Check for plane intersection then make the new section a new transform?

I'll be honest I don't even know where to start with this. I'd like to get the following effect: Effect. The best idea I've had so far is when spawning the blood check if it's mesh (it's a plane) intersects any rigidbody objects and cut it the mesh into two pieces based on that then parent the intersecting piece to the rigidbody object. Could someone show me how to do that?