On the validity of the inductionless approximation and a φ–A0 formulation for transient liquid-metal magnetohydrodynamic flows in fusion blankets

Liquid-metal breeding blankets in fusion reactors operate in strong magnetic fields and may experience rapid electromagnetic forcing during plasma transients. Under normal steady-state reactor operation, the magnetic Reynolds number of the liquid-metal flow in a blanket is much smaller than unity, and the inductionless approximation is well justified. During plasma transients, however, liquid-metal velocities may increase substantially, and a magnetic Reynolds number based on the duct dimension can approach or exceed unity. This work examines whether such liquid-metal flows can nevertheless remain inductionless and develops a reduced φ–A0 formulation for time-dependent imposed magnetic fields, where φ is the electric scalar potential and A0 is the prescribed magnetic vector potential. Analytical and scaling arguments show that, for strong-field flows in which flow-induced magnetic perturbations are controlled by the Hartmann-layer dynamics, the relevant local length scale is the Hartmann-layer thickness rather than the duct dimension. The resulting local magnetic Reynolds number remains well below unity for representative blanket conditions. The analysis further distinguishes the inductionless condition from the quasi-static electromagnetic condition, since sufficiently rapid imposed-field variations may produce magnetic-diffusion or Alfvénic transients even at small local magnetic Reynolds numbers. Within the quasi-static regime, the proposed φ−A0 formulation retains the non-conservative electric field required by Faraday's law without evolving the flow-induced magnetic field. Analytical cross-verification for a two-dimensional rectangular duct with electrically insulating walls yields the same velocity and current-density fields as the corresponding magnetic-field formulation.