I think 身を任せる
(Literally: "leave a body to something/someone", "Leave a body to being under other people/thing's control")
means
"throw away your will/intention and leave yourself to somebody/something"
"let it go as it goes", "go with the flow",
「時の流れに身を任せる」
--> "leave yourself to the flow of the time"
--> No plan, action, intention. Let everything go as it goes.
「愛に身を任せる」
--> "leave yourself to love“
--> Don't think too much. Let yourself go as the feeling of love goes.
「あなたに身を任せる」
--> "leave myself(my body) to you“
--> (Sexual implication)
****
The point is "throwing away your will / intention".