  This paper deals with a two-phase fluid free boundary problem in a slot-film
cooling. We give two well-posedness results on the existence and uniqueness of
the incompressible inviscid two-phase fluid with a jump relation on free
interface. The problem formulates the oblique injection of an incompressible
ideal fluid from a slot into a free stream. From the mathematical point of
view, this work is motivated by the pioneer work [13] by A. Friedman, in which
some well-posedness results are obtained in some special case. Furthermore, A.
Friedman proposed an open problem in [14] on the existence and uniqueness of
the injection flow problem for more general case. The main results in this
paper solve the open problem and establish the well-posedness results on the
physical problem.
