Magnetohydrodynamic flow in a curved pipe

The flow of conductive fluids in highly conductive curved pipes is studied analytically in this paper. The flow is assumed to be steady state, laminar, and fully developed. Coupled continuity, Navier--Stokes, and appropriate Maxwell equations are solved in toroidal coordinates. The dimensionless parameters of the problem are Dean number K and Hartmann number Ha. For low Hartmann numbers [Ha2∼θ(1)], the solution is expanded in a power series of K and Ha2. For intermediate Hartmann numbers [Ha2∼θ(1000)], the solution is expressed as a power series of K. The axial velocity contours are shown to be shifted towards the outer wall. For low Ha, these contours are nearly circular. The effect of a strong transverse magnetic field is to enhance the compression of fluid towards the outer wall. The secondary flow field comprises a symmetric pair of counter-rotating vortices. A strong magnetic field is found to confine the secondary flow streamlines to a thin layer near the tube wall. The secondary flow rate in the near-wall boundary layer is increased by the magnetic field. This increase in flow rate raises the possibility of efficient convective cooling of curved first wall tubes in magnetic confinement fusion reactors (MFCR)