In analysing fluid forces on a moving body, a natural approach is to seek a component due to viscosity and an ‘inviscid’ remainder. It is also attractive to decompose the velocity field into irrotational and rotational parts, and apportion the force resultants accordingly. The ‘irrotational’ resultants can then be identified as classical ‘added mass’, but the remaining, ‘rotational’, resultants appear not to be consistent with the physical interpretation of the rotational velocity field (as that arising from the fluid vorticity with the body stationary). The alternative presented here splits the inviscid resultants into components that are unquestionably due to independent aspects of the problem: ‘convective’ and ‘accelerative’. The former are associated with the pressure field that would arise in an inviscid flow with (instantaneously) the same velocities as the real one, and with the body’s velocity parameters – angular and translational – unchanging. The latter correspond to the pressure generated when the body accelerates from rest in quiescent fluid with its given rates of change of angular and translational velocity. They are reminiscent of the added-mass force resultants, but are simpler, and closer to the standard rigid-body inertia formulae, than the developed expressions for added-mass force and moment. Finally, the force resultants due to viscosity also include a contribution from pressure. Its presence is necessary in order to satisfy the equations governing the pressure field, and it has previously been recognised in the context of ‘excess’ stagnation-point pressure. However, its existence does not yet seem to be widely appreciated.

The cross-slot stagnation point flow is one of the benchmark problems in non-Newtonian fluid mechanics as it allows large strains to develop and can therefore be used for extensional rheometry measurements or, once instability arises, as a mixing device. In such a flow, beyond a critical value for which the ratio of elastic force to viscous force is high enough, elasticity can break symmetry even in the absence of significant inertial forces (i.e. creeping flow), which is an unwanted phenomenon if the device is to be used as a rheometer but beneficial from a mixing perspective. In this work, a passive control mechanism is introduced to the cross-slot by adding a cylinder at the geometric centre to replace the ‘free’ stagnation point with ‘pinned’ stagnation points at the surface of the cylinder. In the current modified geometry, effects of the blockage ratio (the ratio of the diameter of the cylinder to the width of the channel), the Weissenberg number (the ratio of elastic forces to viscous forces) and extensibility parameters ($\unicode[STIX]{x1D6FC}$ and $L^{2}$) are investigated in two-dimensional numerical simulations using both the simplified Phan-Thien and Tanner and finitely extensible nonlinear elastic models. It is shown that the blockage ratio for fixed solvent-to-total-viscosity ratio has a stabilizing effect on the associated symmetry-breaking instability. The resulting data show that the suggested modification, although significantly changing the flow distribution in the region near the stagnation point, does not change the nature of the symmetry-breaking instability or, for low blockage ratio, the critical condition for onset. Using both numerical and physical experiments coupled with a supporting theoretical analysis, we conclude that this instability cannot therefore be solely related to the extensional flow near the stagnation point but it is more likely related to streamline curvature and the high deformation rates towards the corners, i.e. a classic ‘curved streamlines’ purely elastic instability. Our work also suggests that the proposed geometric modification can be an effective approach for enabling higher flow rates to be achieved whilst retaining steady symmetric flow.