Bifurcation phenomena of natural convection in horizontal annulus under self-induced circular magnetic fields

Nomenclature

Roman letters

B

= Magnetic-field flux vector;

F

= Average equivalent conductivity ratio of Case 1 and Case 2;

F_L

= Lorentz force vector;

g

= Gravitational acceleration;

Ha

= Hartmann number;

J

= Electrical current density;

k_eq

= Equivalent conductivity;

L

= Gap size between the inner and outer surfaces;

Nu

= Nusselt number;

n^*

= Coordinate normal to the wall;

Pr

= Prandtl number;

p

= Pressure;

Ra

= Rayleigh number;

r

= Distance from the center of the annulus;

T

= Temperature;

t

= Time;

U, V

= Dimensionless velocity components along the x and y directions;

u, v

= Velocity components along the x and y directions;

V

= Velocity vector;

X, Y

= Dimensionless Cartesian coordinates; and

x, y

= Cartesian coordinates.

Greek letters

α

= Thermal diffusivity;

β

= Thermal expansion coefficient;

θ

= Dimensionless temperature;

ρ

= Density;

σ

= Electrical conductivity;

τ

= Dimensionless time;

υ

= Kinematic viscosity;

φ

= Angle; and

ϕ

= Electric potential.

Subscript

X, Y

= Dimensionless x- and y-directional components;

x, y

= x- and y-directional components;

i

= Inner wall surface;

o

= Outer wall surface; and

ref

= Reference value.

1. Introduction

Natural convection in an annulus is frequently observed in various engineering fields, such as energy storage systems, nuclear reactor systems and the Czochralski crystal growth process. Accurately predicting the heat-transfer rate in these applications is essential for technical maturity. Bifurcation is a phenomenon where an engineering problem has multiple solutions according to the initial condition or external perturbation, even if the same boundary conditions are applied. Natural convection in an annulus is a good example of the bifurcation phenomenon. The bifurcation of the natural convection induces multiple solutions for not only the velocity field but also the resulting heat-transfer rate. Further, bifurcation is closely related to the stability problem. Therefore, a detailed investigation of the bifurcation phenomenon is helpful to improve the technical maturity for the corresponding problem.

In the natural convection in an annulus, two half-moon-shaped flow cells are generally formed symmetrically under low-Rayleigh number conditions. However, as the Rayleigh number increases, the steady solution starts to bifurcate, and two different flow patterns appear depending on the initial...

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