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A finite difference analysis of laminar magneto-hydrodynamic flow in the entrance region of a flat rectangular duct

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Applied Scientific Research, Section B

Summary

The development of the velocity profile, from a uniform one at the entrance, for an electrically conducting fluid entering a semi-infinite flat duct with a transversely applied magnetic field is investigated. It is assumed that the fluid has constant physical properties, the duct walls are electrically nonconducting, a uniform magnetic field is imposed perpendicular to the duct walls, and there can be a net electrical current flow parallel to the walls and perpendicular to the flow direction with a variable external resistance connecting the two side plates, which are displaced at infinity. The basic governing continuity and momentum equations are expressed in finite difference form and solved numerically on a high-speed digital computer with a mesh network superimposed on the flow field. A practical mesh size ratio suitable for computation was determined. Results were obtained for the variations of velocity and pressure distribution between the inlet and the region for the fully developed velocity profile for Hartmann numbers of 0, 4, and 10.

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Abbreviations

a :

duct half-height

B :

magnetic induction, B = μ e H

C :

friction factor defined in equation (15)

D e :

diameter, D e = 4 × area/perimeter

e :

electric field magnitude factor, E 0 = − eu 0 B 0

E :

electric field strength

H :

magnetic field intensity

K :

constant defined in equation (15)

J :

electric current density

M :

Hartmann number, M = μ e H 0 a[σ e /ρν]1/2

p :

fluid pressure

p 0 :

pressure at channel mouth

P :

dimensionless pressure, (p - p 0)/ρ u 0 2

ΔP:

pressure defect, (p 0 - p)/(ρ u 0 2/2)

Re a :

Reynolds number, ρ u 0 a/μ

s :

duct height

t :

time

u :

fluid velocity in x-direction

u 0 :

fluid velocity at inlet

U :

dimensionless u velocity, u/u 0

u :

fluid velocity in y-direction

V :

dimensionless u velocity, av ρ /μ

x :

coordinate along channel

X :

dimensionless x-coordinate μx/ ρ a 2 u 0 = (x/a)/Re a

X′:

dimensionless x-coordinate, as μx/ρ s 2 u 0 = X/4

y :

coordinate across channel

Y :

dimensionless y-coordinate, y/a

v :

kinematic viscosity

ρ :

fluid density

μ :

dynamic viscosity

μ e :

magnetic permeability

σ e :

electric conductivity

References

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Authors and Affiliations

  1. Kansas State University, Manhattan, Kansas, USA

    Ching-Lai Hwang & Liang-Tseng Fan

Authors
  1. Ching-Lai Hwang
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  2. Liang-Tseng Fan
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Hwang, CL., Fan, LT. A finite difference analysis of laminar magneto-hydrodynamic flow in the entrance region of a flat rectangular duct. Appl. sci. Res. 10, 329–343 (1963). https://doi.org/10.1007/BF03177939

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  • DOI: https://doi.org/10.1007/BF03177939

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