To push something or someone gently, especially to push someone with your elbow (= the middle part of your arm where it bends) to attract the person’s attention. (Cambridge Dictionary)
To move slowly and almost reach a higher point or level. (Cambridge Dictionary)
If you nudge someone, you push them gently, usually with your elbow, in order to draw their attention to something. (Collins Dictionary)
If you nudge someone or something into a place or position, you gently push them there. (Collins Dictionary)
To touch or push gently. (Merriam – Webster)
Prod (someone) gently with one’s elbow in order to attract attention.
The Book Nudge and Its Lessons to Companies.
Nudge, not sludge