Laminar flow of a conducting fluid with constant properties entering a semi-infinite flat duct with a transverse applied magnetic field is considered. The fluid is assumed to enter the duct with a uniform velocity profile. The duct walls are nonconducting and a variable external resistance connects the two end plates which are displaced to infinity. The velocity and pressure distributions are then determined for the region between the inlet and fully developed flow. A method developed by Schlichting is used wherein the flow field is divided into two sections and an appropriate analysis utilized in each. In the section near the inlet a boundary-layer formulation of equations is used and a solution developed in a series stream function with Blasius functions as coefficients. When this solution becomes unwieldy, an exponential velocity deviation from the fully developed flow is assumed and joined to the boundary-layer solution to complete the description of the flow.
An Approximate Analysis of Laminar Magnetohydrodynamic Flow in the Entrance Region of a Flat Duct
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Roidt, M., and Cess, R. D. (March 1, 1962). "An Approximate Analysis of Laminar Magnetohydrodynamic Flow in the Entrance Region of a Flat Duct." ASME. J. Appl. Mech. March 1962; 29(1): 171–176. https://doi.org/10.1115/1.3636451
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