The fundamental equations governing the motion of an incompressible, viscous, radiating, and electrically conducting fluid are:
All the physical quantities are described in the list of abbreviations.
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 \(a < b\).
A cylindrical polar coordinate system \(( r,\theta,z )\) is introduced with the axis of the coaxial cylinders as the zaxis. A magnetic field of intensity \(H_{0}\) (constant) is applied in the azimuthal direction. In order to make the physical model idealized, the present investigation is restricted to the following assumptions:

(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.
We recall that the fluid moves parallel to the zaxis, suggesting us to take \(\vec{q}\) as \(( 0,0,V_{z} )\). Equation (1) in \(( r,\theta,z )\) system becomes \(\frac{1}{r}\frac{\partial}{\partial z} ( rV_{z} ) = 0\) which yields \(V_{z} = V_{z} ( r,t )\), due to symmetry of the model. Proceeding with the analysis, the momentum equation takes the form
$$ \rho \frac{\partial V_{z}}{\partial t} =  \frac{\partial p}{\partial z} + \mu \biggl( \frac{\partial^{2}V_{z}}{\partial r^{2}} + \frac{1}{r}\frac{\partial V_{z}}{\partial r} \biggr)  \sigma \mu_{e}^{2}H_{0}^{2}V_{z}  \rho g. $$
(5)
The equation of state on the basis of classical Boussinesq approximation (Bergman et al. [8]) is
$$ \rho_{s} \simeq \rho \bigl[ 1 + \beta ( T  T_{s} ) \bigr]. $$
(6)
In the static condition, (5) renders \(0 =  \frac{\partial p_{s}}{\partial z}  \rho_{s}g\), where \(p_{s}\) is the static fluid pressure. Utilization of this in (5) produces
$$ \rho \frac{\partial V_{z}}{\partial t} =  \frac{\partial ( p  p_{s} )}{\partial z} + \mu \biggl( \frac{\partial^{2}V_{z}}{\partial r^{2}} + \frac{1}{r}\frac{\partial V_{z}}{\partial r} \biggr)  \sigma \mu_{e}^{2}H_{0}^{2}V_{z}  g ( \rho  \rho_{s} ). $$
(7)
With \(p^{*} = p  p_{s}\), the application of (6) leads to the following equation of motion:
$$ \frac{\partial V_{z}}{\partial t} =  \frac{1}{\rho} \frac{\partial p^{*}}{\partial z} + \frac{\mu}{\rho} \biggl( \frac{\partial^{2}V_{z}}{\partial r^{2}} + \frac{1}{r}\frac{\partial V_{z}}{\partial r} \biggr)  \frac{\sigma \mu_{e}^{2}H_{0}^{2}}{\rho} V_{z} + g\beta ( T  T_{s} ). $$
(8)
In lieu of the assumptions (II) and (III), the energy equation takes the form
$$ \rho C_{p}\frac{\partial T}{\partial t} = K_{T} \biggl[ \frac{\partial^{2}T}{\partial r^{2}} + \frac{1}{r}\frac{\partial T}{\partial r} \biggr]  4I ( T  T_{s} ). $$
(9)
On account of the axial symmetry and the annulus being infinite in the zdirection, the temperature field is independent of θ and z. In (9), the rate of radiative heat flux in the optically thin limit for a nonGray gas near equilibrium is due to the following formula attributed to Cogley et al. [9]:
$$ \frac{\partial q_{r}}{\partial r} = 4I ( T  T_{s} ), $$
(10)
where \(I = \int_{0}^{\infty} ( K_{\lambda '} )_{w} ( \frac{\partial e_{\lambda 'h}}{\partial T} )_{w}\, d\lambda '\).
The boundary conditions to be satisfied by (8) and (9) are
$$ \left . \begin{array}{l} V_{z} = 0 \quad \mbox{at } r = a, \\ V_{z} = 0 \quad \mbox{at } r = b, \\ T = T_{s} + T_{s}n_{1}e^{i\alpha_{1}t}\quad \mbox{at } r = a, \\ T = T_{s} + T_{s}n_{2}e^{i\alpha_{1}t}\quad \mbox{at } r = b. \end{array} \right \} $$
(11)
The following nondimensional quantities are introduced in order to normalize the model:
$$ \left . \begin{array}{l} V_{z}^{\prime} = \frac{V_{z}a}{\nu},\qquad r' = \frac{r}{a},\qquad z' = \frac{z}{a}, \qquad p^{*\prime} = \frac{p^{*}a^{2}}{\mu \nu}, \\ t' = \frac{t\nu}{a^{2}},\qquad \lambda = \frac{b}{a},\qquad \alpha = \frac{\alpha_{1}a^{2}}{\nu},\qquad \psi ' = \frac{T  T_{s}}{T_{s}}, \\ \mathit{Gr} = \frac{g\beta a^{3}}{\nu^{2}}T_{s},\qquad \mathit{Pr} = \frac{\mu C_{p}}{K_{T}}, \qquad Q = \frac{4Ia^{2}}{\mu C_{p}}. \end{array} \right \} $$
(12)
The dimensionless forms of (8), (9), and (11) are (removing the primes):
$$\begin{aligned}& \frac{\partial V_{z}}{\partial t} =  \frac{\partial p^{*}}{\partial z} + \biggl( \frac{\partial^{2}V_{z}}{\partial r^{2}} + \frac{1}{r}\frac{\partial V_{z}}{\partial r} \biggr)  M^{2}V_{z}  \mathit{Gr}\psi, \end{aligned}$$
(13)
$$\begin{aligned}& \mathit{Pr}\frac{\partial \psi}{\partial t} = \frac{\partial^{2}\psi}{\partial r^{2}} + \frac{1}{r} \frac{\partial \psi}{\partial r}  Q\mathit{Pr}\psi, \end{aligned}$$
(14)
$$\begin{aligned}& \left . \begin{array}{l} V_{z} = 0 \quad \mbox{at } r = 1, \\ V_{z} = 0\quad \mbox{at } r = \lambda, \\ \psi = n_{1}e^{i\alpha t} \quad \mbox{at } r = 1, \\ \psi = n_{2}e^{i\alpha t} \quad \mbox{at } r = \lambda, \end{array} \right \} \end{aligned}$$
(15)
where \(M = \mu_{e}H_{0}a\sqrt{\frac{\sigma}{\mu}}\) is the Hartmann number and \(1 \le r \le \lambda\).