Apparatus and method for forming nanofibers and nonwoven webs formed of such nanofibers. The nanofibers are formed by changing the rheology in at least first and second flows of a liquid material such that the changed rheology of the first and second flows differs by an amount sufficient to produce a phase separation when the first and second flows are combined to define meltblown fibers. Each meltblown fiber has lengthwise first and second cross-sectional regions formed of the liquid material from the first and second flows, respectively. These regions are separated along a length of at least a majority of the meltblown fibers to form a plurality of nanofibers, which are collected together with any unsplit meltblown fibers to form the nonwoven web. The difference in the changed rheology of the flows may be produced by using two extruders with different barrel diameters to create differ shear histories for the two flows.