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An unsteady MHD duct flow

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Abstract

An exact solution is presented for unsteady laminar flow of a viscous, incompressible, electrically conducting fluid between nonconducting, parallel, flat plates. A constant magnetic field is suddenly applied perpendicular to the plates and the motion is modified by the induced current. Numerical results are given which show how the velocity profile changes from the parabolic profile of hydrodynamics to the Hartmann profile of magnetohydrodynamics.

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Abbreviations

2a :

separation of parallel plates

B :

dimensionless induced field

\(\bar B\) :

Laplace transform of B

E :

electric field vector

H :

magnetic field vector

H x :

induced magnetic field in x-direction

H 0 :

applied transverse magnetic field

j :

current density vector

k :

pressure gradient in x-direction

M :

Hartmann number

m 1, m 2 :

−M/2±(M 2/4+q)1/2

p :

pressure in fluid

q :

Laplace transform parameter

V :

fluid velocity vector

V x :

fluid velocity

W :

dimensionless fluid velocity

\(\bar W\) :

Laplace transform of W

W 0 :

mean fluid velocity

α v :

−π 2 ν 2/4, ν=1, 2, 3,...

η :

dimensionless space coordinate in y-direction

μ :

magnetic permeability

ν :

kinematic viscosity of fluid

ρ :

density of fluid

σ :

electrical conductivity of fluid

τ :

dimensionless time

References

  1. Hartmann, J., Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 15 (1937) No. 6.

  2. Hwang, C. L., K. C. Li and L. T. Fan, Phys. Fluids 9 (1966) 1134.

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  3. Chang, C. C. and T. S. Lundgren, Z. Angew. Math. Phys. 12 (1961) 100.

    MathSciNet  Google Scholar 

  4. Yen, J. T. and C. C. Chang, Phys. Fluids 4 (1961) 1355.

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Authors and Affiliations

  1. Dept. of Mathematics, University of Strathclyde, Glasgow, UK

    D. M. Sloan

Authors
  1. D. M. Sloan
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Sloan, D.M. An unsteady MHD duct flow. Appl. Sci. Res. 25, 126–136 (1972). https://doi.org/10.1007/BF00382289

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  • DOI: https://doi.org/10.1007/BF00382289

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