  We compute, in topological terms, the spectral flow of an arbitrary family of
self-adjoint Dirac type operators with classical (local) boundary conditions on
a compact Riemannian manifold with boundary under the assumption that the
initial and terminal operators of the family are conjugate by a bundle
automorphism. This result is used to study conditions for the existence of
nonzero spectral flow of a family of self-adjoint Dirac type operators with
local boundary conditions in a two-dimensional domain with nontrivial topology.
Possible physical realizations of nonzero spectral flow are discussed.
