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In contrast with the fast solar wind, that originates in coronal holes, the source of the slow solar wind is still debated. Often intermittent and enriched with low FIP elements -- akin to what is observed in closed coronal loops -- the slow wind could form in bursty events nearby helmet streamers. Slow winds also exhibit density perturbations which have been shown to be periodic and could be associated with flux ropes ejected from the tip of helmet streamers, as shown recently by the WISPR white light imager onboard Parker Solar Probe (PSP). In this work, we propose that the main mechanism controlling the release of flux ropes is a flow-modified tearing mode at the heliospheric current sheet (HCS). We use MHD simulations of the solar wind and corona to reproduce realistic configurations and outflows surrounding the HCS. We find that this process is able to explain long ($\sim 10-20$h) and short ($\sim 1-2$h) timescales of density structures observed in the slow solar wind. This study also sheds new light on the structure, topology and composition of the slow solar wind, and could be, in the near future, compared with white light and in situ PSP observations.