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This paper presents the effect of magnetic field, indicated by Hartmann number (
*H*
*a*), on the free convective flow of Magneto-hydro-dynamic (MHD) fluid in a square cavity with a heated cone of different orientation. Although similar studies abound, the novelty of this work lies in the presence of the heated cone, whose orientation is varied at different angles. The mathematical model includes the system of governing mass, momentum and energy equations. The system is solved by finite element method. The calculations are performed for Prandtl number
*Pr* = 0.71; the Rayleigh number
*Ra* = 10, 1000, 100,000; and for Hartmann number
*Ha* = 0, 20, 50, 100. The results are illustrated with streamlines, velocity profiles and isotherms. From the results, it is found that for the present configuration, magnetic field (Hartmann number) has no effect on the shape of the streamlines for low Rayleigh numbers. However, for high values of
*Ra*, the effect of
*Ha* becomes quite visible. Magnetic field affects the flow by retarding the fluid movement, and thus affects convective heat transfer. At low
*Ra*, the fluid movement and heat transfer rate are already slowing, thus impressing a magnetic field does not produce much effect. At high
*Ra*, fluid particles move at high velocity and change the stream lines, in absence of any magnetic force. Impressing magnetic field in this situation produced noticeable effect by slowing down the fluid movement and changing the streamlines back to low
* Ra* situations. It is noted that a combination of low
*Ra* with zero or low
*Ha* produces similar effects with the combination of high
*Ra* and high
* Ha*. It can be concluded that with increasing
*Ha*, heat transfer mode in MHD fluid gradually changes toward conduction from convection. It can be surmised that sufficiently large
*Ha* can potentially stop the fluid movement altogether. In that case, heat transfer would be fully by conduction.

Heat transfer and fluid flow in electrically conductive or Magneto-hydro-dynamic (MHD) fluids is an important area of research. It is connected to many scientific and engineering applications, such as plasma containment, liquid metal processing, power generation, cooling of electrical and electronic equipment, high energy wind tunnels, etc. The theory of MHD was introduced by Alfven Hannes in 1940, for which he was awarded the Nobel Prize for physics in 1970. The first practical application is attributed to Julian Hartmann who designed an electro-magnetic pump for liquid metals in 1937. Due to the theoretical complexity and practical implications, MHD flow attracted many researchers, which is reflected by the volume of literature published on this subject. Chamkha [

Chamkha [

The brief discussion above shows the wide variety of works undertaken by the different researchers. The variations mainly involved thermal conditions of the walls, shape of the object in the cavity, and the fluid properties. It may be mentioned here that all of the above works were accomplished by using numerical methods or finite element methods. However, to the best of the authors’ knowledge, the issue of the present work, i.e., the effect of magnetic field on MHD free convection in a square cavity with a heated cone has not been reported.

_{h}) and low (T_{c}) temperatures respectively. The cone inside the cavity was oriented at three different angles: 1) the cone is vertical, 2) the cone is inclined to the left, and 3) the cone is inclined to the right. A magnetic field of uniform intensity B_{0} is applied on the fluid, perpendicular to the direction of flow. The gravitational force g, acts vertically downward.

All the fluid properties are considered to be constant except the density. Radiation heat transfer and Joule heating effects are neglected. Thus the governing equations for mass, momentum and energy are formulated as follows.

1) Conservation of mass ∂ u ∂ x + ∂ v ∂ y = 0 (Continuity Equation) (1)

2) Conservation of momentum:

a) ρ ( u ∂ u ∂ x + v ∂ u ∂ y ) = − ∂ p ∂ x + μ ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) (x-momentum Equation) (2)

b) ρ ( u ∂ v ∂ x + v ∂ v ∂ y ) = − ∂ p ∂ y + μ ( ∂ 2 v ∂ x 2 + ∂ 2 v ∂ y 2 ) + ρ g β ( T − T c ) − σ B 0 2 ν

(y-momentum Equation) (3)

3) Conservation of Energy: u ∂ T ∂ x + v ∂ T ∂ y = α ( ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 ) (Energy Equation) (4)

The governing equations are made dimensionless using the following dimensionless variables:

X = x L Y = y L U = u L α V = v L α P = p L 2 ρ α 2

θ = T − T c T h − T c σ = ρ 2 α L 2 α = k ρ C p ν = μ ρ

Applying these definitions, the following dimensionless equations are obtained.

