  We develop a hydrodynamic theory for an electron system exhibiting the
anomalous Hall effect, and show that an additional anomalous Hall effect is
induced by a vorticity generated near boundaries. We calculate the momentum
flux and force proportional to the electric field using linear response theory.
The hydrodynamic equation is obtained by replacing the local electric field
with the electric current, focusing on a scale that is sufficiently larger than
the mean free path. It is demonstrated that there is a coupling between a
vorticity of an electric current and a magnetization which generates a pressure
from non-uniform vorticity. Taking into account Hall viscosity and relaxation
forces, a non-uniform flow near a boundary and an additional Hall force are
calculated. The additional anomalous Hall force is opposite to conventional
anomalous Hall force, resulting in a sign reversal in thin systems.
