Magnetohydrodynamic mixed convection flow in vertical concentric annuli with time periodic boundary condition: Steady periodic regime

Abstract

This work reports an analytical solution for fully developed mixed convection flow of viscous, incompressible, electrically conducting fluid in vertical concentric annuli under the influence of a transverse magnetic field, where the outer surface of inner cylinder is heated sinusoidally and the inner surface of outer cylinder is kept at a constant temperature. The analysis is carried out for fully developed parallel flow and steady-periodic regime. The governing dimensionless momentum and energy equations are separated into steady and periodic parts and solved analytically. Closed form solutions are expressed in terms of modified Bessel function of first and second kind. The influence of each governing parameters such as magnetic field parameter, Prandtl number and the dimensionless frequency of heating on flow formation and thermal behaviour are discussed with the aid of graphs. During the course of investigation, it is found that the oscillation amplitude of the friction factor is maximized at a resonance frequency near the surface of the concentric annuli where there is periodic heating. Furthermore, increasing transverse magnetic field decreases the oscillation amplitude of the friction factor.

Appendix

$$ \begin{aligned} & Z_{1} = K_{0} \left( {D_{1} } \right)I_{0} \left( {D_{1} d} \right) - K_{0} \left( {D_{1} d} \right)I_{0} \left( {D_{1} } \right),\quad Z_{2} = \frac{{\left[ {I_{0} \left( {D_{1} } \right) - I_{0} \left( {D_{1} d} \right)} \right]}}{{Z_{1} M^{2} }},\quad Z_{3} = \frac{{\left[ {K_{0} \left( {D_{1} d} \right) - K_{0} \left( {D_{1} } \right)} \right]}}{{Z_{1} M^{2} }}, \\ & Z_{4} = \frac{{\left[ {Z_{3} \left( {dI_{1} \left( {D_{1} d} \right) - I_{1} \left( {D_{1} } \right)} \right) - Z_{2} \left( {dK_{1} \left( {D_{1} d} \right) - K_{1} \left( {D_{1} } \right)} \right)} \right]}}{{D_{1} }},\quad Z_{5} = K_{0} \left( {D_{2} d} \right)I_{0} \left( {D_{2} } \right) - K_{0} \left( {D_{2} } \right)I_{0} \left( {D_{2} d} \right), \\ & Z_{6} = K_{0} \left( {D_{3} } \right)I_{0} \left( {D_{3} d} \right) - K_{0} \left( {D_{3} d} \right)I_{0} \left( {D_{3} } \right),\quad Z_{7} = \frac{{\left[ {I_{0} \left( {D_{3} } \right) - I_{0} \left( {D_{3} d} \right)} \right]}}{{Z_{6} i\Omega }},\quad Z_{8} = \frac{{I_{0} \left( {D_{3} } \right)}}{{\left[ {D_{3}^{2} - D_{2}^{2} } \right]Z_{6} }}, \\ & Z_{9} = \frac{{\left[ {K_{0} \left( {D_{3} d} \right) - K_{0} \left( {D_{3} } \right)} \right]}}{{Z_{6} i\Omega }},\quad Z_{10} = \frac{{K_{0} \left( {D_{3} } \right)}}{{\left[ {D_{2}^{2} - D_{3}^{2} } \right]Z_{6} }},\quad Z_{11} = \frac{{\left[ {dI_{1} \left( {D_{3} d} \right) - I_{1} \left( {D_{3} } \right)} \right]}}{{D_{3} }}, \\ & Z_{12} = \frac{{\left[ {dK_{1} \left( {D_{3} d} \right) - K_{1} \left( {D_{3} } \right)} \right]}}{{D_{3} }}\quad Z_{13} = \frac{{C_{3} }}{{D_{2} }}\left[ {dI_{1} \left( {D_{2} d} \right) - I_{1} \left( {D_{2} } \right)} \right],\quad Z_{14} = \frac{{C_{4} }}{{D_{2} }}\left[ {dK_{1} \left( {D_{2} d} \right) - K_{1} \left( {D_{2} } \right)} \right], \\ & Z_{15} = \frac{1}{{\left[ {D_{3}^{2} - D_{2}^{2} } \right]}},\quad Z_{16} = Z_{15} \left[ {Z_{13} - Z_{14} } \right],\quad \\ & C_{2} = \lambda_{a} Z_{2} ,\quad C_{3} = - \frac{{K_{0} \left( {D_{2} } \right)}}{{Z_{5} }},\quad C_{4} = \frac{{I_{0} \left( {D_{2} } \right)}}{{Z_{5} }}, \\ & C_{5} = Z_{9} \lambda_{b} + Z_{10} ,\quad C_{6} = Z_{7} \lambda_{b} + Z_{8}. \\ \end{aligned} $$

$a$ :

Radius of the inner cylinder

$A(t)$ :

Function of time

$B_{0}$ :

Constant magnetic flux density

$b$ :

Radius of the outer cylinder

$d$ :

Radius ratio

$f$ :

Fanning friction factor

$g$ :

Gravitational acceleration

$Gr$ :

Grashof number

$i$ :

Imaginary unit

$I_{n}$ :

Modified Bessel function of first kind and order $n$

$K_{n}$ :

Modified Bessel function of second kind and order $n$

$k$ :

Thermal conductivity

$n$ :

Integer number

$M$ :

Transverse magnetic field

p :

Pressure

$P$ :

Difference between the pressure and the hydrostatic pressure

$\Pr$ :

Prandtl number

$R$ :

Radial coordinate

$\text{Re}$ :

Reynolds number

${\Re }\text{e}$ :

Real part of a complex number

t :

Time

T :

Temperature

$T_{0}$ :

Mean temperature in concentric annuli defined in Eq. (5)

$T_{1}$ :

Mean temperature of the outer surface of inner cylinder

u :

Dimensionless velocity

$u^{*}$ :

Dimensionless complex-valued function

$u_{a}^{*} ,u_{b}^{*}$ :

Dimensionless complex-valued function

$U$ :

Fluid velocity

$X$ :

Longitudinal coordinate in vertical direction

$\alpha$ :

Thermal diffusivity

$\beta$ :

Volumetric coefficient of thermal expansion

$\Delta T$ :

Amplitude of the temperature oscillations at the outer surface of inner cylinder

$\lambda$ :

Dimensionless parameter defined in Eq. (10)

$\lambda^{*}$ :

Dimensionless complex-valued function defined in Eq. (18)

$\lambda_{a}^{*} ,\lambda_{b}^{*}$ :

Dimensionless complex-valued function defined in Eq. (23)

$\eta$ :

Dimensionless time

$\theta$ :

Dimensionless temperature

$\theta_{a}^{*} ,\theta_{b}^{*}$ :

Dimensionless complex-valued function defined in Eq. (23)

$\mu$ :

Dynamic viscosity

$\nu$ :

Kinematic viscosity

$\Phi$ :

Dimensionless heat flux

$\Phi _{a}^{*} ,\Phi _{b}^{*}$ :

Dimensionless complex-valued function

$\rho$ :

Mass density

$\rho_{0}$ :

Mass density for $T = T_{0}$

$\omega$ :

Frequency of the temperature oscillation

$\Omega$ :

Dimensionless frequency

$\sigma$ :

Electrical conductivity of the fluid