Magnetohydrodynamic stability of the developing laminar flow in a parallel-plate channel

Linear stability of the developing laminar flow of an electrically conducting, incompressible fluid in a parallel-plate channel under a transverse magnetic field is investigated. The case of small magnetic Reynolds numbers is treated. The developing flows whose stability characteristics are studied correspond to those induced by a uniform and a parabolic velocity distribution at the channel inlet. The stability of the fully developed Hartmann flow is also reexamined. Neutral stability curves and axial variations of the critical Reynolds number are presented for a range of Hartmann numbers between 0 and 4. It is found that for Hartmann numbers larger than one, the developing flow induced by a uniform inlet velocity distribution is the least stable, while the flow induced by a parabolic inlet velocity distribution could be the most stable in the entrance region of the channel. A comparison with the available experimental data shows that for a given Hartmann number the transition Reynolds number is about two orders of magnitude lower than the critical Reynolds number.