Regarding the OP doc and UV coordinates. A major area of investigation for us back in the day was finding interesting ways to displace the uv texture coordinates for each corner of the rectangular mesh. We used per-vertex colors, these days one would use a fragment (pixel) shader like those in ShaderToy.
A very interesting process displaces the texture coordinates by advecting them along a flow field. Use any 2D vector field and apply displacement to each coordinate iteratively. Even inaccurate explicit methods give good results.
After the coordinates have been distorted to a far distance, the image becomes unrecognizable. A simple hack is to have a "restore" force applied to the coordinates, and they spring back to their original position, like flattening a piece of mirroring foil.
Just now I am using feedback along with these displacement effects. Very small displacements applied iteratively result in motion that looks quite a bit like fluid flow.