Answer: The droplets are caused by turgor pressure. As roots take up fluids, they move throughout the tissues of a plant, filling the cells with water and salts. Cells can and will squeeze out excess fluids, or they'd become so inflated that they would burst. What you're witnessing is common with grape vines, especially early in the season when the plant is about to go into a vigorous period of growth. It's normal and natural - nothing to worry about. If your grapevines were outdoors and were pruned in early spring, they'd pour out copious amounts a liquid from every pruning cut. It's annoying, but not harmful at all to the plant. Eventually the weeping will stop on its own.
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