This study examines the asymptotic and numerical behaviour of Newtonian fluid flows in geometries with sharp corners and the influence of the Navier slip boundary condition. A new similarity solution for a reentrant corner flow is derived by introducing a modification to the classical Navier slip law, where the slip coefficient is modelled as a function of the radial distance along the walls from the reentrant corner. This spatially dependent slip coefficient interpolates between the well-known no-slip similarity solution and the constant slip coefficient case in which the walls behave locally as free surfaces. The stress and pressure singularities now depend on the slip coefficient and the similarity solution is validated numerically through flow simulations in an L-shaped domain. This modified slip coefficient is then used to numerically investigate the influence of the corner stress singularity on the global flow behaviours of two benchmark problems: the 4:1 planar contraction flow and the 1:4 planar expansion flow. Specifically, its effect on salient vortex size and intensity, Couette correction and the flow type (extensional, shear or rotation). This combined asymptotic and numerical framework provides new insights into the role of boundary conditions in controlling flow behaviour near singular geometries, which has not previously been investigated.