Abstract
An analytical continuum solution of the Rayleigh problem in slip flow with applied magnetic field is obtained using a modified initial condition and slip boundary conditions. The results are uniformly valid for all times and show that the velocity slip and the local skin friction coefficient remain almost unaffected by the imposition of the magnetic field for small times. They increase however with the magnetic field for large times. The present results reduce to the corresponding results of the hydrodynamic case when there is no magnetic field.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- A :
-
constant
- b :
-
characteristic length
- B :
-
magnetic field vector
- B 0 :
-
magntidue of the applied magnetic field normale to the plate
- B x :
-
magnitude of the induced magnetic field parallel to the plate
- C :
-
slip coefficient, (2−f)/f
- C f :
-
skin friction coefficient, \(\frac{{\mu \left( {\frac{{\partial u}}{{\partial y}}} \right)_{y = 0} }}{{\tfrac{1}{2}\rho U^2 }}\)
- C D :
-
average drag coefficient
- erfc(x):
-
complementary error function, \(1 - \frac{2}{{\pi ^{\tfrac{1}{2}} }}\int\limits_0^x {e^{ - \eta ^2 } d\eta }\)
- E :
-
electric field vector
- f :
-
Maxwell's reflection coefficient
- H a :
-
Hartmann number, (σB 20 b 2/μ)1/2
- \(\overline H _a = Kn H_a\) :
-
nondimensional magnetic parameter
- J :
-
current vector
- Kn=L/b :
-
Knudsen number
- L :
-
mean free path
- M :
-
Mach number
- p :
-
constant parameter
- P m :
-
magnetic Prandtl number, Re m/Re=μ 0 σν
- q :
-
velocity vector
- Re :
-
Reynolds number, Ub/ν
- Re m :
-
magnetic Reynolds number, μ 0 σUb
- t :
-
time
- \(\bar t\) :
-
nondimensional time, tU/b
- u :
-
velocity of the fluid parallel to the plate
- ū :
-
nondimensional velocity, u/U
- U :
-
velocity of the plate
- \(\bar V\) :
-
Laplace transform of ū
- x, y :
-
coordinates along and normal to the plate respectively
- y:
-
nondimensional distance, y/b
- Z :
-
nondimensional parameter, 1/Re 1/2 Kn
- γ :
-
ratio of specific heats
- δ :
-
boundary layer thickness
- Δū :
-
velocity slip
- μ :
-
viscosity
- μ 0 :
-
magnetic permeability
- ν :
-
kinematic viscosity
- ξ :
-
nondimensional time parameter, (\(\bar t\)/Re)1/2/Kn
- ρ :
-
density
- σ :
-
electrical conductivity
References
Schaaf, S. A., A note on flat plate drag coefficient, University of California, Institute of Engineering Research, M.E. 150–66, 1950.
Russo, E. P. and O. A. Arnas, ASME Trans., J. Appl. Mech. 34 (1967) 837.
Gross, E. P. and E. A. Jackson, Phys. Fluids 1 (1958) 318.
Yang, H. T. and L. Lees, Rarefied Gas Dynamics, Pergamon Press Inc., New York, 1960.
Sone, Y., J. Phys. Soc. Japan 19 (1964) 1463.
Cercignani, C. and F. Sernagiotto, Rarefied Gas Dynamics, Academic Press, New York, 1965.
Reddy, K. C., ASME Trans., J. Appl. Mech. 34 (1967) 833.
Rossow, Y. J., On flow of an electrically conducting fluid over a flat plate in the presence of transverse magnetic field, NASA Report No. 1358, 1958.
Chang, C. C. and J. T. Yen, Phys. Fluids 2 (1959) 393.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nath, G. Rayleigh problem in slip flow with transverse magnetic field. Appl. Sci. Res. 23, 315–323 (1971). https://doi.org/10.1007/BF00413207
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00413207