Abstract
Transpiration cooling with magnetohydrodynamic effects is studied in a porous-wall Couette flow of a steadily moving, incompressible, electrically conducting fluid. The flow, induced by the motion of the moving wall, is subject to a uniform transverse magnetic field. A coolant fluid, with the same properties as the main-stream fluid, is uniformly injected into (or extracted from) the flow through a porous stationary wall. Solutions are presented for the velocity and induced magnetic field distributions across the parallel walls, the pressure difference induced by the magnetic drag, and the friction factor in such a flow configuration. Considerations are further given to the heat transfer characteristics wherein viscous and Ohmic heating are significant. In this connection, the recovery factor and the Nusselt number are also presented over a wide range of blowing and suction rates.
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Abbreviations
- B :
-
magnetic field vector with components (B x , B y , B z ) in (x, y, z) directions
- B x :
-
dimensionless magnetic field, B x /B 0
- B 0 :
-
uniform applied magnetic field
- C p :
-
specific heat at constant pressure
- E :
-
electric field vector
- Ē :
-
dimensionless electric field, E z /U 1 B 0
- Ek :
-
Eckert number, U 21 /C p (T 1−T 0)
- f :
-
friction factor, τ 0/ρU 21
- h :
-
heat transfer coefficient
- I :
-
total electric current across parallel walls
- J :
-
electric current density
- L :
-
distance between parallel walls
- M :
-
Hartmann number, \(\sqrt {\sigma /\rho v } B_0 L\) B 0 L
- Nu :
-
Nusselt number, hL/κ
- p :
-
pressure
- \(\bar p\) :
-
dimensionless pressure, pL/ρνU 1
- Pr :
-
Prandtl number, ρνC p /κ
- Pm :
-
magnetic Prandtl number, Rm/Re
- q 0 :
-
heat flux density at stationary lower wall
- r :
-
recovery factor, (T r −T 1)/(U 21 /C p )
- Re :
-
Reynolds number, U 1 L/ν
- Rm :
-
magnetic Reynolds number, σμ e U 1 L
- T :
-
temperature, with T 0 and T 1 as temperatures of the lower and upper walls
- \(\bar T\) :
-
dimensionless temperature, (T−T 0)/(T 1−T 0)
- T r :
-
recovery temperature
- U 1 :
-
uniform velocity of upper wall
- ū :
-
dimensionless velocity, u/U 1
- V :
-
fluid velocity vector with components (u, v, w) in (x, y, z) directions
- \(\bar v,\bar v_0\) :
-
dimensionless velocity, v/U 1=v 0/U 1
- v 0 :
-
velocity of injection or suction at the walls
- (x, y, z):
-
rectangular coordinates
- η :
-
dimensionless height, y/L
- Φ :
-
viscous dissipation function
- κ :
-
thermal conductivity
- μ e :
-
magnetic permeability
- ν :
-
kinematic viscosity
- ρ :
-
fluid density
- σ :
-
electrical conductivity
- τ 0 :
-
shear stress at stationary lower wall
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Tan, C.W. Magnetohydrodynamic Couette flow with transpiration cooling. Appl. Sci. Res. 23, 167–184 (1971). https://doi.org/10.1007/BF00413195
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DOI: https://doi.org/10.1007/BF00413195