A dynamical mother body in a Hele-Shaw problem

It is well-known that in the framework of the internal Hele-Shaw problem the fluid cannot be fully removed through a single point sink because of a cusp formation occurring prior to the moving boundary reaching the sink unless the initial domain is a circle with a sink located in its center. We give a definition of a dynamical mother body and use it for developing an algorithm of complete removal of a fluid droplet having an algebraic boundary. To illustrate our theory we consider examples where fluid can be completely removed through the sinks distributed along arcs of curves and/or points.