Abstract
For MHD flows in a rectangular duct with unsymmetrical walls, two analytical solutions have been obtained by solving the governing equations in the liquid and in the walls coupled with the boundary conditions at fluid-wall interface. One solution of ‘Case I’ is for MHD flows in a duct with side walls insulated and unsymmetrical Hartmann walls of arbitrary conductivity, and another one of ‘Case II’ is for the flows with unsymmetrical side walls of arbitrary conductivity and Hartmann walls perfectly conductive. The walls are unsymmetrical with either the conductivity or the thickness different from each other. The solutions, which include three parts, well reveal the wall effects on MHD. The first part represents the contribution from insulated walls, the second part represents the contribution from the conductivity of the walls and the third part represents the contribution from the unsymmetrical walls. The solution is reduced to the Hunt’s analytical solutions when the walls are symmetrical and thin enough. With wall thickness runs from 0 to ∞, there exist many solutions for a fixed conductance ratio. The unsymmetrical walls have great effects on velocity distribution. Unsymmetrical jets may form with a stronger one near the low conductive wall, which may introduce stronger MHD instability. The pressure gradient distributions as a function of Hartmann number are given, in which the wall effects on the distributions are well illustrated.
Similar content being viewed by others
References
Davidson P A. An Introduction to Magnetohydrodynamics. Cambridge: Cambridge University Press, 2001
Molokov S, Moreau R, Moffatt H K. Magnetohydrodynamics-Historical Evolution and Trends. New York: Springer, 2007
Moreau R J. Magnetohydrodynamics. Dordrecht: Kluwer Academic Publishers, 1990
Müller U, Bühler L. Magnetofluiddynamics in Channels and Containers. Berlin: Springer, 2001
Shercliff J A. Steady motion of conducting fluids in pipes under transverse magnetic fields. Proc Camb Phil Soc, 1953, 49: 126–144
Chang C C, Lundgren T S. Duct flow in magnetohydrodynamics. Z Angew Math Phys, 1961, 12: 100–114
Uflyand Y S. Flow stability of a conducting fluid in a rectangular channel in a transverse magnetic field. Soviet Phys Tech Phys, 1961, 5: 1191–1193
Hunt J C R. Magnetohydrodynamic flow in rectangular ducts. J Fluid Mech, 1965, 21: 577–590
Sloan D M, Smith P. Magnetohydrodynamic flow in a rectangular pipe between conducting plates. Z Angew Math Mech, 1966, 46: 439–443
Butler G F. A note on magnetohydrodynamic duct flow. Proc Camb Phil Soc, 1969, 66: 66–77
Gold R R. Magnetohydrodynamic pipe flow. Part 1. J Fluid Mech, 1962, 13: 505–512
Samad S K A. The flow of conducting fluids through circular pipes having finite conductivity and finite thickness under uniform transverse magnetic fields. Int J Eng Sci, 1981, 19: 1221–1232
Ni M-J, Munipalli R, Huang P, et al. A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated mesh. J Comput Phys, 2007, 227: 205–228
Schercliff J A. The flow of conducting fluids in circular pipes under transverse magnetic fields. J Fluid Mech, 1956, 1: 644–666
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tao, Z., Ni, M. Analytical solutions for MHD flow at a rectangular duct with unsymmetrical walls of arbitrary conductivity. Sci. China Phys. Mech. Astron. 58, 1–18 (2015). https://doi.org/10.1007/s11433-014-5518-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11433-014-5518-x