OK one thought:
Make the flowing drop as you have in the image above, with OK next to it.
Add an armature with two bones, parent the “end” or “drop” part of the mesh to the second smaller bone, make this bone a parent of the first bone, so it does NOT inherit scale.
Parent the rest of the “smear” vertices to the main bone, Then scale the main bone in Y so the “smear” gets longer, but not wider or thicker. This will move the “drop” bone and the “drop” part of the mesh away from the tail of the main bone, and the “smear” will get longer.
As an added bonus, you could put a driver or transform constraint on the X and Z scales of the main bone so the “smear” gets thinner or narrower as the “drop” moves away, you could also add drivers to the “drop” bone so it too gets smaller as it moves.
I will look at doing a sample blend file later on today - got to go out now.
The other option would be to use a curve as the path and animate the “smear” as a bevel object, by altering the start and end values, then add a curve modifier a separate “drop” mesh so it moves along the curve as the “smear” progresses, see my website, fourth page of the “Cord on a Drum” tutorial for an example (the snake pictures. You have to go through each page to get the link to the next one.) website is: http://www.lafavinie.f2s.com/blender/index.html