Tiny quantity of liquid around old sink (5)
I believe the answer is:
droop
'sink' is the definition.
(drooping is a kind of sinking)
'tiny quantity of liquid around old' is the wordplay.
'tiny quantity of liquid' becomes 'drop'.
'around' means one lot of letters goes inside another.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'drop' enclosing 'o' is 'DROOP'.
(Other definitions for droop that I've seen before include "Hang down loosely" , "languish" , "Go limp" , "Hang downwards, lose strength or spirit" , "Hang limply" .)