∂ U ∂ X + ∂ V ∂ Y = 0 (5)

U ∂ U ∂ X + V ∂ U ∂ Y = − ∂ P ∂ X + Pr ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) (6)

U ∂ V ∂ X + V ∂ V ∂ Y = − ∂ P ∂ Y + Pr ( ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2 ) + R a Pr θ − H a 2 Pr Pr V (7)

U ∂ θ ∂ X + V ∂ θ ∂ Y = ∂ 2 θ ∂ X 2 + ∂ 2 θ ∂ Y 2 (8)

where,

Prandtl number, Pr = ν α ; (ratio of viscous to thermal diffusion rates, which

indicates the ratio or dominance of heat transfer mode-convection over conduction)

Hartmann number, H a 2 = σ B 0 2 L 2 μ ; (ratio of electromagnetic force to the viscous force)

Grashof number, G r = g β L 3 ( T h − T c ) ν 2 ; (ratio of the buoyancy to viscous force acting on a fluid)

Rayleigh number, R a = g β L 3 ( T h − T c ) Pr ν 2 ; (product of Gr and Pr. It also indicates the ratio or dominance of heat transfer mode-convection over conduction, but incorporates the buoyancy force)

The dimensionless boundary conditions become:

U = V = 0 , θ = 1 at bottom wall and heated conical body (at higher constant temperature)

U = V = 0 , θ = 0 at top wall (at lower constant temperature)

U = V = 0 , ∂ θ ∂ N = 0 at side walls (thermally insulated)

P = 0 Fluid pressure at the inside and on the walls of the cavity

The above system of equations is solved along with the boundary conditions shown above, by finite element method. This technique is described various researchers such as Taylor and Hood [

The calculations are performed for Prandtl number Pr = 0.71, the Rayleigh number Ra = 10, 1000, 100,000; and for Hartmann number Ha = 0, 20, 50, 100. The results are illustrated with streamlines, velocity profiles, and isotherms. Several observations can be made from the results, which are discussed next, according to the orientation of the cone.

Streamlines:

Velocity profiles: The y-component of velocity along two different lines parallel to the x-axis (y = 0.15 and y = 0.50) are presented.

conical body. Here also it is seen that the velocity increases by orders of magnitude with increasing Ra. Velocity peaks at four locations approximately at x = 0.1, 0.3, 0.7, and 0.9 (

Isotherms:

Streamlines:

Isotherms:

The results are very similar to those for the left inclined orientation. Both the streamlines and isotherms look almost like mirror image of the left inclined orientation. Therefore detailed discussions will be redundant. However, the figures are presented for the sake of completeness.

The effects of Ra and Ha on the fluid flow and heat transfer are investigated in a square cavity filled with MHD fluid, with a heated conical object inside the cavity. With increasing Ra, the velocity of the fluid particles increases significantly and the streamlines also show visible changes in shape. Ha affects the flow by retarding the fluid movement. Therefore, increasing Ha influences the streamlines, fluid velocities, and heat transfer. However, the effect of Ha is not significant for low values of Ra. It is also noted that the combination of low or zero Ha with low Ra produces very similar results with high Ha and high Ra. It can be said

that the heat transfer mode in MHD fluid gradually changes toward conduction from convection with increasing Ha. A sufficiently large magnetic field can potentially stop fluid movement altogether. In that case, heat transfer would be fully by conduction.

The author declares no conflicts of interest regarding the publication of this paper.

Mahjabin, S. and Alim, M.A. (2018) Effect of Hartmann Number on Free Convective Flow of MHD Fluid in a Square Cavity with a Heated Cone of Different Orientation. American Journal of Computational Mathematics, 8, 314-325. https://doi.org/10.4236/ajcm.2018.84025