We derive a novel set of analytical solutions of the Navier–Stokes equations describing stationary plane motion of two immiscible fluids emitted from a two-dimensional point source. The solutions are two-fluid generalizations of Jeffery–Hamel flows. The presence of an interface yields an unexpectedly rich and complex structure of solutions. Each set of physical parameter values admits a hierarchy of three different types of solution. The solutions bifurcate when parameters of the flow change sufficiently, with each type of solution having a different bifurcation diagram.