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Linearized analysis of magnetohydrodynamic entrance flow in a vertical channel with combined thermal convection

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Abstract

A linearized analysis is presented for the magnetohydrodynamic entrance flow with combined forced and free convection in a vertical, constant wall temperature parallel-plate channel. Numerical results are obtained for slug velocity profile at the entrance and for various Hartmann and Grashof Numbers. The results agree well with the finite difference numerical solutions obtained elsewhere. They demonstrate that the velocity development and pressure gradient in the channel entrance region are greatly influenced by the Hartmann Number and the Grashof Number. Increasing Hartmann Number decreases velocity entrance length while increasing Grashof Number increases it. Thermal development is also found to be dependent on the above mentioned parameters, but to a relatively minor extent.

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Abbreviations

A m :

constant defined by equation (23)

B 0 :

applied magnetic field

C n :

constant defined by equation (13)

E 0 :

constant electric field

e :

nondimensional electric field parameter, E 0/U mB0

Gr :

Grashof Number, βgL 3(T w−T 0)/ν 2

L :

half-width of the channel

M :

Hartmann Number, B 0 L(σ/μ)1/2

Nu :

Nusselt Number, (∂θ/∂y) y=1/(θ w−θ m)

P :

pressure

Pr :

Prandtl Number, ν/α

p :

nondimensional pressure parameter, (P−P 0+ρ 0 gX)/P 0 U 2m

Re :

Reynolds Number, U m L/ν

T :

temperature

T 0 :

inlet temperature

T w :

wall temperature

U :

velocity, X direction

U m :

average velocity, (1/L)∫ L0 UdY

u :

nondimensional form of U, U/U m

u 0(y):

nondimensional inlet velocity

V :

velocity, Y direction

v :

nondimensional form of V, VL/ν

X :

coordinate, axial direction

x :

nondimensional form of X, vX/L 2 U m

Y :

coordinate perpendicular to the channel

y :

nondimensional form of Y, Y/L

α :

thermal diffusivity

α m :

eigenvalue defined by equation (25)

β :

thermal expansion coefficient

β m :

eigenvalue defined by equation (24)

ε :

stretching factor, weighting function

θ :

nondimensional form of T, (T−T 0)/(T w−T 0)

θ m :

mean nondimensional temperature, ∫ 10 θudy

ν :

kinematic viscosity

μ :

magnetic permeability

ρ :

mass density

σ :

electrical conductivity

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Authors and Affiliations

  1. Dept. of Eng. Sci., Aerospace Eng. and Nuclear Eng., State Univ. of New York at Buffalo, N.Y., USA

    C. P. Yu & C. E. Hendrix

Authors
  1. C. P. Yu
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  2. C. E. Hendrix
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Yu, C.P., Hendrix, C.E. Linearized analysis of magnetohydrodynamic entrance flow in a vertical channel with combined thermal convection. Appl. Sci. Res. 33, 369–384 (1977). https://doi.org/10.1007/BF00383962

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  • DOI: https://doi.org/10.1007/BF00383962

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