"Liquid" distortions on the plane: how to approach the problem?

Question

Algorithms

Given a rectangular image. Distortions are applied to it, as if it were jam or dough spilled on the table: stretch it, smear it with your finger. Somewhere they twisted a "vortex". Somewhere they took the entire edge and pulled it with an inclination. Those. free non-linear deformation without breaks. At the same time, the “fabric” has some elasticity and tends to partially maintain its density. Pulled for a point, let's go and a few neighboring ones, but less.

1. how best to keep the image grid: square cells or triangular?
   
2. as a minimum of data to indicate the introduced distortions?
   
3. "What to read?"
   

Answer 1

In this formulation, they will not answer you, there is not enough initial data about the task.
What's this?
1 fabric simulation?
2 fluid simulation?
3 visual effect?
Is accuracy necessary?
Is retriagulation necessary after the introduction of distortions?
distortion 2D or 3D?
....
If the first two then see the implementation in BLENDER