  We perform a detailed theoretical study of the edge fracture instability,
which commonly destabilises the fluid-air interface during strong shear flows
of entangled polymeric fluids, leading to unreliable rheological measurements.
By means of direct nonlinear simulations, we map out phase diagrams showing the
degree of edge fracture in the plane of the surface tension of the fluid-air
interface and the imposed shear rate, within the Giesekus and Johnson-Segalman
models, for different values of the nonlinear constitutive parameters that
determine the dependencies on shear rate of the shear and normal stresses. The
threshold for the onset of edge fracture is shown to be relatively robust
against variations in the wetting angle where the fluid-air interface meets the
hard walls of the flow cell, whereas the nonlinear dynamics depend strongly on
wetting angle. We perform a linear stability calculation to derive an exact
analytical expression for the onset of edge fracture, expressed in terms of the
shear-rate derivative of the second normal stress difference, the shear-rate
derivative of the shear stress (sometimes called the tangent viscosity), the
jump in shear stress across the interface between the fluid and the outside
air, the surface tension of that interface, and the rheometer gap size. Full
agreement between our analytical calculation and nonlinear simulations is
demonstrated. We also elucidate in detail the mechanism of edge fracture, and
finally suggest a new way in which it might be mitigated in experimental
practice. Some of the results in this paper were first announced in an earlier
letter. The present manuscript provides additional simulation results,
calculational details of the linear stability analysis, and more detailed
discussion of the significance and limitations of our findings.
