A theoretical study of the effects of wall conductivity, non-uniform magnetic fields and variable-area ducts on liquid-metal flows at high Hartmann number

Abstract

Flows of incompressible, electrically conducting liquids along ducts with electrically insulating or weakly conducting walls situated in a strong magnetic field are analysed. Except over a short length along the duct where the magnetic field strength and/or the duct cross-sectional area vary, the duct is assumed to be straight and the field to be uniform and aligned at right angles to the duct. Magnitudes of the field strength B₀ and the mean velocity V are taken to be such that the Hartmann number M [Gt ] 1, the interaction parameter N (= M²/Re) [Gt ] 1 (Re being the Reynolds number of the flow) and the magnetic Reynolds number R_m [Lt ] 1.

For an O(1) change in the product VB₀ along the duct across the non-uniform region, it is shown that:

(i) In the non-uniform region the streamlines and current flow lines follow surfaces containing the field lines satisfying $\int B^{-1}ds = {\rm constant}$, the integration being carried out along the field line within the duct; these surfaces are equipotentials and isobarics. This leads to

(ii) a tube of stagnant, but not current-free fluid at the centre of the duct parallel to the field lines around which the flow divides to bypass it. To accommodate this flow,

(iii) the usual uniform field/straight duct flow is disturbed over very large distances upstream and downstream of this region, the maximum length O(duct radius × M^½) occurring in a non-conducting duct;

(iv) a large pressure drop is introduced into the pressure distribution regardless of the direction of the flow, the effect being most severe in a non-conducting duct, where the drop is O(duct radius × (uniform field/straight duct pressure gradient) × M^½);

(v) in the part of the duct with the lower value of VB₀ a region of reverse flow occurs near the centre of the duct and the stagnant fluid.