  Long duration gamma-ray bursts are powerful sources that can accelerate
particles to ultra-high energies. Acceleration of protons in the forward shock
of the highly relativistic GRB blastwave allows PeV--EeV neutrino production by
photopion interactions of ultra-high energy protons with X-ray to optical
photons of the GRB afterglow emission. Four different blastwave evolution
scenarios are considered: adiabatic and fully radiative blastwaves in a
constant density circumburst medium and in a wind environment with the particle
density in the wind decreasing inversley proportional to the square of the
radius from the center of the burst. The duration of the neutrino flux depends
on the evolution of the blastwave, and can last up to a day in the case of an
adibatic blastwave in a constant density medium. Neutrino fluxes from the three
other blastwave evolution scenarios are also calculated. Diffuse neutrino
fluxes calculated using the observed rate of long-duration GRBs are consistent
with the recent IceCube upper limit on the prompt GRB neutrino flux below PeV.
The diffuse neutrino flux needed to explain the two neutrino events at PeV
energies recently detected by IceCube can partially come from the presented GRB
blastwave diffuse fluxes. Future observations by IceCube and upcoming huge
radio Askaryan experiments will be able to probe the flux models presented here
or constrain the GRB blastwave properties.
