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, non-Gray 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 z-axis. 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 z-axis, 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 z-direction, the temperature field is independent of θ and z. In (9), the rate of radiative heat flux in the optically thin limit for a non-Gray 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 non-dimensional 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 \(1 \le r \le \lambda\).