Left water radical - anchors the meaning. Juice, gravy, broth, ink are all liquids. Anything pressed out of fruit or boiled down from food is 汁. Joins the family of liquid words: 汤 (soup), 油 (oil), 酒 (wine), 泪 (tear).
Right 十 supplies the sound (shí to zhī, with palatalization). The 'ten' meaning has no role here. The simplest two-component phono-semantic compound in this set: water radical + sound key = juice. Same phonetic body in 针 (needle), 计 (count) - all share the historical shi reading.
