On the rectilinear flow of a second-order fluid and the role of the second normal stress difference in edge fracture in rheometry

Assuming that a rectilinear flow is possible in an incompressible simple fluid, the vanishing of the shear stress on a free surface in the flow is shown to lead to one of three restrictions: the second normal stress is zero, or either the velocity gradient is orthogonal to the external, unit normal to the surface, or it is parallel to the unit normal. The consequences of the last two are investigated when the fluid is the second-order fluid and the flow occurs between two parallel plates and the free surface has a small semi-circular indentation in it and when the edge crack in the free surface is almost parallel to the plates. It is found that when there is a small semi-circular indentation, the normal stress at the midway point is tensile, causing the free surface to move into the fluid. The proof depends on obtaining a lower bound to the shear rate at this point and is based on an application of the maximum principle to harmonic functions. Hence, the Tanner-Keentok calculation of the stress at this point is in accord with the present proof; indeed, the magnitude found by them is the lower bound to the true tensile stress and equals the true tensile stress if the ratio of the radius of the indentation to the semi-gap between the parallel plates vanishes. When the edge fracture has moved into the fluid, driven by the above tensile stress and has become almost flat, it is shown that the velocity gradient is parallel to the unit normal to the surface and that the normal stress is compressive, forcing the edges together and preventing the crack from moving further into the fluid.