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.
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