with an analysis of thermoelasticity of initially stressed bodies with asymptotic equipartition of energies. Marin et al. [6] also carried out a theoretical investigation of thermoelasticity taking into account the heat conduction in deformable bodies depending on two different temperatures viz. a conductive temperature and a thermodynamic temperature. Othman and Zaki [7] studied the effect of a vertical magnetic field on the onset of a convective instability in a conducting micropolar fluid layer heated from below and confined between two horizontal planes under the coupled action of the rotation of the system and a vertical temperature gradient.
In this study, the problem of an unsteady MHD flow of a Newtonian nonGray optically thin fluid within the annulus of two infinite concentric vertical cylinders is analyzed. The current is set up within the annulus due to the application of a time dependent periodic pressure gradient and this is subjected to a magnetic field applied in the azimuthally direction. Heat is simultaneously applied to both walls of the annulus in the form of periodic wall temperature whence the thermal radiation too is considered. The magnetic Reynolds number is considered to be small enough, as a result of which the induced magnetic field can be neglected. The introduction of a cylindrical polar coordinate system renders the resulting governing equations of motion and energy to a form which can be solved exactly, thus obtaining the expressions for velocity and temperature fields. Subsequently, the mass flux coefficient, the skin friction coefficient, and the coefficient of heat transfer (Nusselt number) are derived and are depicted graphically.
2 Basic equations

Equation of continuity:$$ \vec{\nabla} \cdot \vec{q} = 0. $$(1)

MHD momentum equation:$$ \rho \biggl[ \frac{\partial \vec{q}}{\partial t} + ( \vec{q} \cdot \vec{\nabla} )\vec{q} \biggr] =  \vec{\nabla} p + \mu \nabla^{2}\vec{q} + \vec{J} \times \vec{B} + \rho \vec{g}. $$(2)

Energy equation:$$ \rho C_{p} \biggl[ \frac{\partial T}{\partial t} + ( \vec{q} \cdot \vec{\nabla} )T \biggr] = K_{T}\nabla^{2}T + \varphi  \vec{\nabla} \cdot \vec{q}_{r}. $$(3)

Ohm’s law for an electrically conducting fluid:$$ \vec{J} = \sigma ( \vec{q} \times \vec{B} ). $$(4)
Consider a laminar, radiative flow of an incompressible, Newtonian, electrically conducting, nonGray and optically thin fluid within an annulus, influenced by a time dependent periodic pressure gradient and a periodic temperature applied to the walls of the annulus. The annulus is assumed to be bounded by two cylinders of radii a and b, where \(a < b\).
 (I)
All the fluid properties are considered constants except the influence of the variation in density in the buoyancy force term.
 (II)
The viscous dissipation of energy is negligible.
 (III)
The radiation heat flux (\(q_{r}\)) in the vertical direction is considered to be negligible in comparison to that in the normal direction.
3 Exact solution
It may be noted that only the real part of \(V_{z}\) contributes to the fluid velocity, so far as the numerical calculations are concerned.
4 Mass flux
5 Skin friction
6 Nusselt number
7 Results and discussion
8 Conclusions

An increase in the Hartmann number M causes the fluid flow to be retarded.

The flow gets decelerated and therefore mass flux gets reduced corresponding to a reduction in the thermal diffusivity of the fluid.

Retardation in the fluid flow and a decrease in mass flux are observed with an increase in thermal radiation.

The buoyancy force causes the fluid flow to accelerate, thereby causing the mass flux to increase proportionately.

A reduction in fluid temperature is directly proportional to the diminution in thermal diffusivity.

Fluid temperature can be reduced by increasing the frequency parameter associated with the fluid flow.

Viscous drags at the inner and outer walls have identical magnitude but they act in opposite directions.

Friction at the walls increases with an increase in thermal Grashof number, but it diminishes with an increase in Prandtl number and radiation parameter, respectively.

The rate of heat transfer at either walls of the annulus with increasing frequency parameter depends on the Prandtl number of the fluid.
Declarations
Acknowledgements
The authors are thankful to the honorable reviewers for their valuable suggestion and comments, which improved the paper. This work is partially supported by unassigned grant of Gauhati University, India.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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