  One of the scenarios for the formation of grand-design spiral arms in disky
galaxies involves their interactions with a satellite or another galaxy. Here
we consider another possibility, where the perturbation is instead due to the
potential of a galaxy cluster. Using $N$-body simulations we investigate the
formation and evolution of spiral arms in a Milky Way-like galaxy orbiting a
Virgo-like cluster. The galaxy is placed on a few orbits of different size but
similar eccentricity and its evolution is followed for 10 Gyr. The tidally
induced, two-armed, approximately logarithmic spiral structure forms on each of
them during the pericenter passages. The spiral arms dissipate and wind up with
time, to be triggered again at the next pericenter passage. We confirm this
transient and recurrent nature of the arms by analyzing the time evolution of
the pitch angle and the arm strength. We find that the strongest arms are
formed on the tightest orbit, however they wind up rather quickly and are
disturbed by another pericenter passage. The arms on the most extended orbit,
which we analyze in more detail, wind up slowly and survive for the longest
time. Measurements of the pattern speed of the arms indicate that they are
kinematic density waves. We attempt a comparison with observations by selecting
grand-design spiral galaxies in the Virgo cluster. Among those, we find nine
examples bearing no signs of recent interactions or the presence of companions.
For three of them we present close structural analogues among our simulated
spiral galaxies.
