Abstract
In the present paper, we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field. The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account. The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity, induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli. The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables. It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder, while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder. These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases. It is also found that as the Hartmann number increases, there is a flattening tendency for both the velocity and the induced magnetic field. The influence of the induced magnetic field is to increase the velocity profiles.
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Abbreviations
- a :
-
Radius of inner cylinder (m)
- b :
-
Radius of outer cylinder (m)
- g :
-
Acceleration due to gravity (m/s2)
- H′ z′ :
-
Induced magnetic field in z′-direction (A/m)
- H :
-
Non-dimensional induced magnetic field in z-direction
- J θ :
-
Induced current density (A/m2)
- M :
-
Hartmann number
- q′:
-
Heat flux (W/m2)
- r′, θ′, z′:
-
Cylindrical coordinates (m), (°), (m)
- r :
-
Non-dimensional radial distance
- T′ f :
-
Ambient temperature (K)
- T′ w :
-
Temperature at outer surface of inner cylinder (at r′ = a), (K)
- T :
-
Temperature of the fluid in non-dimensional form (K)
- u :
-
Fluid velocity in non-dimensional form along axial direction (m/s)
- u′:
-
Velocity of fluid along axial direction (m/s)
- U :
-
Characteristics velocity of fluid (m/s)
- β :
-
Coefficient of thermal expansion
- µ e :
-
Magnetic permeability (H/m)
- η :
-
Magnetic diffusivity
- ϑ :
-
Kinematic viscosity of the fluid (m2/s)
- ρ :
-
Density of fluid (kg/m3)
- λ :
-
Ratio of outer radius and inner radius, b/a
- σ :
-
Conductivity of fluid (A2S−3/kgm3)
- 1:
-
Value at inner cylinder
- λ :
-
Value at outer cylinder
- θ :
-
Along θ-direction
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Singh, R.K., Singh, A.K. Effect of induced magnetic field on natural convection in vertical concentric annuli. Acta Mech Sin 28, 315–323 (2012). https://doi.org/10.1007/s10409-012-0052-4
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DOI: https://doi.org/10.1007/s10409-012-0052-4