  The Brillouin flow is a rectilinear, sheared electron fluid flow in a crossed
electric field (E) and magnetic field (B), in the E x B direction with zero
flow velocity and zero electric field at the surface with which the flow is in
contact. It is broadly considered as the equilibrium electron flow in high
power crossed-field devices including the magnetron and magnetically insulated
transmission line oscillators. This paper provides an examination of Brillouin
flow in two dimensions, in a cylindrical geometry where the anode radius
changes abruptly at a single axial location while the cathode surface has a
constant radius. Our simulation confirms the proof that there is no equilibrium
Brillouin flow solution for such a geometry. It further reveals that this
change in the anode radius introduces novel bunching of the electrons within
the Brillouin hub. This bunching occurs at low frequencies and is very
pronounced if the Brillouin flow is from the small gap region to the large gap
region, but is minimal if the Brillouin flow is from the large gap region to
the small gap region. New insights are provided into the physical processes
that initiate and sustain the bunching processes that are unique for a
crossed-field diode, as compared with a non-magnetized diode. We argue that
this enhanced bunching, and its concomitant formation of strong vortices, is
not restricted to an abrupt change in the anode-cathode gap spacing.
