In induction heaters metal seems to levitate in center of radial axis, as well as longitudinal axis (see video). Is by any chance the reason for that like in this schematic, that magnetic field lines of a metal rod, forming closed loops, move in opposite direction to the coil when going around the sides of the metal? So, repulsion at the sides, and attraction along the poles. Or, is it other reason?
Eddy currents neutralize partially the varying magnetic field.
In electromechanical systems force affects always to the direction where smallest amount of movement causes maximal loss of field energy. There happens to be maximum loss when the heated piece is in the middle of the coil. That shouldn't surprise anyone who knows that inside a coil (which has current) there's magnetic field strength maximum in the middle. The levitation happens if the force caused by field energy reduction vs changing place happens to win the gravity.
BTW forces in DC solenoids and between permanent magnets and pieces of iron are based on the same field energy changing law but the change of the field energy occurs in AC field without magnetic materials, conductivity (+ caused eddy current) is enough.